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The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(F:(\mathbb{K}^{n},a)\rightarrow (\mathbb{K}^{m},0)\) be an analytic mapping, where \(\mathbb{K=R}\) or \(\mathbb{C}\). Then the known Łojasiewicz inequality holds
\[
\left| F(x)\right| \geq C\text{dist}(x,F^{-1}(0))^{\eta },\;\;\;\;\left| x-a\right| <\varepsilon \tag{1}
\]
for some positive constants \(C,\eta ,\varepsilon\). The smallest exponent \( \eta \) in ({1}) is called the Łojasiewicz exponent of \(F\) at \(a\) and is denoted \(\mathcal{L}_{a}^{\mathbb{K}}(F).\) The main results of the paper is that for any linear mapping \(L:\mathbb{K}^{m}\rightarrow \mathbb{K}^{k},\) \ \(n\leq k\leq m,\) we have
\[
\mathcal{L}_{a}^{\mathbb{K}}(F)\leq \mathcal{L}_{a}^{\mathbb{K}}(L\circ F) \tag{2}
\]
and for generic \(L\) there is an equality in ({2}), in the following cases:
1. the point \(a\) is an isolated zero of \(F\),
2. \(F\) is a polynomial mapping.
Similar results for the Łojasiewicz exponent \(\mathcal{L}_{\infty }^{ \mathbb{K}}(F)\) of \(F\) at \(\infty \) for polynomial mappings having a compact set of zeroes are also proved. Lojasiewicz exponent; polynomial mapping; analytic mapping Spodzieja, S; Szlachcińska, A, Łojasiewicz exponent of overdetermined mappings, Bull. Polish Acad. Sci. Math., 61, 27-34, (2013) Real algebraic and real-analytic geometry, Local complex singularities Łojasiewicz exponent of overdetermined mappings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A version of adjoint theory concerns the studying of an adjoint linear systems \(|K_X+r_L|\) for suitably chosen positive integer \(r\). Problems concerning adjoint divisors have drawn a lot of attention, starting from the classical works of Castelnuovo and Enriques, who considered adjoint linear systems on surfaces. The most interesting case concerns the situations when the adjoint divisor \(K_X+rL\) is nef but not ample. Kawamata-Shokurov contraction theorem asserts that \(|m(K_X+rL)|\) is base point free for \(m\gg 0\) and defines an adjoint contraction morphism \(f:X\to Y\) onto a normal projective variety which is called a contraction of extremal face with supporting divisor \(K_X+rL\). Understanding this map seems to be of great importance for any classification theory of higher dimensional varieties. The maps for \(r\geq n-2\) have been well studied. In this paper we study the contraction of extremal face with supporting divisor \(K_X+(n-3)L\). We give a detailed and unified discription of the structure of this contraction morphism. Minimal model program (Mori theory, extremal rays), Surfaces of general type, Adjunction problems On adjoint contractions of extremal faces. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper considers the classical Bernoulli scheme, that is, a sequence of independent random variables identically distributed with respect to the Lebesgue measure \(m\) on the interval \([0,1]\). The space of realizations of this scheme is the infinite-dimensional cube \(\mathcal{X} = ([0,1]^{\mathbb{N}},\mu )\) with Lebesgue measure \(\mu = m^\mathbb{N} \). It is proved that there exists a function \(k (\cdot): (0, 1) \rightarrow \mathbb{R} \) (which can be defined by \(k ( \epsilon ) = C/ \mu^5)\) such that, given any \(n \in \mathbb{N}\) and \(\epsilon \in (0, 1)\), one can choose a measurable set \(\mathcal{X}_{n,\varepsilon } \subset \mathcal{X}\) of measure at least \(1 - \epsilon\) so that the coordinate \(x_n\) of any realization \(x = \{ x_n\}_n \in \mathcal{X}_{n,\varepsilon}\) reaches the first column of the Young \(P\)-tableau after at most \(k ( \epsilon ) n^2\) insertions of the RSK (Robinson-Schensted-Knuth) algorithm. RSK algorithm; Bernoulli scheme; permutations Combinatorial aspects of representation theory, Exact enumeration problems, generating functions, Grassmannians, Schubert varieties, flag manifolds Estimate of time needed for a coordinate of a Bernoulli scheme to fall into the first column of a Young tableau | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let a space curve \(F\) be defined in \(\mathbb{C}^n\) by the system of polynomial equations \(f_i(X)=0\), \(i=1,\dots,n-1\). Let \(X=(x_1,\dots,x_n)=0\) be a singular point of \(F\), i.e. all \(f_i(0)=0\) and rank \((\partial f_i/\partial x_j)<n-1\) in \(X=0\). Then several branches of \(F\) pass through the point \(X=0\). Each branch has its own local uniformization
\[
x_i= \sum^\infty_{k=1} b_{ik}t^{p_{ik}},\quad i=1,\dots,n, \tag{*}
\]
where \(p_{ik}\) are integers, \(0>p_{ik}>p_{ik+1}\) and \(b_{ik}\) are complex numbers and the series converge for large \(|t|\), i.e. \(X\to 0\) for \(t\to\infty\). We offer an algorithm for finding any initial parts of series (*) for all branches of the curve \(F\). A. D. Bruno and A. Soleev, ''Local uniformization of branches of an algebraic curve,'' Preprint34, I.H.E.S., Paris (1990). Computational aspects of algebraic curves, Plane and space curves The local uniformization of branches of a space curve | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(m>d\) be positive integers and let \(k=m-d-1\). Let \(\Lambda=(\Lambda_1,\ldots, \Lambda_m)\) be an \(k\times m\) matrix with real entries such that the following `weak hyperbolicity condition' is satisfied: the origin is not the convex hull of any \(k\) column vectors of \(\Lambda\). Define \(Z=Z(\Lambda)\) be the subspace of unit sphere in \(\mathbb{R}^m\) defined by the system of quadratic equations \(\sum_{1\leq j\leq m} \Lambda_{j}x_j^2=0, ~1\leq i\leq k.\) The weak hyperbolicity implies that \(Z(\Lambda)\) is smooth manifold. The paper under review studies the topology of these manifolds and the related manifold \(Z^\mathbb{C}=Z^\mathbb{C}(\Lambda)\) contained in the unit sphere of \(\mathbb C^m\) defined by the system equations as above with \(x_j\) replaced by \(|z_j|\). Special cases of these have been studied previously in various guises. For example, \(2\)-connected \(Z^\mathbb C\) are the moment-angle manifolds constructed by \textit{M. W. Davis} and \textit{T. Januszkiewicz} [Duke Math. J. 62, No. 2, 417--451 (1991; Zbl 0733.52006)].
Let \(J=(j_1,\ldots, j_m)\) be a sequence of positive integers. Denote by \(\Lambda^J\) the matrix replicating the \(i\)-th column of \(\Lambda\) \(j_i\) times, \(1\leq i\leq m\). The resulting manifold \(Z(\Lambda^J)\) is denoted \(Z^J\). When \(J=(2,\ldots, 2)\), \(Z^J=Z^\mathbb{C}\). One of the main results of this paper is is the following theorem which is proved by an inductive procedure. Call a product \(\mathbb{S}^m\times \mathbb{S}^n\) of two spheres a `sphere product'.
{Theorem 1.3. } Assume \(Z\) is of dimension \(2c\) and \(c-1\) connected. Then (1) if \(c\geq 3\), \(Z\) is a connected sums of certain sphere products, that is, \(Z\cong \#_{i} \mathbb{S}^{m_i}\times \mathbb{S}^{n_i}\) for suitable positive integers \(m_i,n_i\); (2) if \(c\geq 2\) and if \(Z^J\) is of dimension at least \(5\), then \(Z^J\) is a connected sum of certain sphere products.
It has been conjectured by Bosio and Meersseman that if \(Z\) is of dimension \(2c\) and is \((c-1)\)-connected, then \(Z^\mathbb{C}\) is a connected sum of sphere products. The above theorem establishes the conjecture for \(c\geq 2\). The authors point out, the `counterexample' to the conjecture announced by \textit{D. Allen} and \textit{J. La Luz} [Contemporary Mathematics 460, 37--45 (2008; Zbl 1149.14306)], is based on incorrect computation of \(\pi_6(Z)\).
There is a natural action of \(\mathbb{Z}_2^m\) action on \(Z\) with quotient \(P\) being a simple polytope. The polytope \(P\) may be identified with the intersection of \(Z\) with \(\mathbb{R}_{\geq 0}^m\). Then \(Z=Z(P)\) can be recovered from \(P\) at least as a \(PL\)-manifold. The topological effect of cutting off a vertices and edges of \(P\) is described. If \(C_v\) is the \(3\)-dimensional cube \(C\) with one of its vertices cut off, it is shown that the mod \(2\) cohomology algebra of \(Z=Z(C_v)\) is \textit{not} isomorphic as an ungraded ring to the cohomology algebra of \(Z^\mathbb{C}(C_v)\). The authors point out that this contradicts an earlier result of \textit{M. de Longueville} [Math. Z. 233, No. 3, 553--577 (2000; Zbl 0956.52020)]. quadrics Gitler, S., López de Medrano, S.: Intersections of quadrics, moment-angle manifolds and connected sums. Geom. Topol. 17(3), 1497-1534 (2013) Topology of real algebraic varieties, Algebraic topology on manifolds and differential topology, Groups acting on specific manifolds Intersections of quadrics, moment-angle manifolds and connected sums | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(G\) be a complex connected linear algebraic group, \(P\) be a parabolic subgroup of \(G\) and \(\beta\in A_1(G/P)\) be a 1-cycle class in the Chow group of \(G/P\). An \(n\)-pointed genus 0 stable map into \(G/P\) representing the class \(\beta\), consists of data \((\mu:C\to X;\;p_1, \dots,p_n)\), where \(C\) is a connected, at most nodal, complex projective curve of arithmetic genus 0, and \(\mu\) is a complex morphism such that \(\mu_* [C]= \beta\) in \(A_1(G/P)\). In addition \(p_i\), \(i=1,\dots,n\), denote \(n\) nonsingular marked points on \(C\) such that every component of \(C\), which by \(\mu\) maps to a point, has at least 3 points which are either nodal or among the marked points (this we will refer to as every component of \(C\) being stable). The set of \(n\)-pointed genus 0 stable maps into \(C/P\) representing the class \(\beta\) is parametrized by a coarse moduli space \(\overline M_{0,n}(G/P, \beta)\). In general it is known that \(\overline M_{0,n}(G/P,\beta)\) is a normal complex projective scheme with finite quotient singularities. In this paper we will prove that \(\overline M_{0,n}(G/P,\beta)\) is irreducible. It should also be noted that we in addition will prove that the boundary divisors in \(\overline M_{0,n}(G/P,\beta)\), usually denoted by \(D(A,B,\beta_1,\beta_2)\) \((\beta=\beta_1 +\beta_2\), \(A\cup B\) a partition of \(\{1,\dots,n\})\), are irreducible. connected projective curve; irreducibility of \(n\)-pointed genus 0 stable maps; linear algebraic group; coarse moduli space; quotient singularities J. Thomsen : Irreducibility of \(\overline{M}_{0,n}(G/P,\beta)\), Internat. J. Math. 9, no. 3 (1998), 367-376. Families, moduli of curves (algebraic), Homogeneous spaces and generalizations, Fine and coarse moduli spaces, Enumerative problems (combinatorial problems) in algebraic geometry Irreducibility of \(\overline{M}_{0,n}(G/P,\beta)\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We formulate a higher-rank version of the boundary measurement map for weighted planar bipartite networks in the disk. It sends a network to a linear combination of \(\operatorname{SL}_r\)-webs and is built upon the \(r\)-fold dimer model on the network. When \(r\) equals 1, our map is a reformulation of Postnikov's boundary measurement used to coordinatize positroid strata. When \(r\) equals 2 or 3, it is a reformulation of the \(\operatorname{SL}_2\)- and \(\operatorname{SL}_3\)-web immanants defined by the second author. The basic result is that the higher-rank map factors through Postnikov's map. As an application, we deduce generators and relations for the space of \(\operatorname{SL}_r\)-webs, re-proving a result of \textit{S. Cautis} et al. [Math. Ann. 360, No. 1--2, 351--390 (2014; Zbl 1387.17027)]. We establish compatibility between our map and restriction to positroid strata and thus between webs and total positivity. dimer; web; boundary measurement; positroid; Grassmannian Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds, Representations of finite symmetric groups, Planar graphs; geometric and topological aspects of graph theory From dimers to webs | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \({\mathcal F}\) denote a reflexive sheaf of rank r and homological dimension \(\leq 1\) on a smooth algebraic variety of dimension \( n\) over the field of complex numbers. Recall that the singularity locus of \({\mathcal F}\) is defined as \(Sing({\mathcal F})=\{x\in X| \quad F_ x\) not free\}. The reflexive sheaf \({\mathcal F}\) is called smooth, if for every \(x\in Sing({\mathcal F})\) one has \(Ext^ 1({\mathcal F},{\mathcal O})_ x\simeq {\mathcal O}_ x/(t_ 1,...,t_{r+1})\), where \(t_ 1,...,t_ n\) denotes a suitable regular system of parameters of \({\mathcal O}_ x\). In this case Sing(\({\mathcal F})\) is smooth of codimension \(r+1.\)
Theorem 1 is a result of Bertini-type: Suppose \({\mathcal F}\) is generated by a subvectorspace \(V\subseteq H^ 0(X,{\mathcal F})\) of finite dimension and \(r\geq (n-1)/2\). Then for a general \(\sigma\in V\) the zero scheme Z(\(\sigma\)) is either empty or smooth of codimension \(r+1\) and it contains Sing(\({\mathcal F}).\)
Theorem 2 shows an interrelation between smooth reflexive sheaves and degeneracy loci of homomorphisms of vector bundles: Suppose E and F are vector bundles of rank \(e\) and \(f\geq e+2\) on a smooth projective variety X of dimension \(n\leq 2(f-e+2)\). Then there is a general homomorphism u: \(E\to F\) defining an exact sequence
\[
0\quad \to \quad E\quad \to \quad F\quad \to \quad {\mathcal F}\quad \to \quad 0
\]
with \({\mathcal F}\) smooth reflexive having the degeneracy locus \(\{x\in X| \quad u(x)\) not of rank \(e\}\) as singular locus. Some applications are given. Proofs shall appear elsewhere [see e.g. ``Smooth reflexive sheaves'', Bucuresti INCREST, Preprint Ser. Mat. 17 (1989)]. Bertini's theorem; singularity locus; smooth reflexive sheaves; degeneracy loci of homomorphisms Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Algebraic cycles Faisceaux réflexifs lisses. (Smooth reflexive sheaves) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors study the Newton polytope \(\sum (m,n)\) of the product of all minors of an \(m\times n\)-matrix \(A=(a_{ij})\) of indeterminates, that is the convex hull of all exponent vectors obtained when this product is expanded as a polynomial in the \(a_{ij}\). In particular, the outer normal vectors of the facets are investigated. For instance, it is shown that the \(0\)-\(1\) -matrices in which the ones form the union of two ``proper'' rectangles constitute normal vectors of facets. On the other hand, there are facets that need arbitrarily large integers for any integral normal. For the special cases \(\sum (2,n)\) and \(\sum (3,3)\) all the facet normals are determined. Furthermore, formulas for the support function of \(\sum (m,n)\) are derived.
The whole paper is based on the fact that \(\sum (m,n)\) is equal to the secondary polytope of the product of simplices \(\Delta _{m-1}\times \Delta _{n-1}\), this is (up to a scalar multiple) the same as the fiber polytope of the projection of \(\Delta _{mn-1}\) onto \(\Delta _{m-1}\times \Delta _{n-1}\). Newton polytope; secondary polytope; fiber polytope; product of minors Babson E., Billera L.: The geometry of products of minors. Discrete Comput. Geom. 20, 231--249 (1998) Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Toric varieties, Newton polyhedra, Okounkov bodies The geometry of products of minors | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Given a field \(k\) and a faithful action of a finite group \(G\) on a \(k\)-vector space \(V\), \(g \in G\) is a pseudoreflection if \(V^g\) is a hyperplane. The classical Chevalley-Shephard-Todd theorem asserts that if \(|G| \not | \; \, \mathrm{char}(k)\), then \(k[V]^G\) is polynomial if and only if \(G\) is generated by pseudoreflections. In this paper, the author generalizes this theorem to the case of finite linearly reductive groups schemes.
Let \(G\) be a finite linearly reductive group scheme acting faithfully on a \(k\)-vector space \(V\). A subgroup scheme \(N\) of \(G\) is called a pseudoreflection if \(V^N\) has codimension \(1\) in \(V\), and \(G\) is said to be generated by pseudoreflections if \(G\) is the intersection of the subgroup schemes which contain all of the pseudoreflections of \(G\). The Chevalley-Shephard-Todd theorem is generalized (in the case of algebraically closed field) to
Theorem 1.3. If \(k\) is algebraically closed, then \(k[V]^G\) is polynomial if and only if \(G\) is generated by pseudoreflections.
The author prove Theorem 1.3 in a more general context. The ``if'' part of Theorem 1.3 holds for arbitrary fields which are not algebraically closed. The ``only if'' part does not hold for arbitrary fields and linearly reductive group scheme, it holds for arbitrary fields and a smaller class of group schemes, which are called \textit{stable}; see Definition 1.4 for a precise definition. However, over algebraically closed fields, the class of stable group schemes coincides with that of finite linearly reductive group schemes. See Theorem 1.6 for the generalization of Theorem 1.3.
Theorem 1.6 can be generalized to action of a finite linearly reductive group scheme on a smooth scheme. Let \(U\) be a smooth affine scheme, \(x \in U(K)\) a fixed field-valued point. A subgroup scheme \(N\) of \(G\) is a pseudoreflection at \(x\) if \(N_K\) is a pseudoreflection with respect to the induced action of \(G_K\) on the cotangent space at \(x\). Theorem 1.6 thus gives a criterion on the smoothness of an image of \(x\) in \(U/G\). The second main result of the paper is
Theorem 1.9: Let \(U\) be a smooth affine \(k\)-scheme with a faithful action by a stable group scheme \(G\) over \(\mathrm{Spec}k\). Suppose \(K/k\) is a finite separable field extension and \(G\) fixes a point \(x \in U(K)\). Let \(M = U/G\), let \(M^0\) be the smooth locus of \(M\), and let \(U^0 = U \times_M M^0\). If \(G\) has no pseudoreflections at \(x\), then after possibly shrinking \(M\) to a smaller Zariski neighborhood of the image of \(x\), we have that \(U^0\) is a \(G\)-torsor over \(M^0\).
As an application of Theorem 1.9, the author generalizes a well-known result which says that schemes with quotient singularities prime to the characteristic are coarse spaces of smooth Deligne-Mumford stacks, to schemes which are étale-locally the quotient of a smooth scheme by a finite linearly reductive group scheme. The author shows that every such scheme is the coarse space of a smooth tame Artin stack (as defined by Abramovich, Olsson, and Vistoli). Chevalley-Shephard-Todd theorem; pseudoreflection; linearly reductive group; tame stacks Satriano, M, The Chevalley-shephard-Todd theorem for finite linearly reductive group schemes, Algebra Number Theory, 6-1, 1-26, (2012) Generalizations (algebraic spaces, stacks), Group schemes The Chevalley-Shephard-Todd theorem for finite linearly reductive group schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Consider the following system of PDEs:
\[
\frac{\partial{\Psi}}{\partial{t_i}}=P_i(z, u_1,\dots, u_n),\quad i=1,\dots,N,
\]
\noindent where \(t_1, \dots, t_N\) are independent variables, \(u_1,\dots, u_n\) are dependent variables and \(P_i(z,u_1,\dots,u_n), i=1,\dots,N\) are some functions. The compatibility conditions of this system are called Whitham type hierarchy and have the form:
\[
\sum_{l=1}^{n}\Bigg(\bigg(\frac{\partial{P_i}}{\partial{z}}\frac{\partial{P_j}}{\partial{u_l}}-\frac{\partial{P_j}}{\partial{z}}\frac{\partial{P_i}}{\partial{u_l}}\bigg)\frac{\partial{u_l}}{\partial{z}}+\bigg(\frac{\partial{P_j}}{\partial{z}}\frac{\partial{P_k}}{\partial{u_l}}-\frac{\partial{P_k}}{\partial{z}}\frac{\partial{P_j}}{\partial{u_l}}\bigg)\frac{\partial{u_l}}{\partial{t_i}}
\]
\[
+\bigg(\frac{\partial{P_k}}{\partial{z}}\frac{\partial{P_i}}{\partial{u_l}}-\frac{\partial{P_i}}{\partial{z}}\frac{\partial{P_k}}{\partial{u_l}}\bigg)\frac{\partial{u_l}}{\partial{t_j}}\Bigg)=0.
\]
\noindent Under some conditions on the functions \(P_i(z,u_1,\dots,u_n)\) the last system is equivalent to a hydrodynamic type system of the following form:
\[
\sum_{l=1}^{n}\Bigg(a_{rl}(u_1,\dots,u_n)\frac{\partial{u_l}}{\partial{t_i}}+b_{rl}(u_1,\dots,u_n)\frac{\partial{u_l}}{\partial{t_j}}+c_{rl}(u_1,\dots,u_n)\frac{\partial{u_l}}{\partial{t_k}}\Bigg)=0.
\]
\noindent The present paper is devoted to the construction of such hydrodynamic type systems associated with compact Riemann surfaces of arbitrary genus. Potentials of these hierarchies are written explicitly as integrals of hypergeometric type. integrable hierarchies; hypergeometric functions; tau-function Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions, Relationships between algebraic curves and integrable systems A simple construction of integrable Whitham type hierarchies | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The subject of this work is the study of regularity of linear systems of curves through generic fat points on \(K3\) surfaces. The main result in the paper states that on a generic \(K3\) surface \(X\), if we have that \(\forall m,d \in \mathbb {N}\), a fat point \(mP\) imposes \({m+1 \choose 2}\) independent conditions to curves in \(| dH| \) (where \(H\) is a generator for Pic\(X\)), then the same happens for \(n\) generic fat points, all with the same multiplicity, when \(n=4^u9^v\). Moreover they show that this is the case on a generic quartic surface in \(\mathbb {P}^3\). The authors use a degeneration of \(X\) and of linear systems on it, as it is done by \textit{C. Ciliberto} and \textit{R. Miranda} [Matematiche 55, No. 2, 259--270 (2000; Zbl 1165.14301); Trans. Am. Math. Soc. 352, No. 9, 4037--4050 (2000; Zbl 0959.14015)]. This degeneration allows them to work mainly in \(\mathbb {P}^2\), where they can use results of \textit{A. Buckley} and \textit{M. Zompatori} [Trans. Am. Math. Soc. 355, No. 2, 539--549 (2003; Zbl 1038.14014)] who work themselves in the line of Ciliberto and Miranda. Cindy De Volder and Antonio Laface, Degeneration of linear systems through fat points on \?3 surfaces, Trans. Amer. Math. Soc. 357 (2005), no. 9, 3673 -- 3682. Divisors, linear systems, invertible sheaves, \(K3\) surfaces and Enriques surfaces Degeneration of linear systems through fat points on \(K3\) surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(f:V\to W\) be a finite polynomial mapping of algebraic subsets \(V,W\) of \(k^n\) and \(k^m\), respectively, with \(n\leq m\). It is known that
\(f\) can be extended to a finite polynomial mapping \(F:k^n\to k^m\). Moreover, it is known that, if \(V,W\) are smooth of dimension \(k\), \(4k+2\leq n=m\), and \(f\) is dominated on every component (without vertical components) then there exists a finite polynomial extension
\(F:k^n\to k^m\) such that \(\mathrm{gdeg} F\leq (\mathrm{gdeg} f)^{2k+1}\), where \(\mathrm{gdeg} h\) means the number of points in the generic fiber of \(h\). In this note we improve this result. Namely we show that there exists a finite polynomial extension \(F:k^n\to k^m\) such that \(\mathrm{gdeg} F\leq (\mathrm{gdeg} f)^{k+1}\). Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem), Affine geometry, Birational geometry Geometric degree of finite extensions of mappings from a smooth variety | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper is the first in a series in which we offer a new framework for hermitian K-theory in the realm of stable \(\infty\)-categories. Our perspective yields solutions to a variety of classical problems involving Grothendieck-Witt groups of rings and clarifies the behaviour of these invariants when 2 is not invertible. In the present article we lay the foundations of our approach by considering Lurie's notion of a Poincaré \(\infty\)-category, which permits an abstract counterpart of unimodular forms called Poincaré objects. We analyse the special cases of hyperbolic and metabolic Poincaré objects, and establish a version of Ranicki's algebraic Thom construction. For derived \(\infty\)-categories of rings, we classify all Poincaré structures and study in detail the process of deriving them from classical input, thereby locating the usual setting of forms over rings within our framework. We also develop the example of visible Poincaré structures on \(\infty\)-categories of parametrised spectra, recovering the visible signature of a Poincaré duality space. We conduct a thorough investigation of the global structural properties of Poincaré \(\infty\)-categories, showing in particular that they form a bicomplete, closed symmetric monoidal \(\infty\)-category. We also study the process of tensoring and cotensoring a Poincaré \(\infty\)-category over a finite simplicial complex, a construction featuring prominently in the definition of the L- and Grothendieck-Witt spectra that we consider in the next instalment. Finally, we define already here the 0th Grothendieck-Witt group of a Poincaré \(\infty\)-category using generators and relations. We extract its basic properties, relating it in particular to the 0th L- and algebraic K-groups, a relation upgraded in the second instalment to a fibre sequence of spectra which plays a key role in our applications. Grothendieck-Witt groups; Hermitian \(K\)-theory; \(L\)-theory; Poincaré categories Hermitian K-theory for stable \(\infty\)-categories. I: Foundations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper, one can find the construction of a \textit{slit analytic fibre space of complex Lie groups}, according to the terminology of \textit{K. Kato} and \textit{S. Usui} [Classifying spaces of degenerating polarized Hodge structures.Annals of Mathematics Studies 169. Princeton, NJ: Princeton University Press. (2009; Zbl 1172.14002)], which \textit{graphs admissible normal functions with no singularities}, in a similar way as the classical Néron model [\textit{S. Bosch, W. Lütkebohmert} and \textit{M. Raynaud}, Néron models. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, 21. Berlin etc.: Springer-Verlag. (1990; Zbl 0705.14001)] graphs admissible normal functions arising from families of curves (the definition of admissible normal function can be found in: [\textit{M. Saito}, J. Algebr. Geom. 5, No.2, 235--276 (1996; Zbl 0918.14018)].
The contents of this paper are best presented by the authors' abstract: ``We show that the limit of a one-parameter admissible normal function with no singularities lies in a non-classical sub-object of the limiting intermediate Jacobian. Using this, we construct a Hausdorff slit analytic space, with complex Lie group fibres, which `graphs' such normal functions. For singular normal functions, an extension of the sub-object by a finite group leads to the Néron models. When the normal function comes from geometry, that is, a family of algebraic cycles on a semistably degenerating family of varieties, its limit may be interpreted via the Abel-Jacobi map on motivic cohomology of the singular fibre, hence via regulators on \(K\)-groups of its substrata. Two examples are worked out in detail, for families of 1-cycles on CY and abelian 3-folds, where this produces interesting arithmetic constraints on such limits. We also show how to compute the finite `singularity group' in the geometric setting.'' Néron model; slit analytic space; Abel-Jacobi map; admissible normal function; variation of Hodge structure; limit mixed Hodge structure; motivic cohomology; unipotent monodromy; semistable reduction; algebraic cycle; higher Chow cycle; Ceresa cycle; Clemens-Schmid sequence; polarization; slit analytic space Green, Mark; Griffiths, Phillip; Kerr, Matt, Néron models and limits of Abel-Jacobi mappings, Compos. Math., 0010-437X, 146, 2, 288-366, (2010) Algebraic cycles, Transcendental methods, Hodge theory (algebro-geometric aspects), Applications of methods of algebraic \(K\)-theory in algebraic geometry, Fibrations, degenerations in algebraic geometry, Variation of Hodge structures (algebro-geometric aspects), Motivic cohomology; motivic homotopy theory, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Néron models and limits of Abel-Jacobi mappings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In the paper, we study real forms of the complex generic Neumann system. We prove that the real forms are completely integrable Hamiltonian systems. The complex Neumann system is an example of the more general Mumford system. The Mumford system is characterized by the Lax pair \((L^{\mathbb{C}}(\lambda),M^{\mathbb{C}}(\lambda))\) of \(2 \times 2\) matrices, where
\(L^{\mathbb{C}}(\lambda)=\begin{bmatrix} V^{\mathbb{C}}(\lambda) & W^{\mathbb{C}}(\lambda)\\ U^{\mathbb{C}}(\lambda)& -L^{\mathbb{C}}(\lambda)\end{bmatrix}\) and \(U^{\mathbb{C}}(\lambda)\), \(V^{\mathbb{C}}(\lambda)\), \(W^{\mathbb{C}}(\lambda)\) are suitable polynomials. The topology of a regular level set of the moment map of a real form is determined by the positions of the roots of the suitable real form of \(U^{\mathbb{C}}(\lambda)\), with respect to the position of the values of suitable parameters of the system. For two families of the real forms of the complex Neumann system, we describe the topology of the regular level set of the moment map. For one of these two families the level sets are noncompact.
In the paper, we also give the formula which provides the relation between two systems of the first integrals in involution of the Neumann system. One of these systems is obtained from the Lax pair of the Mumford type, while the second is obtained from the Lax pair whose matrices are of dimension \((n+1) \times(n+1)\). integrable systems; Neumann system; Arnold-Liouville level sets; spectral curves; real structures; real forms Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Relationships between algebraic curves and integrable systems, Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics Geometry of real forms of the complex Neumann system | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper studies \(C^r\) splines on closed vertex stars of degree at least \(3r+2\). A vertex star \(\Delta\) is a pure \(n\)-dimensional simplicial complex all of whose \(n\)-dimensional simplices share a common vertex \(\gamma\). We say that the vertex star is closed when the common vertex \(\gamma\) is an interior vertex of \(\Delta\). The link of such a vertex star is the subcomplex of \(\Delta\) consisting of all simplices of \(\Delta\) which do not contain \(\gamma\). Further, a tetrahedral vertex star is a pure hereditary, three-dimensional vertex star whose link is simply connected.
The following notation is used throughout the paper:
\medskip \noindent Notation: Let \(S = \mathbb R[x,y,z]\) be the polynomial ring in three variables and \(S_d\) denote the vector space of homogeneous polynomials of degree \(d\). Similarly, let \(S_{\leq d}\) denote the vector space of polynomials of total degree at most \(d\). Let \(\Delta \subset \mathbb R^3\) be a tetrahedral vertex star and \(r \geq 0\) be a given integer. Then
\begin{itemize}
\item \(\Delta_3\) is the set of faces of \(\Delta\) of dimension 3;
\item \(C^r(\Delta)\) is the set of all functions \(F: \Delta \rightarrow \mathbb R\) which are continuously differentiable of order \(r\);
\item \(\mathcal{S}_d^r(\Delta) = \{F \in C^r(\Delta) : F\mid_{\alpha} \in S_{\leq d} \, \, \text{for all} \, \, \alpha \in \Delta_3\}\);
\item \(\mathcal H_d^r(\Delta) = \{F \in C^r(\Delta) : F\mid_{\alpha} \in S_d \, \, \text{for all} \,\, \alpha \in \Delta_3\}\);
\item \(f_1^{\circ}\) is the number of interior edges in \(\Delta\);
\item \(D_{\gamma} = \begin{cases} 2r & \text{if } f_1^{\circ} = 4 \\
\lfloor (5r+2)/3 \rfloor & \text{if } f_1^{\circ} = 5 \\
\lfloor (3r+1)/2 \rfloor & \text{if } f_1^{\circ} \geq 6. \end{cases}\)
\end{itemize}
The main result of the paper verifies that a formula of \textit{P. Alfeld} et al. [SIAM J. Math. Anal. 27, No. 5, 1482--1501 (1996; Zbl 0854.41030)] is also a lower bound on the dimension of \(\mathcal H_d^r(\Delta)\) for a closed vertex star \(\Delta\) with vertex coordinates general enough and when there are at least six boundary vertices. More precisely:
Theorem. Let \(r \geq 0\) and \(\Delta\) be a closed vertex star with interior vertex \(\gamma\). If \(d > D_{\gamma}\), then \[\mathcal H_d^r(\Delta) \geq \max \left\{ \binom{d+2}{2}, \mathrm{LB}^{\star}(\Delta, d, r) \right\}\] and \[\dim \mathcal S_d^r(\Delta) \geq \binom{D_{\gamma}+3}{3} + \sum_{i = D_{\gamma}+1}^d \max \left\{ \binom{i+2}{2}, \mathrm{LB}^{\star}(\Delta, i, r) \right\},\] where \(\mathrm{LB}^{\star}(\Delta, d, r)\) is an involved formula which can be found in Equation 4.6 of the paper.
The techniques of the proof of this main result involves apolarity and the Waldschmidt constant of sets of points in projective space. In particular, a reduction method from [\textit{S. Cooper} et al., J. Pure Appl. Algebra 215, No. 9, 2165--2179 (2011; Zbl 1221.14063)] is used.
The case where \(\Delta\) is a generic closed vertex star with four or five interior edges is handled in the second result of the paper. In this case, the authors show that for four edges the only homogeneous splines of degree at most \(2r\) are global polynomials. The same conclusion is shown to hold for five edges of degree at most \((5r+2)/3\). spline functions; apolarity; fat point ideals; Waldschmidt constant Syzygies, resolutions, complexes and commutative rings, Numerical computation using splines, Spline approximation, Divisors, linear systems, invertible sheaves A lower bound for splines on tetrahedral vertex stars | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper studies the relation between the measure on the space of function-germs and the motivic measure on the space of arcs. The author demonstrates the way to reduce the integration over the space of functions to the integration over multi-arcs.
Let \(\mathbb{P}O_{(C^2,0)}\) be the projectivized space of function germs on the plane, let \(\mathcal{L}_{(C^2,0)}\) be the space of arcs and \(B\) be the space of branches at the origin (so that \(B=\mathcal{L}/\text{Aut}(C^1,0)\)). Each function-germ defines the germ of curve, thus defining the map: \(Z:\mathbb{P}O_{(C^2,0)}\to\coprod S^kB\).
The measure on \(\mathbb{P}O_{(C^2,0)}\) was constructed by Campillo Delgado and Goussein-Zade (first defined for the cylindric sets then extended to the algebra of measurable sets). The motivic measure on \(\mathcal{L}\) was constructed by Kontsevich, Denef and Loeser. It descends to the measure on the space of branches \(B\).
These measures extend naturally to the symmetric products: \(S^k\mathcal{L}\) and \(S^kB\). The relation between the corresponding integrals is given in Lemma 3.2.
Finally, the measures on \(S^kB\) and \(\mathbb{P}O_{(C^2,0)}\) are related by the formula (Theorem 3.5):
\[
\mu(N)=\int_M\mathbb{L}^{\delta(\gamma)-k-P(\gamma)}d\chi_g
\]
Here: \(M\subset S^kB\), \(N=Z^{-1}(M)\), \(\mathbb{L}\in K_0(\text{Var}_C)\) is the class of \(C^1\) in the Grothendieck ring of varieties, \(\delta(\gamma)\) is the delta invariant of the plane curve defined by \(\gamma\) (aka the genus discrepancy, aka the virtual number of nodes) and \(P(\gamma)\) is the kappa invariant of the plane curve (the local degree of intersection of the curve with its generic polar).
Several examples are considered (the spaces of curves with ordinary multiple points, \(A_k\)'s and \(x^p+y^q\) for \(\text{gcd}(p,q)=1\) and some other integrals). Finally, the correspondence for Euler characteristic is discussed. motivic integration; measure Birational geometry, Complex singularities, Local theory in algebraic geometry, Group- or semigroup-valued set functions, measures and integrals On the motivic measure on the space of functions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We study the moduli space of the spectral curves \(y^2 = W^\prime (z)^2 + f(z)\) which characterize the vacua of \(\mathcal N = U(n)\) supersymmetric gauge theories with an adjoint Higgs field and a polynomial tree level potential \(W(z)\). The integrable structure of the Whitham equations is used to determine the spectral curves from their moduli. An alternative characterization of the spectral curves in terms of critical points of a family of polynomial solutions \(\mathbb W\) to Euler-Poisson-Darboux equations is provided. The equations for these critical points are a generalization of the planar limit equations for one-cut random matrix models. Moreover, singular spectral curves with higher order branch points turn out to be described by degenerate critical points of \(\mathbb W\). As a consequence we propose a multiple scaling limit method of regularization and show that, in the simplest cases, it leads to the Painlevè-I equation and its multi-component generalizations. Konopelchenko, B.; Martí nez Alonso, L.; Medina, E., Spectral curves in gauge/string dualities: integrability, singular sectors and regularization, J. Phys. A: Math. Theor., 46, (2013) String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Yang-Mills and other gauge theories in quantum field theory, Supersymmetric field theories in quantum mechanics, Special quantum systems, such as solvable systems, Selfadjoint operator theory in quantum theory, including spectral analysis Spectral curves in gauge/string dualities: integrability, singular sectors and regularization | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(K\) be a \(p-\)adic field (i.e. \([K:\mathbb{Q}_p]<\infty\)), \(R_K\) be the valuation ring of \(K\), \(f_1,\dots,f_l\) be polynomials in \(K[x_1,\dots,x_n]\), or, more generally, \(K-\) analytic functions on an open set \(U\subset K^n\) and let \(\mathbf{f}\) be the mapping \(\mathbf{f}=(f_1,\dots,f_l):K^n\rightarrow K^l\), respectively, \(U\rightarrow K^l\). Let \(\Phi:K^n\rightarrow \mathbb{C}\) be a Schwartz-Bruhat function (with support in \(U\) in the second case). The Igusa local zeta function associated to the above data is defined as
\[
Z_{\Phi}(s,\mathbf{f})=Z_{\Phi}(s,\mathbf{f},K)=\int_{K^n}\Phi(x)\left\|\mathbf{f}(x)\right\|^s_{K}|dx|,
\]
for \(s\in \mathbb{C}\) with \(Re(s)>0\), where \(|dx|\) is the Haar measure on \(K^n\) normalized in such a way that \(R_K^n\) has measure \(1\).
The function \(Z_{\Phi}(s,\mathbf{f})\) admits a meromorphic continuation to the complex plane as a rational function of \(q^{-s}\), where \(q\) is the cardinality of the residue field of \(K\). One of the problem which is studied extensively is the problem of determining the poles of the meromorphic continuation of the \(Z_{\Phi}(s,\mathbf{f})\) for \(Re(s)<0\). Its relevance is due to the existence of several conjectures relating the poles of \(Z_{\Phi}(s,\mathbf{f})\) with the structure of the singular locus of \(\mathbf{f}\).
In Theorem 2.4 the authors give a new proof of the meromorphic continuation of the \(Z_{\Phi}(s,\mathbf{f})\). Especially, a geometric description of the candidate poles of \(Z_{\Phi}(s,\mathbf{f})\), \(l\geq 1\), in terms of a log-principalization of the \(K[x]-\)ideal \(\mathcal{I}_{\mathbf{f}}=(f_1,\dots,f_l)\) it is provided. One should mention here that in [\textit{W. A. Zuniga-Galindo}, Bull. Lond. Math. Soc. 36, No. 3, 310--320 (2004; Zbl 1143.11356)], it is given an algorithm for computing a list of possible poles of \(Z_{\Phi}(s,\mathbf{f})\), \(l\geq 1\), in terms of an embedded resolution of singularities of the divisor \(\bigcup_{i=1}^{l}f_i^{-1}(0)\). The method from the current paper provides a much shorter list of candidates poles compared with the previous known results, as one can see from the given examples.
The largest real part, \(-\lambda(\mathcal{I}_\mathbf{f})\), of the poles of the Igusa zeta function is determined from a log-principalization of the ideal \(\mathcal{I}_\mathbf{f}\). As a corollary, the authors obtain an asymptotic estimation for the number of solutions of an arbitrary system of polynomial congruences in the terms of the log-canonical threshold of a log-principalization.
Moreover, the authors associate to an analytic mapping \(\mathbf{f}=(f_1,\dots,f_l)\) a Newton polyhedron \(\Gamma(\mathbf{f})\) and a new notion of a non-degeneracy with respect of \(\Gamma(\mathbf{f})\). The novelty of this notion resides in the fact that it depends on \textit{one} Newton polyhedra of \(f_1,\dots,f_l\).
Also, in Theorem 3.11 it is given a generalization to the case \(l\geq 1\) of a well-known result that describes the poles of the local zeta function associated to a non-degenerate polynomial in terms of the corresponding Newton polyhedron. By constructing a log-principalization, it is given an explicit list of the candidate poles of \(Z_{\Phi}(s,\mathbf{f})\), \(l\geq 1\), in the case in which \(\mathbf{f}\) is non-degenerate with respect to \(\Gamma(\mathbf{f})\). Igusa zeta functions; congruences in many variables; topological zeta functions; motivic zeta functions; Newton polyhedra; toric varieties; log-principalization of ideals Willem Veys & Wilson A. Zúñiga-Galindo, Zeta functions for analytic mappings, log-principalization of ideals, and Newton polyhedra, Trans. Am. Math. Soc.360 (2008), p. 2205-2227 Zeta functions and \(L\)-functions, Congruences in many variables, Toric varieties, Newton polyhedra, Okounkov bodies, Modifications; resolution of singularities (complex-analytic aspects) Zeta functions for analytic mappings, log-principalization of ideals, and Newton polyhedra | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Interpolation problems for homogeneous polynomials (forms) over points with given multiplicities are object of many recent deep investigations. Yet, basic properties of the structure of interpolating linear systems are still unknown. Even for ternary forms (which correspond to plane curves), the fundamental Segre's conjecture, which suggests the existence of a base locus for special interpolating systems, is open. For quaternary forms (corresponding to space surfaces), even less is known. There are (weaker) extensions of Segre's conjecture, which suggest that the speciality of some interpolating systems should depend on the existence of \textit{special effect varieties} through the interpolating points, which force a base locus.
The authors extend to some new initial case our knowledge on the behaviour of special linear systems of quaternary forms. They produce a characterization of special systems of degree \(2m\) and \(2m+1\), with \(8\) interpolating points of multiplicity \(m\) and one extra interpolating point of multiplicity \(a\leq m\) (quasi-homogeneous case). The special effect varieties that appear in these cases are quadric surfaces. The characterization produces, as a consequence, the proof of the Laface-Ugaglia conjecture on the structure of special Cremona-reduced systems, when the number of interpolating points is up to \(9\) and the multiplicities are \(\leq 8\). fat points; degeneration techniques; Laface-Ugaglia conjecture; base locus; quadric surface DOI: 10.1007/s10231-015-0528-5 Divisors, linear systems, invertible sheaves, Hypersurfaces and algebraic geometry, Rational and ruled surfaces On linear systems of \(\mathbb P^3\) with nine base points | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The main result of the article is an existence theorem on geometric quotients for proper actions of algebraic group schemes on algebraic spaces. Let \(G\) be an affine algebraic group scheme of finite type over some excellent base scheme \(S\). Suppose that \(G\) acts morphically on the algebraic space \(X\) of finite type over \(S\). The notion of a geometric quotient for the action of \(G\) on \(X\) is defined by the author basically in analogy to D. Mumford's concept for the case of schemes. Whereas in the category of schemes a geometric quotient is necessarily categorical and hence unique, these two properties are no longer satisfied in the category of algebraic spaces, as the author shows in an explicit example. Now assume that the action of \(G\) on \(X\) is proper.
The main result of the article ensures that there exists a geometric quotient \(q : X \to Y\) for the action of \(G\) on \(X\) if one of the following conditions is valid:
(a) \(G\) is reductive over \(S\),
(b) \(S\) is the spectrum of a field of positive characteristic.
Moreover, then the algebraic space \(Y\) over \(S\) is separated and \(q\) is in fact categorical.
The author obtains also important properties of quotient morphisms. In fact, he works under slightly weaker assumptions and considers the notion of an approximate quotient for the action of \(G\) on \(X\). For \(G\) universally open over \(S\) it is shown that such a quotient \(p: X \to Y\) is an affine morphism. If furthermore \(G\) is flat over \(S\), then for every coherent \(G\)-sheaf \(F\) on \(X\) the sheaf \((p_*F)^G\) of invariants on \(Y\) is coherent. The main result yields the existence of certain moduli spaces that was known before only in characteristic zero. In the meantime \textit{S. Keel} and \textit{S. Mori} [Ann. Math., II. Ser. 145, No. 1, 193-213 (1997; see the following review)] presented a proof for the existence of geometric quotients by proper actions of flat algebraic group schemes with finite stabilizer on algebraic spaces in a more general framework. algebraic group actions; geometric quotients; algebraic spaces \textsc{J. Kollár}, \textit{Quotient spaces modulo algebraic groups}, Ann. of Math. (2) 145 (1997), no. 1, 33--79. DOI 10.2307/2951823; zbl 0881.14017; MR1432036; arxiv alg-geom/9503007 Homogeneous spaces and generalizations, Group actions on varieties or schemes (quotients), Geometric invariant theory Quotient spaces modulo algebraic groups | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X \subset Y \subseteq \mathbb P^n_k\) be closed subschemes of \(\mathbb P^n\). Let \(R\) be the homogeneous coordinate ring of \(Y\), and denote \(I_X\subset R\) the ideal of \(X\) in \(Y\). A scheme \(Z \subset Y\) of codimension at least \(s\) in \(Y\) is called an \(s\)-residual intersection of \(X\) in \(Y\) if \(Z\) is defined by an ideal of the form \((f_1,\dots, f_s):_R I_X\), with \(f_1, \dots, f_s\) homogeneous elements in \(I_X\). Denote \(\omega_R\) the graded canonical module of \(R\). The main result of the paper (Theorem 1.4) is a formula for the Hilbert series of \(Z\) in terms of the degrees of the \(f_i\) and the Hilbert series of finitely many modules of the form \(\omega_R/I^j \omega_R\) under certain technical conditions on \(X\) (which is satisfied when \(Y = \mathbb P^n\) and \(X\) is smooth). An interesting application of the result is a criterion for when a subscheme \(X \subset \mathbb P^n\) can be defined by \(s\) equations, for some \(s \geq \text{codim} X\). residual intersection; Hilbert series; conormal modules DOI: 10.1112/S0010437X15007289 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Linkage, complete intersections and determinantal ideals, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Linkage Hilbert series of residual intersections | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let $(R,m,K)$ be local ring. F-signature of this ring is introduced implicitly in the work on rings of differential operators by \textit{K. E. Smith} and \textit{M. Van den Bergh} [Proc. Lond. Math. Soc. (3) 75, No. 1, 32--62 (1997; Zbl 0948.16019)]. Its formal definition was formulated and given by \textit{C. Huneke} and \textit{G. J. Leuschke} [Math. Ann. 324, No. 2, 391--404 (2002; Zbl 1007.13005)] by a limit process provided that the limit exists. Some other people developed and improved this concept [\textit{K. Tucker}, Invent. Math. 190, No. 3, 743--765 (2012; Zbl 1271.13012)] and [\textit{Y. Yao}, J. Algebra 299, No. 1, 198--218 (2006; Zbl 1102.13027)]. Although The F-signature has a number of properties that relate to different aspects of the singularity of \(R\), however,computing the F-signature is a very difficult task. The F-signature in its nature is not Defined for rings of characteristic zero. The authors of the paper under review by looking at different aspects of the F-signature define a numerical invariant for rings of characteristic zero or $p>0$ that carries many of the useful properties of the F-signature. Their new numerical Invariant is more Computational and they compute many examples of this invariant, including cases where the F-signature is not known. The paper is long one and consists of very well motivated approach. They are able to obtain some results on symbolic powers and Bernstein-Sato polynomials. \(D\)-modules; singularities; numerical invariant; rings of invariant Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Rings of differential operators (associative algebraic aspects), Singularities in algebraic geometry, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure Quantifying singularities with differential operators | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Some geometric and combinatorial aspects of the solution to the full Kostant-Toda (f-KT) hierarchy are studied, when the initial data is given by an arbitrary point on the totally non-negative (tnn) flag variety of \(\text{SL}_n(\mathbb{R})\). The f-KT flows on the tnn flag variety are complete, and it is shown that their asymptotics are completely determined by the cell decomposition of the tnn flag variety given by Rietsch. These results represent the first results on the asymptotics of the f-KT hierarchy (and even the f-KT lattice); moreover, these results are not confined to the generic flow, but cover non-generic flows as well. The f-KT flow on the weight space via the moment map is defined, and it is shown that the closure of each f-KT flow forms an interesting convex polytope which is called a Bruhat interval polytope. In particular, the Bruhat interval polytope for the generic flow is the permutohedron of the symmetric group \(\mathfrak{G}_n\). An analogous results for the full symmetric Toda hierarchy, by mapping f-KT solutions to those of the full symmetric Toda hierarchy is also proved. In the appendix, it is shown that Bruhat interval polytopes are generalized permutohedra. Kostant-Toda lattice; Grassmannian; flag variety; moment polytope Kodama, Y.; Williams, L., The full Kostant--Toda hierarchy on the positive flag variety, Commun. Math. Phys., 335, 247-283, (2015) Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions, Grassmannians, Schubert varieties, flag manifolds, Special polytopes (linear programming, centrally symmetric, etc.) The full Kostant-Toda hierarchy on the positive flag variety | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a smooth scheme of finite type over a field \(k\), equipped with a projective birational map \(\pi : X\rightarrow Y\) onto a normal irreducible affine scheme of finite type over \(k\). The main result of the paper under review asserts that if \(\text{char}\, k = 0\) and \(X\) is symplectic (i.e., it admits a non-degenerate closed 2-form \(\Omega \in H^0(X,{\Omega}^2_X)\)) then every point \(y\in Y\) has an étale neighborhood \(U_y \rightarrow Y\) such that there exists a vector bundle \(\mathcal E\) on the pullback \(X_y = X{\times}_YU_y\) which is a tilting generator of the derived category \(D^b_{\text{coh}}(X)\). The last assertion means that: (i) \({\text{Ext}}^i({\mathcal E},{\mathcal E}) = 0\), \(\forall i > 0\), and (ii) \(\forall {\mathcal F}^{\bullet} \in \text{Ob}\, D^-_{\text{coh}}(X_y)\), \({\text{RHom}}^{\bullet}({\mathcal E},{\mathcal F}^{\bullet}) \simeq 0\) implies \({\mathcal F}^{\bullet} \simeq 0\). In this case, \({\mathcal F}^ {\bullet} \mapsto {\text{RHom}}^{\bullet}({\mathcal E},{\mathcal F}^{\bullet})\) is an equivalence of categories \(D^b_{\text{coh}}(X_y) \rightarrow D^b_{\text{coh}}(R\text{-mod})\), where \(R = \text{End}({\mathcal E})\) and \(R\)-mod is the category of finitely generated left \(R\)-modules (which is abelian since it turns out that \(R\) is left noetherian).
The author also shows that if \({\pi}^{\prime} : X^{\prime} \rightarrow Y\) is another resolution of singularities and if the canonical bundles \(K_X\) and \(K_{X^{\prime}}\) are trivial, then every point \(y \in Y\) admits an étale neighborhood \(U_y \rightarrow Y\) such that \(D^b_{\text{coh}}(X_y)\) and \(D^b_{\text{coh}}(X^{\prime}_y)\) are equivalent.
The author proves these results by reduction to positive characteristic. He uses his results from \textit{D. Kaledin} [Math. Res. Lett. 13, No. 1, 99--107 (2006; Zbl 1090.53064)] about the existence of twistor deformations of line bundles on \(X\), and a quantization result in positive characteristic based on the techniques developed in \textit{R. Bezrukavnikov} and \textit{D. Kaledin} [J. Am. Math. Soc. 21, No. 2, 409--438 (2008; Zbl 1138.53067)]. Resolution of singularities; symplectic manifold; derived category; Poisson structure; quantized algebra; Frobenius endomorphism D. Kaledin, ''Derived equivalences by quantization,'' Geom. Funct. Anal., vol. 17, iss. 6, pp. 1968-2004, 2008. Global theory and resolution of singularities (algebro-geometric aspects), Poisson manifolds; Poisson groupoids and algebroids, Deformation quantization, star products, Derived categories, triangulated categories, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure Derived equivalences by quantization | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(f: ({\mathbb{C}}^{n+1},0)\to ({\mathbb{C}},0)\) be a holomorphic function germ with an isolated singularity at 0. Take \(\epsilon\),\(\eta\in {\mathbb{R}}\) with \(0<\eta \ll \epsilon \ll 1\) and put
\[
X=\{z\in {\mathbb{C}}^{n+1}/\quad | z| <\epsilon \text{ and } | f(z)| <\eta \},\quad X_ t=X\cap f^{-1}(t)\text{ and } X^*=X\setminus X_ 0.
\]
Then \(f| X^*: X^*\to \{t\in {\mathbb{C}}| 0<| t| <\eta \}\) is a \(C^{\infty}\) fiber bundle. Let \(X_ f\) denote a typical fiber of this. Then \(X_ f\) has the homotopy type of a wedge of \(\mu\) n- spheres, where \(\mu\) is the Milnor number of f. Let h: \(X_ f\to X_ f\) denote the characteristic homeomorphism of this Milnor fibration. The monodromy transformation T of f is defined as \(T=h^*{}^{-1}: \tilde H^ n(X_ f,{\mathbb{C}})\to \tilde H^ n(X_ f,{\mathbb{C}}).\)
The following theorem is well-known. Monodromy Theorem: The eigenvalues of T are roots of unity. The Jordan blocks of T are of size at most \(n+1\). Jordan blocks for eigenvalue 1 of T are of size at most n.
In this paper the following supplement to this theorem is given. Theorem 1: If T has a Jordan block of size \(n+1\) (necessarily for an eigenvalue \(\neq 1)\), then T also has a Jordan block of size n for the eigenvalue 1.
Two proofs of this supplement are given. The first works only for nondegenerate functions with respect to their Newton diagram, but has independent interest, because it expresses the Hodge-Steenbrink numbers \(h^{n,n}_{\neq 1}\) and \(h_ 1^{n,n}\) in terms of the Newton diagram. The second proof works in the general case and Steenbrink's theory of the mixed Hodge structure on the cohomology group of \(X_ f\) is applied in it.
In the latter half of this paper a related conjecture is given and a proof of the conjecture in some restricted cases is given.
The reviewer thinks that the second proof of Theorem 1 does not work if the set V of \((n+1)\)-fold points of \(D\cup \tilde X_ 0\) is not sufficiently large. (We follow the notation in the paper.) Moreover, in the proof of the conjecture the inequality \(c,d>1\) on the line 13 from below on page 232 does not always hold. For example for the plane curve singularity \(x^ 3+xy^ 3=0,\) we have \(e_ 1=9\), \(e_ 2=5\), \(e_ 3=3\) and thus \(c=\delta_ 2=\gcd (e_ 2,e_ 1)=1\ngtr 1\) and \(d=\delta_ 3=\gcd (e_ 3,e_ 1)=3\). However, with some minor supplements, we can easily make all the proofs complete. singularity; monodromy; Newton diagram Athanasiadis, C., Savvidou, C.: A symmetric unimodal decomposition of the derangement polynomial of type \(B\). Preprint. arXiv:1303.2302 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Singularities in algebraic geometry, Milnor fibration; relations with knot theory A supplement to the monodromy theorem | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author shows an equivalent formulation of the Pierce-Birkhoff conjecture in terms of the set of separating functions of points of the real spectrum. Precisely, let A denote a commutative ring with unit and SA(A) its ring of abstract continuous semialgebraic functions. Set PW(A)\(\subset SA(A)\) for the subring of functions which are piecewise defined by elements of A. Then it is shown that the following are equivalent:
(a) A verifies the Pierce-Birkhoff conjecture, that is, any \(t\in PW(A)\) is of the form \(t=\sup_{i}\{\inf_{j}(b_{ij})\}\) for some \(b_{ij}\in A.\)
(b) For any \(t\in PW(A)\) and \(\gamma,\delta \in Spec_ R(A)\), there is \(h\in A\) such that h(\(\gamma)\)\(\geq t(\gamma)\) and h(\(\delta)\)\(\leq t(\delta).\)
(c) For any \(t\in PW(A)\) and \(\gamma,\delta \in Spec_ R(A)\), \(t_{\gamma}-t_{\delta}\in <\gamma,\delta >\subseteq A\) is the ideal generated by all functions \(a\in A\) with a(\(\gamma)\)\(\geq 0\) and a(\(\delta)\)\(\leq 0.\)
Then the author applies this characterization to show that fields and Dedekind domains satisfy the Pierce-Birkhoff conjecture, improving a result of C. N. Delzell. polynomial functions; Pierce-Birkhoff conjecture; separating functions of points of the real spectrum; Dedekind domains J. Madden , Pierce-Birkhoff rings , Archiv der Math. \((\)Basel\()\) 53(6) ( 1989 ), 565 - 70 . MR 1023972 | Zbl 0691.14012 Real algebraic and real-analytic geometry, Polynomials over commutative rings, Relevant commutative algebra Pierce-Birkhoff rings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In the paper under review the authors study the so-called interpolation problem for fat point schemes in \(\mathbb{P}^{1} \times \mathbb{P}^{1}\). A very natural motivation for this study is the celebrated Alexander-Hirschowitz theorem which provides us a complete classification of ideals of double points in general position in the \(n\)-dimensional complex projective space. In particular, this classification contains all the cases when double points fail to impose independent conditions on hypersurfaces of some degree.
Let \(S = \mathbb{C}[x_{0},x_{1},y_{0},y_{1}] = \bigoplus_{i,j} S_{i,j}\) be the bigraded coordinate ring of \(\mathbb{P}^{1} \times \mathbb{P}^{1}\). Let \(\mathcal{P} = \{P_{1}, \dots, P_{s}\}\) be a set of points in \(\mathbb{P}^{1} \times \mathbb{P}^{1}\) in general position and for each point \(P_{i}\) we denote by \(\mathfrak{p}_{i}\) the prime ideal defining that point. The scheme of fat points of multiplicity \(m\geq 1\) with the support at \(\mathcal{P}\) is the scheme \(\mathbb{X}\) defined by the ideal \(I_{\mathbb{X}} = \mathfrak{p}_{1}^{m} \cap \dots \cap \mathfrak{p}_{s}^{m}\). Now for any bihomogeneous ideal \(I\) in \(S\), we define the Hilbert function of \(S/I\) as
\[\mathrm{HF}_{S/I}(a,b) : = \dim_{\mathbb{C}}(S/I)_{(a,b)} = \dim_{\mathbb{C}}S_{(a,b)} -\dim_{\mathbb{C}} I_{(a,b)} \quad \text{for} \quad (a,b) \in \mathbb{N}^{2}.\]
For short, we denote by \(\mathrm{HF}_{\mathbb{X}}\) be the Hilbert function of the quotient ring \(S / I_{\mathbb{X}}\).
The key question which gives a motivation for the paper under review can be formulated as follows.
Question: Let \(\mathbb{X}\) be a scheme of fat points of multiplicity \(m\) in \(\mathbb{P}^{1} \times \mathbb{P}^{1}\). What is the bigraded Hilbert function of \(\mathbb{X}\)?
The main contribution of the present paper is the following general result.
Theorem A. Let \(a\geq b\) and we assume that \(b\geq m\). Consider the fat point scheme \(\mathbb{X} = mP_{1} +\dots+ mP_{s} \subset \mathbb{P}^{1} \times \mathbb{P}^{1}\). Then
\[\mathrm{HF}_{\mathbb{X}}(a,b) =\min \bigg\{ (a+1)(b+1), s\binom{m+1}{2} -s\binom{m-b}{2}\bigg\},\]
except if \(s = 2k+1\) and \(a=bk+c+s(m-b)\) with \(c = 0,\dots, b-2\), where
\[\mathrm{HF}_{\mathbb{X}}(a,b) = (a+1)(b+1) - \binom{c+2}{2}.\]
Moreover, in the case of triple points, the authors are able to provide a complete description of the associated Hilbert functions.
Theorem B. Let \(\mathbb{X} = 3P_{1} + \dots + 3P_{s} \subset \mathbb{P}^{1} \times \mathbb{P}^{1}\). Then
\[\mathrm{HF}_{\mathbb{X}}(a,b) =\min\{(a+1)(b+1),6s\},\]
except for the following situations:
1) \(b=1\) and \(s < \frac{2}{5}(a+1)\), where \(\mathrm{HF}_{\mathbb{X}}(a,1) = 5s\);
2) \(s=2k+1\) and
i) \((a,b) =(4k+1,2)\), where \(\mathrm{HF}_{\mathbb{X}}(4k+1,2) = (a+1)(b+1) - 1\);
ii) \((a,b) = (3k,3)\), where \(\mathrm{HF}_{\mathbb{X}}(3k,3) = (a+1)(b+1)-1\);
iii) \((a,b) = (3k+1,3)\), where \(\mathrm{HF}_{\mathbb{X}}(3k+1,3) = 6s-1\);
3) \(s=5\) and \((a,b)=(5,4)\), where \(\mathrm{HF}_{\mathbb{X}}(5,4)=29\). fat point schemes; bihomogeneous ideals; projective spaces Divisors, linear systems, invertible sheaves, Multilinear algebra, tensor calculus, Graded rings, Syzygies, resolutions, complexes and commutative rings, Classical problems, Schubert calculus On the Hilbert function of general fat points in \(\mathbb{P}^1\times\mathbb{P}^1\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Hausel and Rodriguez-Villegas (2015, \textit{Astérisque} 370, 113-156) recently observed that work of Göttsche, combined with a classical result of Erdös and Lehner on integer partitions, implies that the limiting Betti distribution for the Hilbert schemes \((\mathbb{C}^2)^{[n]}\) on \(n\) points, as \(n\rightarrow +\infty\), is a \textit{Gumbel distribution}. In view of this example, they ask for further such Betti distributions. We answer this question for the quasihomogeneous Hilbert schemes \(((\mathbb{C}^2)^{[n]})^{T_{\alpha ,\beta}}\) that are cut out by torus actions. We prove that their limiting distributions are also of Gumbel type. To obtain this result, we combine work of Buryak, Feigin, and Nakajima on these Hilbert schemes with our generalization of the result of Erdös and Lehner, which gives the distribution of the number of parts in partitions that are multiples of a fixed integer \(A\geq 2\). Furthermore, if \(p_k(A;n)\) denotes the number of partitions of \(n\) with exactly \(k\) parts that are multiples of \(A\), then we obtain the asymptotic
\[
p_k(A,n)\sim \frac{24^{\frac k2-\frac14}(n-Ak)^{\frac k2-\frac34}}{\sqrt2\left(1-\frac1A\right)^{\frac k2-\frac14}k!A^{k+\frac12}(2\pi)^k}e^{2\pi\sqrt{\frac1{6}\left(1-\frac1A\right)(n-Ak)}},
\]
a result which is of independent interest. Betti numbers; Hilbert schemes; partitions Parametrization (Chow and Hilbert schemes), (Co)homology theory in algebraic geometry, Analytic theory of partitions Limiting Betti distributions of Hilbert schemes on \(n\) points | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In order to apply nonstandard methods to modern algebraic geometry, as a first step in this paper we study the applications of nonstandard constructions to category theory. It turns out that many categorical properties are well behaved under enlargements.
From the introduction: The plan of the paper is as follows: In the second chapter, we begin by introducing ``\(\widehat{S}\)-small'' categories for a superstructure \(\widehat{S}\) (categories whose morphisms form a set which is an element of \(\widehat{S}\)) and functors between these. Given an enlargement \(\widehat{S}\to \widehat{{}^*S}\), we describe how to ``enlarge'' \(\widehat{S}\)-small categories and functors between them: we assign an \(\widehat{{}^*S}\)-small category \({}^*{\mathcal C}\) to every \(\widehat{S}\)-small category \({\mathcal C}\) and a functor \({}^*F:{}^*{\mathcal C}\to {}^*{\mathcal D}\) to every functor \(F:{\mathcal C}\to{\mathcal D}\) of \(\widehat{S}\)-small categories. Then we show that most basic properties of given \(\widehat{S}\)-small categories and functors between them, like the existence of certain finite limits or colimits, the representability of certain functors or the adjointness of a given pair of functors, carry over to the associated enlargements.
In the third chapter, we study the enlargements of filtered and cofiltered categories and the enlargements of filtered limits and colimits. Using the ``saturation principle'' of enlargements, we will be able to prove that all such limits in an \(\widehat{{}^*S}\)-small category \({\mathcal C}\) are ``dominated'' (in a sense that will be made precise) by objects in \({}^*{\mathcal C}\).
In the fourth chapter, we study \(\widehat{S}\)-small additive and abelian categories (and additive functors between them), and their enlargements (which will be additive respectively abelian again), and we give an interpretation of the enlargement of a category of \(R\)-modules (for a ring \(R\) that is an element of \(\widehat{S}\)) as a category of internal \({}^*R\)-modules.
The fifth chapter is devoted to derived functors between \(\widehat{S}\)-small abelian categories. We will see that taking the enlargement is compatible with taking the \(i\)-th derived functor (again in a sense that will be made precise).
In the sixth chapter, we look at triangulated and derived \(\widehat{S}\)-small categories and exact functors between them and study their enlargements. The enlargement of a triangulated category will be triangulated again, the enlargement of a derived functor will be exact, and we prove several compatibilities between the various constructions.
In the seventh chapter, we add yet more structure and study \(\widehat{S}\)-small fibred categories and \(\widehat{S}\)-small additive, abelian and triangulated fibrations and their enlargements. This is important for most applications we have in mind. applications of nonstandard constructions to category theory; enlargements; superstructure; small categories; limits; colimits; derived functors; abelian categories; derived categories; exact functors; triangulated category; fibred categories; triangulated fibrations Brünjes, Lars; Serpé, Christian: Enlargements of categories, Theory appl. Categ. 14, 357-398 (2005) Nonstandard models in mathematics, Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.), Derived categories, triangulated categories, Foundations of algebraic geometry Enlargements of categories | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The aim of this paper is to prove two solvability theorems for the multidimensional moment problem based on growth condition. The first one deals with the Stieltjes moment problem:
Suppose $L$ is a linear functional on \(\mathbb{R}[x_1,\dots,x_d]\) such that \(L(p^2)\ge 0\) and \(L(x_jp^2)\ge 0\) for \(p \in \mathbb{R}[x_1,\dots,x_d]\) and \(j=1,\dots,d\) and
\[
\sum_{n=1}^{\infty}L(x_j^n)^{-1/2n}=\infty \text{ for } j=1,\dots,d.
\]
Then \(L\) is a Stieltjes moment functional which has a unique representing measure.
This result is applied to derive the second theorem on \(\mathcal{K}\)-moment problem for unbounded semi-algebraic sets. More precisely, the author proves:
Let \(f=\{f_1,\dots,f_m\}\) be a finite set of polynomials that generate the polynomial algebra \(\mathbb{R}[x_1,\dots,x_d]\) and \(L\) is a linear functional \(\mathbb{R}[x_1,\dots,x_d]\) such that \(L(p^2)\ge 0\) and \(L(f_jp^2)\ge 0\) for \(p \in \mathbb{R}[x_1,\dots,x_d]\) and \(j=1,\dots,m\) and
\[
\sum_{n=1}^{\infty}L(f_j^n)^{-1/2n}=\infty \text{ for } j=1,\dots,m.
\]
Then \(L\) is a moment functional which has a unique representing measure. Furthermore, the measure is supported on the basic closed semi-algebraic \(\mathcal{K}(f)\) associated to~\(f\),
\[
\mathcal{K}(f)=\{ x \in \mathbb{R}^d \; : \; f_1(x)\ge 0, \dots, f_m(x)\ge0\}.
\] semi-algebraic set; moment problem; Stieltjes vector Moment problems, Linear symmetric and selfadjoint operators (unbounded), Semialgebraic sets and related spaces The Stieltjes condition and multidimensional \({\mathcal{K}} \)-moment problems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The setting of the article under review is twofold: (i) a smooth algebraic stack \(X\) with a smooth divisor \(D \subset X\), (ii) a degeneration \(\pi : W \to B\) over a curve with special fibre given by the union of two smooth algebraic stacks meeting transversally along a smooth divisor.
In the setting (i) the authors construct the relative configuration space \(X_D^{[n]}\), which is a natural compactification of the cartesian product \((X \setminus D)^n\). In the setting (ii) they construct the stack \(W^{[n]}_\pi\), which compactifies the space of \(n\) ordered points on the smooth fibres of \(\pi\). Their methods rely on the universal approach to expanded pairs and expanded degenerations, developed in [\textit{D. Abramovich} et al., Commun. Algebra 41, No. 6, 2346--2386 (2013; Zbl 1326.14020)]. If the spaces in (i) are schemes, an alternative construction of \(X_D^{[n]}\) already appeared in [\textit{B. Kim} and \textit{F. Sato}, Sel. Math., New Ser. 15, No. 3, 435--443 (2009; Zbl 1177.14029)].
In the second part of the paper, the authors apply their constructions to prove the properness of the stack \(K_{\mathrm{pd}}\) of predeformable expanded stable maps in the sense of \textit{J. Li} [J. Differ. Geom. 57, No. 3, 509--578 (2001; Zbl 1076.14540)], in both the settings (i) and (ii). When the spaces in (i) and (ii) are schemes, the properness of \(K_{\mathrm{pd}}\) was already proved in [loc. cit.] with different methods. This kind of properness result is the first ingredient in the statement and the proof of degeneration formulas in Gromov-Witten theory. moduli spaces; Gromov-Witten invariants; algebraic stacks; configuration spaces Stacks and moduli problems, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Configurations of points on degenerate varieties and properness of moduli spaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors describe the construction of an algebra of functions from a combinatorial gadget called a consistent wall structure. Versions of such structures have appeared particularly in the context of mirror symmetry -- see [\textit{M. Kontsevich} and \textit{Y. Soibelman}, Prog. Math. 244, 321--385 (2006; Zbl 1114.14027); \textit{M. Gross} and \textit{B. Siebert}, Ann. Math. (2) 174, No. 3, 1301--1428 (2011; Zbl 1266.53074); \textit{M. Gross} et al., Publ. Math., Inst. Hautes Étud. Sci. 122, 65--168 (2015; Zbl 1351.14024)].
A wall structure is a collection of walls which are codimension one polyhedral subsets called walls in an integral affine manifold with singularities \(B\), decorated by functions. Given a wall structure, the authors describe combinatorial objects called ``broken lines'' which in rough terms are unions of line segments in \(B\) that may bend when crossing a wall. Using such broken lines, the authors define what it means for a wall structure to be consistent. In this paper, the authors define consistent wall structures in a big generality and consider not only integral affine manifolds but more general pseudomanifolds on which wall structures are supported.
The authors explain given a consistent wall structure how to construct an affine formal family of varieties from it, whose ring of functions admit a basis of so called theta functions indexed by integral tropical points of the pseudo-manifold on which the wall structure is defined.
When the consistent wall structure is conical affine they construct a polarized projective formal family whose homogeneous coordinate ring admit a basis of theta functions. For particularly simple consistent wall structures one recovers Mumford degenerations of abelian varieties and in this situation the theta functions generating the homogeneous coordinate ring coincide with the classical theta functions of abelian varieties.
Gross-Siebert in their aforementioned previous work have constructed consistent wall structures in the context of mirror symmetry for toric degenerations. As an application in the current paper, the authors show that the homogeneous coordinate ring of the mirror toric degeneration admits a canonical basis of theta functions.
The general construction of theta functions from a consistent wall structure plays key role in mirror symmetry [\textit{M. Gross} and \textit{B. Siebert}, Invent. Math. 229, No. 3, 1101--1202 (2022; Zbl 1502.14090)]. mirror symmetry; theta functions; wall structures Research exposition (monographs, survey articles) pertaining to algebraic geometry, Mirror symmetry (algebro-geometric aspects), Calabi-Yau manifolds (algebro-geometric aspects), Fano varieties Theta functions on varieties with effective anti-canonical class | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Here the author gives several uper bounds --- both new and classical ones (but with a new proof) for the dimension of the tangent space of the Hilbert scheme of a subvariety \(X\) of a fixed variety \(V\subset\mathbb{P}^ n\); usually \(X\) is a linear space or at least a complete intersection in \(\mathbb{P}^ n\). Sample result:
Theorem: Set \(\delta(r,m):=(m+1)(r-m)\); let \(X\) be an integral complete intersection in \(\mathbb{P}^ n\) with \(m=\dim(X)\); let \(V\) be an integral variety with \(X\subset V\subset\mathbb{P}^ n\), \(X\cap V_{reg}\neq\emptyset\), and \(r=\dim(V)\); let \(t_{X,V}\) the dimension of the tangent space at \(X\) of the Hilbert scheme of subvarieties of \(V\). Let \(L\hat{}\) be the intersection of all tangent spaces to \(V\) (seen as subspaces of \(\mathbb{P}^ n)\) at the points of \(X\cap V_{reg}\). Then \(t_{X,V}\leq\delta(r,m)\) and equality holds if and only if \(\dim(L\hat {})=r\) (i.e. the tangent spaces are the same at all such points). Assume \(X\) linear; if \(t_{X,V}=\delta(r,m)-m\), then \(\dim(L\hat {})=r-1\); if \(\dim(L\hat {})=r-1\), then \(\delta(r,m)-1\leq t_{X,V}\leq \delta(r,m)\).
The proofs use the comparison between the deformation functor of \(X\) in \(V\) and of their affine cones and explicit computations (using the very simple cohomology of \(X\)) of the tangent space to the second functor. dimension of the tangent space of the Hilbert scheme; subvariety; deformation functor; affine cones Parametrization (Chow and Hilbert schemes), Formal methods and deformations in algebraic geometry, Projective techniques in algebraic geometry, Divisors, linear systems, invertible sheaves Some observations on the singularities of the Hilbert scheme which parametrizes the linear varieties contained in a projective variety | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors prove the following
Theorem. Let \(M\) and \(N\) be two compact, connected, closed, irreducible graph 3-manifolds with infinite fundamental group. Let \(f:M\to N\) be a homology equivalence (all homologies are with integer coefficients) such that for any finite covering of \(N\) (regular or not), the induced map \({\widetilde f} :{\widetilde M}\to {\widetilde N}\) is still a homology equivalence. Then \(f\) is homotopic to a homeomorphism.
As an application, they prove the following
Corollary: Let \(M\) and \(N\) be as above and suppose that there exists a cobordism \(W^4\) between \(M\) and \(N\) such that (i) \(\pi_1(N)\to \pi_1(W)\) is surjective; (ii) \(W\) is obtained from \(N\) by adding handles of index \(\leq 2\); (iii) the inclusions \(M\to W\) and \(N\to W\) are homology equivalences. Then, \(M\) and \(N\) are homeomorphic.
The authors say that these results were motivated by the ``\(\mu\)-constant problem in complex dimension 3'', and they give a review of this problem. They recall in particular a theorem due to Lê Dung Trang and C. P. Ramanujan, stating that if \(f_t:(U,0)\subset (C^n,0)\to (C,0)\) is a one-parameter family of holomorphic functions with isolated singularity at \(0\), for each \(t\in V(0)\subset R\), \(C^{\infty}\) in the \(t\) variable, with the same Milnor number at \(0\) (such a family is called a \(\mu\)-constant family) then, for \(n\geq 3\), \(f_0\) and \(f_t\) have the same topological type for each \(t\) in a neighborhood \(W(0)\subset V(0)\subset R\). The results of this paper imply that the Lê-Ramanujan theorem holds also in dimension 3, provided that \(\pi_1(K_t)\to \pi_1(W)\) is surjective. graph manifold; complex algebraic surface; singularity; Seifert fibered space; covering space; homology equivalence; homeomorphism; Milnor fiber; \(\mu\) constant problem B. Perron and P. Shalen, Homeomorphic graph manifolds: a contribution to the \(\mu \) constant problem, Topology Appl. 99 (1999), no. 1, 1-39. Topology of general 3-manifolds, \(h\)- and \(s\)-cobordism, Covering spaces and low-dimensional topology, Singularities of surfaces or higher-dimensional varieties Homeomorphic graph manifolds: A contribution to the \(\mu\) constant problem | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author considers the Monge-Ampère type equation of bidegree \((n-1,n-1)\) on a compact complex manifold with \(\dim_{\mathbb C}X=n>2\). Let \(\omega\) be a Gauduchon metric on \(X\) which means that \(\partial\overline{\partial}\omega^{n-1}=0\). Then the author considers the equation
\[
[ (\omega^{n-1}+i\partial\overline{\partial}\phi\wedge \omega^{n-2}+\frac i2(\partial\phi\wedge \overline{\partial}\omega^{n-2}-\overline{\partial}\phi\wedge \partial\omega^{n-2}))^{\frac1{n-1}}]^n=e^f\omega^n\eqno{(\ast)}
\]
subject to the positivity and normalization conditions
\[
\omega^{n-1}+i\partial\overline{\partial}\phi\wedge \omega^{n-2}+\frac i2(\partial\phi\wedge \overline{\partial}\omega^{n-2}-\overline{\partial}\phi\wedge \partial\omega^{n-2})>0
\]
and \(\sup_X\phi=0\) where for a \((n-1,n-1)\)-form \(\Gamma\), the form \(\Gamma^{\frac{1}{n-1}}\) is the unique \((1,1)\)-form \(\gamma\) such that \(\Gamma=\gamma^{n-1}\). The author proves that the solution \(\phi\) to \((\ast)\) under the positivity and normalization conditions is unique. The author also proves that the principal part of the linearization of Equation \((\ast)\) is the Laplacian associated with a certain Hermitian metric on \(X\). The investigation of \((\ast)\) is associated with the question to which extent there is an Aeppli-Gauduchon analogue of Yau's theorem of the Calabi conjecture. In the paper the author also studies the Bott-Chern and Aeppli cohomologies in particular on \(\partial\overline{\partial}\) complex manifolds. He also introduces the Gauduchon and sG cones in \(H_A^{n-1,n-1}(X,\mathbb R)\) and studies their properties. positivity in bidegree \((n-1,n-1)\); Gauduchon and SG cones; duality between the Bott-Chern and Aeppli cohomologies; equation of the Monge-Ampére type in bidegree \((n-1,n-1)\) Popovici, D.: Aeppli cohomology classes associated with Gauduchon metrics on compact complex manifolds. Bulletin de la Société Mathématique de France \textbf{143}(4), 763-800 Global differential geometry of Hermitian and Kählerian manifolds, Calabi-Yau theory (complex-analytic aspects), Deformations of complex structures, Transcendental methods, Hodge theory (algebro-geometric aspects) Aeppli cohomology classes associated with Gauduchon metrics on compact complex manifolds | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a smooth projective curve of genus \(g\geq 2\) over \(\mathbb{C}\). Recently \textit{Sh.-W. Zhang} [Invent. Math. 179, No. 1, 1--73 (2010; Zbl 1193.14031)] and \textit{N. Kawazumi} [``Johnson's homomorphisms and the Arakelov-Green function'', preprint, \url{arxiv:0801.4218}] introduced independently a new real-valued invariant \(\phi(X)\) of \(X\), which is defined as follows. Choose an orthonormal basis \((\eta_1,\cdots,\eta_g)\) of \(H^0(X,\omega_X)\) with respect to the natural hermitian inner product, and put \(\mu_X=\frac{i}{2g}\sum_{k=1}^{g}\eta_k\wedge \bar{\eta}_k\). Let \(\Delta_{\mathrm{Ar}}\) be the Laplacian on \(L^2(X,\mu_X)\) determined by the equality \(\frac{\partial\bar{\partial}}{\pi i}f=\Delta_{\mathrm{Ar}}(f)\cdot\mu_X\), and let \((\phi_\ell)_{\ell=0}^{\infty}\) be an orthonormal basis of real eigenfunctions of \(\Delta_{\mathrm{Ar}}\) with eigenvalues \(0=\lambda_0<\lambda_1\leq \lambda_2\leq\cdots\). The above mentioned invariant \(\phi(X)\), called the \(\phi\)-invariant, is defined by the formula \(\phi(X)=\sum_{\ell >0}\frac{2}{\lambda_{\ell}}\sum_{m.n=1}^g|\int_X\phi_{\ell}\cdot\eta_m\wedge\bar{\eta}_n|^2\). In this paper the author determines its behavior in a neighborhood of the boundary of the moduli space \(\mathcal{M}_g\) of complex curves of genus \(g\) in the Deligne-Mumford compactification \(\overline{\mathcal{M}}_g\), and he calculates \(\phi\) for hyperelliptic curves. moduli space of curves; Hodge structure; hyperelliptic curves de Jong, R., Second variation of Zhang's \({\lambda }\) -invariant on the moduli space of curves, Am. J. Math., 135, 275-290, (2013) Families, moduli of curves (algebraic), Coverings of curves, fundamental group Second variation of Zhang's \(\lambda \)-invariant on the moduli space of curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper discuses the Ihara zeta functions of regular graphs and certain coverings of regular graphs. As a warm-up example, we calculate the zeta functions of the platonic solids. We outline our recent results, describing a ramified covering of graphs \(Y\to X\), where \(Y\) is formed by identifying a single vertex on \(k\) copies of a \((q+1)\)-regular graph \(X\). We find that the reciprocal of the zeta function for \(X\) ``almost divides'' the reciprocal of the zeta function for \(Y\), in the sense that the reciprocal of the zeta function of \(X\) divides the product of the reciprocal of the zeta function of \(Y\) and some polynomial of bounded degree (which depends only on the graph \(X\)). Two specific examples show that in fact ``almost divisibility'' is the best that can be hoped for. These coverings give examples of the zeta functions of infinite families of non-regular graphs. Non-regular complete bipartite graphs are also considered, and we provide a new proof of Hashimoto's result on the zeta functions of these graphs. Graphs and linear algebra (matrices, eigenvalues, etc.), Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Ramified covers of graphs and the Ihara zeta functions of certain ramified covers | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The article contributes to the discussion on resolution of singularities
in characteristic \(p>0\). Examples are given where after blowing ups centered
at points, certain singularities are transformed into a kind of cycle
leading to an infinite increase of the residual order.
Residual order
is a candidate for adapting the string of invariants originating
from the classical inductive characteristic-zero resolution procedure
to positive characteristic: The authors announce a new proof for
the embedded resolution of surfaces which makes use of it [\textit{H. Hauser} and \textit{S. Perlega}, ``A new proof for the embedded resolution of surface singularities'', manuscript (2018)].
In higher dimension the situation appears to be different.
Let \(f(z,x)=z^{p^e} + F(x_1,\dots ,x_n)\) be purely inseparable, \(e>0\),
\(F\) a formal power series having
\(\mathrm{ord} (F)\geq p^e\). A change of parameters \(z_1=z-g(x_1,\dots ,x_n)\)
allows to replace
\(f\) by \(z_1^{p^e} + g(x_1,\dots , x_n)^{p^e} + F(x_1,\dots ,x_n)\)
and thus to remove
any \(p^e\)-th powers in the expansion of \(F\). This procedure is refered to
as \textit{cleaning} of \(F\) and is supposed to be done now.
Let \(E\) be a strictly normal crossing divisor given as the product of \(x_i\),
\(i\in \Delta \) where \(\Delta \subseteq \{ 1, \dots ,n\}\).
Write \(F=\prod _{i\in \Delta} x_i^{r_i}\cdot G(x_1,\dots ,x_n)\) such that
\(r_i=\mathrm{ord}_{(x_i)} (F)\). Then the residual order is defined to be
\(\mathrm{residual.order}_E(f):= \mathrm{ord} (G)\).
The authors give examples for the increase of \(\mathrm{residual.order}_E(f)\)
after consecutive blowing up of points, where the shape of the
equation remains the same
(apart from the increase of exponents). Thus the residual order can attain
arbitrarily large values. Calculations are done explicitely for certain equations of the following type:
(1)
\(p=2\), \(\mathrm{ord} (f) = 8\) in dimension 5,
(2)
\(p\geq 3\), \(\mathrm{ord} (f) = p^3\) in dimension 4.
The examples are not intended to
disprove the existence of a resolution in positive
characteristic since admissible centers of the blowing ups may have
positive dimension as well. resolution of singularities in positive characteristic; residual order; examples for infinite increase of residual order Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Positive characteristic ground fields in algebraic geometry, Formal power series rings Cycles of singularities appearing in the resolution problem in positive characteristic | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems \textit{N. Hindman} and \textit{I. Leader} [Semigroup Forum 59, No. 1, 33--55 (1999; Zbl 0942.22003)]
first introduced the notion of the semigroup of ultrafilters converging to zero for a dense subsemigroup of \(((0, \infty), +)\). Using the algebraic structure of the Stone-Čech compactification, \textit{M. A. Tootkaboni} and \textit{T. Vahed} [Topology Appl. 159, No. 16, 3494--3503 (2012; Zbl 1285.54017)]
generalized and extended this notion to an idempotent instead of zero, that is a semigroup of ultrafilters converging to an idempotent \(e\) for a dense subsemigroup of a semitopological semigroup \((R, +)\) and they gave the combinatorial proof of the Central Sets Theorem near \(e\). Algebraically one can define quasi-central sets near \(e\) for dense subsemigroups of \((R, +)\). In a dense subsemigroup of \((R, +)\), C-sets near \(e\) are the sets, which satisfy the conclusions of the Central Sets Theorem near \(e\). \textit{S. K. Patra} [Topology Appl. 240, 173--182 (2018; Zbl 1392.37008)]
gave dynamical characterizations of these combinatorially rich sets near zero. In this paper, we shall establish these dynamical characterizations for these combinatorially rich sets near \(e\). We also study minimal systems near \(e\) in the last section of this paper. Stone-Čech compactification; dense subsemigroup; idempotent; dynamical system; uniform recurrence near idempotent; proximality near idempotent Notions of recurrence and recurrent behavior in topological dynamical systems, Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.), Dynamics in general topological spaces, Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.), Čech types, Compactness, Extensions of spaces (compactifications, supercompactifications, completions, etc.), Compactifications; symmetric and spherical varieties, Structure of topological semigroups Dynamics near an idempotent | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Consider a system in Lax form, \(dA/dt=[A,B]\) where \(A\) and \(B\) are \((r\times r)\) matrix functions, polynomial in a variable \(x(t)\). The coefficient \(A_d\) of the leading term \(A_dx^d\) in the \(x\) polynomial defining \(A\) is a constant of motion. Define the function \(Q(x,y)=\det [A(x)-yI]\), and the isospectral manifold
\[
V_Q^M= \bigl\{A(x): Q(x,y)=0,\;A_d=M\bigr\}
\]
Consider the affine curve \(X=\{(x,y)\in \mathbb{C}^2: Q(x,y)=0\}\) and its compactification \(\overline X\), called the spectral curve for \(A\). Also, denote as \(\mathbb{P}_M\) the centralizer of \(M\) in the projective group \(\mathbb{P} GL_r (\mathbb{C})\).
If \(X\) is smooth, then so is \(V_Q^M\), and a number of results are available in the literature. In particular, \textit{P. van Moerbeke} and \textit{D. Mumford} [Acta Math. 143, 93-154 (1979; Zbl 0502.58032] showed that, in this case, the quotient \(V_Q^M/ \mathbb{P}_M\) corresponds to a Zariski open subset in the Jacobian \(J(X)\); and that if \(X\) is non-smooth, \(V_Q^M/\mathbb{P} M\) corresponds to a Zariski open subset in a generalized Jacobian. Work by \textit{M. R. Adams}, \textit{J. Harnad} and \textit{J. Hurtubise} [Commun. Math. Phys. 134, 555-585 (1990; Zbl 0717.58051)] and by \textit{A. Beauville} [Acta Math. 164, No. 3/4, 211-235 (1990; Zbl 0712.58031)] showed that \(V_Q^M/ \mathbb{P}_M\) can also be described in terms of an affine part of the standard Jacobian \(J(X)\).
Assume \(X\) is smooth and at \(q_i\in \overline X\), with \(x(q_i)=\infty\), \(\overline X\) is locally a normal crossing of several branches. Denote by \({\mathcal F}=\{(\mathbb{C}^r,M)\), \((E,K)\}\) the data at infinity of \(\overline X\), with \(M\in\text{End} (\mathbb{C}^r)\), \(E\) a vector space and \(K\in\text{End}(E)\); we denote by \(\pi\) the projection on \(E\). Assume \(M\) is diagonalizable, and \(K\) is diagonalizable with distinct eigenvalues. With this, one can introduce
\[
{\mathcal M}_Q^{\mathcal F}=\bigl\{A\in V_Q^M:[\pi A_{d-1}]_E= K\bigr\}.
\]
Then the author considers the (singular) curve \(Y\) obtained from the smooth compactification of \(X\) by identifying the ``infinite'' points on \(X\), and denote by \(\Theta(Y)\) the theta divisor formed by special line bundles on \(Y\) of degree equal to the arithmetic genus of \(Y\).
The main result of the paper is that the manifold \({\mathcal M}^{\mathcal F}_Q\) is smooth and bi-holomorphic to a Zariski open subset of the generalized Jacobian \(J(Y)-\Theta(Y)\). integrable systems; spectral curve; Lax form; isospectral manifold Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Relationships between algebraic curves and integrable systems, Jacobians, Prym varieties, Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics Jacobians of singular spectral curves and completely integrable systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\mathbb{P}^n\) be the \(n\)-dimensional projective space (over a field of characteristic \(0\)). Consider \(r\) general points \(p_1, \dots, p_r \in \mathbb{P}^n\) and a \(r\)-uple \(m=(m_1,\dots, m_r)\) of positive integers. It is a classical task to compute the dimension of the linear system \(V^n(d,m)\) of hypersurfaces of degree \(d\) passing by \(p_i\) with multiplicity at least \(m_i\) (\(i=1, \dots, r\)). This computation is, in fact, a matrix rank computation and the paper under review (extending previous methods by Dumnicki) proposes a method for it based on partitioning the monomial basis of the vector space \(H^0({\mathcal O}_{\mathbb{P}^n}(d))\) of degree \(d\) homogeneous polynomials.
Several applications of the method are described:
(1) There is a natural \textit{expected dimension} for these linear systems, say:
\[
\dim V ^n(d,m)=\max \{ {d+n \choose n} -\Sigma{m_i+n-1 \choose n}, 0\}
\]
and \((n,d,m)\) is \textit{special} when the dimension of \(V^n(d,m)\) fails to be the expected one. In Section 4 new algorithms for determining speciality are provided.
(2) The \textit{multipoint Seshadri constant} of \(\mathbb{P}^2\) at \(p_1, \dots, p_r\) is the infimum over the set of plane curves \(C\) of the quotients of the degree of \(C\) over the sum of the mutliplicities of \(C\) at \(p_i\). New lower bounds on multipont Seshadri constants of \(\mathbb{P}^2\) are produced (see Figure 5.2).
(3) A classical conjecture in the field is Nagata's one: if \(m_1=\dots =m_r=M\) and \(r>9\) then \(d^2 \leq rM^2\) implies \(\dim V^2(d,m)=0\). This conjecture was generalized to any dimension by Iarrobino (see Section 5.2 for details) and is known to be true in the perfect-power cases (cf. \textit{L. Evain} [J. Algebra 285, No. 2, 516--530 (2005; Zbl 1077.14012)]). A new proof of this result is also given. linear systems; fat points; Seshadri constants Paul, S, New methods for determining speciality of linear systems based at fat points in \({\mathbb{P}}^n\), J. Pure Appl. Algebra, 217, 927-945, (2013) Divisors, linear systems, invertible sheaves New methods for determining speciality of linear systems based at fat points in \(\mathbb P^n\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author studies the ramification sets of finite analytic mappings and applications of his results and methods to punctual Hilbert schemes and to finite analytic maps. He uses essentially the technique of ``chaining'' which consists in associating to a finite map a sequence (or chain) of sets, which are components of ramification loci of increasing multiplicity, and then in controlling their dimensions. For that purpose he uses a theorem of Grothendieck about the order of connexity of subgerms of an irreducible analytic germ, and also in the projective case the Fulton-Hansen theorem and a theorem of Deligne.
In {\S} 1, the author introduces three different notions of multiplicity. The topological multiplicity, the stable multiplicity and the algebraic one. In {\S} 2, he gives a fairly general lower bound to the dimension of the ramification set \(T^{d+1}(f)\), the set of points at which the multiplicity is at least \(d+1\). For the topological case he needs a hypothesis about f, called weak multitransversality which guarantees the additivity of multiplicity under deformation. This theorem is proved by a complicated induction involving multiproducts of ramification sets and the theorem of Grothendieck.
In {\S} 3, the author gives applications of {\S} 2, and of the chaining technique to the punctual Hilbert scheme \(Hilb'{\mathcal O}_{X,x}\) which parametrizes in \(Hilb'(X)\) the punctual schemes concentrated at \(x\in X\). The idea consists in identifying the germ of \(Hilb'({\mathcal O}_{X,x})\) at a smoothable element z with the ramification loci an appropriate map obtained by unfolding the equation of z. He thus obtains a lower bound for the local dimension at z of the open set U of smoothable points in X. This bound is (n-1)(\(\ell -1)\) with \(n=\dim (X)\) in the easiest case (X everywhere irreducible). Various, and more complicated results are obtained when we drop the irreducibility hypothesis or consider instead of U the open set of weakly smoothable (i.e. smoothable in a smooth ambient space) element.
Finally in {\S} 4, the author proves similar results for a finite projective morphism \(f:\quad X^ n\to P^ p.\) He generalizes a previous joint result of himself with Lazarsfeld (case \(n=p)\). This consists again in giving cases of non-emptiness for \(T^{d+1}(f)\) under some complicated numerical conditions. ramification sets of finite analytic mappings; punctual Hilbert schemes; ramification loci of increasing multiplicity T. Gaffney, ''Multiple points, chaining and Hilbert schemes,'' Amer. J. Math., vol. 110, iss. 4, pp. 595-628, 1988. Parametrization (Chow and Hilbert schemes), Singularities in algebraic geometry Multiple points, chaining and Hilbert schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(f: (\mathbb{C}^{n+1},0) \to(\mathbb{C},0)\) be a germ of a holomorphic function with an isolated critical point at the origin, \(V_\varepsilon =f^{-1} (\varepsilon) \cap B_\delta\) -- its Milnor fiber \((0<|\varepsilon |\ll \delta\); \(B_\delta\) is the ball of radius \(\delta\) with center at the origin in \(\mathbb{C}^{n+1})\). The matrix of the intersection form on the homology group \(H_n (V_\varepsilon, \mathbb{Z})\) in a basis of a special type (so-called distinguished basis) or the Dynkin diagram (the graphic representation of this matrix) is of interest for study of hypersurface singularities. For calculation of Dynkin diagrams the method of real morsifications is known. It can be applied only to singularities of functions of two variables (or the case \(n=1)\), and expresses a Dynkin diagram in terms of the geometry of a real plane curve with simple self-intersections.
In this article the generalization of the method of real morsifications for one-dimensional isolated complete intersection singularities is explained. The generalization is almost parallel to the hypersurface case except that a reasonable analogue of a distinguished basis is not a basis of \(H_n (V_\varepsilon, \mathbb{Z})\) but a set of generators. Several concrete examples are given in the latter half. Milnor fiber; Dynkin diagram; real morsifications; complete intersection singularities S.M. Gusein-Zade, ''Dynkin diagrams of some complete intersections and real morsifications,''Tr. Mat. Inst. Russ. Akad. Nauk (in press). Local complex singularities, Milnor fibration; relations with knot theory, Singularities of curves, local rings, Complete intersections, Complex surface and hypersurface singularities, Singularities in algebraic geometry Dynkin diagrams of some complete intersections and real morsifications | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(P_{1},\dots, P_{s}\) be a finite set of \(s\) distinct points in a projective space and let \(m_{1}, \dots, m_{s}\) be positive integers. Consider homogeneous polynomials that vanish at \(P_{i}\) to order \(m_{i}\) for all \(i \in \{1,\dots, s\}\). The set of all such polynomials is the homogeneous ideal \(I_{X}\) of the fat point scheme \(X = \sum_{i=1}^{s} m_{i}P_{i}\). The vector space dimension of the degree \(d\) polynomials in \(I_{X}\) is know if \(d\) is large enough. In geometric terms, we can say that the fat point scheme \(X\) imposes independent conditions on forms of degree \(d>>0\). The least integer \(d\) such that this is true for degree \(d\) forms is called the regularity index \(X\), denoted here by \(r(X)\). It was conjectured that \(r(X) \leq \mathrm{Seg}(X)\), where
\[\mathrm{Seg}(X) := \max\bigg\{ \frac{-1 + \sum_{P_{i} \in L}m_{i}}{\dim L} \, : \, L \subset \mathbb{P}^{n}\text{ is a positive dimensional linear subspace}\bigg\}.\]
The number \(\mathrm{Seg}(X)\) is called the Segre bound. In this extremely interesting paper the authors show that for every fat point scheme \(X\) of some projective space one always has \(r(X) \leq\mathrm{Seg}(X)\). The proof of the authors is based on a new partition result for matroids (Proposition 2.6 therein) that might be of independent interest. fat point schemes; regularity Divisors, linear systems, invertible sheaves, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Configurations and arrangements of linear subspaces, Syzygies, resolutions, complexes and commutative rings, Combinatorial aspects of matroids and geometric lattices Segre's regularity bound for fat point schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Conditional Sums-of-AM/GM-Exponentials (conditional SAGE) is a decomposition method to prove nonnegativity of a signomial or polynomial over some subset \(X\) of real space. In this article, we undertake the first structural analysis of conditional SAGE signomials for convex sets \(X\). We introduce the \(X\)-circuits of a finite subset \({\mathcal{A}}\subset{\mathbb{R}}^n\), which generalize the simplicial circuits of the affine-linear matroid induced by \({\mathcal{A}}\) to a constrained setting. The \(X\)-circuits serve as the main tool in our analysis and exhibit particularly rich combinatorial properties for polyhedral \(X\), in which case the set of \(X\)-circuits is comprised of one-dimensional cones of suitable polyhedral fans. The framework of \(X\)-circuits transparently reveals when an \(X\)-nonnegative conditional AM/GM-exponential can in fact be further decomposed as a sum of simpler \(X\)-nonnegative signomials. We develop a duality theory for \(X\)-circuits with connections to geometry of sets that are convex according to the geometric mean. This theory provides an optimal power cone reconstruction of conditional SAGE signomials when \(X\) is polyhedral. In conjunction with a notion of reduced \(X\)-circuits, the duality theory facilitates a characterization of the extreme rays of conditional SAGE cones. Since signomials under logarithmic variable substitutions give polynomials, our results also have implications for nonnegative polynomials and polynomial optimization. sums of arithmetic-geometric exponentials; positive signomials; exponential sums; sums of nonnegative circuit polynomials (SONC); positive polynomials; multiplicative convexity; log convex sets Real algebraic sets, Polynomial optimization, Nonlinear programming, Combinatorial aspects of matroids and geometric lattices, Convex sets in \(n\) dimensions (including convex hypersurfaces) Sublinear circuits and the constrained signomial nonnegativity problem | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author is interested in studying multiplicity and regularity on an analytic setting and, in this context, he introduces a weaker variation of the concept of blow-spherical equivalence introduced by \textit{A. Fernandes} et al. [Indiana Univ. Math. J. 66, No. 2, 547--557 (2017; Zbl 1366.14051)], which he also names blow-spherical equivalence or equivalence under a blow-spherical homeomorphism. Equivalence under this concept lives strictly between topological equivalence and sub analytic bi-Lipschitz equivalence and, also, between topological equivalence and differential equivalence.
Recently, the author conjectured that if \(X\) and \(Y\) are two real analytic sets included in \(\mathbb{R}^n \) and there is a bi-Lipschitz homeomorphism: \( \varphi: (\mathbb{R}^n, X, 0) \rightarrow (\mathbb{R}^n, Y, 0)\), then the multiplicities at origin of \(X\) and \(Y\) satisfy \(m(X) \equiv m(Y) \bmod 2\). He proved that this conjecture is true when \(n=3\). In this paper, the author proposes the same conjecture but replacing ``bi-Lipschitz'' with ``blow-spherical''. Proposition 5.2, Theorems 5.5, 5.7 and 5.16 and Corollary 5.18 in the paper prove that the conjecture holds whenever \(n \leq 3\) or whenever \(\varphi\) is also image arc-analytic. The author also gives a real analogue of the Gau-Lipman's Theorem.
Finally, and concerning regularity of complex analytic sets, the author proves that if a real analytic set \(X \subseteq \mathbb{R}^n\) is blow-spherical regular at \(0 \in X\), then \(X\) is \(C^1\) smooth at \(0\) if and only if the dimension \(d\) of \(X\) is \(1\). blow-spherical equivalence; real analytic sets; multiplicity Singularities in algebraic geometry, Topology of real algebraic varieties, Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants Multiplicity, regularity and blow-spherical equivalence of real analytic sets | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(f\in k[x_0,\dots,x_m]_d\) be a generic element which can be written as \(f=\sum_{i=1}^{t-2}L_i + L_{t-1}^{d-1}L_t\), where the \(L_i\)'s are linear forms. For \(m\geq2\), \(d\geq 7\) and certain values of \(t\) (\(3\leq t\leq \lfloor {m+d-2\choose m}/(m+1)\rfloor \)), it is proved in this paper that the linear forms \(L_i\) are uniquely determined (up to scalar multiplications).
This can be expressed in a more geometric way, as follows: Let \(X_{m,d}\subset \mathbb{P}^N\), \(N={m+d\choose m}-1\), be the \(d\)-uple Veronese embedding of \(\mathbb{P}^m\) (\(X_{m,d}\) parameterizes the forms \(f\) of degree \(d\) of type \(f=L^d\), for \(L\in k[x_0,\dots,x_m]_1\)) and let \(\tau(X_{m,d})\) be its tangential developable. Consider the variety \(\tau(X_{m,d},t)\) which is the join of \(\tau(X_{m,d})\) and \(t-2\) copies of \(X_{m,d}\); then the previous result amounts to saying that given a generic \(P\in \tau(X_{m,d},t)\), there exist and are uniquely determined (for \(d,m,t\) satisfying the bounds above) points \(P_1,\dots,P_{t-2}\in X_{m,d}\) and a tangent line \(l\) to \(X_{m,d}\), such that \(P\in <P_1,\dots,P_{t-2},l>\).
The proof is based on the possibility of transposing these geometric properties into statements about subschemes of \(\mathbb{P}^m\) and their Hilbert function. Veronese variety; tangential variety; join; polynomial decomposition DOI: 10.1090/S0002-9939-2012-11191-8 Projective techniques in algebraic geometry, Homogeneous spaces and generalizations Symmetric tensor rank with a tangent vector: a generic uniqueness theorem | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This article is a continuation of our earlier work [\textit{C. Chen} et al., J. Symb. Comput. 49, 3--26 (2013; Zbl 1260.14070)], which introduced triangular decompositions of semi-algebraic systems and algorithms for computing them. Our new contributions include theoretical results based on which we obtain practical improvements for these decomposition algorithms.
We exhibit new results on the theory of border polynomials of parametric semi-algebraic systems: in particular a geometric characterization of its ``true boundary'' (Definition 2). In order to optimize these algorithms, we also propose a technique, that we call relaxation, which can simplify the decomposition process and reduce the number of redundant components in the output. Moreover, we present procedures for basic set-theoretical operations on semi-algebraic sets represented by triangular decomposition. Experimentation confirms the effectiveness of our techniques. border polynomial; effective boundary; regular semi-algebraic system; relaxation; triangular decomposition Chen, C., Davenport, J.H., Moreno Maza, M., Xia, B., Xiao, R.: Computing with semi-algebraic sets represented by triangular decomposition. In: Proceedings of ISSAC 2011, pp. 75--82. ACM, New York (2011) Semialgebraic sets and related spaces, Symbolic computation and algebraic computation Computing with semi-algebraic sets represented by triangular decomposition | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This is the second part of the authors, ibid., 109-160 (1997; Zbl 0879.58007), see the review above. It presents numerous results about mappings of cusp type or, in other words, mappings \(F\), which in a suitable nonlinear system of ``coordinates'' \(\mathbb{R}^2\times E\) can be represented in the form \(F(s,t,v) =(s^3-ts,t,v)\); some other singularities are also presented in this part. The account is also based on the abstract global characterization of the cusp maps which was obtained by the authors in 1993.
The contents of this part are: (13) Introduction; (14) Critical values of Fredholm mappings; (15) Applications of critical values to nonlinear differential equations; (16) Factorization of differentiable maps; (17) Local structure of cusps; (18) Some local cusp results (Lazzeri-Micheletti cusp study, Cafagna-Tarantello multiplicity results, Lupo-Micheletti cusp, other local cusp results); (19) von Kármán equation; (20) Abstract global characterization of the cusp map; (21) Mandhyan integral operator cusp map; (22) Pseudo-cusp; (23) Cafagna and Donati theorems on ordinary differential equations (Cafagna-Donati global cusp map, Donati pendulum cusp, Cafasgna-Donati generalized Riccati equation); (24) Micheletti cusp-like map; (25) Cafagna Dirichlet example; (26) \(u^3\) Dirichlet map -- initial results; (27) \(u^3\) Dirichlet map -- the singular set and its image; (28) \(u^3\) Dirichlet map -- the global results; (29) Ruf \(u^3\) Neumann cusp map; (30) Ruf's higher order singularities; (31) Damon's work in differential equations.
The second part of this survey is written with the same accuracy and fullness as the first one; the acquaintence with both parts of this survey is undoubtedly useful to all specialists in the field and all who study Nonlinear Analysis. survey; mappings of cusp type; singularities Church, P. T.; Timourian, J. G.: Global structure for nonlinear operators in differential and integral equations. I. folds; II. Cusps: topological nonlinear analysis. (1997) Differentiable maps on manifolds, Theory of singularities and catastrophe theory, Equations involving nonlinear operators (general), Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) Global structure for nonlinear operators in differential and integral equations. II: Cusps | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper sets out the theory of regular singularities for \( p\)-adic differential modules in the several variables case. Even if regular singularities seem less basic in the \( p\)-adic case than in the complex case, this work is the first (and until now the only one) significant step toward a \( p\)-adic \( D\)-module theory.
Let \( K \) be a complete extension of \( \mathbb{Q}_{p}\), let \( (I_{i})_{1\leq i\leq n} \) be intervals of \( \mathbb{R}^{+}\), let \( I=\prod I_{i} \) and let \( {\mathcal A}_{K}(I) \) be the ring of functions analytic in the (poly)annulus \( {\mathcal C}(I)\)=\(\{(x_{1},\cdots,x_{n});x_{i}\in I_{i}\} \) with coefficients in \( K\). A differential module, namely a free \( {\mathcal A}_{K}(I)\)-module of finite type endowed with an integrable connection, is said to have the Robba property if its local solutions near any point \( t \) in \( {\mathcal C}(I) \) do converge in the maximal polydisk \( (|x_{i}-t_{i}|<|t_{i}|)_{1\leq i<n} \) centered in \( t \) and contained in \( {\mathcal C}(I)\). The basic step in this paper is to give the definition of exponents for differential modules with the Robba property. One of the striking properties of these exponents is their invariance when specializing variables. The main result of the paper is that any differential module with the Robba property whose exponents have non-Liouville differences is Fuchsian, namely has a basis in which the matrices of the \( x_{i} {d \over dx_{i}} \) are constant. Both results and proofs are generalizations of the one variable theory as it is explained in Dwork's paper [\textit{B. Dwork}, J. Reine Angew. Math. 484, 85-126 (1997; Zbl 0870.12008)]. However this generalization is not at all straightforward. It needed, among other, to extend the explicit majoration of Dwork-Robba in the several variables setting and, above all, to prove a \( p\)-adic Hartogs theorem (analyticity with respect to each variable implies global analyticity). regular singularity; Fuchsian differential module; Hartogs theorem Gachet, F.: Structure fuchsienne pour des modules différentiels sur une polycouronne ultramétrique. Rend. sem. Mat. univ. Padova 102, 157-218 (1999) \(p\)-adic differential equations, Modules of differentials, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials Fuchsian structure for differential modules on an ultrametric polyannulus | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(S\) be a hyperelliptic surface defined over an algebraic number field \(k\), and for each finite extension \(K\) of \(k\) and an ample invertible sheaf \({\mathcal L}\) on \(S/K\) let \(N_{\mathcal L} (S(K); M)\) denote the number of \(K\)-rational points \(P\in S(K)\) on \(S\) with \(h_{\mathcal L} (P)\leq M\), where \(M\) is a positive number and \(h_{\mathcal L}\) is the logarithmic height function associated to \({\mathcal L}\). In this paper the authors prove that there exists a finite extension \(k'\) of \(k\) satisfying the following property. For any finite extension \(K\) of \(k'\) and for any ample invertible sheaf \({\mathcal L}\) on \(S/K\), \(N_{\mathcal L} (S(K); M)\) is equal to \(cM^{r/2}+ O(M^{(r-1)/ 2})\) as \(M\to \infty\), where \(r\) is a non-negative integer depending only on \(K\) and \(c\) is a positive number depending on both \(K\) and the equivalence class of \({\mathcal L}\).
As an application the authors also show that the Dirichlet series
\[
Z_{\mathcal L} (S(k); s)= \sum_{P\in S(k)} (\exp\circ h_{\mathcal L} (P))^{-s}
\]
converges absolutely and uniformly for \(\text{Re } (s)\geq \delta\) for any positive number \(\delta\). convergence of Dirichlet series; hyperelliptic surface; logarithmic height Y. MORITA AND A. SATO, Distribution of rational points on hyperelliptic surfaces, Tohoku Math J. 44 (1992), 345-358. Rational points, Elliptic surfaces, elliptic or Calabi-Yau fibrations, Varieties over global fields Distribution of rational points on hyperelliptic surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Consider a finite system of nonstrict real polynomial inequalities and suppose its solution set \(S\subseteq\mathbb R^n\) is convex, has nonempty interior, and is compact. Suppose that the system satisfies the Archimedean condition, which is slightly stronger than the compactness of \(S\). Suppose that each defining polynomial satisfies a second order strict quasiconcavity condition where it vanishes on \(S\) (which is very natural because of the convexity of \(S\)) or its Hessian has a certain matrix sums of squares certificate for negative-semidefiniteness on \(S\) (fulfilled trivially by linear polynomials). Then we show that the system possesses an exact Lasserre relaxation. In their seminal work of 2009, \textit{J. W. Helton} and \textit{J. Nie} [ibid. 20, No. 2, 759--791 (2009; Zbl 1190.14058)] showed under the same conditions that \(S\) is the projection of a spectrahedron, i.e., it has a semidefinite representation. The semidefinite representation used by Helton and Nie arises from glueing together Lasserre relaxations of many small pieces obtained in a nonconstructive way. By refining and varying their approach, we show that we can simply take a Lasserre relaxation of the original system itself. Such a result was provided by Helton and Nie with much more machinery only under very technical conditions and after changing the description of \(S\). moment relaxation; Lasserre relaxation; basic closed semialgebraic set; sum of squares; polynomial optimization; semidefinite programming; linear matrix inequality; spectrahedron; semidefinitely representable set Semidefinite programming, Semialgebraic sets and related spaces, Convex sets in \(n\) dimensions (including convex hypersurfaces), Convex functions and convex programs in convex geometry, Nonconvex programming, global optimization, Real algebra On the exactness of Lasserre relaxations for compact convex basic closed semialgebraic sets | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author shows in this paper that the Lie cohomology for a Lie algebroid over a noetherian separated scheme can be obtained as a derived functor. He then implements this rather general result to obtain two spectral sequences: an Hochschild-Serre type of spectral sequence and a local-to-global type one.
The notion of a Lie algebroid has been of extensive use in mathematics, especially for questions related to the ``higher'' versions of the notion of a Lie algebra: Lie algebroids correspond to infinitesimal Lie groupoids, which are internal groupoids in the category of smooth manifolds. On the algebraic side, the notion of a Lie-Rinehart algebra encodes the same properties of a Lie algebroid: they correspond to Lie algebroids over affine schemes. For general (non affine) schemes Lie algebroids are obtained by gluing along affine patches local Lie-Rinehart algebras. The most remarkable example of a Lie algebroid over a scheme \(X\) is given by the Atiyah algebroid associated with a coherent \(\mathcal{O}_X\)-module \(\mathcal{E}\) determined by the sheaf of first-order differential operators on \(\mathcal{E}\).
For Lie algebroids there is an analogue construction of the Chevalley-Eilenberg complex, defined in a similar fashion as for Lie algebras, that computes and defines Lie algebroid cohomology. However, the expert knows that Lie algebra cohomology can be defined, more canonically, as the derived functor of the functor of invariants. Even more canonically, Lie algebra cohomology is computed by the \(\mathrm{Ext}\) groups in category of module over the (non commutative) enveloping algebra of a Lie algebra. The Chevalley-Eilenberg complex appears only when an actual computation is implemented for a special resolution, known as the Bar resolution.
A similar approach is followed by the author for Lie algebroid cohomology: he shows with rather basic homological algebra techniques that the such cohomology can be computed as the derived functor of the functor of invariants and, relying on the construction of the enveloping algebra for a Lie algebroid [\textit{I. Moerdijk} and \textit{J. Mrčun}, Proc. Am. Math. Soc. 138, No. 9, 3135--3145 (2010; Zbl 1241.17014)], he shows that it is also computed by the \(\mathrm{Ext}\) groups in the category of modules such algebra. A crucial step in the proof is the existence of a particular projective resolution of the local ring \(\mathcal{O}_{X,x}\) due to \textit{G. S. Rinehart} [Trans. Am. Math. Soc. 108, 195--222 (1963; Zbl 0113.26204)] which allows him to show the main result of this paper.
The author then provides two nice applications of this general statement: first, he shows that an Hochschild-Serre spectral sequence can be obtained quite naturally from this general approach and, second, that a local-to-global spectral sequence exists for Lie algebroid cohomology.
The proof presented appear to be correct. The significance of the results presented should be framed in terms of providing a nice complement to a rather established theory. It is moreover interesting to notice that the result holds for a quite general category of schemes without needing, for instance, any smoothness assumptions. Possible follow ups of this work could be to establish a similar result for the homology theory and understand how this result could be useful to study objects related to Lie algebroids on singular varieties. Lie algebroid cohomology; derived functors; spectral sequences Bruzzo, U., Lie algebroid cohomology as a derived functor, J. Algebra, 483, 245-261, (2017) de Rham cohomology and algebraic geometry, Spectral sequences, hypercohomology, Sheaves and cohomology of sections of holomorphic vector bundles, general results, Other homology theories in algebraic topology, General theory of spectral sequences in algebraic topology Lie algebroid cohomology as a derived functor | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems If \((X, x)\) is a germ of complex analytic space, \(A=F{\mathcal O}_{X,x}\) and \(M\) is a submodule of \(A^p\), such that the length of \((A^p/M)\) is finite, then there is a notion of multiplicity \(e(M)\) of the submodule \(M\). This concept has important applications in equisingularity theory. For instance, it allows us to give numerical criteria to control different equisingularity conditions (e.g, conditions \(A_f\), \(W_f\), \(W\)) when dealing with a family of isolated singularities which are complete intersections. To extend the theory to more general singularities one needs some analog of the multiplicity, but dropping the finite co-length condition.
The present paper presents such an extension. Indeed, in case \(X\) is equidimensional (of dimension \(d\)) and generally reduced, given an \(A\)-module \(M\) (the stalk of a coherent subsheaf (still denoted by \(M\) of \({\mathcal O}_{X^p}\))), the author constructs a filtration
\[
M\subset H_d(M)\subset\cdots\subset H_0(M)\subset{\mathcal O}_{X^p}= F
\]
by coherent \({\mathcal O}_X\)-modules, and introduces numbers \(e_n(M, x)\), \(0\leq n\leq n\). The sheaves \(H_n(X)\) are integrally closed, \(H_d(M)\) is the integral closure of \(M\), and the number \(e_n(M, x)\) is defined as intersection multiplicitiy of suitable restrictions of \(H_n(X)\) and \(M\), in the sense of \textit{S. Kleiman} and \textit{A. Thorup} [J. Algebra 167, No. 1, 168--231 (1994; Zbl 0815.13012)]. If the lengthh of \(F/M\) is finite, then \(e_d(M, x)\) is the usual multiplicity of \(M\).
Then the author proves a generalization of the celebrated theorem of Rees: If \(M\subset N\subset F\) are \({\mathcal O}_X\)-modules (where \((X, x)\) and the notation are as above), and \(e_n(M, x)= e_n(N, x)\) for \(n= 0,\dots,d\), then both \(M\) and \(N\) have the same integral closure.
Some applications of this result to equisingularity theory are given, e.g., in the hypersurface case, a criterion for a hyperplane through \(x\) to be a limit of tangent hyperplanes and a one for Whitney equisingularity for the family of sections of \(X\) by linear subspaces of dimension \(k\).
The author says that in another paper the present results will be applied to a generalization of the principle of specialization of the integral closure. multiplicity; integral closure; local ring; singularity Terence Gaffney, ``Generalized Buchsbaum-Rim Multiplicities and a Theorem of Rees'', Commun. Algebra31 (2003) no. 8, p. 3811-3828 Equisingularity (topological and analytic), Singularities in algebraic geometry, Multiplicity theory and related topics, Invariants of analytic local rings Generalized Buchsbaum-Rim multiplicities and a theorem of Rees | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The integers \(n\geq m-1\geq 0\), a Birkhoff interpolation problem \({\mathcal B}\) (interpolation of a polynomial and certain of its derivatives of order \(\leq n\) at \(m\) points of a field) induced by a matrix \(E= [e_{i,k}]\), \(1\leq i\leq m\), \(0\leq k\leq n\), \(e_{i,k}\in\{0,1\}\), a prime \(p>n\) and a \(p\)-power \(q\). Here we prove the regularity of \({\mathcal B}\) at \((t_1,\dots,t_m)\in\mathbb F_q^m\) if it is regular at \((t_1^{q/p},\dots, t_1^{q/p})\in\mathbb F_p^n\). The regularity over \(\mathbb F_p\) was recently studied by T. Tassa to solve a cryptographic model (hierarchical threshold secret sharing) . Finite ground fields in algebraic geometry, Approximation by polynomials, Applications to coding theory and cryptography of arithmetic geometry, Multidimensional problems, Projective techniques in algebraic geometry Birkhoff interpolation over a finite field | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper studies a moment map \(\mu\) associated to the partial Grothendieck-Springer resolution for \(G=GL_n(\mathbb{C})\). Let \(P \subset G\) be a parabolic Lie subgroup, and consider its natural action on \(\mathfrak{p} \times \mathbb{C}^n\), where \(\mathfrak{p}=\operatorname{Lie}(P)\) is the parabolic Lie subalgebra of \(\mathfrak{gl}_n(\mathbb{C})\). The aforementioned map \(\mu\) is the moment map \(\mu\colon T^*(\mathfrak{p} \times \mathbb{C}^n) \to \mathfrak{p}^*\) corresponding to this action, and the quotient \(\mu^{-1}(0)/P\) is isomorphic to the partial Grothendieck-Springer resolution. The first main result of this paper is that if the block size vector \(\alpha=(\alpha_1,\dots,\alpha_\ell)\) of the parabolic subgroup \(P\) is of length \(\ell \le 5\), then \(\mu^{-1}(0)\) is a complete intersection (Theorem 1.1). The paper further gives a description of the irreducible components of \(\mu^{-1}(0)\), which yields that \(\mu^{-1}(0)\) is reduced and equidimensional (Theorem 1.2). The paper also studies (variation of) Wilson's Calogero-Moser space in relation with the fiber \(\mu^{-1}(0)\) (Theorem 1.5). Grothendieck-Springer resolution; moment map; complete intersection Complete intersections, Momentum maps; symplectic reduction, Coadjoint orbits; nilpotent varieties, Group actions on varieties or schemes (quotients), Geometric invariant theory, Linear algebraic groups over the reals, the complexes, the quaternions The regularity of almost-commuting partial Grothendieck-Springer resolutions and parabolic analogs of Calogero-Moser varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors develop a Mayer-Vietoris spectral sequence for cohomology of local systems over spaces equipped with certain types of combinatorial stratifications. This is applied in several settings: complements of arrangements of hyperplanes in \({\mathbb{CP}}^n\); complements of elliptic arrangements, that is, arrangements of kernels of homomorphisms \(E^n \to E\), \(E\) an elliptic curve; and toric complexes: unions of coordinate subtori of \((S^1)^n\) determined by simplicial complexes. The purpose is to establish general conditions under which the local system cohomology vanishes except in a single degree. The spectral sequence applies to local systems of modules over arbitrary fields or the integers, with no assumption of finite generation or other restrictions. In particular it applies with group ring coefficients, in which case the vanishing results are used to establish duality and abelian duality properties. These in turn have consequences for propagation of resonance, a nesting phenomenon for cohomology jump loci treated in a subsequent paper by the same authors, see [Sel. Math., New Ser. 23, No. 4, 2331--2367 (2017; Zbl 1381.55005)].
Roughly speaking, a combinatorial cover of a space \(X\) is a cover of \(X\) and an auxiliary ranked poset \(P\), with an order-preserving map from the nerve of the cover to \(P\) that respects homotopy types in a certain sense. Given such data, and a locally constant sheaf \({\mathcal F}\) of modules over \(X\), the authors build a spectral sequence abutting to \(H^\cdot(X,{\mathcal F})\) whose \(E_2\) term is described in terms of cohomology of local systems over the order complexes of intervals in \(P\). Such covers are constructed for \(X\) the complement of a complex projective hyperplane arrangement using the DeConcini-Procesi wonderful model, with \(P\) being the poset of nested sets. Certain free abelian subgroups of \(\pi_1(X)\) are identified, coming from centers of local fundamental groups at strata corresponding to nested sets. For cohomology vanishing the operative assumptions are that \({\mathcal F}\) comes from a module which is maximal Cohen-Macaulay over the (Laurent polynomial) group rings of these subgroups -- this replaces the condition on nonvanishing residues that appears in earlier results -- and that the strata are Stein manifolds. With these assumptions the spectral sequence implies the vanishing results for hyperplane arrangements, which are then used to establish related vanishing results for complements of elliptic arrangements, implying in particular that they are duality and abelian duality spaces. Consequently the pure braid group of an elliptic curve is a duality and abelian duality group. The methods also apply to show that a toric complex is an abelian duality space if and only if the associated simplicial complex is Cohen-Macaulay.
Serious readers are advised of a smattering of occasionally confusing typos and notational inconsistencies. combinatorial cover; cohomology with local coefficients; spectral sequence; hyperplane arrangement; elliptic arrangement; toric complex; Cohen-Macaulay property; Stein manifold Spectral sequences in algebraic topology, Vanishing theorems in algebraic geometry, Relations with arrangements of hyperplanes, Homology with local coefficients, equivariant cohomology, Cohen-Macaulay modules in associative algebras, Group rings Combinatorial covers and vanishing of cohomology | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper introduces an algebra-geometric setting for the space of bifurcation functions involved in the local 16th Hilbert's problem on a period annulus. The authors consider one-parameter deformations \(X_{\lambda (\varepsilon )} \) of a polynomial planar vector fields \(X_{\lambda } \) having a center at the origin. This approach allows the authors to build a parametric family in the Nash space of arcs \(\mathrm{Arc}(B_{I}\mathbb C^{n} ,E)\). An arc \(\mathrm{Arc}(B_{I} \mathbb C^{n} ,E)\) in the Nash space is a perturbation \(X_{\lambda (\varepsilon )} \) on the blow-up of the Bautin ideal associated with the family \(X_{\lambda } \). The notion of a ``complete set of essential perturbations'' of irreducible components of the arc space \(\mathrm{Arc}(B_{I}\mathbb C^{n} ,E)\) is central to this paper. This term was first introduced by \textit{I. D. Iliev} [Bull. Sci. Math. 122, No. 2, 107--161 (1998; Zbl 0920.34037)]. Iliev determines the essential perturbations, which can realize the maximum number of limit cycles produced by the entire class of systems. Authors of the paper proves that the complete set of essential perturbations is the set of one-parameter deformations \(X_{\lambda (\varepsilon )} \), which, considered as arcs, form the set of irreducible components of the Nash space of arcs \(\mathrm{Arc}(B_{I}\mathbb C^{n} ,E)\). Each irreducible component is an essential perturbation. Bifurcation functions are put in bijective correspondence with points on the exceptional divisor of the blow-up of the Bautin ideal. The number of essential perturbations is not smaller than the number of irreducible components of the center variety. As an example, the problem on a period annulus for quadratic vector fields is considered. In this case, the number of irreducible components of the central set is four, and the number of essential perturbations is five. A dictionary of the correspondence of terms of algebraic geometry and bifurcation theory is given. For example: exceptional divisor -- set of all bifurcation functions, essential set of irreducible components of the Nash space -- complete set of essential perturbations. local Hilbert's 16th problem; Nash space of arcs; planar quadratic vector fields; essential perturbation Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications) for ordinary differential equations, Arcs and motivic integration, Bifurcation theory for ordinary differential equations Hilbert's 16th problem on a period annulus and Nash space of arcs | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author analyzes the implicitization problem of the image of a finite rational map \(\phi: \mathcal{X} \dashrightarrow \mathbb{P}^{n}\), where \(\mathcal{X}\) is a non-degenerate toric variety over a field \(K\) of dimension \(n-1\). Let \(R\) be the Cox ring of \(\mathcal{X}\) and \(\mathrm{Cl}(\mathcal{X})\) the divisor group of \(\mathcal{X}\). Assume that \(\phi\) is defined by \(n+1\) homogeneous elements \(f_{0},\dots,f_{n} \in R\) and write \(I=(f_{0},\dots,f_{n})\) for the homogeneous \(R\)-ideal generated by the \(f_{i}\). The author provides under suitable assumptions, resolutions \(\mathcal{Z}_{\bullet}\) for the symmetric algebra \(\mathrm{Sym}_{R}(I)\) graded by \(\mathrm{Cl}(\mathcal{X})\) such that the determinant of a graded strand, \(\det((\mathcal{Z}_{\bullet})_{\mu})\), gives a multiple of the implicit equation, for suitable \(\mu \in \mathrm{Cl}(\mathcal{X})\). Indeed, he computes a region in \(\mathrm{Cl}(\mathcal{X})\) which depends on the regularity of \(\mathrm{Sym}_{R}(I)\) where to choose \(\mu\). Furthermore, he gives a geometrical interpretation of the possible other factors appearing in \(\det((\mathcal{Z}_{\bullet})_{\mu})\). A detailed description is given when \(\mathcal{X}\) is a multiprojective space. implicitization; implicit equation; hypersurfaces; toric variety; elimination theory; Koszul complex; approximation complex; resultant; graded ring; multigraded ring; graded algebra; multigraded algebra; Castelnuovo-Mumford regularity Nicolás Botbol, The implicit equation of a multigraded hypersurface, J. Algebra 348 (2011), 381 -- 401. Toric varieties, Newton polyhedra, Okounkov bodies, Computational aspects of algebraic surfaces, Effectivity, complexity and computational aspects of algebraic geometry The implicit equation of a multigraded hypersurface | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \((A,{\mathfrak m})\) be a two-dimensional regular local ring. A linear system \(S\) in \(A\) is the projectivization of a vector space \(V(S)\subset A\) free of common factors. \(S\) has a base point if \(S\subset{\mathfrak m}\). The process of resolution of the (base point of the) linear system by quadratic transformations is studied. Especially there is a description of the behaviour at the first infinitesimal neighborhood in terms of the Plücker formulas. An effective characterization of the Zariski open set of equisingular elements in the linear system, i.e. elements \(f\in {\mathfrak m}\) with the same weighted desingularization tree, is given. resolution of linear system; quadratic transformations; Plücker formulas; equisingular elements Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves, Regular local rings, Equisingularity (topological and analytic) Plücker formula and the equisingularity set of a linear system | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We define an algebraic derivation on an affine domain \(B\) defined over an algebraically closed field \(k\) of characteristic 0, which is called a locally finite derivation by \textit{A. van den Essen} [Polynomial automorphisms and the Jacobian conjecture. Progress in Mathematics (Boston, Mass.). 190. Basel: Birkhäuser. xviii, 329 p. (2000; Zbl 0962.14037)], for example, and has appeared in commutative and non-commutative contexts in other references. We, without being aware of this existing definition and related results, introduced the term of algebraic derivation by extracting a property analogous to algebraic actions of algebraic groups.
The first section is devoted to the graded ring structure which the algebraic derivation \(D\) defines on \(B\) in a natural fashion. The graded ring structure is indexed by an Abelian monoid which is a submonoid of the additive group of the ground field \(k\). This structure is already observed by \textit{J. Krempa} [Bull. Lond. Math. Soc. 12, 374--376 (1980; Zbl 0418.16020)]. But our approach is more computational and straightforward. If the monoid indexing the graded ring structure of \(B\) is rather restricted (see Theorem 1.9 below), the derivation \(D\) is close to what is called an Euler derivation mixed with a locally nilpotent derivation. We observe this fact when \(B\) is a polynomial ring mostly in dimension two.
In fact, the results in section two give various characterization of a polynomial ring \(k[x,y]\) in terms of algebraic derivations. The third section gives a remark on singularities which can coexist with algebraic derivations. The results given in sections two and three are new. algebraic derivation; polynomial ring Derivations and commutative rings, Group actions on affine varieties Algebraic derivations on affine domains | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Given an algorithm for resolution of singularities that satisfies certain conditions (``a good algorithm''), natural notions of simultaneous algorithmic resolution, and of equi-resolution, for families of embedded schemes (parametrized by a reduced scheme \(T\)) are defined. It is proved that these notions are equivalent. Something similar is done for families of sheaves of ideals, where the goal is algorithmic simultaneous principalization. A consequence is that given a family of embedded schemes over a reduced \(T\), this parameter scheme can be naturally expressed as a disjoint union of locally closed sets \(T_j\), such that the induced family on each part \(T_j\) is equi-resolvable. In particular, this can be applied to the Hilbert scheme of a smooth projective variety; in fact, our result shows that, in characteristic zero, the underlying topological space of any Hilbert scheme parametrizing embedded schemes can be naturally stratified in equi-resolvable families. resolution of singularities; algorithmic resolution; simultaneous resolution; Hilbert schemes Encinas, S., Nobile, A. and Villamayor, O.: On algorithmic equi-resolution and stratification of Hilbert schemes. Proc. London Math. Soc. 86 (2003), no. 3, 607-648. Global theory and resolution of singularities (algebro-geometric aspects) On algorithmic equi-resolution and stratification of Hilbert schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Given an integral scheme \(X\) over a non-archimedean valued field \(k\), we construct a universal closed embedding of \(X\) into a \(k\)-scheme equipped with a model over the field with one element \(\mathbb{F}_1\) (a generalization of a toric variety). An embedding into such an ambient space determines a tropicalization of \(X\) by previous work of the authors, and we show that the set-theoretic tropicalization of \(X\) with respect to this universal embedding is the Berkovich analytification \(X^{\mathrm{an}}\). Moreover, using the scheme-theoretic tropicalization we previously introduced, we obtain a tropical scheme \(\mathit{Trop}_{univ}(X)\) whose \(\mathbb{T}\)-points give the analytification and that canonically maps to all other scheme-theoretic tropicalizations of \(X\). This makes precise the idea that the Berkovich analytification is the universal tropicalization. When \(X=\operatorname{Spec}A\) is affine, we show that \(\mathit{Trop}_{univ}(X)\) is the limit of the tropicalizations of \(X\) with respect to all embeddings in affine space, thus giving a scheme-theoretic enrichment of a well-known result of Payne. Finally, we show that \(\mathit{Trop}_{univ}(X)\) represents the moduli functor of semivaluations on \(X\), and when \(X=\operatorname{Spec}A\) is affine there is a universal semivaluation on \(A\) taking values in the idempotent semiring of regular functions on the universal tropicalization. tropical geometry; tropical schemes; idempotent semirings; Berkovich analytification; semivaluation Foundations of tropical geometry and relations with algebra, Rigid analytic geometry The universal tropicalization and the Berkovich analytification. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Two distinct projections of finite rank \(m\) are adjacent if their difference is an operator of rank two or, equivalently, the intersection of their images is \((m-1)\)-dimensional. We extend this adjacency relation on other conjugacy classes of finite-rank self-adjoint operators which leads to a natural generalization of Grassmann graphs. Let \(\mathcal{C}\) be a conjugacy class formed by finite-rank self-adjoint operators with eigenspaces of dimension greater than 1. Under the assumption that operators from \(\mathcal{C}\) have at least three eigenvalues we prove that every automorphism of the corresponding generalized Grassmann graph is the composition of an automorphism induced by a unitary or anti-unitary operator and the automorphism obtained from a permutation of eigenspaces with the same dimensions. The case when the operators from \(\mathcal{C}\) have two eigenvalues only is covered by classical Chow's theorem which says that there are graph automorphisms induced by semilinear automorphisms not preserving orthogonality. Grassmann graph; conjugacy class of finite rank self-adjoint operators; graph automorphism Group actions on combinatorial structures, Isomorphism problems in graph theory (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.), Hermitian and normal operators (spectral measures, functional calculus, etc.), Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects), Homomorphism, automorphism and dualities in linear incidence geometry, Hermitian, skew-Hermitian, and related matrices, Transformers, preservers (linear operators on spaces of linear operators), Homogeneous spaces and generalizations Generalized Grassmann graphs associated to conjugacy classes of finite-rank self-adjoint operators | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper is divided into 21 sections which we consider separately.
Section 1 The article starts with an example, the moduli of complex elliptic curves as a locally symmetric space. After the usual identification with \(X=\text{GL}_2(\mathbb Z)/H\); the author reminds us of the Eichler-Shimura theorem, which calculates the parabolic cohomology \(H_P(X^{*},{\mathcal E})\) of a compactification \(X^{*}\) of \(X\) with coefficients on a sheaf \({\mathcal E}\) induced by a representation.
Section 2 In the same example, in order to compare the parabolic cohomology with the \({\mathcal L}^2\)-cohomology, the author uses the Hodge decomposition of modular forms. This is possible because the metric behaves well when approaching the infinite cusp point.
Sections 3 It is here where we find for the first time a definition of locally symmetric spaces, in terms of a connected reductive real algebraic group \(G\) over \(\mathbb Q\), a maximal compact subgroup \(K\) and a symmetric space \(D\) induced by them. Also here we find the Zucker conjecture, relating the \({\mathcal L}^2\)-cohomology of \(X\) to the middle intersection cohomology of a compactification \(X^{*}\).
Section 4 is an introduction to sheaf theoretic intersection cohomology. The author shows the relation between the middle intersection cohomology and the \({\mathcal L}^2\)-cohomology of a stratified pseudomanifold.
Sections 5 presents, with the aid of many interesting examples, several compactifications of a locally symmetric space \(X\); namely the Borel-Serre compactification \(\overline{X}\), the reductive Borel-Serre compactification \(\widehat{X}\) and the Satake compactification \(X^{*}\); all of them in the context of stratified pseudomanifolds. The author shows the different nature of these three constructions and how this difference can be detected in the links of the boundary. One interesting feature is that, when passing from \(\overline{X}\) to \(\widehat{X}\) and to \(X^{*}\) we somehow simplify the boundary, but we pay the prize by a complication of the links.
Section 6 Along the following three sections the author describes in more detail the construction of the three compactifications. We start with the Borel-Serre compactification \(\overline{X}\), which is done in the context of algebraic groups and parabolic subgroups, liftings and rational characters of Levi quotients over \(\mathbb Q\), unipotent radicals and so on. This is a topological compactification, each stratum of \(\overline{X}\) is a flat bundle over its corresponding stratum on \(X\) with fiber a compact nilmanifold.
Section 7 The author obtains the reductive Borel-Serre compactification as a quotient \(q:\overline{X}\rightarrow \widetilde{X}\), the projection is a stratified morphism.
Section 8 The Satake compactification is obtained, again, as a quotient \( \pi:\widehat{X}\rightarrow X^{*}. \) The projection is also a stratified morphism and will appear in all the results of the article. The construction of \(X^{*}\) is done in an algebraic way; it depends on the choice of an irreducible representation of \(G\) and the suitable extention of the action of \(G\) over \(D\) to a space with real boundary components.
Section 9 provides a résumé of all the work in sections 5 to 8. Since the boundary links of \(X^{*}\) are more complicated, the local calculations of intersection cohomology for \(X^{*}\) get harder. A possible way is to compare \(X^{*}\) with \(\widehat{X}\). This is precisely what does the Rapoport/Goresky-MacPherson conjecture, relating the middle intersection cohomologies of both compactifications. The precise statement compares the sheaves instead, through the pushforward induced by \(\pi\). It says that \( IC_{{\overline {p}}}(X^{*},\mathcal E)=\pi_{*}(IC_{{\overline {p}}}(\widehat{X},\mathcal E)) \) where \({\overline {p}}\) is the middle perversity.
Sections 10 and 11 make a brief disgregation to weighted cohomology, this is motivated by the possibility of showing the Rapoport/Goresky-MacPherson conjecture in some cases, through Nair's theorem that compares these sheaves with the middle weighted cohomology and the \({\mathcal L}^2\)-cohomology. The proof is false in general, and the author shows that there is a condition on the rank of the real boundary components of \(X^{*}\). This condition is far from being evident, so the reader will have to wait until section 19 for a justification.
Section 12 This and the following six sections constitute the body of the article and the main contribution of the author. We find a new algebraic tool, the \(\mathcal L\)-modules, which allow us to compare sheaves simplifying some technical calculations. The author explains how these are in some way the combinatoric analogues of complexes of constructible sheaves. They also admit some operations that are the analogues of push-forward, pull-back, truncation, etc. Each \(\mathcal L\)-module \(\mathcal M\) has a sheaf realization \(S(\mathcal M)\) and the custom known sheaves are realizations of some \(\mathcal L\)-modules. This is also related to other combinatoric constructions such as the sheaves of fans or the moment graphs.
Sections 13 to 17 are devoted to the main feature of \(\mathcal L\)-modules; those are the micro-supports, a family of sets of irreducible representations of Levi quotients. The micro-support \(SS(\mathcal M)\) of an \(\mathcal L\)-module \(\mathcal M\) detects locally the degrees for which the fibers of the (hyper)cohomology of the associated sheaf \(S(\mathcal M)\) vanish. In particular, if \(SS(\mathcal M)=\emptyset\) then \(H(\widehat{X},\mathcal M)=0\) identically; this is theorem 14.1. Some of the tools involved here are the Hodge theory for \(\mathcal L^2\)-forms, and a suitable extension of the Arthur-Langland's partition of \(X\) to \(X^{*}\) taking into account the boundary strata, so as the metric while approaching to the boundary.
Section 18 deals with the micro-supports of the middle intersection cohomology. The results of this section are obtained with the help of a condition on the system of roots of the group \(G\); the author hopes to remove it soon. The author doesn't show the general proof, which can be found in the bibliography, which makes use of a lot of technical details such as comparing some spectral sequences. We find a scheme of the proof for the low-rank cases (1 and 2), which uses Kostant's theorem about the decomposition of the fiber cohomology in terms of its irreducible components. For the rank \(\geq3\) case, the author proposes a geometric approach. The local cohomology lives on the boundary strata for which there is a truncation.
Section 19 shows the functoriality of micro-supports and the way they change when the \(\mathcal L\)-module is modified by the custom operations. The main result of this section is theorem 19.1, which provides a range for the degrees with non-vanishing micro-supports over an equal-rank Satake compactification. It is here where the rank condition is used, so.
Section 20 proves the Rapoport/Goresky-MacPherson conjecture in terms of the local properties of the sheaves.
Section 21 provides a generalization of the Goresky-Harder-MacPherson theorem. This is a result comparing the middle weighted cohomology of \(\widehat{X}\) with the middle intersection cohomology of \(X^{*}\). According to the author, the proof is simpler since the local calculations do not require an inductive process.
{Comments.}
This article is written in a broad, quite encyclopedic spirit. Between the various research fields that are touched, we find the theory of modular forms, stratified pseudomanifolds, sheaf theoretic intersection cohomology, etc. There is also a mentioning of themes such as elliptic curves, representation graphs for groups, Riemannian geometry.
It is my opinion that, in a work with such a span, sometimes it is not so easy to establish which is the custom or starting level of the reading. The broad fields of interests developed in the article can make it a bit difficult for a non familiar reader, or even for a reader whose current research interests are only a part of these fields. Sometimes, the richness of examples conspires to digress our attention for the main part of the article, which is between the sections 12 to 19, where the author finally presents us the theory of \(\mathcal L\)-modules. Probably, there is a little trouble for deciding if this should be a paper or a book. I would have preferred the second option since the author would have no restrictions of space anymore.
In spite of these few remarks, I must say that the present work is the sum of a huge effort in mathematical research. All the examples are deep and beautiful. From all the exhaustive bibliography given by the author I would just recommend, for a non familiar reader, the references below; especially [\textit{A. Borel} and \textit{L. Ji}, in: Lie theory. Unitary representations and compactifications of symmetric spaces, Prog. Math. 229, 69--137 (2005; Zbl 1088.53034)] for the construction of the Satake compactifications. Also a friendly source for mathematicians who usually do not work with algebraic geometry is the web-page of jmilne.org. In [\textit{H. C. King}, Topology Appl. 20, 149--160 (1985; Zbl 0568.55003)] and [\textit{M. Saralegi}, Ill. J. Math. 38, 47--70 (1994; Zbl 0792.57009)] the reader will find two references for intersection cohomology without sheaves. locally symmetric spaces; sheaf; intersection cohomology; \({\mathcal L}^2\)-cohomology; Borel-Serre compactification; Satake compactification; stratified pseudomanifolds; \(\mathcal L\)-modules; Rapoport/Goresky-MacPherson conjecture; weighted cohomology Leslie Saper, On the cohomology of locally symmetric spaces and of their compactifications, Current developments in mathematics, 2002, Int. Press, Somerville, MA, 2003, pp. 219 -- 289. Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Modular and Shimura varieties, Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects), Cohomology of arithmetic groups, Intersection homology and cohomology in algebraic topology On the cohomology of locally symmetric spaces and of their compactifications | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In the paper under review the authors prove that, roughly speaking, the continuous internal Hom between dg-categories of equivariant factorizations in the homotopy category of \(k\)-linear dg-categories is again a category of factorizations.
Let \(X\) be a smooth variety over an algebraically closed field \(k\) of characteristic zero equipped with an action of an affine algebraic group \(G\). Write \(\pi. G\times X\to X\) for the projection and \(\sigma: G\times X\to X\) for the action. Recall that a \(G\)-equivariant quasi-coherent sheaf on \(X\) is a quasi-coherent sheaf \(\mathcal{F}\) together with an isomorphism \(\theta: \sigma^*\mathcal{F}\to \pi^*\mathcal{F}\) satisfying a cocycle condition. This gives an abelian category \(\mathrm{Qcoh}_GX\). If \(\mathcal{F}\) is coherent or locally free, we accordingly call \((\mathcal{F},\theta)\) coherent or locally free. Let \(w\) be a \(G\)-invariant global section of an invertible equivariant sheaf \(\mathcal{L}\). A quasi-coherent matrix factorization of \(w\) is given by equivariant quasi-coherent sheaves \(\mathcal{E}_{-1}\), \(\mathcal{E}_0\) together with maps \(\phi_0^{\mathcal E}: \mathcal E_{-1}\to \mathcal{E}_{0}\) and \(\phi_{-1}^{\mathcal E}: \mathcal E_{0}\to \mathcal{E}_{-1}\otimes \mathcal{L}\) satisfying
\[
\phi_{-1}^{\mathcal E}\circ\phi_{0}^{\mathcal E}=w=(\phi_{0}^{\mathcal E}\otimes\mathcal{L})\circ\phi_{-1}^{\mathcal E}.
\]
Denote such an object by \(\mathcal{E}\). The morphisms between two matrix factorizations form a complex, where, for instance,
\[
\Hom^{2l}(\mathcal{E},\mathcal{F}):=\mathrm{Hom}(\mathcal{E}_{-1},\mathcal{F}_{-1}\otimes\mathcal{L}^l)\oplus {\Hom}(\mathcal{E}_{0},\mathcal{F}_{0}\otimes\mathcal{L}^l)
\]
for \(l\in \mathbb{Z}\) and the differential is given by an explicit formula. Hence, matrix factorizations form a dg-category, denoted by \(\mathrm{Fact}(X,G,w)\). Considering matrix factorizations with injective components gives a dg-category \(\mathrm{Inj}(X,G,w)\).
The main result of the paper under review can be described as follows. Let \((X,G,w)\) be as above. Let \(H\) be an affine algebraic group acting on a smooth variety \(Y\) and let \(v\) be a \(G\)-invariant section of an \(H\)-equivariant line bundle \(\mathcal{L}'\) on \(Y\). Let \(U(\mathcal{L})\) be the complement of the zero section in the geometric vector bundle corresponding to \(\mathcal{L}\), and denote the same construction for \(\mathcal{L}'\) by \(U(\mathcal{L}')\). Denote the functions induced by \(w\) and \(v\) on \(U(\mathcal{L})\) and \(U(\mathcal{L}')\) by \(f_w\) and \(f_v\), respectively. Set \(-f_w\boxplus f_v=-f_w\otimes_k 1+1\otimes_k f_v\). Note that \(G\times H\) acts on \(U(\mathcal{L})\times U(\mathcal{L}')\) and allow \(\mathbb{G}_m\) to scale the fibres of \(U(\mathcal{L})\times U(\mathcal{L}')\) diagonally. The authors prove that there is an equivalence
\[
\mathrm{RHom}_c(\mathrm{Inj}(X,G,w),\mathrm{Inj}(Y,H,v))\cong\mathrm{Inj}(U(\mathcal{L})\times U(\mathcal{L}'), G\times H\times \mathbb{G}_m,-f_w\boxplus f_v)
\]
in the homotopy category of \(k\)-linear dg-categories. So, roughly speaking, the functors between categories of matrix factorizations are again given by matrix factorizations. The above result is used to compute the (extended) Hochschild cohomology of \((X,G,w)\) when \(X\) is affine and \(G\), \(w\) satisfy some technical assumptions.
The story begins with a thorough introduction to equivariant sheaves. The authors define the abelian category of quasi-coherent equivariant sheaves, restriction and inflation functors, pullback etc. They also study the global dimension of \(\mathrm{Qcoh}_GX\).
Section 3 is devoted to equivariant factorizations. The dg-categories mentioned above and their variants (for instance, one involving coherent sheaves) are defined and studied. For example, the authors define a dg-functor \(\mathrm{Fact}(X,G,w)\otimes_k \mathrm{Fact}(X,G,v)\to \mathrm{Fact}(X,G,w+v)\), a version of the \(\mathcal{H}om\)-functor in this setting, an appropriate notion of box product, restriction and inflation functors for the case of a subgroup \(H\) of \(G\), and establish results relating various of these functors.
In the same section the authors also introduce and study the absolute derived category of matrix factorizations, following work of Positselski. The idea of the construction is roughly as follows. To the dg-category \(\mathrm{Fact}(X,G,w)\) one can associate an abelian category having the same objects, but where the morphisms between two factorizations are given by closed degree-zero morphisms between them in \(\mathrm{Fact}(X,G,w)\). The resulting category is denoted by \(Z^0\mathrm{Fact}(X,G,w)=\mathcal{A}\). Given a complex with objects from \(\mathcal{A}\), one can define a matrix factorization, called its totalization. Considering the subcategory of \(\mathrm{Fact}(X,G,w)\) consisting of totalizations of bounded exact complexes from \(\mathcal{A}\) gives the subcategory of acyclic factorizations. The absolute derived category \(\mathrm{D}^{\mathrm{abs}}[\mathrm{Fact}(X,G,w)]\) is then defined as the Verdier quotient of \([\mathrm{Fact}(X,G,w)]\), the homotopy category of \(\mathrm{Fact}(X,G,w)\) (which is a triangulated category), by the homotopy category of acyclic factorizations. Of course, there are variants of this definition if one works with coherent or locally free factorizations.
One of the good properties \(\mathrm{D}^{\mathrm{abs}}[\mathrm{Fact}(X,G,w)]\) has is that it is a compactly-generated triangulated category and equivalent to \([\mathrm{Inj}(X,G,w)]\). Furthermore, the authors show that the idempotent completion of the coherent version \(\mathrm{D}^{\mathrm{abs}}[\mathrm{Fact}(X,G,w)]\) is equivalent to the homotopy category of those factorizations of \(\mathrm{Inj}(X,G,w)\) which are compact in \([\mathrm{Inj}(X,G,w)]\). Another advantage of the absolute derived category is the existence of an essentially surjective functor from a certain singularity category to it, allowing one to use geometry when proving statements about the absolute derived category. More precisely, let \(Y\) be the vanishing locus of \(w\). Under some technical conditions, there is an essentially surjective functor
\[
\mathrm{D}^{\mathrm{sg}}_G(Y):=\mathrm{D}^{\mathrm{b}}(\mathrm{Coh}_GY)/\mathrm{Perf}_GY\to \mathrm{D}^{\mathrm{abs}}[\mathrm{Fact}(X,G,w)],
\]
where \(\mathrm{Perf}_GY\) is the subcategory of perfect complexes, that is, bounded complexes of locally free \(G\)-equivariant sheaves of finite rank. The functor actually exists in greater generality, since one can consider categories supported on a closed \(G\)-invariant subset of \(Y\).
In Section 4 generation of equivariant derived categories is studied. More precisely, one wants to find a set of generators for the bounded derived category of coherent \(G\)-equivariant sheaves \(\mathrm{D}^{\mathrm{b}}(\mathrm{Coh}_G X)\), where \(X\) is a singular variety equipped with a \(G\)-action. The rough idea is to focus on \(\mathrm{D}^{\mathrm{b}}(\mathrm{Qcoh}_G X)\). Using the essentially surjective functor mentioned above, the authors can then produce a set of generators for an appropriate absolute derived category.
In the following section the main result is proved. For this the authors need to recall some facts from the theory of dg-categories and, in particular, results by Toën concerning the construction of the internal (continuous) Hom in the homotopy category of dg-categories. One of the main steps towards the proof of the main result is an equivalence between some absolute derived category and the derived category of dg-modules over the tensor product of matrix factorizations categories, and this is where the generators from the previous section come in, since the proof involves showing that a compact generating set is sent to a compact generating set and is fully faithful on these. In the same section the authors define and study the (extended) Hochschild (co)homology and compute it in the case mentioned above.
The last section is devoted to two applications. Firstly, combining the computation of the extended Hochschild cohomology with the Hochschild-Kostant-Rosenberg isomorphism and a theorem of Orlov relating the bounded derived category of coherent sheaves on a smooth complex hypersurface \(Z\) in projective space with equivariant matrix factorizations allows the authors to recover Griffiths' description of the primitive cohomology of \(Z\) in terms of homogeneous pieces of the Jacobian algebra of the polynomial defining \(Z\). As a second application the authors give a new proof of the Hodge conjecture for self-products of a particular \(K3\) surface closely related to the Fermat cubic fourfold. Ballard, M.; Favero, D.; Katzarkov, L., A category of kernels for equivariant factorizations and its implications for Hodge theory, Publ. Math. Inst. Hautes Études Sci., 120, 1-111, (2014) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Transcendental methods, Hodge theory (algebro-geometric aspects), Derived categories, triangulated categories, Nonabelian homotopical algebra A category of kernels for equivariant factorizations and its implications for Hodge theory | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Fix \(p_1,\dots, p_h \in \mathbb{P}^r\) distinct points and fix \(m_1, \dots ,m_h\) positive integers. Let \({\mathcal{L}}_{r,d}\) be the linear system of hypersurfaces of \(\mathbb{P}^r\) of degree \(d\) and consider
\[
{\mathcal{L}}:={\mathcal{L}}_{r,d}(m_1, \dots, m_d)
\]
the subsystem of those divisors of \({\mathcal{L}}_{r,d}\) having multiplicity at least \(m_i\) at \(p_i\), \(i=1, \dots, n\). Its virtual dimension is defined to be
\[
\nu({\mathcal{L}}):={r+d \choose r}-1-\sum{i=1}^n {r+m_i-1 \choose r}
\]
i.e., the virtual dimension of \({\mathcal{L}}_{r,d}\) minus the number of conditions imposed by all multiple points \(p_i\). This number cannot be less than \(-1\), hence we define the expected dimension to be
\[
\epsilon({\mathcal{L}}):=\text{max}\{\nu({\mathcal{L}}),-1\}.
\]
If the conditions imposed by the assigned points are not linearly independent, the effective dimension of \({\mathcal{L}}\) is greater than the expected one: in that case we say that \({\mathcal{L}}\) is special. Otherwise, if the effective and the expected dimension coincide, we say that \({\mathcal{L}}\) is non-special.
What is known, as yet, is essentially concentrated in the Alexander-Hirschowitz Theorem which says that a general collection of double points in \(\mathbb{P}^r\) gives independent conditions on the linear system \({\mathcal{L}}\) of the hypersurfaces of degree \(d\), with a well known list of exceptions.
In this paper the author presents a new proof of this theorem which consists in performing degenerations of \(\mathbb{P}^r\) and analyzing how \({\mathcal{L}}\) degenerates.
The degenerations used here were introduced by \textit{C. Ciliberto} and \textit{R. Miranda} [J. Reine Angew. Math. 501, 191--220 (1998; Zbl 0943.14002)] and, originally proposed by Z. Ran, to study higher multiplicity interpolation problem. The original approach consists in degenerating the plane to a reducible surface, with two components intersecting along a line, and simultaneously degenerating the linear system \({\mathcal{L}}\) to a linear system \({\mathcal{L}}_0\) obtained as fibered product of linear systems on the two components over the restricted system on their intersection. The limit linear system \({\mathcal{L}}_0\) is somewhat easier than the original one, in particular this degeneration argument allows to use induction either on the degree or on the number of imposed multiple points. This contruction provides a recursive formula for the dimension of \({\mathcal{L}}_0\) involving the dimensions of the systems on the two components.
In this paper the author generalizes this approach to the case with \(r \geq 3\) and completes the proof of Alexander-Hirschowitz Theorem with this method, exploiting induction on both \(d\) and \(r\).
A tricky point of this approach is the study of the transversality of the restrictions of the systems on the intersection of the two components. In the planar case, Ciliberto and Miranda proved it using the finiteness of the set of inflection points of linear systems on \(\mathbb{P}^1\). In higher dimension transversality is more complicated. In Section 2.2 and in Section 3.1, the author presents a new approach to this problem: if at least one of the two restricted systems is a complete linear system, then the dimension of the intersection is easily computed. Anyhow, this is not sufficient to finish the proof of Alexander-Hirschowitz Theorem. For instance, it does not work in the cubic case. The solution to this obstacle is to blow up a codimension three subspace \(L\) of \(\mathbb{P}^r\), instead of a point. This approach to the cubic case is not so different from the one proposed by \textit{M. C. Brambilla} and \textit{G. Ottaviani} [J. Pure Appl. Algebra 212, No. 5, 1229--1251 (2008; Zbl 1139.14007)] where they give another alternative, and short, proof of Alexander-Hirschowitz Theorem, in the case \(d\geq 4\), and propose a new and simpler degeneration argument in the cubic case.
Also the quartic case must be analysed separately. Indeed, twisting by a negative multiple of the exceptional component of the central fiber, one reduces to quadrics that are special. The proof, in this case, involves a geometric argument that exploits the property of cubics of containing all lines through two distinct double points.
The constructions in this paper, besides its intrinsic intent, gives hope for further extensions to greater multiplicities. degenerations; polynomial interpolation; linear systems; double points Postinghel, E.; A new proof of the Alexander-Hirschowitz interpolation theorem; Ann. Mat. Pura Appl.: 2012; Volume 191 ,77-94. Divisors, linear systems, invertible sheaves, Fibrations, degenerations in algebraic geometry, Projective techniques in algebraic geometry A new proof of the Alexander-Hirschowitz interpolation theorem | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The \(\mathcal{K}\)-equivalence (or contact equivalence) is an important notion in the study of smooth mappings. Geometrically, contact equivalence compares the singularities resulting from the intersection of the graphs of two germs having a singular point at the origin with the source axis. Whereas in the existing literature mostly \(C^\infty\)-\(\mathcal{K}\)-equivalence is investigated, the authors consider in the paper under review \(C^0\)-\(\mathcal{K}\)-equivalence and bi-Lipschitz \(\mathcal{K}\)-equivalence. We give the definitions.
Definition. Two maps \(f,g:(\mathbb{R}^n;0)\to(\mathbb{R}^m;0)\) are \(C^0\)-\(\mathcal{K}\)-equivalent if there are germs of homeomorphisms \(h:(\mathbb{R}^n;0)\to(\mathbb{R}^n;0)\) and \(H:(\mathbb{R}^{n+m};0)\to(\mathbb{R}^{n+m};0)\) such that
\[
\pi_n\circ H=h\circ\pi_n, \pi_m\circ H(x,0)=0\;\mathrm{ and }\;H\circ(\mathrm{id}_{\mathbb{R}^n},f)=(\mathrm{id}_{\mathbb{R}^n},g)\circ h,
\]
where \(\pi_n:\mathbb{R}^n\times\mathbb{R}^m\to\mathbb{R}^n\) is the orthogonal projection onto the first factor and \(\pi_m:\mathbb{R}^n\times\mathbb{R}^m\to\mathbb{R}^m\) is the orthogonal projection onto the second factor.
The germs \(f\) and \(g\) are bi-Lipschitz \(\mathcal{K}\)-equivalent if \(h\) and \(H\) can be chosen bi-Lipschitz.
Definition. Let \(F:U\times[0,1]\to\mathbb{R}^m\), \((x,t)\to F_t(x),\) be a deformation where \(U\) is an open neighbourhood of the origin in \(\mathbb{R}^n\). The deformation \(F\) is called \(C^0\)-\(\mathcal{K}\)-trivial if there exist mappings \(H:\mathbb{R}^{n+m}\times [0,1]\to\mathbb{R}^{n+m}\) and \(h:\mathbb{R}^n\times[0,1]\to\mathbb{R}^n\) such that for any \(t\) the pair \((h_t,H_t)\) is a \(C^0\)-\(\mathcal{K}\)-equivalence between \(F_0\) and \(F_t\).
It is called bi-Lipschitz (resp. semialgebraically) \(\mathcal{K}\)-trivial if the homeomorphisms can be chosen bi-Lipschitz (resp. semialgebraic).
The authors show results for the general case and for the semialgebraic case. In the general situation they prove the following.
Theorem. Let \(F:U\times[0,1]\to \mathbb{R}^m\) be a \(C^0\) deformation. If \(F_t^{-1}(0)\) is locally topologically trivial at the origin then \(F\) is \(C^0\)-\(\mathcal{K}\)-trivial.
Moreover, they give a new proof of a degree criterion for \(C^0\)-\(\mathcal{K}\)-equivalence of \(C^\infty\)-germs by \textit{T. Nishimura} [Paris: Éditeurs des Sciences et des Arts. Trav. Cours. 54, 83--93 (1997; Zbl 0896.58008)], using the second Thom-Mather's isotopy theorem instead of the Poincaré conjecture.
In the semialgebraic situation they show a semialgebraic version of the above theorem, avoiding integration of vector fields in the proof.
Theorem. Let \(F:U\times[0,1]\to \mathbb{R}^m\) be a semialgebraic deformation. If \(F_t^{-1}(0)\) is semialgebraically topologically trivial at the origin then \(F\) is semialgebraically \(C^0\)-\(\mathcal{K}\)-trivial.
For polynomial deformations a criterion for semialgebraic bi-Lipschitz \(\mathcal{K}\)-triviality is given involving the notion of the real spectrum. Given a polynomial family of maps the members are piecewise bi-Lipschitz \(\mathcal{K}\)-equivalent. \(C^0\)-\(\mathcal{K}\)-equivalence; bi-Lipschitz \(\mathcal{K}\)-equivalence; \(\mathcal{K}\)-triviality Ruas, M.A.S.; Vallete, G., \(C\)\^{}\{0\} and bi-Lipschitz \({\mathcal{K}}\) -equivalence of mappings, Math. Z., 269, 293-308, (2011) Classification; finite determinacy of map germs, Semialgebraic sets and related spaces, Germs of analytic sets, local parametrization \(C ^{0}\) and bi-Lipschitz \({\mathcal{K}}\)-equivalence of mappings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(f:X\rightarrow\mathbb{R}\) be a function defined on a connected nonsingular real algebraic set \(X\) in \(\mathbb{R}^n \). We prove that regularity of \(f\) can be detected by controlling the restrictions of \(f\) to either algebraic curves or algebraic surfaces in \(X\). If \(\operatorname{dim}X\geq 2\) and \(k\) is a positive integer, then \(f\) is a regular function whenever the restriction \(f|_C\) is a regular function for every algebraic curve \(C\) in \(X\) that is a \(\mathcal{C}^k\) submanifold homeomorphic to the unit circle and is either nonsingular or has precisely one singularity. Moreover, in the latter case, the singularity of \(C\) is equivalent to the plane curve singularity defined by the equation \(x^p=y^q\) for some primes \(p<q\). If \(\operatorname{dim}X\geq 3\), then \(f\) is a regular function whenever the restriction \(f|_S\) is a regular function for every nonsingular algebraic surface \(S\) in \(X\) that is homeomorphic to the unit 2-sphere. We also have suitable versions of these results for \(X\) not necessarily connected. Real algebraic sets, Semialgebraic sets and related spaces, Real rational functions, Real-analytic manifolds, real-analytic spaces Hartogs-type theorems in real algebraic geometry. I | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper studies the Reeb space of a continuous map \(f:X \to Y\) definable in an o-minimal expansion of an ordered real closed field. The Reeb space of \(f\) denoted by \(\operatorname{Reeb}(f)\) is the topological space \(X/\sim\) equipped with the quotient topology, where the equivalence relation \(\sim\) is defined so that \(x \sim x'\) if and only if \(f(x)=f(x')\) and both \(x\) and \(x'\) are contained in the same definably connected component of \(f^{-1}(f(x))\).
The first contribution of this paper is the assertion that the Reeb space of \(f\) exists as a definably proper quotient of \(X\) when \(X\) is closed and bounded in its ambient space. Its proof is constructive, but its known complexity is at least doubly exponential. This paper does not provide a singly exponential algorithm for constructing the Reeb space, but alternatively gives the upper bounds of the sum of its Betti numbers \(b(\operatorname{Reeb}(f))\) of the Reeb space. The paper firstly introduces the negative result that \(b(\operatorname{Reeb}(f_n))\) is arbitrarily larger than \(b(X_n)\) for some sequences of maps \(( f _n: X _n \to Y_n)_{n>0}\). Its second result is a positive one. It gives a singly exponential upper bound on the sum of the Betti numbers of the Reeb space of a proper semi-algebraic map in terms of the number and degrees of the polynomials defining the map. Reeb spaces; o-minimal structures; Betti numbers; semi-algebraic maps Model theory of ordered structures; o-minimality, Semialgebraic sets and related spaces On the Reeb spaces of definable maps | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\mathbb Q[X_1,\dots,X_n]\) be the polynomial ring in \(n\) variables \(X_1,\dots,X_n\) over the rationals \(\mathbb Q\), let \(f,f_1,\dots,f_d\in \mathbb Q[X_1,\dots,X_n]\) non-zero polynomials with \(f=f_1\cdots f_d\) and let \(a_1,\dots,a_d\in\mathbb Q\) be given rational numbers. Let \(\mathbb C\) denote the complex numbers and let \(\mathcal{U}:=\mathbb C^n\setminus \{f=0\}\), and consider \(\mathcal{U}\) as a \(2n\)-dimensional real \(C^\infty\)-manifold. Let \(\mathbb C_\mathcal{U}\) be the constant sheaf on \(\mathcal{U}\) with stalk \(\mathbb C\) and let \(\mathcal{V}\) be the locally constant sheaf of rank 1 defined by the multidimensional function \(f_1^{a_1}\cdots f_d^{a_d}\) on \(\mathcal{U}\).
The paper describes an algorithm which computes for any non-negative integer \(k\) the abstract structure of the cohomology groups \(H^k(\mathcal{U},\mathbb C_\mathcal{U})\) and \(H^k(\mathcal{U},\mathcal{V})\). This abstract structure is expressed by means of generators and relations (algorithms 1.2 and 2.3). Moreover, for \(X\) a non-singular closed algebraic subvariety of \(\mathbb C^n\) given by generators of its vanishing ideal and for \(Y:=\{f_1=0,\dots,f_d=0\}\), the paper contains a procedure which computes the cohomology group \(H^k(X\setminus Y,\mathbb C_X)\) if all local cohomology groups of \(Y\) in \(X\) vanish except possibly one of them (theorem 7.6). The algorithm is based on rewriting techniques in the free Weyl algebra \(A_n:=\mathbb Q[X_1,\dots,X_n,\partial_1,\dots,\partial_n]\).
A technical problem appears since the localized \(\mathbb Q\)-algebra \(\mathbb Q[X_1,\dots,X_n,\frac{1}{f}]\) is typically not anymore a finite \(\mathbb Q[X_1,\dots,X_n]\)-module. This problem is circumvented by representing \(\mathbb Q[X_1,\dots,X_n,\frac{1}{f}]\) as a finite \(A_n\)-module. This requires to compute generators for the corresponding annihilator ideal \(I\) of \(A_n\) which can be done by the calculation of the Bernstein operator \(L\) and the Bernstein polynomial \(b(s)\) associated to \(f\) and by determining the minimal integer roots of \(b(s)\) (procedure 1.4). Then the cohomology groups of \(A_n/\partial_1A_1+\cdots+\partial_nA_n\otimes_{A_n}^L A_n/I\) are determined. These cohomology groups tensored by \(\mathbb C\) yield the cohomology groups \(H^k(\mathcal{U},\mathbb C_\mathcal{U})\) (this is a consequence of the Grothendieck-Deligne comparison theorem, namely theorem 5.1 of the paper which is applied in combination with theorem 6.1).
The computation of the cohomology groups \(H^k(\mathcal{U},\mathcal{V})\) follows a similar way. complex sheaf cohomology; Weyl algebra; D-module; Bernstein polynomial; Gröbner basis; Grothendieck-Deligne comparison Oaku, T.; Takayama, N., An algorithm for de Rham cohomology groups of the complement of an affine variety via \(D\)-module computation, J. Pure Appl. Algebra, 139, 201-233, (1999) de Rham cohomology and algebraic geometry, Symbolic computation and algebraic computation, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Computational aspects of higher-dimensional varieties An algorithm for de Rham cohomology groups of the complement of an affine variety via \(D\)-module computation | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(W\) be a finite group acting in \(\mathbb{R}^l\) and generated by reflections. In the algebra of \(W\)-invariant polynomials, there exists a basis \(p=(p_1,\dots,p_t)\), which is used to define the mapping \(p: \mathbb{R}^l \to\mathbb{R}^l\). Denote by \(C^r(\mathbb{R}^l)^W\) the class of \(C^r\)-smooth functions that are \(W\)-invariant on \(\mathbb{R}^l\). Any function \(f\in C^r (\mathbb{R}^l)^W\) can be represented as \(f=F\circ p\), where the smoothness of \(F\) is smaller than that of \(f\). This phenomenon is called the loss of smoothness.
Let \(W\) be generated by the reflections with respect to the hyperplanes \(H_1, \dots, H_n\), and let \(W_i\) be the subgroup of \(W\) generated by the reflections with respect to \(H_1, \dots, H_{i-1}\), \(H_{i+1}, \dots,H_n\). It was proved in [\textit{G. Barbançon}, Duke Math. J. 53, No. 3, 563-584 (1986; Zbl 0619.20026)] that, if \(f\in C^r(\mathbb{R}^l)^W\), then \(F\in C^{[r/\mu]} (\mathbb{R}^l)\), where
\[
\mu=1+ \text{card} R(W)-\min_{1\leq i\leq n}\text{card} R(W_i), \tag{1}
\]
and \(R(W)\) is the set of all reflections from \(W\). Despite the fact that estimate (1) was viewed in the cited paper as optimal, this cannot be accepted as final. The fact is that the magnitude \(\mu_j\) of the smoothness loss of \({\partial f\over\partial p_j}\circ p\) as compared to \(f\) is generally different for different \(j\); therefore, the loss of smoothness is characterized more precisely by the vector \(\overline\mu=(\mu_1,\dots,\mu_l)\) rather than by the number \(\mu=\max_{1\leq j\leq l}\mu_j\).
Let us give the following definition. A function \(F\) is said to belong to the class \(C^r_{\overline\mu}\) at a point or in a domain if its derivatives \(D^\alpha F\), where \((\alpha, \overline \mu)\leq r\), are continuous in a neighborhood of that point or in that domain, respectively. The classes \(C^r_{\overline\mu}\) were used in [\textit{A. O. Gokhman}, Funkts. Anal. Prilozh. 28, No. 4, 82-84 (1994; Zbl 0845.58015)].
This paper provides a complete description of smoothness loss. Specifically, a theorem on smoothness loss is formulated in terms of \(C^r_{\overline \mu}\) for an arbitrary group \(W\), and the irreducible groups \(W=A_l,B_l,D_l\), \({\mathcal D}_m\) are analyzed separately in detail. loss of smoothness Symmetries, equivariance on manifolds, Differentiable maps on manifolds, Geometric invariant theory, Reflection and Coxeter groups (group-theoretic aspects) Anisotropic loss of smoothness in the representation of differentiable invariants of groups generated by reflections | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \({\mathcal C}\to\Delta\) be a flat family of smooth plane quartic curves which degenerates to a singular quartic \(C_0\). This family induces a map \(\Delta-\{0\}\to {\mathcal M}_3\) which can be completed to a map \(\varphi:\Delta\to\overline{\mathcal M}_3\), \(\overline{\mathcal M}_3\) being the usual stable compactification of the moduli space. The stable curve \(\varphi(0)\) is called the ``stable limit of \({\mathcal C}\)'' and it depends on the singular quartic \(C_0\) as well as on the choice of the particular smoothing \({\mathcal C}\); however, when \(C_0\) is reduced, the stable limit obtained from a generic smoothing of \(C_0\) is well determined.
The author provides a description of the stable limits that one obtains starting with a singular plane quartic \(C_0\) and taking a generic smoothing. In some cases it turns out that the stable limit is smooth. family of smooth plane quartic curves; moduli space; stable limits Pyung-Lyun Kang, ``On singular plane quartics as limits of smooth curves of genus three'', J. Korean Math. Soc.37 (2000) no. 3, p. 411-436 Families, moduli of curves (algebraic), Singularities of curves, local rings, Plane and space curves On singular plane quartics as limits of smooth curves of genus three | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In the paper under review, the authors prove a representability theorem in derived analytic geometry, analogous to
Lurie's generalization of Artin's representablility criteria to derived algebraic geometry.
This is an important, standard type result for the study of moduli problems and
a crucial step towards a solid theory of derived analytic geometry.
More specifically, the authors show that a derived stack for the étale site of derived analytic spaces is a derived analytic stack
if and only if
it is compatible with Postnikov towers, has a global analytic cotangent complex, and its truncation is an analytic stack in the classical
(underived) sense.
The result applies both to complex analytic geometry and non-archimedean analytic geometry.
Central to representability results as in the present paper is deformation theory, which the authors develop here for the derived analytic setup.
The authors define an analytic version of the cotangent complex which controls the deformation theory of the derived stack.
As in the algebraic setting, the cotangent complex represents a functor of derivations.
One key step in order to define the analytic cotangent complex is the elegant description of the \(\infty\)-category of modules over a
derived analytic space \(X\) as the \(\infty\)-category of spectrum objects of a certain \(\infty\)-category associated with \(X\).
Another important construction is the analytification functor which they establish in the derived setting.
To apply derived geometry to classical moduli problems, one may try to enrich classical moduli spaces with derived structures. The paper under review is an important tool in verifying when such enrichments are indeed the correct ones. representability; deformation theory; analytic cotangent complex; derived geometry; rigid analytic geometry; complex geometry; derived stacks Stacks and moduli problems, Rigid analytic geometry, Complex-analytic moduli problems, Fundamental constructions in algebraic geometry involving higher and derived categories (homotopical algebraic geometry, derived algebraic geometry, etc.) Representability theorem in derived analytic geometry | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper the author studies the asymptotic expansion of real functions which are finite compositions of globally subanalytic maps with the exponential function and the logarithmic function.
One of the main results is a preparation theorem for these log-exp functions following the work of \textit{J.-M. Lion} and \textit{J.-P. Rolin} [Ann. Inst. Fourier 47, 859-884 (1997; Zbl 0873.32004)]. This result allows to derive the convergence of some asymptotic expansions of log-exp functions in certain scales of real functions such as the scale of real power functions. It must be noticed that some results in the same spirit have appeared in the paper of \textit{L. van den Dries, A. Macintyre} and \textit{D. Marker} [J. Lond. Math. Soc., II Ser. 56, 417-434 (1997; Zbl 0924.12007)] using model theory but here only arguments from analytic geometry are used.
\textit{L. van den Dries} and \textit{C. Miller} conjectured in [Duke Math. J. 84, 497-540 (1996; Zbl 0889.03025)] that there is no o-minimal structure lying stricly between \(\mathbb R_{\text{an}}\) and \(\mathbb R_{\text{an,exp}}\). As a nice consequence of the results of this paper the author proves this conjecture in the one dimensional case. preparation theorem; asymptotic expansions; log-exp functions R. Soufflet, Asymptotic expansions of logarithmic-exponential functions , Bull. Braz. Math. Soc. (N.S.) 33 (2002), 125-146. Analytic subsets of affine space, Semi-analytic sets, subanalytic sets, and generalizations, Asymptotic expansions of solutions to ordinary differential equations, Analytic algebras and generalizations, preparation theorems, Real-analytic and semi-analytic sets Asymptotic expansions of logarithmic-exponential functions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper, the authors compute the sum of Betti numbers of smooth Hilbert schemes on the complex projective spaces. For fixed positive integer \(n\) and polynomial \(p(t)\), let \(\mathbb P^{n[p]}\) denote the Hilbert scheme of closed subschemes of the complex projective space \(\mathbb P^{n}\) with Hilbert polynomial \(p(t)\). It is previously known that the Hilbert scheme \(\mathbb P^{n[p]}\) is nonempty if and only if \(p(t)= \sum_{i=1}^r \binom{t+\lambda_i - i}{\lambda_i - 1}\) for some integer partition \(\lambda = (\lambda_1, \ldots, \lambda_r)\) satisfying \(\lambda = (n+1)\), \(r = 0\), or \(n \ge \lambda_1 \ge \ldots \ge \lambda_r \ge 1\). Moreover, \(\mathbb P^{n[p]}\) is smooth if and only if one of the following seven cases is true:
\begin{itemize}
\item[1.] \(n \le 2\);
\item[2.] \(\lambda_r \ge 2\);
\item[3.] \(\lambda = (1)\) or \(\lambda = (n^{r-2}, \lambda_{r-1}, 1)\) where \(r \ge 2\);
\item[4.] \(\lambda = (n^{r-s-3}, \lambda_{r-s-2}^{s+2}, 1)\) where \(r \ge s+3\);
\item[5.] \(\lambda = (n^{r-s-5}, 2^{s+4}, 1)\) where \(r \ge s+5\);
\item[6.] \(\lambda = (n^{r-3}, 1^3)\) where \(r \ge 3\);
\item[7.] \(\lambda = (n+1)\) or \(r = 0\).
\end{itemize}
The main theorem of the paper presents explicit formulas for the sum \(H_{n, \lambda}\) of the Betti numbers of \(\mathbb P^{n[p]}\) in the above seven cases except Case~2. For instance,
\[
H_{n, \lambda} = \binom{n+r-2}{r-2} \binom{n+1}{\lambda_{r-1}} (n + 1 - \lambda_{r-1})(\lambda_{r-1} + 1)
\]
when \(\lambda = (n^{r-2}, \lambda_{r-1}, 1)\) is in Case 3 with \(n > \lambda_{r-1} > 1\).
The main ideas in the proofs are to use the \(\mathrm{PGL}(n + 1)\)-action on \(\mathbb P^{n[p]}\) induced from the \(\mathrm{PGL}(n + 1)\)-action on \(\mathbb P^{n}\) and to apply the classical theorem of A.~Bialynicki-Birula. These ideas enable the authors to translate the computation of the ranks of the homology groups into counting saturated monomial ideals and then to translate that into counting choices of orthants in an \((n + 1)\)-dimensional lattice. Hilbert scheme; Betti number; cohomology; homology; saturated monomial ideal Parametrization (Chow and Hilbert schemes), Syzygies, resolutions, complexes and commutative rings The sum of the Betti numbers of smooth Hilbert schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors study some properties of the ring of abstract semialgebraic functions over a constructible subset of the real spectrum of an excellent ring. To be more precise, let \(X\) be a constructible subset of the real spectrum of a ring \(A\). The ring \({\mathcal S}(X)\) of abstract semialgebraic functions over \(X\) was introduced bz \textit{N. Schwartz} [see Mem. Am. Math. Soc. 397 (1989; Zbl 0697.14015)], as a generalization of continuous functions with semialgebraic graph to the context of real spectra. Unfortunately the utility of this functions is not yet quite established. The main result of the paper states that if \(A\) is excellent, the Krull dimension of \({\mathcal S}(X)\) equals the dimension of \(X\) (defined as the maximum of the heights of the supports of points lying in \(X\)), which in turn, as \textit{J. M. Ruiz} showed in C. R. Acad. Sci. Paris, Sér. I 302, 67-69 (1986; Zbl 0591.13017) coincides with its topological dimension.
This was first shown by \textit{M. Carral} and \textit{M. Coste} [J. Pure Appl. Algebra 30, 227-235 (1983; Zbl 0525.14015)] for the particular case of \(X\) being a `true' semialgebraic subset which is locally closed, and the result extends readily to abstract locally closed constructible sets. Then the authors use the compactness of the constructible topology of real spectra and the properties of excellent rings to reduce the general case to the locally closed one.
The paper finishes by characterizing the finitely generated prime ideals of \({\mathcal S}(X)\), namely they are the ideals of the open constructible points of \(X\) whose closure in \(X\) is open of dimension \(\neq 1\). semialgebraic functions; constructible subset of the real spectrum of an excellent ring Gamboa, On rings of semialgebraic functions, Math. Z. 206 (4) pp 527-- (1991) Semialgebraic sets and related spaces On rings of semialgebraic functions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Given a real function \(f(X,Y)\), a box region \(B\) and \(\varepsilon>0\), we want to compute an \(\varepsilon\)-isotopic polygonal approximation to the curve \(C: f(X,Y)=0\) within \(B\). We focus on subdivision algorithms because of their adaptive complexity. \textit{S. Plantinga} and \textit{G. Vegter} [``Isotopic approximation of implicit curves and surfaces'', in: Proc. Eurographics. Symposium on Geometry Processing. New York: ACM Press. 245--254 (2004)] gave a numerical subdivision algorithm that is exact when the curve \(C\) is non-singular. They used a computational model that relies only on function evaluation and interval arithmetic. We generalize their algorithm to any (possibly non-simply connected) region \(B\) that does not contain singularities of \(C\). With this generalization as subroutine, we provide a method to detect isolated algebraic singularities and their branching degree. This appears to be the first complete numerical method to treat implicit algebraic curves with isolated singularities.
For Part I, see [\textit{C. K. Yap}, in: Proceedings of the 22nd annual symposium on computational geometry, SCG'06, Sedona, Arizona, USA, June 5--7, 2006. New York, NY: Association for Computing Machinery (ACM). 217--226 (2006; Zbl 1153.65324)]. Numerical aspects of computer graphics, image analysis, and computational geometry, Computer graphics; computational geometry (digital and algorithmic aspects), Computational aspects of algebraic curves, Singularities of curves, local rings Complete subdivision algorithms. II: Isotopic meshing of singular algebraic curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Given a cohomological field theory (CohFT) \(\Omega_{g,n}:(V^*)^{\otimes n}\to H^*(\overline M_{g,n})\), under convergence assumptions, it is possible to realize \(V\) as the tangent space of a point of a Frobenius manifold, and \(\Omega_{g,n}\) can be extended to a family of CohFTs over this Frobenius manifold. For such CohFTs (called convergent CohFTs) the reconstruction theorem of Givental-Teleman reconstructs \(\Omega_{g,n}\) from its underlying Frobenius manifold near a semisimple point (up to choice of integration constants), and also extends a neighborhood of this point to a convergent CohFT.
By a result of Hertling, the germ of an \(N\)-dimensional semisimple Frobenius manifold near a smooth point \(p\) on the discriminant locus is of the form \(I_2(m)\times A_1^{N-2}\) for some integer \(m\geq 3\), so in particular all but two idempotent vector fields extend to \(p\).
The main result of the paper under review combines the Givental-Teleman reconstruction theorem with Herting's result to prove that, for \(m=3\), a neighborhood of a smooth point on the discriminant of a semisimple Frobenius manifold can be extended to a convergent CohFT.
As an application the paper shows that a large set of tautological relations obtained from the Givental-Teleman classification for semisimple CohFTs follow from Pixton's generalized Faber-Zagier relations. Frobenius manifolds; discriminant; cohomological field theories; tautological ring Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Families, moduli of curves (algebraic) Frobenius manifolds near the discriminant and relations in the tautological ring | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(G\) be the group of polynomial automorphisms of the complex plane \(\mathbb{C}^{2}.\) By the Jung-van der Kulk theorem each \(\sigma\in G\) can be represented as a composition
\[
\sigma=\alpha_{1}\circ\beta_{1}\circ\ldots\alpha_{k}\circ\beta_{k}\circ \alpha_{k+1},
\]
where \(\alpha_{i}\) are affine automorphisms and \(\beta_{j}\) are triangular automorphisms. If this composition is reduced i.e. \(\alpha_{i}\) (\(2\leq i\leq k\)) are not triangular and \(\beta_{j}\) are not affine automorphisms then the number \(k\) is unique and it is called the length of \(\sigma\) and denoted \(l(\sigma).\) The multidegree \(d(\sigma)\) of \(\sigma\) is defined as
\[
d(\sigma)=(\deg\beta_{1},\ldots,\deg\beta_{k}).
\]
It is a finite sequence of integers \(\geq2.\) Let us denote by \(G_{d}\) the set of automorphisms whose multidegree is \(d=(d_{1},\ldots,d_{k}),\) \(d_{i}\geq2.\) In the Zariski topology of \(G\) (it is an infinite dimensional algebraic variety) the length \(l(\sigma)\) is a lower semicontinuous function on. It gave rise to the question whether for any \(G_{d}\) there exists a subset \(E(d)\) of multidegrees such that
\[
\overline{G_{d}}= {\displaystyle\bigcup\limits_{e\in E(d)}} G_{e}.
\]
The authors prove that this conjecture is not true. In particulary they show that
\[
G_{(19)}\cap\overline{G_{(11,3,3)}}\neq\emptyset\quad \text{and}\quad G_{(19)}\not \subset \overline{G_{(11,3,3)}}.
\]
length of automorphism; multidegree of automorphism Edo, E.; Furter, J.-P., Some families of polynomial automorphisms, J. Pure Appl. Algebra, 194, 263-271, (2004) Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) Some families of polynomial automorphisms | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Given a separated scheme of finite type over a finite field of characteristic \(p\) and an adic \(\mathbb{Z}_l\)-algebra \(\Lambda\), one may attach, to any perfect complex of \(\Lambda\)-sheaves \(\mathcal{F}^{\bullet}\) on the étale site of X, \(L\)-functions \(L(\mathcal{F}^{\bullet},T),L(Rf_{!}\mathcal{F}^{\bullet},T)\in K_1(\Lambda[[T]])\). Here, \(T\) is a formal variable commuting with the elements of \(\Lambda\). When \(l\neq p\), the ratio of these \(L\)-functions is \(1\in K_1(\Lambda[[T]])\). The paper under review is concerned with the case that \(l=p\).
For commutative adic \(\mathbb{Z}_p\)-algebras \(\Lambda\), the ratio was shown by Emerton and Kisin to be a unit in the Jac\((\Lambda)\)-adic completion \(\Lambda\langle T\rangle\) of the polynomial ring \(\Lambda[T]\). The unit group \(\Lambda\langle T\rangle^{\times}\) can be identified with the first completed \(K\)-group \(\hat{K}_1(\Lambda\langle T\rangle)\) and is a subgroup of \(K_1(\Lambda[[T]])\). More generally one has a canonical homomorphism from the first completed \(K\)-group to \(K_1(\Lambda[[T]])\), though when \(\Lambda\) is non-commutative it fails to be injective in general.
The first theorem of this paper is the existence of a unique pre-image in \(\hat{K}_1(\Lambda\langle T\rangle)\) of the ratio \(L(\mathcal{F}^{\bullet},T)\) over \(L(Rs_{!}\mathcal{F}^{\bullet},T)\) in \(K_1(\Lambda[[T]])\), where \(s:X\rightarrow\text{Spec}(\mathbb{F})\) is a separated scheme of finite type, \(\Lambda\) is an adic \(\mathbb{Z}_p\)-algebra, and \(\mathcal{F}^{\bullet}\) is a perfect complex of \(\Lambda\)-sheaves. This pre-image is shown to be multiplicative on exact sequence of perfect complexes, depend only on the quasi-isomorphism class of \(\mathcal{F}^{\bullet}\), and be compatible with change of \(\Lambda\).
The proof proceeds by reducing to the case of the group ring \(\Lambda=\mathbb{Z}_p[G]\) for a finite group \(G\). Cited work of Chinburg-Pappas-Taylor then paves the way to further reductions to the commutative case proved by Emerton-Kisin. The first theorem implies a version of the noncommutative Iwasawa main conjecture for varieties over finite fields, stated in the text as Theorem 1.2, which is a development of a similar result proved elsewhere by Burns.
The paper breaks down as follows. Section 1 offers an introduction to the two main theorems of this paper. Section 2 proves preliminary statements about completed \(K\)-theory, which is followed by the specific case of \(\Lambda=\mathbb{Z}_p[G]\) in section 3. Section 4 explains the construction of \(L\)-functions associated to perfect complexes of adic sheaves. The first main theorem is proved in section 5, and the second in section 6. unit \(L\)-Functions; Grothendieck trace formula; Iwasawa main conjecture; varieties over finite fields Witte, M.: Unit L-functions for étale sheaves of modules over noncommutative rings Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Iwasawa theory, Varieties over finite and local fields, Finite ground fields in algebraic geometry Unit \(L\)-functions for étale sheaves of modules over noncommutative rings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(L\) be an ample \(\mathbb{Q}\)-line bundle over a projective manifold \(X\). Let \(m\in\mathbb{N}\) be sufficiently divisible so that \(mL\) is very ample. Given any basis \(\{s_i\}_{i=1}^{d_m}\) of \(H^0(X,mL)\), \(D=\frac{1}{md_m}\sum_{i=1}^{d_m}(s_i)\) is called an \(m\)-basis divisor of \(L\). Denote by \(\delta_m(L)\) the infimum of log canonical threshold (lct) \(lct(X,D)\) of all \(m\)-basis divisors of \(L\). It's known that \(\delta(L):=\limsup_m\delta_m(L)=\lim_m\delta_m(L)\) characterizes uniform \(K\)-stability on a Fano variety, and \(\delta(L)=1\) is a threshold for the existence of twisted Kähler-Einstein metric.
One goal of this paper is to find an analytic interpretation of these algebraically defined \(\delta_m\)- and \(\delta\)-invariant. Associated to the Monge-Ampère energy \(E(\cdot)\) on the space of Kähler potentials or smooth \(\omega\)-strictly plurisubharmonic functions where \(\omega\) is given by the curvature of \(L\), one may define an analytic counterpart \(\delta^A(L)\) of \(\delta(L)\). It is conjectured that \(\delta(L)\) equals to the threshold \(\delta^A(L)\).
Motivated by this conjecture, and inspired by the relation of \(\alpha\)-invariant and lct, the authors introduce the \(m\)th analytic stability threshold \(\delta_m^A(L)\) associated to a functional \(E_m(\cdot,\cdot)\) that quantizes \(E(\cdot)\), which can be regarded as the optimal constant in a quantized Moser-Trudinger inequality. The functional \(E_m(\cdot,\cdot)\) is defined on the space of all Hermitian inner products on the complex vector space \(H^0(X,mL)\). Then the authors prove that \(\delta_m^A(L)=\delta_m(L)\) (i.e. Theorem 2.8), a quantized version of the above conjecture. As a consequence, the authors obtain \(\delta(L)=\lim_m\delta_m^A(L)\), which may serve as an analytic interpretation of the \(\delta\)-invariant.
The ingredients of the proof include the lower semi-continuity of complex singularity exponents, a relation between \(E_m\)-functional and the \(m\)th expected vanishing order of \(L\) along divisors, Bergman geodesics, and a valuative description of \(\delta_m(L)\).
Given a smooth form \(\theta\) cohomologous to \(c_1(X)-c_1(L)\). As another main result, the authors prove that \(\delta_m(L)=1\) is a threshold for the existence of \(\theta\)-balanced metrics of level \(m\) (i.e. Theorem 2.3).
This quantization approach also gives a way to calculate \(\delta_m\)-invariant (see Theorem 2.11). In particular, the authors are able to compute \(\delta_m(-K_X)\) for toric Fano manifolds \(X\), for example, \(\delta_m(-K_{\mathbb{P}^n})=1\). basis divisor; \(\delta\)-invariant; log canonical threshold; Kähler-Einstein metric; balanced metric Fano varieties, Notions of stability for complex manifolds, Global differential geometry of Hermitian and Kählerian manifolds Basis divisors and balanced metrics | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We express conjugation-invariant functions on the space of equivalence classes of pairs of \(k\times k\) matrices via theta functions. Our approach is based on the well-known interplay between algebraic geometry and the theory of integrable systems. Every equivalence class of pairs of matrices defines the so-called spectral curve \(S\) and a line bundle \(L\) on \(S\). Both together give a complete invariant for equivalence classes of pairs of matrices. This yields an identification between the set of equivalence classes of pairs of matrices with fixed spectral curve and the affine Jacobian of this curve, i.e. the Jacobian without a theta divisor. By using Painlevé-analysis we describe subvarieties of the theta divisor by families of formal Laurent series of matrix polynomials. This leads to a description of conjugation invariant functions by theta functions modulo constant terms. To determine the explicit constant term we develop concrete extensions of sections of appropriate vector bundles of arbitrary order. Finally we present an algorithm for describing conjugation invariant functions on pairs of matrices via theta functions and carry out the calculations on the example \(\mathrm{tr}([A,B]^2)\). Research exposition (monographs, survey articles) pertaining to algebraic geometry, Theta functions and curves; Schottky problem, Vector bundles on curves and their moduli Theta functions and conjugation-invariant functions on pairs of matrices | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Consider a polynomial map \(f :{\mathbb C}^{n+1} \longrightarrow {\mathbb C}\). For \(r\) a big enough real number the map \(1/f : Z:= Z_r :={\mathbb C}^{n+1} \setminus f^{-1}(D_r) \rightarrow D^*_{1-r}\) is a locally trivial \(C^{\infty}\) fibration. The map \(1/f\) can be compactified, and a fiber over 0 can be added, so that we get a projective map \(p : X \rightarrow D_{1-r}\). Write \(Y = p^{-1}(0)\), and \(X \setminus Z = Y \cup \Delta\), where \(\Delta\) is the union of irreducible components of \(X \setminus Z\) not contained in \(Y\). Consider the universal covering \({\mathbb H}\) of \(D^*_{1-r}\) and let \(\widetilde{Z}:= Z \times_{D_{1/r}} {\mathbb H}\), which has the homotopy type of a generic fiber of \(f\). The cohomology groups of \(\widetilde{Z}\) with rational coefficients carry a limit mixed Hodge structure. The one on \(n\)--th cohomology group is called the limit MHS at infinity.
The authors study the equivariant Hodge numbers of this limit MHS in case that for all \(t \in {\mathbb C}\) which is not a critical value of \(f\), the closure of \({f = t}\) in projective space is non-singular. mixed Hodge structures; polynomial maps; hypersurface singularities R. García López and A. Némethi, Hodge numbers attached to a polynomial map, Ann. Inst. Fourier (Grenoble) 49 (1999), no. 5, 1547 -- 1579 (English, with English and French summaries). Mixed Hodge theory of singular varieties (complex-analytic aspects), Global theory and resolution of singularities (algebro-geometric aspects), Variation of Hodge structures (algebro-geometric aspects) Hodge numbers attached to a polynomial map | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\delta\) be a locally nilpotent derivation on \(\mathbb C[x_1,\dots,x_n]\) such that the associated \(G_a\)-action is fixed point free. Suppose that the ring of invariants \(R := \mathbb C[x_1,\dots,x_n]^{\delta}\) is finitely generated. Let \(\mathfrak m \subset R\) be a maximal ideal such that \(R_{\mathfrak m}\) is a regular local ring and \(S := R \smallsetminus \mathfrak m\). It is proved that \(\Omega_{S^{-1}\mathbb C[x_1,\dots,x_n]| R_{\mathfrak m}}\) is free of rank one. A consequence is the following result:
If \(R\) is regular then the associated \(G_a\)-action is locally trivial if and only if it is proper.
Finally it is proved that a set of polynomials \(\{f_1,\dots,f_{n-1}\}\) is part of a coordinate system for \(\mathbb C^n\) (i.e. there exist \(f_n\) such that \(\mathbb C[x_1,\dots,x_n] = \mathbb C[f_1,\dots,f_n]\)) if and only if \(\mathbb C[f_1,\dots,f_{n-1}]\) is the ring of invariants for a proper \(G_a\)-action (the action is generated by the derivation \(\delta\) defined by \(\delta(h) = \lambda \det(J(f_1,\dots,f_{n-1},h))\) for some \(\lambda \in \mathbb C^*\)). additive group action; ring of invariants; nilpotent derivation James K. Deveney and David R. Finston, Regular \?\? invariants, Osaka J. Math. 39 (2002), no. 2, 275 -- 282. Actions of groups on commutative rings; invariant theory, Polynomial rings and ideals; rings of integer-valued polynomials, Group actions on varieties or schemes (quotients), Linear algebraic groups over the reals, the complexes, the quaternions Regular \(G_a\) invariants. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The Adams-Novikov spectral sequence provides a close connection between stable homotopy theory and the homological algebra of $BP_*BP$-comodules. History has taught us that the cleanest way to do homological algebra is often to work directly on the derived level. The classical derived category $\mathcal{D}_{BP_*BP}$ of the abelian category of $BP_*BP$-comodules has unpleasant properties (as e.g.\ the tensor unit $BP_*$ is not compact in it). Thus the authors replace it by the stable category $\mathrm{Stable}_{BP_*BP}$ of $BP_*BP$-comodules as introduced by Hovey. In their formulation it is defined as the ind-$\infty$-category on the dualizable objects in $\mathcal{D}_{BP_*BP}$. \par Besides basic formal properties about stable categories of Hopf algebroids, the results of the authors fall into two classes: \par Firstly, they prove algebraic analogues of the Ravenel conjectures: An algebraic nilpotence theorem, an algebraic telescope conjecture and an algebraic chromatic convergence result. The nilpotence result has necessarily to be weaker than its topological counterpart though, essentially because the vanishing curve on the $E_{\infty}$-page of the Adams-Novikov spectral sequence is much better than on the $E_2$-term. \par Secondly, they give cleaner proofs for some results in the homological algebra of $BP_*BP$-comodules, which simultaneously generalize them. We mention in particular their version of the chromatic spectral sequence, which is just a Bousfield-Kan spectral sequence associated to the tower of algebraic chromatic localizations. As a consequence of this spectral sequence, they prove in particular a nice comparison result between the $BP$-based Adams-Novikov spectral sequence for a bounded below spectrum whose $BP$-homology has finite projective dimension and its $E_n$-based one. \par We remark that due to work of Isaksen, Gheorghe, Wang and Xu the $\infty$-category $\mathrm{Stable}_{BP_*BP}$ embeds into motivic homotopy theory as modules over a certain motivic spectrum $C\tau$. Thus, the algebraic analogues of the Ravenel conjectures in the present paper are promising to play a role in the exploration of chromatic homotopy theory in the motivic setting. Hopf algebroid; chromatic localization; chromatic spectral sequence Bordism and cobordism theories and formal group laws in algebraic topology, Abstract and axiomatic homotopy theory in algebraic topology, Localization and completion in homotopy theory, Stacks and moduli problems, Motivic cohomology; motivic homotopy theory Algebraic chromatic homotopy theory for \(BP_\ast BP\)-comodules | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(F|\mathbb{Q}_2\) be a finite extension. In this paper, we construct an RZ-space \(\mathcal{N}_{E}\) for split \(\operatorname{GU}(1,1)\) over a ramified quadratic extension \(E|F\). For this, we first introduce the naive moduli problem \(\mathcal{N}_E^{\operatorname{naive}}\) and then define \(\mathcal{N}_{E} \subseteq \mathcal{N}_E^{\operatorname{naive}}\) as a canonical closed formal subscheme, using the so-called straightening condition. We establish an isomorphism between \(\mathcal{N}_{E}\) and the Drinfeld moduli problem, proving the 2-adic analogue of a theorem of Kudla and Rapoport. The formulation of the straightening condition uses the existence of certain polarizations on the points of the moduli space \(\mathcal{N}_E^{\operatorname{naive}}\). We show the existence of these polarizations in a more general setting over any quadratic extension \(E|F\), where \(F|\mathbb{Q}_p\) is a finite extension for any prime \(p\). Rapoport-Zink spaces; Shimura varieties; bad reduction; unitary group; exceptional isomorphism Arithmetic aspects of modular and Shimura varieties, Modular and Shimura varieties Construction of a Rapoport-Zink space for \(\mathrm{GU}(1,1)\) in the ramified 2-adic case | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \({\mathcal D}\) be a base point free linear system on an n-dimensional complex manifold X. The author introduces the notion of a linear system of Lefschetz type and he proves the Lefschetz type theorems for the section of a generic member of \({\mathcal D}\). He shows that this notion behaves well under the pull back \(f^*| {\mathcal D}\), where f: \(Y\to X\) is a cyclic branched covering, branched along a smooth codimension one submanifold \({\mathcal S}\) such that \({\mathcal D}| {\mathcal S}\) is also of Lefschetz type. He uses these results to construct simply connected algebraic surfaces \(Y_ k(x_ 1,...,x_ k)\), \(x_ i\in {\mathbb{N}}\) which give infinitely many homeomorphic but not diffeomorphic algebraic surfaces of general type. The surface \(Y_ k(x_ 1,...,x_ k)\) is the composition of k finite triple cyclic coverings starting from \({\mathbb{P}}^ 1\times {\mathbb{P}}^ 1\) with a smooth branch locus which is linearly equivalent to \(3x_ 1C\) with \(C=pt\times {\mathbb{P}}^ 1+{\mathbb{P}}^ 1\times pt\). linear system on complex manifold; linear system of Lefschetz type; cyclic branched covering; algebraic surfaces; homeomorphic but not diffeomorphic Moishezon, B.: Analogs of Lefschetz theorems for linear systems with isolated singularities. J. Diff. Geometry31 47--72 (1990) Topology of Euclidean 4-space, 4-manifolds, Special surfaces, Differentiable structures in differential topology, Homotopy theory and fundamental groups in algebraic geometry Analogs of Lefschetz theorems for linear systems with isolated singularities | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a smooth projective complex manifold. Then it is a fundamental problem to understand the behavior of a multiple linear system, and this problem has been studied by many authors. The purpose of this paper is to give effective versions of some well-known theorems on multiple linear systems for \(\dim X=2\). The main result of this paper is the following:
Let \(A\) be a nef and big divisor on a smooth projective complex surface \(X\), let \(T\) be any fixed divisor, and let \(k\) be a nonnegative integer. Assume that either \(n>k+\mathcal{M}(A,T)\), or \(n\geq \mathcal{M}(A,T)\) when \(k=0\) and \(T\sim K_{X}+\lambda A\) for some rational number \(\lambda\), where \(\mathcal{M}(A,T):=((K_{X}-T)A+2)^{2}/4A^{2}-(K_{X}-T)^{2}/4\). Suppose that there exists a zero dimensional subscheme \(\Delta\) on \(X\) with minimal degree \(\deg\Delta \leq k\) such that it does not give independent conditions on \(| nA+T| \). Then there is an effective divisor \(D\not=0\) containing \(\Delta\) such that \(TD-D^{2}-K_{X}D\leq k\) and \(DA=0\).
By direct applications of the above result for various \(T\), we obtain the effective version of known theorems. projective complex surfaces; \(k\)-very ampleness; pseudo-effective divisor; nef divisor Tan S L. Effective behavior of multiple linear systems. Asian J Math, 2004, 8: 287--304 Divisors, linear systems, invertible sheaves, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Riemann-Roch theorems, Surfaces and higher-dimensional varieties Effective behavior of multiple linear systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a smooth hypersurface of degree \(d\) in \(\mathbb{P}^n_{\mathbb{C}}\). The Fano scheme \(F(X)\) of lines on \(X\) is the family of all lines contained in \(X\). Let \(N = \binom{n +d}{d} -1\), and \(U\) be an open subscheme in \(\mathbb{P}^N_{\mathbb{C}}\) parametrizing degree \(d\) smooth hypersurfaces \(X\) in \(\mathbb{P}^n_{\mathbb{C}}\). The Universal Fano scheme is defined by
\[
\Sigma = \lbrace (L, X) \in G (1, n) \times U \vert L \subset X \rbrace.
\]
\noindent Since the normal bundle \(N_{L/X}\) of line \(L\) in \(X\) is a rank \(n-2\) vector bundle on \(\mathbb{P}^1_{\mathbb{C}} = L\), it can be identified with a direct of sum of line bundles \(\mathcal{O}(a_1) \oplus \cdots \oplus \mathcal{O}(a_{n-2})\), for some nondecreasing integers \(\vec{a} = (a_1, \dots , a_{n-2})\), called the splitting type. We denote \(\mathcal{O}(a_1) \oplus \cdots \oplus \mathcal{O}(a_{n-2})\) by \(\mathcal{O}(\vec{a})\), where \(\vec{a} = (a_1, \dots, a_{n-2})\) with \(a_1 \leq \cdots \leq a_{n-2}\).
Let \(F_{\vec{a}}(X) := \{ L \in F(X) \vert N_{X/L} \cong \mathcal{O}(\vec{a}) \}\), and \(\sum_{\vec{a}} := \{ (L, X) \in \sum \vert N_{X/L} \cong \mathcal{O}(\vec{a}) \}.\)
The main result of this article concerns the fibres of natural projection map \(\sum_{\vec{a}} \rightarrow \mathbb{P}_{\mathbb{C}}^N\), which are the strata \(F_{\vec{a}}(X)\). More precisely, it is shown that for general hypersurfaces, these strata have the expected dimension. Also in this case, the class of the closure of the strata in the Chow ring of the Grassmannian of lines in projective space is computed. Further, for certain splitting types, the upper bounds on the dimension of the strata that hold for all smooth \(X\), is provided. algebraic cycles; Fano varieties Algebraic cycles, Fano varieties, Grassmannians, Schubert varieties, flag manifolds Normal bundles of lines on hypersurfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author surveys some recent progress on the Kato conjecture, by himself and Jannsen and more definitely by the author and Kerz.
The Kato conjecture is a generalization of the Hasse principle from local and global fields to schemes over a finite field, or the integer ring of a number field or a local field. The Kato homology of a scheme is defined as the homology of the Kato complex, and the Kato conjecture asserts that the Kato homology is trivial for a proper regular scheme for all positive homology dimensions. In the case of a scheme of dimension 1, the conjecture recovers the Hasse principle.
There had been proofs of the Kato conjecture when the dimension of the scheme or the of the Kato homology is small, more precisely \(\leq 3\). Jannsen and the author proved the Kato conjecture for homology dimension 4, under certain assumptions. Kerz and the author further improved the result to all dimensions, for both the finite field and the arithmetic cases, at least if we are restricted to the part prime to the characteristic of the field.
The methods of the proof is then explained in steps. The author interprets the Kato homology as the \(E^2\) terms of the niveau filtration, utilizes the edge homomorphism, analyzes log pairs to incur induction, and at the end uses Gabber's result to reduce the general situation to divisors with simple normal crossings.
At the end, some applications to the finiteness of Chow groups are presented. Kato conjecture; Hasse principle; Kato homology; niveau filtration; log pairs; Gabber's refined uniformization S. Saito, Recent progress on the Kato conjecture, in Quadratic Forms, Linear Algebraic Groups, and Cohomology, Developments in Math., vol. 18, pp. 109--124, 2010. Étale and other Grothendieck topologies and (co)homologies, Motivic cohomology; motivic homotopy theory, Positive characteristic ground fields in algebraic geometry, Global ground fields in algebraic geometry, Galois cohomology Recent progress on the Kato conjecture | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors give a characterization of the atypical values at infinity for a real rational function \(f/g\) defined on a real algebraic surface \(V\) in \(\mathbb{R}^n\). In order to do so, the authors define a 1--dimensional semi--algebraic subset \(\Gamma\) of \(V\), called tangency curve of \((f/g)|_V\) and consider the half--branches of \(\Gamma\) at infinity. Let \(H=0\) be an algebraic equation for \(\Gamma\) (\(H\) depends on \(f,g\) and the real polynomials defining \(V\)). For each half--branch \(C\) of \(\Gamma\), \(f/g\) is monotonous (increasing, decreasing or constant) over \(C\) and \(H\) might or might not change sign along \(C\). With this information and a point \(K\) to make a start (the authors choose \(K\) to be a connected component of \(V\setminus \{g=0\}\) intersected with a big enough sphere), critical clusters belonging to \(\lambda\in\mathbb R\), bands, valleys and crests are defined. Then, for each critical cluster, an integer in \(\{0,\pm 1,\pm 2\}\) is computed. The main theorem of the paper asserts that \(\lambda\in\mathbb R\) is an atypical value at infinity for \((f/g)|_V\) if and only if there exists a critical cluster belonging to \(\lambda\) such that the corresponding integer is non--zero. The second main result provides an upper bound for the number of atypical values for \((f/g)|_V\), in terms of the degrees of \(f,g\) and the polynomials defining \(V\). The paper finishes with three worked out examples.
The techniques used in the proofs are standard of algebraic geometry and real algebraic geometry. Definitions and arguments closely follow those in a paper by \textit{M. Coste} and \textit{M. J. de la Puente} [J. Pure Appl. Algebra 162, No. 1, 23--35 (2001; Zbl 1042.14038)], as the authors themselves say.
The paper is rather interesting. Some questions arise, though. For instance, in all the examples contained in the paper, \(g=1\). Is the case \(g\) general much more complicated? Another question is the following. In all the examples in the paper, \(V\) is a hypersurface in \(\mathbb{R}^3\). Is the general case much more involved? And, finally, are the bounds sharp? Polynomial function; rational function; algebraic surface; atypical value at infinity; critical cluster Topology of real algebraic varieties, Semialgebraic sets and related spaces, Nash functions and manifolds Atypical values at infinity of polynomial and rational functions on an algebraic surface in \(\mathbb R^n\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Recall that a linear system of hypersurfaces of some degree \(d\) in the projective space \(\mathbb{P}^n\) is called special if its dimension is larger than its expected dimension. For special linear systems \({\mathfrak L}(d, m_1,\dots,m_r)\) of hypersurfaces of degree \(d\) passing through a scheme \(Z=m_1p_1+\cdots+m_rp_r\) of fat points in general position in \(\mathbb{P}^2\), there exists a well-studied and partially proven characterization due to B. Harbourne and A. Hirschowitz [see \textit{B. Harbourne}, Can. Math. Soc. Conf. Proc. 6, 95--111 (1986; Zbl 0611.14002) and \textit{A. Hirschowitz}, J. Reine Angew. Math. 397, 208--213 (1989; Zbl 0686.14013)]. The authors study the analogous linear systems in \(\mathbb{P}^3\). Their main tool is the cubic Cremona transformation Cr :\((x_0:x_1: x_2:x_3)\mapsto(x_0^{-1}:x_1^{-1};x_1^{-1}:x_3^{-1})\). They describe the action of Cr on the Picard group of the blow-up \(X\) of \(\mathbb{P}^3\) at \(\{p_1,\dots,p_r\}\) and use it to bring the linear systems into a standard form. For linear systems in standard form, they present a conjectural characterization of the special ones. In [\textit{C. De Volder} and \textit{A. Laface}, J. Algebra 310, No. 1, 207--217 (2007; Zbl 1113.14036)], this conjecture has been verified for \(r\leq 8\) fat points. The authors also apply their conjecture to the ``homogeneous case'' \({\mathfrak L}(d,m,\dots,m)\) and provide further evidence for it in various other cases. Cremona transformation; virtual dimension; fat point scheme; linear systems Laface, A.; Ugaglia, L., On a class of special linear systems of \(\mathbb{P}^3\), Trans. Amer. Math. Soc., 358, 5485-5500, (2006), (electronic) Divisors, linear systems, invertible sheaves On a class of special linear systems of \(\mathbb{P}^3\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author defines the notions of connected component and of irreducible component for an algebraic stack \(\mathcal{X}\) of finite presentation over a scheme \(S\) and proves representability theorems for the functors which to such a stack associate their connected or irreducible components. In a first time the author defines an open connected component of an algebraic stack \(\mathcal{X}\) of finite presentation over a scheme \(S\) to be an open substack \(\mathcal{C} \subset \mathcal{X}\), faithfully flat and of finite presentation over \(S\) such that for any geometric point \(s\) of \(S\), \(\mathcal{C}_s\) is a connected component of \(\mathcal{X}_s\), and he denotes by \(\pi_0 (\mathcal{X}/S)\) the functor which to any \(S\)-scheme \(T\) associates the set of all such substacks of \(\mathcal{X}_T\) over \(T\). He proves that if in addition the stacks \(\mathcal{X}\) above are flat with fibers geometrically reduced over \(S\), then such a functor is representable by an étale algebraic space, quasi-compact over \(S\), that can be expressed as the quotient of an algebraic stack by an étale equivalence relation. In the event that the fibers of \(\mathcal{X}\) are non-geometrically reduced, the author defines a relative closed connected component of \(\mathcal{X}\) over \(S\) to be a closed substack \(\mathcal{C} \subset \mathcal{X}\), flat, of finite presentation over \(S\) such for any geometric point \(s\) of \(S\) the support of \(\mathcal{C}_s\) is a connected component of \(\mathcal{X}_s\) and he denotes by \(\pi_0(\mathcal{X}/S)^{\mathfrak{f}}\) the functor of all such closed substacks of \(\mathcal{X}\) over \(S\). Further, for \(\mathcal{C}\) such a substack of \(\mathcal{X}\), it is said to be reduced if its fibers are geometrically reduced and he denotes by \(\pi_0(\mathcal{X}/S)^{\mathfrak{r}}\) the corresponding functor. Then \(\pi_0(\mathcal{X}/S)^{\mathfrak{f}}\) is representable by a formal algebraic space, locally of finite presentation and separated over \(S\) and \(\pi_0(\mathcal{X}/S)^{\mathfrak{r}}\) is representable by a quasi-finite, separated formal scheme over \(S\).
If it turns out \(\mathcal{X}\) is an algebraic stack of finite presentation, flat over \(S\) with geometrically reduced fibers, then M. Romagny proves \(\pi_0(\mathcal{X}/S)^{\mathfrak{f}} = \pi_0(\mathcal{X}/S)^{\mathfrak{r}} \subset \pi_0(\mathcal{X}/S)\). If in addition \(\mathcal{X}\) is proper and \(\mathcal{X} \rightarrow \text{St}(\mathcal{X}/S) \rightarrow S\) is the Stein factorization, then we have isomorphisms \(\text{St}(\mathcal{X}/S) \simeq \pi_0(\mathcal{X}/S)^{\mathfrak{f}} = \pi_0(\mathcal{X}/S)\).
In the same manner that the author discusses connected components he also tackles irreducible components: for \(\mathcal{X}\) an algebraic stack of finite presentation and with geometrically reduced fibers over a scheme \(S\), M. Romagny defines an open irreducible component of \(\mathcal{X}\) over \(S\) to be an open substack \(\mathcal{I} \subset \mathcal{X}\), faithfully flat and of finite presentation over \(S\) such that for any geometric point \(s\) of \(S\) \(\mathcal{I}_s\) is an open irreducible component of \(\mathcal{X}_s\). He denotes by \(\text{Irr}(\mathcal{X}/S)\) the functor that associates to an \(S\)-scheme \(T\) the set of open irreducible components of \(\mathcal{X}_T\) over \(T\). He proves that if in addition \(\mathcal{X}\) above is flat then \(\text{Irr}(\mathcal{X}/S)\) is representable by an étale and quasi-compact algebraic space over \(S\), quotient of an algebraic stack by an étale equivalence relation and there is a surjective morphism \(\text{Irr}(\mathcal{X}/S) \rightarrow \pi_0(\mathcal{X}/S)\). In the event that \(\mathcal{X}\) is with fibers non-geometrically reduced, M. Romagny defines the notion of relative closed irreducible components of \(\mathcal{X}\) of finite presentation over \(S\) as a closed substack \(\mathcal{I} \subset \mathcal{X}\), flat of finite presentation over \(S\) such that the support for \(\mathcal{I}_s\) is an irreducible component of \(\mathcal{X}_s\) for any geometric point \(s\) of \(S\) and he denotes by \(\text{Irr}(\mathcal{X}/S)^{\mathfrak{f}}\) the functor of such closed substacks of \(\mathcal{X}\) over \(S\). If in addition \(\mathcal{X}\) is flat, with fibers geometrically reduced, there is a monomorphism of functors \(\text{Irr}(\mathcal{X}/S)^{\mathfrak{f}} \hookrightarrow \text{Irr}(\mathcal{X}/S)\). Further if \(\mathcal{X}\) is proper, \(\text{Irr}(\mathcal{X}/S)^{\mathfrak{r}}\) is an étale separated scheme. stacks; moduli stacks Matthieu Romagny, ``Composantes connexes et irréductibles en familles'', Manuscripta Math.136 (2011) no. 1-2, p. 1-32 Stacks and moduli problems, Generalizations (algebraic spaces, stacks), Fibrations, degenerations in algebraic geometry, Fine and coarse moduli spaces, Connections of general topology with other structures, applications, Connected and locally connected spaces (general aspects) Connected and irreducible components in families | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The main result of the paper under review proves a Darboux theorem for derived schemes. It says that if \(\mathbf{X}\) is a derived scheme and \(\omega_{\mathbf{X}}\) is a \(k\)-shifted symplectic structure on \(\mathbf{X}\) for \(k < 0\) with \(k \not \equiv 2\bmod 4\), then \((\mathbf{X},\omega_{\mathbf{X}})\) is Zariski locally equivalent to \((\text{Spec } A,\omega)\), for \(\text{Spec } A\) an affine derived scheme in which the commutative differential graded algebra \(A\) is smooth in degree 0 and quasi-free in negative degrees, and that has Darboux-like coordinates \(x^i_j\) , \(y^{k-i}_j\) with respect to which the symplectic form \(\omega=\sum_{i,j}d_{dR}x^i_jd_{dR}y^{k-i}_j\) is standard, and in which the differential in \(A\) is given by a Poisson bracket with a Hamiltonian function of degree \(k + 1\). When \(k < 0\) with with \(k \equiv 2 \bmod 4\) then the same statment holds étale locally instead (and a variation of the statment holds Zariski locally). In the case \(k =-1\), this result in particular shows that a \(-1\)-shifted symplectic derived scheme \((\mathbf{X},\omega_\mathbf{X})\) is Zariski locally equivalent to the derived critical locus \(\text{Crit}(\Phi)\) of a regular function \(\Phi: U \to \mathbb A^1\) on a smooth scheme \(U\). In this case, the underlying classical scheme \(X = t_0(\mathbf{X})\) extends naturally to an algebraic \(d\)-critical locus \((X,s)\) (defined by the last author of the paper) in which the geometric structure \(s\) records information on how \(X\) may locally be written as a classical critical locus \(\text{Crit}(\phi)\) of a regular function \(\Phi : U \to \mathbb A^1\) on a smooth scheme \(U\). This means that the last statement defines a truncation functor from \(-1\)-shifted symplectic derived schemes to algebraic \(d\)-critical loci. One of the applications of these results will be to categorified and motivic Donaldson-Thomas theory of Calabi-Yau 3-folds Darboux theorem; derived schemes; shifted symplectic structure; \(d\)-critical locus Generalizations (algebraic spaces, stacks), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Stacks and moduli problems, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) A Darboux theorem for derived schemes with shifted symplectic structure | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A codimension \(r+1\) scheme \(X \subset {\mathbb P}^n\) is called a standard determinantal scheme if \(I_X\) is the ideal generated by the maximal minors of a homogeneous \(t\times (t+r)\)-matrix \(A\). The determinantal scheme \(X\) will be called good if after performing some row operations on \(A\), the resulting matrix contains a \((t-1)\times (t+r)\)-submatrix whose ideal of maximal minors defines a scheme of codimension \(r+2\). This paper gives several characterizations of standard and good determinantal schemes.
The main results can be summarized as follows. Let \(X\) be a subscheme of \({\mathbb P}^n\) with codimension \(\geq 2\). Then the following conditions are equivalent:
(a) \(X\) is a good determinantal scheme of codimension \(r+1\),
(b) \(X\) is the zero-locus of a regular section of the dual of a first Buchsbaum-Rim sheaf of rank \(r+1\),
(c) \(X\) is standard determinantal and locally a complete intersection outside a subscheme \(Y \subset X\) of codimension \(r+2\).
Furthermore, for any good determinantal subscheme \(X\) of codimension \(r+1\) there is a good determinantal subscheme \(S\) of codimension \(r\) such that \(X\) sits in \(S\) in a nice way.
As an application, the authors show that for a zero-scheme \(X \subset {\mathbb P}^3\), being good determinantal is equivalent to the existence of an arithmetically Cohen-Macaulay curve \(S\) which is a local completion such that \(X\) is a subcanonical Cartier divisor on \(S\). This generalizes an earlier result of \textit{M. Kreuzer} on 0-dimensional complete intersection [Math. Ann. 292, No. 1, 43-58 (1992; Zbl 0741.14030)]. The paper closes with a number of examples. standard determinantal scheme; Buchsbaum-Rim sheaf; Cartier divisor; good determinantal schemes; arithmetically Cohen-Macaulay curve; complete intersection Kreuzer, M; Migliore, J; Nagel, U; Peterson, C, Determinantal schemes and Buchsbaum-rim sheaves, J. Pure Appl. Algebra, 150, 155-174, (2000) Determinantal varieties, Complete intersections, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Syzygies, resolutions, complexes and commutative rings, Divisors, linear systems, invertible sheaves, Linkage, complete intersections and determinantal ideals Determinantal schemes and Buchsbaum-Rim sheaves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems If \(\{f_\lambda\}_\Lambda\) is a one-parameter family of discrete dynamical systems on \(\mathbb{R}\) indexed by an interval \(\Lambda\) of \(\mathbb{R}\), then a parameter value \(\lambda_0\) is called orbit-creating if, at \(\lambda_0\), periodic orbits are created and no periodic orbits are annihilated. Similarly, \(\lambda_0\) is called orbit-annihilating if periodic orbits are annihilated at \(\lambda_0\) and no new periodic orbits are created; \(\lambda_0\) is called neutral if periodic orbits are neither created nor annihilated at \(\lambda_0\). The family \(\{f_\lambda\}_\Lambda\) is called monotone increasing (respectively, decreasing) if every parameter value in \(\Lambda\) is neutral or orbit-creating (respectively, annihilating). The logistic family \(\{f_\lambda (x)= \lambda x(1-x)\); \(x\in [1,4]\}\) is monotone increasing as is its topological entropy.
The author of this paper considers the family \(\{mf (x)\}_m\) where \(f\) is a real quadratic rational function. This family can be monotone or non-monotone depending on the choice of \(f\). The author explains how variations in the bifurcation process occur. The primary tool is analysis of the family in the moduli space of the quadratic rational maps. chaotic dynamical systems; monotone bifurcations; moduli space of quadratic rational functions Local and nonlocal bifurcation theory for dynamical systems, Strange attractors, chaotic dynamics of systems with hyperbolic behavior, Low-dimensional dynamical systems, Algebraic moduli problems, moduli of vector bundles Non-monotone bifurcations for quadratic rational functions \(\{mf(x)\}_m\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\mathcal{S}^2\) be an oriented two-sphere and \(\phi:\mathcal{S}^2\longrightarrow \mathcal{S}^2\) an orientation-preserving branched covering whose post-critical set \[P=\{\phi^n(x)\mbox{ such that } x \mbox{ is a critical point of } \phi \mbox{ and } n>0\},\]
is finite. By work of \textit{S. Koch} [Adv. Math. 248, 573--617 (2013; Zbl 1310.32016)], the Thurston pullback map induced by \(\phi\) on Teichmüller space descends to a multivalued self-map (a Hurwitz correspondence \(H_\phi\)) of the moduli space \(\mathcal{M}_{0,P}\). The author studies the dynamics of Hurwitz correspondences via numerical invariants called dynamical degrees. He shows that the sequence of
dynamical degrees of \(H_\phi\) is always non-increasing and that the behavior of this sequence is constrained by the behavior of \(\phi\) at and near points of its post-critical set. This paper is organized as follows: The first section is an introduction to the subject. In the second section the author gives background on meromorphic multivalued maps (henceforth referred to as rational correspondences), the moduli space \(\mathcal{M}_{0,P}\) and its compactification \(\mathcal{\overline{M}}_{0,P}\), Hurwitz spaces and Hurwitz correspondences. The third section, contains the proof of the main result and the fourth section deals with an application of the main theorem to enumerative algebraic geometry. In the fifth and sixth sections the author gives examples of correspondences specific to Hurwitz. dynamical degrees; correspondences; moduli spaces Families, moduli of curves (algebraic), Projective and enumerative algebraic geometry, Special varieties, Dynamical systems involving relations and correspondences in one complex variable Dynamical degrees of Hurwitz correspondences | 0 |
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