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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 In this paper, we iterate the algebraic étale-Brauer set for any nice variety \(X\) over a number field \(k\) with \(\pi_1^{\text{ét}}(\overline{X})\) finite and we show that the iterated set coincides with the original algebraic étale-Brauer set. This provides some evidence towards the conjectures by Colliot-Thélène on the arithmetic of rational points on nice geometrically rationally connected varieties over \(k\) and by Skorobogatov on the arithmetic of rational points on K3 surfaces over \(k\); moreover, it gives a partial answer to an ``algebraic'' analogue of a question by Poonen about iterating the descent set. rational points; weak approximation; Brauer-Manin obstruction; Étale-Brauer obstruction; universal torsors; algebraic groups Varieties over global fields, Brauer groups of schemes, Rational points, Global ground fields in algebraic geometry, Linear algebraic groups over global fields and their integers Iterating the algebraic étale-Brauer set | 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 complex algebraic surfaces \(S\), a line bundle \(L\) on \(S\), and integer \(\delta\), the Severi degree \(n^{(S,L),\delta}\) counts the number of \(\delta\)-nodal curves in the linear system \(|L|\) passing through \(\dim|L|-\delta\) general points. The Göttsche multiplicativity conjecture says that for \(L\) sufficiently ample with respect to \(\delta\), the Severi degrees \(n^{(S,L),\delta}\) are coefficients of \(t^\delta\) in the product \(a_1(t)^{L^2}a_2(t)^{LK_S}a_3(t)^{K^2_S}a_4(t)^{\chi({\mathcal O}_S}\), where \(a_1(t),a_2(t),a_3(t),a_4(t)\) are some universal power series. The authors analyze recent proofs of this conjecture for the plane and some other toric surfaces given by \textit{F. Block} et al. [Bull. Braz. Math. Soc. (N.S.) 45, No. 4, 625--647 (2014; Zbl 1323.14027)] and by \textit{F. Liu} and \textit{B. Osserman} [``Severi degrees on toric surfaces'', Preprint, \url{arXiv:1401.7023}], and which are based on the tropical geometry techniques (so-called long-edge graphs, or floor diagrams). In these proofs, the Severi degrees are expressed as the sums of weights of suitable long-edge graphs, where the weight of a long-edge graph turns to be the product of multiplicities of its edges. In the present paper, the authors replace the integer multiplicities \(m\) of the edges by refined multiplicities \(\frac{y^{m/2}-y^{-m/2}}{y^{1/2}-y^{-1/2}}\) (as it was done before for refined counting of plane tropical curves by \textit{F. Block} and \textit{L. Göttsche} [Compos. Math. 152, No. 1, 115--151 (2016; Zbl 1348.14125)] and introduce refined Severi degrees \(N_\delta((S,L),y)\) that appear to be Laurent polynomials in \(y\). The main result states the Göttsche multiplicativity of the refined Severi degrees for the plane and for ruled toric surfaces. For example, in the planar case, the generating function for the refined Severi degrees is the product \(a_1(y,t)a_2(y,t)^da_3(y,t)^{d^2}\), where \(a_1,a_2,a_3\in{\mathbb Q}[y,y^{-1}][[t]]\), and \(d\) the given degree of curves. At \(y=1\) this formula turns into the multiplicativity formula for the usual plane Severi degrees. At \(y=-1\) one obtains a generating function for Welschinger numbers that count real plane \(\delta\)-nodal curves of degree \(d\) passing through specific configurations of real points in the plane. The authors also discuss possible generalizations of the main result to counting curves on singular surfaces and to counting curves with prescribed multiple points. enumerative geometry; Severi degrees; node polynomials; refined curve counting; tropical geometry Göttsche, Lothar; Kikwai, Benjamin, Refined node polynomials via long edge graphs, Commun. Number Theory Phys., 10, 2, 193-224, (2016) Enumerative problems (combinatorial problems) in algebraic geometry, Relations with algebraic geometry and topology, Theta functions and abelian varieties, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Refined node polynomials via long edge graphs | 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 For an ideal \(I\) in a polynomial ring one can associate a family of jet scheme ideals \(\mathcal{J} = \{I^{(k)}\}_{k \in \mathbb{N}_+}\). In his paper (see [Invent. Math. 145, No. 3, 397--424 (2001; Zbl 1091.14004)]) \textit{M. Mustaţă} has shown (among others) that the jet schemes of a determinantal variety of \(r \times r\)-matrices of rank at most \(r-1\) are irreducible. The authors start to investigate jet scheme ideals of Pfaffians. Let \(X\) denote a \(n \times n\)-skew symmetric matrix of indeterminates over an algebraically closed field \(K\). Let \(I_r \subset K[X]\) denote the Pfaffian ideal of size \(2r\). Then they investigate the jet scheme ideals \(I^{n,k}_r\). Among others it is shown: (1) \(I^{2r,k}_r\) is prime of codimension \(k\). (2) \(I^{2r+1,k}_r\) is prime of codimension \(3k\). (3) \(I^{n,k}_r\) is reducible and not pure for \(n \geq 2r+2\) and \(r \geq 2\). -- The main idea of their proofs is a reduction technique from the study of \(I^{n,k}_r\) to that of some \(I^{n_1,k_1}_{r_1}\) with \(n_1 \leq n, k_1 \leq k, r_1 \leq r\) and results about Gröbner bases. jet schemes; Pfaffian varieties; determinantal ideals; irreducible varieties Linkage, complete intersections and determinantal ideals, Determinantal varieties, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) On jet schemes of Pfaffian ideals | 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 general, any snake can be associated with Morse polynomials in one variable, and the number of snakes is equal to the number of topologically nonequivalent Morse polynomials (see [\textit{V. I. Arnol'd}, Russ. Math. Surv. 47, No. 1, 1 (1992; Zbl 0791.05001); translation from Usp. Mat. Nauk 47, No. 1, 3--45 (1992)]). The author considers snakes associated with alternating permutations, given by the relative positions of critical values of a Morse polynomial (cf. [\textit{S. K. Lando}, Lectures on generating functions. Transl. from the Russian by the author. Providence, RI: American Mathematical Society (AMS) (2003; Zbl 1032.05001)]), and calls them Arnold snakes. Then using the notion of separable permutation (see [\textit{P. Bose} et al., Inf. Process. Lett. 65, No. 5, 277--283 (1998; Zbl 1338.68304); \textit{S. Kitaev}, Patterns in permutations and words. Berlin: Springer (2011; Zbl 1257.68007)]) and some other combinatorial objects and considerations, she shows how to construct explicitly polynomials in one variable with preassigned critical values configurations for a special class of Arnold snakes associated with separable permutations.
The author emphasizes also that the paper is based on the first chapter of her PhD thesis [The shapes of level curves of real polynomials near strict local minima. Université de Lille (2018), \url{https://hal.archives-ouvertes.fr/tel-01909028v1}], defended at Paul Painlevé Laboratory in Lille. snakes; Morse polynomials; critical values; minima and maxima; valuations; contact trees; bifurcation vertices; binary trees; alternating permutations; separable permutations Topology of real algebraic varieties, Permutations, words, matrices, Singularities of curves, local rings, Real algebraic sets, Trees, Computational aspects of algebraic curves, General theory for finite permutation groups, Real polynomials: location of zeros, Critical points of functions and mappings on manifolds, Algorithms on strings Constructing separable Arnold snakes of Morse polynomials | 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 toric Hilbert scheme is a parameter space for all ideals with the same multigraded Hilbert function as a given toric ideal. Unlike the classical Hilbert scheme, it is unknown whether toric Hilbert schemes are connected. We construct a graph on all the monomial ideals on the scheme, called the flip graph, and prove that the toric Hilbert scheme is connected if and only if the flip graph is connected. These graphs are used to exhibit curves in \(\mathbb P^4\) whose associated toric Hilbert schemes have arbitrary dimension. We show that the flip graph maps into the Baues graph of all triangulations of the point configuration defining the toric ideal. Inspired by the recent discovery of a disconnected Baues graph, we close with results that suggest the existence of a disconnected flip graph and hence a disconnected toric Hilbert scheme. Maclagan D., ''Combinatorics of the toric Hilbert scheme'' (1999) Parametrization (Chow and Hilbert schemes), Toric varieties, Newton polyhedra, Okounkov bodies Combinatorics of the toric Hilbert scheme | 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 \(Y\subseteq \mathbb P^n\) be a pure one-dimensional locally Cohen-Macaulay closed subscheme such that for every general hyperplane \(H\) there is a zero-dimensional subscheme \(Z_H\) of \(Y\cap H\) and a subscheme \(C_H\) of \(H\) with \(Z_H\subseteq C_H\). We prove, under certain assumptions on \(Y\), length \((Z_H)\) and \(C_H\), the existence of a subcurve \(Y'\subseteq Y\) and a scheme \(S\subseteq\mathbb P^n\) with \(Y'\cap H=Z_H\) and \(S\cap H=C_H\), for general \(H\). The main cases we study are: \(C_H\) is a rational normal curve and \(C_H\) is a linear space. We prove also a lifting theorem for the property arithmetically Gorenstein, which generalizes a lifting theorem by \textit{R. Strano} [Proc. Am. Math. Soc. 104, No. 3, 711--715 (1988; Zbl 0693.14020)] and \textit{C. Huneke} and \textit{B. Ulrich} [J. Algebr. Geom. 2, 487--505 (1993; Zbl 0808.14041)]. Ballico E., J. Pure and Appl. Algebra 183 pp 1-- (2003) Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Projective techniques in algebraic geometry, Plane and space curves Projective curves: multisecant schemes and lifting 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 In a previous paper [Jpn. J. Math. (3) 13, No. 2, 235--271 (2018; Zbl 1401.14209)], the authors constructed all polynomial tau-functions of the \(1\)-component KP hierarchy, namely, they showed that any such tau-function is obtained from a Schur polynomial \(s_\lambda(t)\) by certain shifts of arguments. In the present paper they give a simpler proof of this result, using the (\(1\)-component) boson-fermion correspondence. Moreover, they show that this approach can be applied to the \(s\)-component KP hierarchy, using the \(s\)-component boson-fermion correspondence, finding thereby all its polynomial tau-functions. The authors also find all polynomial tau-functions for the reduction of the \(s\)-component KP hierarchy, associated to any partition consisting of \(s\) positive parts.
The paper is organized as follows. The first section is an introduction to the subject. Section 2 is devoted to the fermionic formulation of the KP hierarchy and Section 3 to the bosonic formulation of KP. Section 4 deals with polynomial solutions of KP. In Section 5 the authors introduce the \(s\)-component KP, where \(s\) is a positive integer. Section 6 is devoted to the \(n\)-KdV where \(n\) is an integer, \(n\geq2\). In Section 7 the authors consider a reduction of the \(s\)-component KP hierarchy, which describes the loop group orbit of \(SL_n\), where \(n=n_1+n_2+\cdots+n_s\), with \(n_1\geq n_2\geq\cdots\geq n_s\geq1\). The case \(s=1\) is the nth Gelfand-Dickey hierarchy. The case \(n=s=2\), i.e., \(n_1=n_2=1\), is the AKNS (or nonlinear Schrödinger) hierarchy. Section 8 is devoted to the AKNS hierarchy. KP hierarchy; multicomponent KP; tau-functions; Schur polynomials Grassmannians, Schubert varieties, flag manifolds, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Kac-Moody groups, Applications of Lie groups to the sciences; explicit representations, KdV equations (Korteweg-de Vries equations) Polynomial tau-functions for the multicomponent KP hierarchy | 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, reductive Lie group, and \(P\) a parabolic subgroup. The cohomology of the generalized flag variety \(G/P\) has a geometric basis, represented by so-called Schubert varieties. It is a longstanding goal in combinatorial algebraic geometry to find visibly positive combinatorial rules that calculate the structure constants of the cohomology ring with respect to this basis. The paper under review provides such a concise, root system uniform rule for the structure constants, in selected cases of \(G\) and \(P\) (called minuscule, and cominuscule). The main result is the extension of the well-known \textsl{jeu de taquin} formulation of these structure constants in the Grassmannian case. This result seems to be the first uniform generalization of the classical Littlewood-Richardson rule that involves both classical and exceptional Lie types. Although the proof of the main result is not type free (in particular, does not provide a new proof in the classical cases), the new viewpoint of the paper is an important contribution to the connections between Schubert calculus and representation theory. Schubert calculus; Littlewood-Richardson rule; minuscule; cominuscule; jeu de taquin Hugh Thomas & Alexander Yong, ``A combinatorial rule for (co)minuscule Schubert calculus'', Adv. Math.222 (2009) no. 2, p. 596-620 Enumerative problems (combinatorial problems) in algebraic geometry A combinatorial rule for (co)minuscule Schubert calculus | 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 \(k\) denote an algebraically closed field of arbitrary characteristic. Varieties are integral separated finite type schemes over \(k\). If \(S\) and \(X\) are varieties, then \(\text{\textbf{Hom}}(S,X)\) is the functor whose value at the variety \(T\) is \(\text{Hom}(S\times T,X)\). If \(A\) is a (commutative) group scheme over the variety \(S\) then \(\text{\textbf{Sect}}(A/S)\) is the functor whose value at \(T\) is \{sections of \(A\times T\to S\times T\}\). The author proves that (for \(k\) the complex numbers) if \(U\to S\) is an étale morphism of varieties and if the universal cover of the variety \(X\) is a Stein space then \(\text{\textbf{Hom}}(S,X)\to\text{\textbf{Hom}}(U,Y)\) is a closed immersion of functors. Also (still for \(k\) the complex numbers) if \(G\) is a commutative flat group scheme locally of finite type over the variety \(X\) such that \(G\to X\) has connected fibres, and if \(U\to X\) is étale, then \(\text{\textbf{Sect}}(G/X)\to\text{\textbf{Sect}}(G_ U/U)\) is a closed immersion. group scheme sections; varieties Jaffe, D.B. : Relative representability of group scheme sections , Amer. J. Math. 112 (1990), 461-479. Varieties and morphisms, Group varieties Relative representability of group scheme sections | 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 Gelfand, Graev, Kapranov, and Zelevinsky defined certain linear systems of partial differential equations, now known as \(A\)-\textit{hypergeometric} or \textit{GKZ hypergeometric} systems \(H_A(\beta)\), whose solutions generalize the classical hypergeometric series. These holonomic systems are constructed from discrete input consisting of an integer \(d \times n\) matrix \(A\) along with continuous input consisting of a complex vector \(\beta\in\mathbb C^d\). Assume that the convex hull conv\((A)\) of the columns of \(A\) does not contain the origin. The matrix \(A\) determines a semigroup ring \({\mathbb C}[\mathbb NA]\), and the dimension rank\((H_A(\beta))\) of the space of analytic solutions of \(H_A(\beta)\) is independent of \(\beta\) whenever \({\mathbb C}[\mathbb NA]\) is Cohen-Macaulay. In this note, the authors use the combinatorics of \(\mathbb Z^d\)-graded local cohomology to characterize the set of parameters \(\beta\) for which the rank goes up, in the simplicial case. The premise is the standard fact that a semigroup ring \({\mathbb C}[\mathbb NA]\) fails to be Cohen-Macaulay iff a local cohomology module \(H_{\mathfrak m}^i({\mathbb C}[\mathbb NA])\) is nonzero for some cohomological index \(i\) strictly less than the dimension \(d\) of \({\mathbb C}[\mathbb NA]\). After gathering some facts about \(A\)-hypergeometric systems, the authors prove the simplicial case of the following: Assume that conv\((A)\) has dimension \(d-1\). The set of parameters \(\beta\in \mathbb C^d\) such that rank\((H_A(\beta))\) is greater than the generic rank equals the Zariski closure (in \(\mathbb C^d\)) of the set of \(\mathbb Z^d\)-graded degrees where the local cohomology \(\bigoplus_{i<d}H_{\mathfrak m}^i({\mathbb C}[\mathbb NA])\) is nonzero. Using a different approach, the authors prove the full conjecture in the paper \textit{L. F. Matusevich, E. Miller}, and \textit{U. Walther} [Homological methods for hypergeometric families, J. Am. Math. Soc. 18, No. 4, 919-941 (2005; Zbl 1095.13033)]. \(A\)-hypergeometric systems; simplicial; combinatorics of \(\mathbb Z^d\)-graded local cohomology; local cohomology for semigroup rings; \(A\)-hypergeometric module; local cohomology module; rank-jumping parameter Laura Felicia Matusevich and Ezra Miller, Combinatorics of rank jumps in simplicial hypergeometric systems, Proc. Amer. Math. Soc. 134 (2006), no. 5, 1375 -- 1381. Other hypergeometric functions and integrals in several variables, Toric varieties, Newton polyhedra, Okounkov bodies, Commutative rings of differential operators and their modules, Local cohomology and commutative rings, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Cohen-Macaulay modules, Ordinary and skew polynomial rings and semigroup rings, Semigroup rings, multiplicative semigroups of rings Combinatorics of rank jumps in simplicial hypergeometric 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 \(R = \mathbb{C}[x_{1},\dots,x_{N}]\) and let \(F =\{ f_{1},\dots,f_{t}\} \subset R\) be a set of generators for an ideal \(I\). Let \(Y =\{ y_{1},\dots,y_{\ell}\} \subset {\mathbb{C}}^{N}\) be a subset of the set of isolated solutions of the zero locus of \(F\). Let \(\mathfrak{m}_{y_{i}}\) denote the maximal ideal of \(y_i\) and let \(\mathcal{P}_{y_{i}}\) denote the \(\mathfrak{m}_{y_{i}}\)-primary component of \(I\). Let \(J = \cap _{i=1}^{l}\mathcal{P}_{y_{i}}\) and let \(\mathcal{Z}\) denote the corresponding zero dimensional subscheme supported on \(Y\). This article presents a numerical algorithm for computing the Hilbert function and the regularity of \(\mathcal{Z}\). In addition, the algorithm produces a monomial basis for \(R\slash J\). The input for the algorithm is the polynomial system \(F\) and a numerical approximation of each element in \(Y\). Griffin, Zachary A.; Hauenstein, Jonathan D.; Peterson, Chris; Sommese, Andrew J.: Numerical computation of the Hilbert function of a zero-scheme. Springer Proceedings in mathematics \& statistics (2011) Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Numerical computation of the Hilbert function and regularity of a zero dimensional scheme | 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 Schubert varieties \(X, X', \ldots, X''\) of a flag variety the authors are concerned with the following question: when is the intersection of the translates \( gX \cap g'X' \cap \ldots \cap g'' X''\) non-empty? It is well known that for the Grassmannian it is non-empty if and only if the indices of the Schubert varieties satisfy the linear Horn inequalities. The authors answer to this question is based upon the work of \textit{A. A. Klyachko} [Sel. Math. 4, 419--445 (1998; Zbl 0915.14010)], \textit{A. Knutson} and \textit{T. Tao} [J. Am. Math. Soc. 12, No. 4, 1055--1090 (1999; Zbl 0944.05097)] and \textit{A. Horn} [Pac. J. Math. 225--241 (1962; Zbl 0112.01501)].
The authors' main theorem, theorem 4, states that for a collection \(M(P)\) of parabolic subgroups \(Q \subset L\) , assuming that \(G/P\) is a cominuscule flag variety, such an intersection is non-empty if and only if for every \(Q \in M(P) \) and every list of Schubert varieties \(X, X', \ldots, X''\) of \( L/Q\) whose general translates have non-empty intersection a set of necessary inequalities here numbered as (8) which depend on the combinatorial condition on the tangent space to \(G/R\), the corresponding lie algebra to \(R\) and the natural embedding of the center of the nilradical of R must be satisfied.
The paper is organized as follows. Section one gives the basic definitions and notations used throughout the paper, starting with section 1.1 about linear algebraic groups and their flag varieties.~Section 1.2 deals with Schubert varieties and their tangent spaces. Section 1.3 is about transversality and section 1.4 is about cominuscule flag varieties; stating four equivalent characterizations of cominuscule flag varieties. In subsection 2.1, the main theorem, here theorem 4 is precisely stated and derives a very general inequality here numbered (7) in theorem 2 of this section and an inequality derived from (7) here numbered as (8) used to formulate theorem 4. The set of inequalities which determine the non-emptiness of the Schubert varieties are recursive in the sense that they come from similar non-empty intersections on smaller cominuscule varieties namely those given by \(L/Q\) where \( Q \in M(P)\) where the latter is the set of standard parabolic subgroups of \(L\) which are equal to the stabilizer of the tangent space to some \(L\)-orbit on the lie algebra of \(G/P\).
Section 3 is devoted to the proof of theorem 4 relying upon technical results about root systems which are given in appendix A of this paper.
In section 4 the authors examine the cominuscule relation in more detail, describing it on a case-by-case basis. It is interesting to note that the inequalities given by the inequality (8) might be redundant and do not in fact give a set of irredundant inequalities as the authors show in the examples of this section. As for example, in section 4.3 the case for type \(B_n\), \(G = SO(2n+1)\), \(G/P\) an odd-dimensional quadric they show that there is only one inequality needed namely the basic codimension inequality. In section 5, the authors discuss how the classical Horn inequalities arise form the inequalities from theorem 2 and show how to modify the proof of theorem 4 to prove their sufficiency in proposition 24 and 25.
Proposition 24 in fact recalls the Horn recursion formula obtained already by \textit{W. Fulton} [Bull. Am. Math. Soc. 37, No. 3, 209--249 (2000; Zbl 0994.15021)]. The authors show that both proposition 24 and 25 give exactly the same recursion formulas and in theorem 26 the authors obtain a different set of necessary inequalities for feasibility on \(G/P\) which they call naive inequalities using proposition 11. The naive inequalities were already given by \textit{P. Belkale} and \textit{S. Kumar} [Invent. Math. 166, No. 1, 185--228 (2006; Zbl 1106.14037)]. Finally in section 5.3, the authors obtain a set of necessary inequalities for the Lagrangian Grassmannian which are derived from the naive inequalities of theorem 26. The obtained inequalities are different from those of corollary 5 to main theorem 4 of this paper. Grassmannians; Schubert varieties; flag manifolds; classical problems; Schubert calculus; combinatorial problems concerning the classical groups; Horn inequalities; cominuscule flag variety; Littlewood-Richardson rule Purbhoo, K., Sottile, F.: The recursive nature of cominuscule Schubert calculus. Adv. Math. 217, 1962--2004 (2008) Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Classical problems, Schubert calculus The recursive nature of cominuscule Schubert calculus | 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 starting point of this paper was an observation: the spectral theory of triangular operators is naturally connected with a special reduction of the 2D Toda hierarchy. The authors present new reductions of the 2D Toda equations associated with low-triangular difference operators with explicit Hamiltonian description. The main goal of this paper is: 1) to construct a bi-Hamiltonian theory for some systems and to show that the space of strictly low diagonal difference operators admits two different Poisson structures; 2) to identify the corresponding Hamiltonians. integrable system; bi-Hamiltonian theory; Baker-Akhiezer function Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Relationships between algebraic curves and integrable systems, Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions Triangular reductions of the \(2D\) Toda hierarchy | 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 Auszug aus einer grösseren Arbeit, welche von den Ausnahmen handeln wird, die der Satz von Cayley über Schnittpunktsysteme erleidet, und von einer Ausdehnung desselben auf den Fall von Curven, deren Schnittpunkte vielfache Punkte einer von ihnen sind. U. a. findet der Verfasser, dass der Satz, wonach eine Curve \(C_m\), die durch \(pq-\delta\) Schnittpunkte
\[
[\delta = \tfrac{1}{2} (p+q -n-1) \; (p+q -n-2)]
\]
einer \(C_p\) und einer \(C_q\) geht \((p+q >n \geqq p \geqq q),\) auch die \(\delta\) übrigen Schnittpunkte derselben enthalten müsse, nur im Allgemeinen richtig ist und seine Giltigkeit verliert, wenn diese \(\delta\) Punkte auf einer Curve von der Ordnung \(p+q -n-3\) liegen. plane curves; Cayley-Bacharach theorem; intersection of plane curves Plane and space curves, Questions of classical algebraic geometry On intersection point systems of 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 Let V be a smooth complex projective threefold of general type and \(K_ V\) the canonical divisor on V. In this paper we study in detail the case when there exists an integer \(m>0\) such that the m-canonical system \(| mK_ V|\) has no fixed components, and the corresponding rational map \(\phi_{mK}\) is generically finite; in particular we study the base locus of this linear system. It is easy to see that under the above assumptions, \(K_ V\cdot C\geq 0\) for all curves C on V. The base curves of \(| mK_ V|\) which are of particular importance turn out to be those base curves C with \(K_ V\cdot C=0.\) It is shown in the paper that such a curve C must be isomorphic to \({\mathbb{P}}^ 1\) and its normal bundle must be one of \({\mathcal O}_ C(-1)\oplus {\mathcal O}_ C(- 1),\quad {\mathcal O}_ C(-2)\oplus {\mathcal O}_ C\) or \({\mathcal O}_ C(- 3)\oplus {\mathcal O}_ C(1).\)- Suppose now that we blow up C, say \(f_ 1:V_ 1\to V\) with exceptional divisor \(E_ 1\), and let \(C_ 1\) denote the minimal section of the ruled surface \(E_ 1\). Now blow up \(C_ 1\), obtaining \(f_ 2\), \(V_ 2\), \(E_ 2\) and \(C_ 2\), etc. The paper goes on to describe and interpret the sequence of normal bundles \(N_{C/V}\), \(N_{C_ 1/V_ 1}\), \(N_{C_ 2/V_ 2},..\). obtained in this way. The new case here is the case when (with the obvious notation) \(N_{C/V}=(-3,1)\); we discover here that the sequence of normal bundles must be (-3,1), (-3,0),...,(-3,0), (-2,-1), (-1,-1),... (where there is a finite number of (-3,0)'s in the sequence). Finally this information is used to obtain results concerning the possible configurations of those base curves C with \(K_ V\cdot C=0\). base curves of multicanonical systems; normal bundles of base curves; configurations of curves; threefold of general type Wilson, P.: Base curves of multicanonical systems on threefolds. Compositio Math.52, 99-113 (1984) \(3\)-folds, Families, moduli, classification: algebraic theory, Divisors, linear systems, invertible sheaves Base curves of multicanonical systems on threefolds | 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 an expository paper, presenting the background to the author's major results in J. Number Theory 23, 1-54 (1986; Zbl 0584.10022) and Ann. Inst. Fourier 36, No. 4, 1-30 (1986; Zbl 0597.32004). Definitions of knots, links, torus links and iterated torus links, and of their Eisenbud-Neumann diagrams are given and illustrated by examples. Of particular interest to the author are examples obtained by intersection of the zero locus of a polynomial in 2 complex variables with a sphere of very small radius (the local link) or very large radius (the link at infinity). The traditional relations with Newton polygons, and with Puiseux exponents, and the calculation of the Alexander polynomial are described, and extended to the case of the link at infinity. The Seifert form of a link is used to define its multisignature, and this in turn is used in the definition of the ``spectrum''. An extensive set of calculations is presented here: several particular examples, torus knots, the general algebraic knot, the local link for a Newton nondegenerate function, and the global link in the corresponding case. Finally the author considers the integrals
\[
\begin{aligned} I_ \infty(f,\beta)(s)&=\int^ \infty_ 1\int^ \infty_ 1x_ 1^{\beta_ 1-1}\cdot x_ 2^{\beta_ 2-1}\cdot f(x_ 1,x_ 2)^{-s}dx_ 1dx_ 2\quad\text{ and } \\ I_ 0(f,\beta)(s)&=\int^ 1_ 0\int^ 1_ 0x_ 1^{\beta_ 1-1}\cdot x_ 2^{\beta_ 2-1}\cdot f(x_ 1,x_ 2)^ sdx_ 1dx_ 2.\end{aligned}
\]
These converge in half-planes \(\text{Re}(s)>\sigma\) and admit meromorphic extension to \(\mathbb{C}\). The poles of \(I_ 0\) where known to be related to the spectrum of the local link of \(f\) at (0,0). The new results give a relation between poles of \(I_ \infty\) and the spectrum of the global link. The details are too complicated to summarise here; the main results assume that the coefficients in \(f\) are real and positive. local link; link at infinity; Newton polygons; Alexander polynomial; Seifert form Cassou-Noguès, P. : Entrelacs toriques itérés et intégrales associées à une courbe plane , Séminaire de théorie de nombres, Bordeaux 2 (1990) 237-331. Pencils, nets, webs in algebraic geometry, Singularities of curves, local rings, Other hypergeometric functions and integrals in several variables Iterated toric webs and integrals associated to a plane 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 We study the resonance variety of a line combinatorics. We introduce the concept of combinatorial pencil, which characterizes the components of this variety and their dimensions. The main theorem in this paper states that there is a correspondence between components of the resonance variety and combinatorial pencils. As a consequence, we conclude that the depth of a component of the resonance variety is determined by its dimension; and that there are no embedded components. This result is useful to study the isomorphisms between fundamental groups of the complements of line arrangements with the same combinatorial type. The definition of combinatorial pencil generalizes the idea of net given by Yuzvinsky and others. line arrangement; resonance variety; pencil Miguel, Á.; Buzunáriz, M., A description of the resonance variety of a line combinatorics via combinatorial pencils, Graphs Combin., 25, 469-488, (2009) Projective techniques in algebraic geometry, Arrangements of points, flats, hyperplanes (aspects of discrete geometry), Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Pencils, nets, webs in algebraic geometry A description of the resonance variety of a line combinatorics via combinatorial pencils | 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 geometric theory of dessins d'enfants is used to make explicit calculations on curves. In particular, an algorithmic procedure for the construction of ramified covering of curves over number fields with prescribed ramifications and for the explicit construction of Jenkins-Strebel differentials are developed. dessins d'enfants; ramified covering; Jenkins-Strebel differentials Arithmetic aspects of dessins d'enfants, Belyĭ theory, Coverings of curves, fundamental group, Elliptic curves, Dessins d'enfants theory, Computational aspects of algebraic curves On computations with dessins d'enfants | 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 \(\Bbbk\) be an algebraically closed field. We prove that a polynomial \(\Bbbk\)-derivation \(D\) in two variables is locally nilpotent if and only if the subgroup of polynomial \(\Bbbk\)-automorphisms which commute with \(D\) admits elements whose degree is arbitrary big. locally nilpotent derivations; polynomials Derivations and commutative rings, Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) A characterization of local nilpotence for dimension two polynomial derivations | 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 Pro-categories, as such, were introduced by Grothendieck as representing objects for certain (possibly non-representable) functors occurring in algebraic geometry. They generalized categories of inverse systems that were traditionally indexed by directed sets and had occurred much earlier, even before categories \textit{per se}, in work on Čech homology. They also naturally occurred in the study of profinite groups and thus in Galois theory. Via the introduction of Grothendieck's fundamental group of a scheme and étale cohomology theory, they became an essential tool in both algebraic geometry and algebraic topology via the Artin-Mazur lecture notes volume on étale homotopy [\textit{M. Artin} and \textit{B. Mazur}, Étale homotopy, Lect. Notes Math. 100 (1969; Zbl 0182.26001)]. There the category pro-Ho(SS) was studied, i.e., the category of pro-objects in the homotopy category of simplicial sets and it was noted that this was not a homotopy category of a category of prosimplicial sets.
The initial applications of Artin-Mazur's theory were in localization theory for spaces, but also, from 1973 onwards in shape theory. It became obvious that there was a much better structured strong shape theory lurking just beneath the surface, but it was hard to make this precise. Shape theory embedded problems in pro-Ho(SS) or similar categories, strong shape embedded problems in Hopro(SS). Several homotopy structures emerged at that time, notably those due to \textit{D. A. Edwards} and \textit{H. M. Hastings} [Čech and Steenrod homotopy theories with applications to geometric topology, Lect. Notes Math. 542 (1976; Zbl 0334.55001)] and in a restricted case to \textit{J. W. Grossman} [Trans. Am. Math. Soc. 201, 161-176 (1975; Zbl 0266.55007)] (Grossman's structure was motivated by problems, not in shape theory, but in proper homotopy theory and a beautiful geometric interpretation of his structure exists in that context).
More applications of pro-categories within algebraic topology have been found, in particular in the study of profinite completions, and there are suggestions that they may be useful in the study of motives. There was thus a need for this area to be revisited and the old results refined and extended. It is this that this paper sets out to do. The theory of Grossman was restricted to towers of spaces or simplicial sets, here it is extended to pro-simplicial sets of sufficient generality for the intended applications. Many results known from other settings are refined and the author uses to the full the insights into abstract homotopy theory obtained since the previous assault on the possible homotopy structures of pro-categories. closed model structures; pro-spaces; étale homotopy; homotopy category of simplicial sets; prosimplicial sets; shape theory; abstract homotopy theory; homotopy structures of pro-categories D. C. Isaksen, ''A model structure on the category of pro-simplicial sets,'' Trans. Amer. Math. Soc., vol. 353, iss. 7, pp. 2805-2841, 2001. Abstract and axiomatic homotopy theory in algebraic topology, Localization of categories, calculus of fractions, Nonabelian homotopical algebra, Homotopy theory and fundamental groups in algebraic geometry, Localization and completion in homotopy theory, Étale and other Grothendieck topologies and (co)homologies, Shape theory A model structure on the category of pro-simplicial 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 From the abstract: The main purpose of this paper is to obtain the Hilbert-Samuel polynomial of a module via blowing-up and applying intersection theory rather than employing associated graded objects. The result comes in the form of a concrete Riemann-Roch formula for the blow-up of a nonsingular affine scheme at its closed point. To achieve this goal, we note that the blow-up sits naturally between two projective spaces, one over a field and one a regular local ring, and then apply the Grothendieck-Riemann-Roch Theorem to each containment. Riemann-Roch Theorem; Hilbert-Samuel polynomial; Blow-up C.-Y. Jean Chan and Claudia Miller, A Riemann-Roch formula for the blow-up of a nonsingular affine scheme, J. Algebra 322 (2009), no. 9, 3003 -- 3025. Riemann-Roch theorems, Complete intersections A Riemann-Roch formula for the blow-up of a nonsingular affine scheme | 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 \(\mathbb{R}^\times:=\mathbb{R}\setminus\{0\}\) and let \(F:(\mathbb{R}^\times)^n\to\mathbb{R}\) be a map defined by Laurent polynomials having \(n+\ell\) distinct monomials, none of which is constant. In the paper under review a method to approximate the real solutions of the fewnomial system \(F(x)=b\), for a given \(b\in\mathbb{R}^n\), is described.
Denote by \(S_F\subset\mathbb{R}^n\) the set of real solutions of the system \(F(x)=b\), and suppose that it is finite. To find approximations to all the points of \(S_F\), the authors rely on the notion of Gale duality for polynomial systems, which is a correspondence between systems of polynomials defining complete intersections in the torus and systems of master functions defining complete intersections on a complement of certain hyperplane arrangements (see [\textit{F. Bihan} and \textit{F. Sottile}, Ann. Inst. Fourier 58, No. 3, 877--891 (2008; Zbl 1243.14044)]).
Applying Gale duality, the authors obtain a dual Gale system \(G(y)=1\), defined by certain rational functions. Then approximations \(S_G^*\) of the real solutions \(S_G\) of the system \(G(y)=1\) in the positive chamber \(\Delta\subset\mathbb{R}^\ell\) of the hyperplane arrangement under consideration are computed using a Khovanskii-Rolle continuation algorithm ([\textit{D. J. Bates} and \textit{F. Sottile}, Found. Comput. Math. 11, No. 5, 563--587 (2011; Zbl 1231.14047)]). This method traces real curves connecting the real solutions of certain start system to the points of \(S_G^*\). Finally, these approximations are used to obtain approximations \(S_F^*\) of the points of the set \(S_F\).
As the number of curves traced by the Khovanskii-Rolle method is essentially bounded above by any fewnomial bound for real solutions, the authors obtain an improved fewnomial bound for the kind of systems under consideration. Then they describe the first and third steps of the method above, and comment on an implementation in the software package \texttt{galeDuality}. fewnomial; Gale duality; Descartes' rule; Khovanskii-Rolle continuation; polynomial system; real algebraic geometry Real algebraic sets, Toric varieties, Newton polyhedra, Okounkov bodies, Computational aspects in algebraic geometry, Numerical computation of solutions to systems of equations, Global methods, including homotopy approaches to the numerical solution of nonlinear equations Software for the Gale transform of fewnomial systems and a Descartes rule for fewnomials | 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 a new formulation of the \(K\)-theoretic Pieri rule regarding multiplication of stable Grothendieck polynomials using iterated residues. They also use theirs method to establish straightening laws to transform Grothendieck polynomials corresponding to general integer sequences to linear combinations of those corresponding to partitions.
The introduction is very well written and they lay down the notations in a very clear matter. The proofs involves Young tableau, residue of meromorphic functions, among other notions and techniques. Although the computations and proofs deal with a considerable amount of indexes and have a naturally convoluted writing due to the subject the authors make a very clear exposition and as a result the work is very pleasant to read. Grothendieck polynomial; \(K\)-theory; Pieri rule; iterated residue Classical problems, Schubert calculus, Applications of methods of algebraic \(K\)-theory in algebraic geometry \(K\)-theoretic Pieri rule via iterated residues | 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 \(f: X\to S\) and \(g: Y\to S\) be projective cohomologically flat schemes over a scheme S of finite type over a field or over an excellent Dedekind domain. The author shows that the functor of divisorial correspondences \(T\rightsquigarrow Pic(X\times_ sY\times_ sT)/f^*(Pic(X\times_ sT)\times g^*(Pic(Y\times_ sT)\) is representable by a separated locally finitely presented and quasi-finite S-scheme of groups C(X,Y). The representabiliy of this functor follows directly from the Artin criterion. The other properties follow easily from the Néron-Severi theorem. Picard scheme; functor of divisorial correspondences; representabiliy Picard groups, Rational and birational maps, Group schemes Divisor correspondences between relative 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 We give a new algorithm computing local system cohomology groups for complexified real line arrangements. Using it, we obtain several conditions for the first local system cohomology to vanish and to be at most one-dimensional, which generalize a result by Cohen-Dimca-Orlik. The conditions are described in terms of discrete geometric structures of real figures. The proof is based on a recent study on minimal cell structures. We also compute the characteristic variety of the deleted \(B_3\)-arrangement. complexified real line arrangements; minimal cell structures Yoshinaga, M, Resonant bands and local system cohomology groups for real line arrangements, Vietnam J. Math., 42, 377-392, (2014) Relations with arrangements of hyperplanes, Configurations and arrangements of linear subspaces, Arrangements of points, flats, hyperplanes (aspects of discrete geometry) Resonant bands and local system cohomology groups for real line arrangements | 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 complex analytic varieties whose singularities can be resolved by blow-ups along smooth centers, intersections of smooth normal crossing divisors. An explicit (based on matrices) algorithm is given in order to replace the finite sequence of blow-ups by just one blow-up along a suitable coherent sheaf of ideals locally generated by monomials in some system of coordinates. resolving singularities by blow-ups (C.) ( Grant Melles ) & (P.) ( Milman ) .- Single-Step Combinatorial Resolution via Coherent Sheaves of Ideals , Singularities in Algebraic and Analytic Geometry, Contemporary Mathematics , American Mathematical Society , 2000 , p. 77 - 88 MR 1792150 | Zbl 0973.14007 Global theory and resolution of singularities (algebro-geometric aspects), Modifications; resolution of singularities (complex-analytic aspects), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Single-step combinatorial resolution via coherent sheaves of ideals | 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 of this paper establishes a Quillen-Gersten type spectral sequence for the \(K\)-theory of schemes with endomorphisms and proves an analogue of Gersten's conjecture in the \(K\)-theory of schemes with endomorphisms for the equal characteristic case. \(K\)-theory of schemes; schemes with endomorphisms; Quillen-Gersten type spectral sequence \(K\)-theory of schemes, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Spectral sequences, hypercohomology A Quillen-Gersten type spectral sequence for the \(K\)-theory of schemes with endomorphisms | 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 \(\mathcal{A}\) be a non-empty finite subset of \(\mathbb{Z}^2\) and let \(\mathbb{K}\) be any valuation field; consider a planar algebraic curve \(C\) with equation
\[
F (x, y) =\sum_{(i,j)\in{\mathcal A}} a_{ij}x^iy^j=0,\quad a_{ij}\in\mathbb{K}^{\ast}
\]
The author investigates how to extract geometric properties of \(C\) from the knowledge of the valuations of the coefficients \(a_{ij}\).
To this aim he considers some combinatorial objects attached to \(C\), namely: {\parindent=0.7cm \begin{itemize}\item[--] the convex hull of \({\mathcal A}\) in \(\mathbb{R}^2\), i.e. the Newton polygon \(\Delta({\mathcal A})\), with the subdivision induced by the projection along \(\mathbb{R}\) of all the faces of the convex hull in \(\mathbb{R}^2\times \mathbb{R}\) of the set \(\{((i,j),s)| (i,j)\in\mathcal{A},s\leq \mathrm{val} (a_{ij})\}\) \item[--] the non-Archimedean amoeba \(\mathrm{Val}(C) = \{(\mathrm{val}(x), \mathrm{val}(y)) | (x, y) \in C\}\) \item[--] the set of non-smooth points of the function \(\max_{(i,j)\in\mathcal{A}}(iX +jY +\mathrm{val}(a_{ij}))\), i.e. the tropical curve \(\mathrm{Trop}(C)\).
\end{itemize}} If the curve \(C\) has one \(m\)-fold point, then it is known that the coefficients of \(F\) are subject to a certain set of \({{m(m+1)}\over{2}}\) linear constraints: the author shows how these constraints influence the properties of the subdivision of \(\Delta(\mathcal{A})\), involving \(\mathrm{Val}(C)\) and \(\mathrm{Trop}(C)\). He also summarizes what it is known about \(m\)-fold tropical points and discusses three different definitions of them. tropical singular point; m-fold point; extended Newton polygon; tropical curve Kalinin, N, The Newton polygon of a planar singular curve and its subdivision, J. Comb. Theory Ser. A, 137, 226-256, (2016) Plane and space curves, , Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) The Newton polygon of a planar singular curve and its subdivision | 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 a summary of the author's results on finite flat commutative group schemes. The properties of the generic fibre functor are discussed. A complete classification of finite local flat commutative group schemes over mixed characteristic complete discrete valuation rings in terms of their Cartier modules (defined by Oort) is given. We also state several properties of the tangent space of these schemes. These results are applied to the study of reduction of abelian varieties. A finite \(p\)-adic semistable reduction criterion is formulated. It looks especially nice in the ordinary reduction case. The plans of the proofs are described. Bondarko, M. V., Finite flat commutative group schemes over complete discrete valuation fields: classification, structural results; application to reduction of abelian varieties, (Mathematisches Institut, Georg-August-Universität Göttingen: Seminars Winter Term 2004/2005, (2005), Universitätsdrucke Göttingen Göttingen), 99-108, MR 2206881 Group schemes Finite flat commutative group schemes over complete discrete valuation fields: classification, structural results; application to reduction of abelian 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 Let \(\boldsymbol{f} = ( f_1, \ldots, f_s)\) be a sequence of polynomials in \(\mathbb{Q} [ X_1, \ldots, X_n]\) of maximal degree \(D\) and \(V \subset \mathbb{C}^n\) be the algebraic set defined by \textbf{f} and \(r\) be its dimension. The real radical \(\sqrt[ r e]{ \langle \boldsymbol{f} \rangle}\) associated to \textbf{\(f\)} is the largest ideal which defines the real trace of \(V\). When \(V\) is smooth, we show that \(\sqrt[ r e]{ \langle \boldsymbol{f} \rangle} \), has a finite set of generators with degrees bounded by \(\deg V\). Moreover, we present a probabilistic algorithm of complexity \(( s n D^n )^{O ( 1 )}\) to compute the minimal primes of \(\sqrt[ r e]{ \langle \boldsymbol{f} \rangle} \). When \(V\) is not smooth, we give a probabilistic algorithm of complexity \(s^{O ( 1 )} ( n D )^{O ( n r 2^r )}\) to compute rational parametrizations for all irreducible components of the real algebraic set \(V \cap \mathbb{R}^n\).
Let \(( g_1, \ldots, g_p)\) in \(\mathbb{Q} [ X_1, \ldots, X_n]\) and \(S\) be the basic closed semi-algebraic set defined by \(g_1 \geq 0, \ldots, g_p \geq 0\). The \(S\)-radical of \(\langle \boldsymbol{f} \rangle \), which is denoted by \(\sqrt[ S]{ \langle \boldsymbol{f} \rangle} \), is the ideal associated to the Zariski closure of \(V \cap S\). We give a probabilistic algorithm to compute rational parametrizations of all irreducible components of that Zariski closure, hence encoding \(\sqrt[ S]{ \langle \boldsymbol{f} \rangle} \). Assuming now that \(D\) is the maximum of the degrees of the \(f_i\)'s and the \(g_i\)'s, this algorithm runs in time \(2^p ( s + p )^{O ( 1 )} ( n D )^{O ( r n 2^r )} \). Experiments are performed to illustrate and show the efficiency of our approaches on computing real radicals. polynomial system; real radical; \(S\)-radical ideal; semi-algebraic set; real algebraic geometry Computational real algebraic geometry, Semialgebraic sets and related spaces, Symbolic computation and algebraic computation Computing real radicals and \(S\)-radicals of polynomial 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 Elliptic curve cryptosystems (ECC) are suitable for memory-constraint devices like smart cards due to their small key-size. A standard way of computing elliptic curve scalar multiplication, the most frequent operation in ECC, is window methods, which enhance the efficiency of the binary method at the expense of some precomputation. The most established window methods are sliding window on NAF (NAF+SW), \(w\)NAF, and \(w\)MOF, where NAF and MOF are acronyms for nonadjacent form and mutually opposite form, respectively. A common drawback of these schemes is that only a small portion of the numbers are possible sizes for precomputation tables. Therefore, in practice, it is often necessary to waste memory because there is no table fitting exactly the available storage. In the case of \(w\)NAF, there exists a variant that allows arbitrary table sizes, the so-called fractional \(w\)NAF (Frac--\(w\)NAF). In this paper, we give a comprehensive proof using Markov theory for the estimation of the average nonzero density of the Frac-\(w\)NAF representation. Then, we propose the fractional \(w\)MOF (Frac-\(w\)MOF), which is a left-to-right analogue of Frac-\(w\)NAF. We prove that Frac-\(w\)MOF inherits the outstanding properties of Frac-\(w\)NAF. However, because of its left-to-right nature, Frac-\(w\)MOF is preferable as it reduces the memory consumption of the scalar multiplication. Finally, we show that the properties of all discussed previous schemes can be achieved as special instances of the Frac-\(w\)MOF method. To demonstrate the practicability of Frac-\(w\)MOF, we develop an on-the-fly algorithm for computing elliptic curve scalar multiplication with a flexibly chosen amount of memory. signed binary representations K. Schmidt-Samoa, O. Semay, T. Takagi, analysis of fractional window recoding methods and their application to elliptic curve cryptosystems. IEEE Trans. Comput. 55(1), 48--57 (2006) Cryptography, Applications to coding theory and cryptography of arithmetic geometry Analysis of fractional window recoding methods and their application to elliptic curve cryptosystems | 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 present algorithms to approximate higher order algebraic curves by piecewise rational curves which then can be parametrized (cf. \textit{Camille Jordan}, Cours d'Analyze, 2nd ed., Gauthier-Villars Paris 1893, Vol. 1, Section 607-612). For nonsingular curves the procedure admits a manageable form of error control. For singularities, they propose resolution by quadratic transformations. The practical implementation of a computer analysis of higher singularities is not attempted. higher order algebraic curves; piecewise rational curves; nonsingular curves; error control; quadratic transformations W.N. Waggenspack, Jr. and C.C. Anderson, Piecewise parametric approximations for algebraic curves, Comp. Aided Geom. Design 6 (1989) 33--53. Algorithms for approximation of functions, Curves in Euclidean and related spaces, Algebraic functions and function fields in algebraic geometry Piecewise parametric approximations for 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 Let \(S\) be the spectrum of a discrete valuation ring of residue characteristic \(p>0\). Let \(Z\) be a seperated scheme which is of finite type over \(S\) and \(\ell\) a prime different from \(p\). The author studies sheaves of \(\mathbb F_\ell\)-modules which are smooth on every stratum of a suitable partition of \(Z\). The key problem is to study the difference of two such sheaves \(F_1\) and \(F_2\) which have the same wild ramification at infinity. The author defines a subgroup \(K_c(Z, \mathbb F_\ell)_t^0\subset K_c(Z, \mathbb F_\ell)\) of virtual sheaves with virtual wild ramification zero which is generated by the classes of these differences. Here \(K_c(Z, \mathbb F_\ell)\) is the Grothendieck group of \(\mathbb F_\ell\)-constructible sheaves on the étale site of \(Z\). The main result is that this subgroup is preserved by the formalism of the six operations ``à la Grothendieck''. This generalizes results of Deligne and the author over a field. Deligne, P.: Théorèmes de finitude en cohomologie \(\mathcall \)-adique, Chomologie étale, SGA\(4\frac{1}{2}\), Springer Lecture Notes in Mathematics, p. 569 Arithmetic ground fields for curves, \(K\)-theory of schemes Nodal curves and virtual wild ramification | 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 Using a new technique to determine obstructions for deforming space curves, we exhibit non-reduced components of the Hilbert scheme of space curves such that
(1) the general curve \(C\) in a component is smooth,
(2) the imbedding codimension of a component at a general point may be arbitrarily large, and
(3) the degree of the minimal degree surface containing \(C\) may be arbitrarily large.
In contrast all earlier examples of non-reduced componenents have consisted of curves lying on cubic surfaces. --- The non-reducedness of a component is established by calculating the obstructions to lifting first order deformations of a general curve \(C\) in the component. The obstructions are calculated by using some recent theory by Walter showing how to calculate coboundaries of some spectral sequences associated to space curves.
As a part of the deformation theory we develop, we also study deformations of coherent sheaves on \(\mathbb{P}^ n\) where parts of its cohomology vary flatly. non-reduced components of the Hilbert scheme of space curves; deformations of coherent sheaves Fløystad, G., Determining obstructions for space curves, with applications to nonreduced components of the Hilbert scheme, J. Reine Angew. Math., 439, 11-44, (1993) Plane and space curves, Parametrization (Chow and Hilbert schemes), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Families, moduli of curves (algebraic), Formal methods and deformations in algebraic geometry Determining obstructions for space curves, with applications to nonreduced components of the Hilbert scheme | 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 \(X \subseteq \mathbb{P}^n\) denote a 0-dimensional subscheme. It is called a Cayley-Bacharach (CB) scheme, if every subscheme \(Y \subseteq X\) of \(\text{degree }Y =\text{degree} X-1\) has the Hilbert function \(H_Y = \min \{H_X, \text{degree} Y\}\). In the case of a CB scheme \(X\) such that it is a not possible to add one point to it and still get a CB scheme \(X\) is called a maximal CB scheme. The paper is concerned with the study of the properties of maximal CB schemes.
On one side it is known that generic sets of points are always CB schemes. On the other side it turns out that (with a few minor exceptions) complete intersections are maximal CB schemes. This follows because CB schemes with all of their minimal generators in low degree (e.g. complete intersections) are maximal. More general it is shown that a CB scheme not contained in a quadric hypersurface with the `tail of a complete intersection' is in fact a maximal CB scheme. Final results are concerned with the liaison of maximal CB schemes. -- For an expository survey on Cayley-Bacharach theorems see also \textit{D. Eisenbud}, \textit{M. Green}, \textit{J. Harris}, ``Cayley-Bacharach theorems and conjectures'', Bull. Am. Math. Soc., New Ser. 33, No. 3, 295-324 (1996). Cayley-Bacharach scheme; maximal CB schemes; liaison Kreuzer M., On maximal Cayley-Bacharach schemes 23 pp 3357-- (1995) Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Complete intersections, Linkage On maximal Cayley-Bacharach 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 \(X\) be a complex \(K3\) surface with an effective symplectic action of a group \(G\). Then \(G\) is a subgroup of the Mathieu group \(M_{23}\subset M_{24}\) [\textit{S. Mukai}, Invent. Math. 94, No. 1, 183---221 (1988; Zbl 0705.14045)], and all possible actions are classified by the 82 topological types of the quotient \(X/G\) [\textit{G. Xiao}, Ann. Inst. Fourier 46, No. 1, 73---88 (1996; Zbl 0845.14026)].
The \(G\)-invariant Hilbert scheme of \(X\) (of length \(n\)) parametrizes the \(G\)-invariant length \(n\) subschemes \(Z\subset X\) of \(X\). It can be identified with the \(G\)-fixed point locus in the Hilbert scheme of points \(\operatorname{Hilb}^n(X)^G\subset \operatorname{Hilb}^n(X)\).
In the paper under review, the authors define the \(G\)-\textit{fixed partition function} of \(X\) as the function \(Z_{X,G}\colon \mathbb{H}\to \mathbb{C}\) of the upper-half plane \(\mathbb{H}\) sending \(\tau\in\mathbb{H}\) to
\[
Z_{X,G}(q):=\sum_{n=0}^\infty e(\operatorname{Hilb}^n(X)^G)q^{n-1},
\]
where \(q=\operatorname{exp}(2\pi i \tau)\) and \(e(-)\) is the topological Euler characteristic.
The authors prove that \(Z_{X,G}(q)^{-1}\) is a modular cusp form of weight \(\frac{1}{2}e(X/G)\) with respect to the congruence subgroup \(\Gamma_0(|G|)\).
They also give an explicit formula of \(Z_{X,G}(q)\) in terms of the Dedekind eta function. The formula is a product in which the factors are related to the singular points of \(X/G\). Using this result, the form \(Z_{X,G}(q)^{-1}\) is explicitly given for all 82 possible types of \((X,G)\). \(K3\) surfaces; modular forms; Hilbert schemes; group actions \(K3\) surfaces and Enriques surfaces, Holomorphic symplectic varieties, hyper-Kähler varieties, Parametrization (Chow and Hilbert schemes), Fourier coefficients of automorphic forms \(G\)-fixed Hilbert schemes on \(K3\) surfaces, modular forms, and eta products | 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 perfect field of characteristic \(p\), let \(A=W(k)\) be the ring of Witt vectors over \(k\), and let \(D_k\) be the associated Dieudonné ring. A \(p\)-group scheme over \(k\) is a formal group scheme \(G\) over \(k\) for which \(G\) is isomorphic to the direct limit \(\displaystyle{\lim_{\longrightarrow} G[p^i]}\) where \(G[p^i]\) denotes the kernel of the map \(p^i: G\rightarrow G\), \(i\geq 0\). Let \(h\geq 0\) be an integer. A \(p\)-divisible group scheme over \(k\) of height \(h\) is a \(p\)-group scheme \(G\) for which \(G[p^i]\) has order \(p^{ih}\), \(\forall i\) (equivalently, the representing algebra of \(G[p^i]\) is free of rank \(p^{ih}\) over \(k\)).
There is a categorical equivalence \({\mathcal M}\) between the category of \(p\)-divisible groups over \(k\) and the category of Dieudonné modules, that is, \(D_k\)-modules which are free over \(A\). Given a Dieudonné module \({\mathcal M}(G)\) we can recover the \(p\)-divisible group as follows. From the relations among the Frobenius, \({\mathcal F}\) and the Verschiebung, \({\mathcal V}\) operators, one defines a finite dimensional formal group which in turn produces a \(p\)-divisible group.
Let \(K=\text{Frac}(A)\), let \(L\) be a finite totally ramified extension of \(K\) with ring of integers \(S\). Let \(m\) denote the maximal ideal of \(S\) and put \(S_n=S/m^n\). Choose \(\pi\) so that \(m=(\pi)\) and let \(e\) be so that \((p)=\pi^e\), \(e<p-1\). Observe that \(k\cong A/(m\cap A)\). In one of the main results of this paper (Theorem 2.8), the author shows that there is a categorical duality between the category of \(p\)-divisible groups over \(S_n\) and the category whose objects are certain triples \((L_n,M,\rho)\). In the duality, \(M={\mathcal M}(G_k)\).
Next, suppose that \(k\) is algebraically closed and let \(d,h\) be positive integers which satisfy \(d<h\) and \(\text{gcd}(d,h)=1\). Let \(\Gamma_0\) be a \(p\)-divisible group over \(k\) whose Dieudonné module \({\mathcal M}(\Gamma_0)\) is \(D_k/({\mathcal F}^d-{\mathcal V}^{h-d})\), and is therefore associated to a certain \(p\)-divisible group \(G_{d,h-d}\) over \({\mathbb F}_p\) of dimension \(d\) and height \(h\). One has \(\Gamma_0\cong G_{d,h-d}\times \text{Spec}\;k\). Let \(L\) be a degree \(h\) extension of \({\mathbb Q}_p\) with ring of integers \({\mathcal O}\). Suppose that the ramification index of \(p\) in \(L\) satsifies \(e<p-1\) and suppose that \(S\) is the ring of integers in a totally ramified degree \(e\) extension of \(K\). An \({\mathcal O}\)-lifting of \(\Gamma_0\) is a \(p\)-divisible group \(\Gamma\) over \(S\) so that \(\Gamma_k\cong \Gamma_0\) and \(\text{End}(\Gamma)={\mathcal O}\). In another main result in the paper (Theorem 4.4) the author shows that there exists an \({\mathcal O}\)-lifting of \(\Gamma_0\) if and only if \(h\geq ed\). group scheme; \(p\)-divisible group; almost canonical lifting DOI: 10.5802/aif.2494 Group schemes, Formal groups, \(p\)-divisible groups Group schemes over Artinian rings and applications | 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 polynomial system \(f\), this article provides a general construction for homotopies that yield at least one point of each connected component on the set of solutions of \(f=0\). This algorithmic approach is then used to compute a superset of the isolated points in the image of an algebraic set which arises in many applications, such as computing critical sets used in the decomposition of real algebraic sets. An example is presented which demonstrates the efficiency of this approach. numerical algebraic geometry; homotopy continuation; projections Computational real algebraic geometry, Topological properties in algebraic geometry Homotopies for connected components of algebraic sets with application to computing critical 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 This article gives an introduction for mathematicians interested in numerical computations in algebraic geometry and number theory to some recent progress in algorithmic number theory, emphasising the key role of approximate computations with modular curves and their Jacobians. These approximations are done in polynomial time in the dimension and the required number of significant digits. We explain the main ideas of how the approximations are done, illustrating them with examples, and we sketch some applications in number theory. Algebraic number theory computations, Software, source code, etc. for problems pertaining to number theory, Galois representations, Arithmetic aspects of modular and Shimura varieties, Computational aspects of algebraic curves, Modular and Shimura varieties, Global methods, including homotopy approaches to the numerical solution of nonlinear equations Approximate computations with modular 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 Let \(A\) be a finite-dimensional associative algebra over an algebraically closed field \(k\). Denote by \(\mathfrak{Aut}(A)\) the affine group scheme of automorphisms of \(A\) and by \(\mathfrak{Inn}(A)\) the smooth characteristic subgroup of inner automorphisms. Then \(\mathfrak{Out}(A)=\mathfrak{Aut}(A)/\mathfrak{Inn}(A)\) is the affine group scheme of outer automorphisms. By the theorem of Cartier, \(\mathfrak{Aut}(A)\) is always reduced if \(\text{char}(k)=0\), but this is not the case if \(\text{char}(k)=p>0\). The authors prove that the group schemes \(\mathfrak{Aut}(A)\) and \(\mathfrak{Out}(A)\) are not reduced if \(A\) admits a \(\mathbb{Z}_p\)-grading which can not be lifted to a \(\mathbb{Z}\)-grading. The proof is based on the following result: an infinitesimal multiplicative subgroup \(\mathfrak{M}\) of a smooth algebraic group \(\mathfrak{G}\) over an algebraically closed field \(k\) of characteristic \(p>0\) is contained in a maximal torus of \(\mathfrak{G}\).
This applies in particular when \(A\) is the group algebra of a \(p\)-group. Another example, where the above result shows that \(\mathfrak{Aut}(A)\) is not reduced, is illustrated by the direct calculation of the automorphism group. finite-dimensional associative algebras; affine group schemes; automorphism groups; inner automorphisms Group schemes, Group rings of finite groups and their modules (group-theoretic aspects), Automorphisms and endomorphisms Non-reduced automorphism 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 \(S\) be a smooth projective surface over the complex numbers; let \(S^{(r)}\) be its \(r\)-fold symmetric product and \(S^{[r]}\) the Hilbert scheme of 0-dimensional subschemes of length \(r\).
In case \(K_S\) is trivial, the deformation theory of \(S^{[r]}\) has been studied by Beauville and Fujiki in order to construct examples of higher-dimensional symplectic manifolds. In that case \(S^{[r]}\) has deformations which are not Hilbert schemes of points on a surface. -- We prove that under suitable hypotheses (e.g. if \(S\) is of general type) this cannot happen; every (small) deformation of \(S^{(r)}\) and \(S^{[r]}\) is induced naturally by a deformation of \(S\) (in particular, all deformations of \(S^{(r)}\) are locally trivial). Hilbert schemes of points; \(r\)-fold symmetric product of projective surface; deformation theory Fantechi, B., Deformation of Hilbert schemes of points on a surface, Compos. Math., 98, 2, 205-217, (1995), MR 1354269 Formal methods and deformations in algebraic geometry, Parametrization (Chow and Hilbert schemes), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Deformation of Hilbert schemes of points on a surface | 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 be a general k-gonal curve, M an image of A in some \({\mathbb{P}}^ n\), and fix integers g, d, x, a and b verifying that \(k\geq n+2\) and \(\rho(d,g,n)\geq 0\) (the Brill-Noether number).
Consider each of the following conditions: (i) M is an embedding of degree d and genus g; (ii) M is an embedding of degree \(d+3x\) and genus \(g+2x\) having a x-secant line; (iii) M is an embedding of degree \(d+a\) and genus g except for b nodes and \((n+1)a\geq nb.\)- Then the author proves that, in each of the above cases, M is a smooth point of the component W of \(Hilb({\mathbb{P}}^ n)\) where it lies. Also, in a neighborhood of M in W, all embeddings of A in the same conditions form a subscheme of the right dimension.
Using the same techniques, the author also proves that a set of points Z in \({\mathbb{P}}^ n\) is the intersection of \({\mathbb{P}}^ n\) with some k- dimensional scroll in \({\mathbb{P}}^{n+k}\) if and only if Z is curvilinear. prints of projective space as intersection with a scroll; k-gonal curve; Brill-Noether number --, On special linear systems on curves.Comm. in Algebra 18 (1990), 279--284. Special algebraic curves and curves of low genus, Divisors, linear systems, invertible sheaves, Projective techniques in algebraic geometry On special linear systems on 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 Let \(S\) be a smooth complex algebraic surface. Given positive integers \(n_1<n_2<\cdots <n_k\), let \(S^{[n_1, n_2, \dots, n_k]}\) denote the nested Hilbert scheme parameterizing nested 0-dimensional sub-schemes of \(S\): \(\xi_{n_1}\subseteq \xi_{n_2}\subseteq \cdots \xi_{n_k}\) of length \(n_i\). The nested Hilbert schemes are natural analogues for the Hilbert schemes \(S^{[n]}\) of points, and some of them have played an important role in the study of syzygies. The present well-written paper gives a quite comprehensive study of \(S^{[n, n+1, n+2]}\).
The first main result is a new proof of the irreducibility of \(S^{[n, n+1, n+2]}\) due to \textit{N. Addington} [Algebr. Geom. 3, No. 2, 223--260 (2016; Zbl 1372.14009)]. The idea here is to realize \(S^{[n, n+1, n+2]}\) as \(\mathbb{P}(\mathscr{I}_{Z_{[n, n+1]}})\), where \(Z_{[n, n+1]}\) is the subscheme of \(S\times S^{[n, n+1]}\) parameterizing triples \((p, \xi_{n}, \xi_{n+1})\) with \(p\in \text{Supp}(\xi_{n+1})\), and use a criterion of \textit{G. Ellingsrud} and \textit{S. A. Strømme} [Trans. Amer. Math. Soc. 350, No. 6, 2547--2552 (1998; Zbl 0893.14001)]. Along the way, with more care the authors establish an estimate on codimension of certain strata of \(Z_{[n, n+1]}\) in \(S\times S^{[n, n+1]}\), which is quadratic in the minimal number \(i\) of generators for the localized ideal. While a linear estimate is sufficient in the criterion mentioned above, the quadratic one is of great interest on its own. Via forgetful and residual point maps, the irreducibility of \(S^{[n, n+2]}, S^{[1, n, n+1, n+2]}, S^{[1, n+1, n+2]}\) and \(S^{[1, n, n+2]}\) are deduced from that of \(S^{[n, n+1, n+2]}\).
The second one is an explicit construction of a family of nested subschemes, indicating that \(S^{[1, 2, \dots, 23]}\) is reducible. As a corollary, \(S^{[n_1, n_2, \dots, n_k]}\) is reducible whenever \(k\ge 23\).
The third one is that \(S^{[n, n+1, n+2]}\) is a local complete intersection and has klt singularities. The proof involves showing that a two-step blowup gives a (small) resolution of singularities of \(S^{[n, n+1, n+2]}\).
In the end, the Picard group and the canonical divisor of \(S^{[n, n+1, n+2]}\) are computed in case \(S\) is regular. nested Hilbert schemes; irreducibility; singularities; Picard group Parametrization (Chow and Hilbert schemes), Singularities in algebraic geometry Irreducibility and singularities of some nested 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 $k$ be a complete non-archimedean valued field and $X$ be a compact quasi-smooth strictly $k$-analytic space. The article under review shows that, after performing a finite extension of the base field and a quasi-étale covering, one may always find a space that admits a strictly semistable formal model. (The word ``altered'' in the title of the article refers to the fact that one can conjecturally replace the quasi-étale morphism by an embedding of a disjoint union of affinoid domains.) This theorem is used by the author in [in: Nonarchimedean and tropical geometry. Based on two Simons symposia, Island of St. John, March 31 -- April 6, 2013 and Puerto Rico, February 1--7, 2015. Cham: Springer. 195--285 (2016; Zbl 1360.32019)] where he investigated pluricanonical forms on quasi-smooth Berkovich spaces.
In the case where the base field $k$ is discretely valued, the theorem had formerly been proved by \textit{U. T. Hartl} [Manuscr. Math. 110, No. 3, 365--380 (2003; Zbl 1099.14010)], building on techniques introduced by \textit{A. J. de Jong} in [Publ. Math., Inst. Hautes Étud. Sci. 83, 51--93 (1996; Zbl 0916.14005)] and involving moduli spaces of proper curves.
In order to go beyond the discretely valued case, the author uses the stable modification theorem from [J. Algebr. Geom. 19, No. 4, 603--677 (2010; Zbl 1211.14032)] that applies to arbitrary relative curves, with no properness assumption. This allows him to first prove an algebraic version of the result (Section 2), whereas Hartl's method needed to use analytic methods from the start in order to obtain relative compactifications. Formal and analytic versions of the result are then deduced in Sections 3 and 4. local uniformization; Berkovich spaces Rigid analytic geometry Altered local uniformization of Berkovich 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 We formulate a conjecture on actions of the multiplicative group in motivic homotopy theory. In short, if the multiplicative group \(G_m\) acts on a quasi-projective scheme \(U\) such that \(U\) is attracted as \(t\) approaches 0 in \(G_m\) to a closed subset \(Y\) in \(U\), then the inclusion from \(Y\) to \(U\) should be an \(A^1\)-homotopy equivalence. We prove several partial results. In particular, over the complex numbers, the inclusion is a homotopy equivalence on complex points. The proofs use an analog of Morse theory for singular varieties. Application: the Hilbert scheme of points on affine \(n\)-space is homotopy equivalent to the subspace consisting of schemes supported at the origin. torus action; Morse homology; Hilbert scheme of points; motivic homotopy theory Group actions on varieties or schemes (quotients), Parametrization (Chow and Hilbert schemes), Motivic cohomology; motivic homotopy theory, Discriminantal varieties and configuration spaces in algebraic topology Torus actions, Morse homology, and the Hilbert scheme of points on affine space | 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 embedded resolution of an algebroid surface over an algebraically closed field of characteristic zero, that is the spectrum of a ring \( K[[X, Y,Z]]/(F)\).
The main combinatorial object associated to \(F\) is Hironaka's characteristic polygon \(\Delta(F)\). The original motivation of this work is: can the combinatorics bound, in some effective sense, the resolution process?
The paper studies in detail the resolution process for prepared equations, that \(F\) is a generic Weierstrass-Tchirnhausen equation, of the form \(Z^n +\sum_{k=0}^{n-2}a_k(X,Y)Z^k\) with \(a_k\) regular in \(X\) of order \(\nu_k=\nu(a_k)\geq n-k\).
The resolution strategy used is the following: (1) if \((Z,X)\) or \((Z, Y )\) are permissible curves, a monoidal transformation centered at them is performed, (2) otherwise, a quadratic transformation. For prepared equations bounds are given for the number of blow-ups needed before the multiplicity drops.
The paper contains many examples. resolution of surface singularities; Newton-Hironaka polygon; equimultiple locus; blowing-up Singularities of curves, local rings, Complex surface and hypersurface singularities Combinatorics and their evolution in resolution of embedded algebroid 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 In this article, the authors study spline spaces of non-uniform degree defined on T-meshes. They derive new combinatorial lower and upper bounds on the spline space dimension and subsequently outline sufficient conditions for the bounds to coincide.
These results generalize the framework presented by \textit{B. Mourrain} [Math. Comput. 83, No. 286, 847--871 (2014; Zbl 1360.65052)] for uniform bi-degree splines on T-meshes. smooth splines; T-meshes; non-uniform degrees; dimension formula Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.), Geometric aspects of numerical algebraic geometry, Numerical computation using splines Polynomial spline spaces of non-uniform bi-degree on T-meshes: combinatorial bounds on the dimension | 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 under review reports on the progress concerning the classification problems of finite group schemes with additional structure.
Let \(k\) be an algebraically closed field of characteristic \(p>0\). Let \(D\) be a finite-dimensional semisimple \(\mathbb{F}_p\)-algebra. The first classification problem addressed in the paper is (GE): Classify pairs (\(\mathcal G, \iota\)) where \(\mathcal G\) is a \(\text{BT}_1\) over \(k\) and \(\iota\colon D\to\text{End}_k(\mathcal G)\) defines an action of \(D\) on \(\mathcal G\) (with \(\iota(1)=\text{id}_{\mathcal G}\)). (Here \(\text{BT}_1\) stands for a truncated Barsotti-Tate group of level \(1\).)
Associated to such a group scheme \(\mathcal G\), there is the Dieudonné module \((N, F_N, V_N)\). There are two discrete invariants which play essential roles in the problem (GE). Let \(G:= G_D(N)\), and consider the structure of \(N\) as a \(D\)-module. Another discrete invariant, which is called the multiplication type of the pair \((\mathcal G,\iota)\), is the structure of \(\text{Ker}(F)\subset N\) as a \(D\)-module. The multiplication type corresponds to a conjugacy class \(X\) of parabolic subgroups of \(G\), which gives rise to a subgroup \(W_X\) of the Weyl group \(W_G\) of \(G\). The coset space \(W_X\backslash W_G\) is a finite set which can be described explicitly.
Given a pair \((\mathcal G,\iota)\) with discrete invariants \(N\) and \(X\), define an element \(\underline{w}(\mathcal G,\iota)\in W_X \backslash W_G\). The first main result on the problem (GE) follows. Theorem 1: Associating \(\underline{w}(\mathcal G,\iota)\) to the pair \((\mathcal G,\iota)\) gives a bijection between the isomorphism classes of pairs (\(\mathcal G,\iota\)) with invariants \(N\) and \(X\), and the coset \(W_X\backslash W_G\).
The second classification problem (GPE) concerns triples \((\mathcal G,\lambda,\iota)\), where \(\mathcal G\) is a \(\text{BT}_1\) over \(k\), \(\lambda: \mathcal G \buildrel\sim\over\rightarrow \mathcal G^D\) is a principal quasi-polarization and \(\iota\) is defined in the same way as (GE). Let \(G=\text{Sp}_D(N,\psi)\), where \(\psi\) is the symplectic form on the Dieudonné module \(N\) corresponding to the polarization \(\lambda\). As for (GE), consider the multiplication type. To a triple \((\mathcal G, \lambda, \iota)\), associate an element \(\underline{w}(\mathcal G, \lambda, \iota)\in W_X\backslash W_G\). The inclusion \(f: \text{Sp}_D(N,\psi)\to\text{GL}_D(N)\) induces an injective map \(W(f): W_X\backslash W_G\hookrightarrow W_X\backslash W_{\text{GL}_D(N)}\) which sends \(\underline{w}(\mathcal G,\lambda,\iota)\) to \(\underline{w}(\mathcal G,\iota)\). The main result on the problem (GPE) is the following.
Theorem 2: Assume that \(k\) is an algebraically closed field of characteristic \(>2\). (i) Let \(\mathcal G\) be a \(\text{BT}_1\) over \(k\) with an action \(\iota\) of the semisimple algebra \(D\). Then there exists a principal quasi-polarization \(\lambda\) such that \((\mathcal G,\lambda,\iota)\) is a triple as in (GPE) if and only if \(\underline{w}(\mathcal G,\iota)\in\break W_X\backslash W_{\text{GL}_D(N)}\) is in the image of the map \(W(f)\).
(ii) If a form \(\lambda\) as in (i) exists, then it is unique up to isomorphism (respecting the \(D\)-action). In other words: associating \(\underline{w}(\mathcal G,\lambda,\iota)\in W_X\backslash W_G\) to a triple \((\mathcal G,\lambda,\iota)\) gives a bijection between isomorphism classes of triples \((\mathcal G,\lambda,\iota)\) with invariants \(N\) and \(X\), and cosets\break \(W_X\backslash W_G\). Ben Moonen. ``Group schemes with additional structures and Weyl group cosets''. In: \emph{Moduli of abelian varieties (Texel Island, 1999)}. Ed. by Carel Faber, Gerard van der Geer, and Frans Oort. Progress in Mathematics, Vol. 195. Basel, Switzerland: Birkhäuser, 2001, pp. 255--298. zbl 1084.14523; MR1827024 Group schemes Group schemes with additional structures and Weyl group cosets | 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 \(\mathcal M_{g,n}\) be the moduli space of genus \(g\) curves with \(n\) marked points. For a tuple of integers \((k; a_1, \ldots, a_n)\), the double ramification cycle \(\mathrm{DR}\subset \mathcal M_{g,n}\) is defined to be the locus of pointed curves \((C, p_1, \ldots, p_n)\) such that \(a_1 p_1 + \cdots + a_np_n \sim \omega_C^{\otimes k}\).
Recently, there have been great interests and efforts in studying compactification of the double ramification cycle, aiming at computing the related cycle class from the viewpoints of Gromov-Witten theory and tautological ring structure. For instance, the first-named author described a compactification using modifications of the stack of stable curves [\textit{D. Holmes}, ``Extending the double ramification cycle by resolving the Abel-Jacobi map'', \url{arXiv:1707.02261}] and the last-two named authors described another compactification using an extended Brill-Noether locus on a suitable compactified universal Jacobian [\textit{J. L. Kass} and \textit{N. Pagani}, ``The stability space of compactified universal Jacobians'', \url{arXiv:1707.02284}].
The main result of this paper shows that these two compactifications coincide. Moreover, the authors also relate these compactifications to that of Li and Graber-Vakil which uses a virtual fundamental class on a space of rubber maps [\textit{J. Li}, J. Differ. Geom. 57, No. 3, 509--578 (2001; Zbl 1076.14540); \textit{J. Li}, J. Differ. Geom. 60, No. 2, 199--293 (2002; Zbl 1063.14069); \textit{T. Graber} and \textit{R. Vakil}, Duke Math. J. 130, No. 1, 1--37 (2005; Zbl 1088.14007); \textit{F. Janda} et al., Publ. Math., Inst. Hautes Étud. Sci. 125, 221--266 (2017; Zbl 1370.14029)]. Jacobians; moduli of curves; double ramification cycle Jacobians, Prym varieties, Picard schemes, higher Jacobians, Algebraic moduli problems, moduli of vector bundles Extending the double ramification cycle using Jacobians | 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 conchoid to an algebraic affine plane curve \(\mathcal C\) from the perspective of algebraic geometry, analyzing their main algebraic properties. Beside \(\mathcal C\), the notion of conchoid involves a point \(A\) in the affine plane (the focus) and a non-zero field element \(d\) (the distance). We introduce the formal definition of conchoid by means of incidence diagrams. We prove that the conchoid is a 1-dimensional algebraic set having at most two irreducible components. Moreover, with the exception of circles centered at the focus \(A\) and taking \(d\) as its radius, all components of the corresponding conchoid have dimension 1. In addition, we introduce the notions of special and simple components of a conchoid. Furthermore we state that, with the exception of lines passing through \(A\), the conchoid always has at least one simple component and that, for almost every distance, all the components of the conchoid are simple. We state that, in the reducible case, simple conchoid components are birationally equivalent to \(\mathcal C\), and we show how special components can be used to decide whether a given algebraic curve is the conchoid of another curve. Sendra J., Sendra J.R.: An algebraic analysis of conchoids to algebraic curves. Appl. Algebra Eng. Commun. Comput. 19, 413--428 (2008) Plane and space curves An algebraic analysis of conchoids to 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 The classical Monodromy Theorem states that the eigenvalues of the monodromy on the cohomology of the Milnor fiber of a germ of a holomorphic function are roots of unity. The authors generalize it as follows. For a germ of a complex analytic set \((\mathcal{X},0)\subset (\mathbb{C}^n,0)\) let \(B\) be a small open ball at the origin in \(\mathbb{C}^n\) and \(U=B\backslash \mathcal{X}\); \(U\) is the small ball complement of \(\mathcal{X}\). Then the cohomology jump loci of rank one of the local systems on this complement are finite unions of torsion translates of subtori. Milnor fiber; monodromy; complex analytic set; local systems; Riemann-Hilbert correspondence; cohomology; jump loci Budur, N; Wang, B, Local systems on analytic germ complements, Adv. Math., 306, 905-928, (2017) Local cohomology and algebraic geometry, Singularities in algebraic geometry, Local complex singularities Local systems on analytic germ complements | 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{H. Tsuji} [``Numerical trivial fibrations'', \url{arxiv:math/0001023}] stated several very interesting assertions on the structure of pseudo-effective line bundles \(L\) on a projective manifold \(X\). In particular he postulated the existence of a meromorphic ``reduction map'', which essentially says that through the general point of \(X\) there is a maximal irreducible \(L\)-flat subvariety. Moreover the reduction map should be almost holomorphic, i.e. has compact fibers which do not meet the indeterminacy locus of the reduction map. However, his proofs are extremely difficult to follow.
The purpose of this note is to establish the existence of a reduction map in the case where \(L\) is nef and to prove that it is almost holomorphic -- this was also stated explicitly in Tsuji's paper [loc. cit.]. Our proof is completely algebraic while Tsuji works with deep analytic methods. Finally, we show by a basic example that in the case where \(L\) is only pseudo-effective, the postulated reduction map cannot be almost holomorphic -- in contrast to a claim of Tsuij [loc. cit.]. maximal irreducible \(L\)-flat subvariety; almost holomorphic; pseudo-effective line bundles Bauer, T.; Campana, F.; Eckl, T.; Kebekus, S.; Peternell, T.; Rams, S.; Szemberg, T.; Wotzlaw, L., \textit{A reduction map for nef line bundles}, Complex geometry (Göttingen 2000), 27-36, (2002), Springer, Berlin Rational and birational maps, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Divisors, linear systems, invertible sheaves A reduction map for nef line bundles | 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 See the preview in Zbl 0723.14035. postulation; arithmetically Cohen-Macaulay 0-dimensional subschemes of a smooth quadric; linear system of divisors; Hilbert function Giuffrida S., Maggioni R., Ragusa A.: On the postulation of 0-dimensional subschemes on a smooth quadric. Pacific J. Math. 155(2), 251--282 (1992) Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Projective techniques in algebraic geometry, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series On the postulation of 0-dimensional subschemes on a smooth quadric. | 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 Hilbert scheme \({\text{Hilb}}^n_{X/S}\) parametrizing length \(n\) closed subschemes of \(X\) over \(S\) continues to draw great interest from algebraic geometers. The case \(X = \mathbb P^N\) is already interesting, as \({\text{Hilb}}^n_{\mathbb P^N/k}\) is smooth and irreducible for \(N=2\), but reducible for \(N = 3\) and \(n\) large [\textit{A. Iarrobino}, Invent. Math. 15, 72--77 (1972; Zbl 0227.14006)] and hence singular by \textit{R. Hartshorne}'s connectedness theorem [Publ. Math. Inst. Hautes Études Sci. 29, 5--48 (1966; Zbl 0171.41502)]. Motivated by \textit{Haiman's} construction of \({\text{Hilb}}^n_{\mathbb A^2/\mathbb C}\) as the blow-up of \(\text{Sym}^n (\mathbb A^2)\) at a concrete ideal [Discrete Math. 193, No. 1--3, 201--224 (1998; Zbl 1061.05509)], the authors construct the \textit{good component} \(G^n_{X/S} \subset {\text{Hilb}}^n_{X/S}\), the closure of subschemes consisting of \(n\) distinct points, as a concrete blow-up of a symmetric product for separated morphisms \(f:X \to S\) of algebraic spaces.
Working at this level of generality, the authors need to show existence of the Hilbert scheme as an algebraic space, extending \textit{M. Artin's} result [in: Global Analysis, Papers in Honor of K. Kodaira 21--71 (1969; Zbl 0205.50402)] for \(f\) locally of finite presentation. In this context, the \(n\)th symmetric product need not exist, so instead they use the \(n\)th divided power product \(\Gamma^n_{X/S}\) (the affine model is due to \textit{N. Roby} [C. R. Acad. Sci., Paris, Sér. A 290, 869--871 (1980; Zbl 0471.13008)]), which is homeomorphic to \(\text{Sym}^n X\) in general and isomorphic when \(f\) is flat [\textit{D. Rydh}, ``Families of zero cycles and divided powers: I. Representability'', \url{arXiv:0803.0618}]. When \(X\) and \(S\) are affine, the authors define the \textit{ideal of norms} \(I\) in the ring corresponding to \(\Gamma^n_{X/S}\) and show that these patch together to define a closed subscheme \(\Delta_X \subset \Gamma^n_{X/S}\). With this machinery in place, the authors prove that if \(f: X \to S\) is a separated morphism of algebraic spaces, then \(G^n_{X/S} \subset {\text{Hilb}}^n_{X/S}\) is isomorphic to the blow-up of \(\Gamma^n_{X/S}\) along the closed subspace \(\Delta_X\). In the important case that \(f\) is flat, \(G^n_{X/S}\) is obtained by blowing up the geometric quotient \(X^n_S/S_n\). As a byproduct of their method, they show that \(G^n_{X/S} = {\text{Hilb}}^n_{X/S}\) is smooth for \(f\) smooth and separated of relative dimension two, extending the result of \textit{J. Fogarty} [Am. J. Math. 90, 511--521 (1968; Zbl 0176.18401)]. Hilbert schemes; algebraic spaces Torsten Ekedahl and Roy Skjelnes, Recovering the good component of the Hilbert scheme, Preprint, May 2004, arXiv:math.AG/0405073. Parametrization (Chow and Hilbert schemes), Generalizations (algebraic spaces, stacks) Recovering the good component of the Hilbert scheme | 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 theory of virtual fundamental classes has played a key role in algebraic geometry, defining important invariants such as Gromov-Witten and Donaldson-Thomas invariants. It has been generalized to cosection localized virtual cycles which have applications in Seiberg-Witten, Fan-Jarvis-Ruan-Witten and other invariants. In this paper, the authors prove the formulas of virtual pullback, torus localization and wall crossing for cosection localized virtual cycles. When the cosection is trivial, these formulas coincide with those for ordinary virtual cycles. virtual cycles; cosection localization; virtual pullback; torus localization; wall crossing Huai-Liang Chang, Young-Hoon Kiem & Jun Li, ``Torus localization and wall crossing for cosection localized virtual cycles'', Adv. Math.308 (2017), p. 964-986 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Local deformation theory, Artin approximation, etc. Torus localization and wall crossing for cosection localized virtual cycles | 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 algebraic geometry M. Artin, Etale coverings of schemes over Hensel rings, Amer. J. Math. 88 (1966), 915-934. Algebraic geometry Etale coverings of schemes over Hensel 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 We continue the investigation of coherent systems of type \((n, d, k)\) on the projective line which are stable with respect to some value of a parameter \(\alpha\). We consider the case \(k=1\) and study the variation of the moduli spaces with \(\alpha\). We determine inductively the first and last moduli spaces and the flip loci, and give an explicit description for ranks 2 and 3. We also determine the Hodge polynomials explicitly for ranks 2 and 3 and in certain cases for arbitrary rank.
For part I, II, cf. Int. J. Math. 15, No. 4, 409--424 (2004; Zbl 1072.14039); Int. J. Math. 18, No. 4, 363--393 (2007; Zbl 1114.14022). Lange, Coherent systems of genus 0. III. Computation of flips for k = 1, Internat. J. Math. 19 (9) pp 1103-- (2008) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Algebraic moduli problems, moduli of vector bundles, Sheaves and cohomology of sections of holomorphic vector bundles, general results Coherent systems of genus 0. III. Computation of flips for \(k=1\) | 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 In this article, we use quite elementary and simple ideas to rebuild and study the patch and flat topologies on the prime spectrum from a natural point of view (this new approach is based on the significant applications of the power set ring). Especially, the proof of a major result in the literature on the comparison of topologies greatly simplified and shortened. Also a new characterization for the finiteness of the minimal primes of a ring is given. Then as an application, all of the related results of Kaplansky, Anderson, Gilmer-Heinzer, Bahmanpour-Khojali-Naghipour and Naghipour on the finiteness of the minimal primes are easily deduced as special cases of this result. Another finiteness result due to Matlis is also easily obtained which states that a given ring has finitely many minimal primes if and only if no minimal prime is contained in the union of the remaining minimal primes. flat topology; ind-power topology; maximal spectrum; minimal spectrum; power set ring Ideals and multiplicative ideal theory in commutative rings, Relevant commutative algebra, Extension theory of commutative rings, Several topologies on one set (change of topology, comparison of topologies, lattices of topologies), Rings of fractions and localization for commutative rings, Stone spaces (Boolean spaces) and related structures A new approach to the patch and flat topologies on a spectral space with applications | 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 prove invariance for the number of planar tropical curves enhanced with polynomial multiplicities recently proposed by \textit{F. Block} and \textit{L. Göttsche} [``Refined curve counting with tropical geometry'', \url{arXiv:1407.2901}]. This invariance has a number of implications in tropical enumerative geometry. Itenberg, Ilia; Mikhalkin, Grigory, On Block-Göttsche multiplicities for planar tropical curves, Int. Math. Res. Not. IMRN, 23, 5289-5320, (2013) , Enumerative problems (combinatorial problems) in algebraic geometry On Block-Göttsche multiplicities for planar tropical 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 For a variation of Hodge structure coming from a geometric family with singular fibers over discriminant locus, Deligne observed that the holomorphic vector bundle fibered by cohomology of smooth projective varieties can be extended holomorphically across a singular locus of simple-normal-crossing type, see [\textit{P. Deligne}, AMS/IP Stud. Adv. Math. 1, 683--699 (1997; Zbl 0939.14005)]. \textit{W. Schmid} [Invent. Math. 22, 211--319 (1973; Zbl 0278.14003)] proved the fundamental result that the degeneration of Hodge structures, and the resulting \emph{limiting mixed Hodge structures}, have essential relations with the unipotent monodromy of the family, as described by the nilpotent and \(\mathrm{SL}_2\)-orbit theorems. A result, by Baily and Borel, states that for a Hermitian symmetric domain \(D\) and an arithmetic subgroup \(\Gamma\leq Hol(D)^+\), the space \(\Gamma\backslash D\) admits a compactification by adding rational boundary components (or ``cusps'') [\textit{W. L. Baily jun.} and \textit{A. Borel}, Ann. Math. (2) 84, 442--528 (1966; Zbl 0154.08602)]. The resulting \emph{Baily-Borel Compactification} \(\Gamma\backslash D^*\) is a projective variety, usually highly singular. To improve the properties of this compactification, [\textit{A. Ash} et al., Smooth compactifications of locally symmetric varieties. With the collaboration of Peter Scholze. 2nd ed. Cambridge: Cambridge University Press (2010; Zbl 1209.14001)] introduced the \emph{toroidal compactification}, which depends on a choice of \(\Gamma\)-admissable polyhedral fan \(\Sigma\).
\textit{K. Kato} and \textit{S. Usui} [Classifying spaces of degenerating polarized Hodge structures. Princeton, NJ: Princeton University Press (2009; Zbl 1172.14002)] developed a theory of partial compactifications for quotients of period domains \(D\) by arithmetic groups \(\Gamma\), in an attempt to generalize the toroidal compactifications of Ash-Mumford-Rapoport-Tai to non-classical cases. Their partial compactifications, which aim to fully compactify the images of period maps, rely on a choice of fan which is strongly compatible with \(\Gamma\). In particular, they conjectured the existence of a \textit{complete} fan, which would serve to simultaneously compactify all period maps of a given type. A long standing question, resolved by [\textit{K. Kato} and \textit{S. Usui}, Classifying spaces of degenerating polarized Hodge structures. Princeton, NJ: Princeton University Press (2009; Zbl 1172.14002)],
has been to enlarge \(D\) to a space \(D^{\Sigma}\) of extended Hodge structures with a set of natural properties. This construction, which is based on the work of Cattani-Kaplan-Schmid, constructs
\(D^{\Sigma}/\Gamma\)as a ``fine moduli space for polarized logarithmic Hodge structures with \(\Gamma\)-level structures''. Although not an
analytic variety in the usual sense, holomorphic maps to it are well-defined and \(D^{\Sigma}/\Gamma\) does have an infinitesimal structure that allows one to define the differential of the period map at the boundary.
The basic idea is to define boundary components of period domains as parameter spaces for multi-variable nilpotent orbits. By putting proper topological structures on the resulting spaces (to ``glue in'' these boundary components), one can provide compactifications whose global structure is encoded by some polyhedral fan of nilpotent elements. Kato and Usui also made conjectures about the complete fan and complete weak fan which, if proved to be true, will guarantee compactification for the image of any period map into \(\Gamma\backslash D\), as such fan or weak fan is, intuitively speaking, the collection of all \(\Gamma\)-compatible nilpotent cones. The conjecture about the existence of complete fan has been verified by Kato and Usui themselves (based on [\textit{A. Ash} et al., Smooth compactifications of locally symmetric varieties. With the collaboration of Peter Scholze. 2nd ed. Cambridge: Cambridge University Press (2010; Zbl 1209.14001)]) for classical cases. However, for non-classical cases, Watanabe provided a counterexample in [\textit{K. Watanabe}, J. Math. Kyoto Univ. 48, No. 4, Article ID 11, p. 951--962 (2008; Zbl 1185.14007)]. The paper constructs a fan which compactifies the image of a period map arising from a particular two-parameter family of Calabi-Yau threefolds studied by Hosono and Takagi, with Hodge numbers \((1,2,2,1)\). On the other hand, we disprove the existence of complete fans in some general cases, including the \((1,2,2,1)\) case. Hodge theory; period map; nilpotent orbit; complete fan; toroidal compactification; Kato-Usui theory Variation of Hodge structures (algebro-geometric aspects), Homogeneous spaces and generalizations, Lie algebras of linear algebraic groups, Linear algebraic groups and related topics, Homogeneous complex manifolds, Affine algebraic groups, hyperalgebra constructions Extension of period maps by polyhedral fans | 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 Schubert polynomials are refined by the key polynomials of Lascoux-Schützenberger, which in turn are refined by the fundamental slide polynomials of Assaf-Searles [\textit{S. Assaf} and \textit{D. Searles}, Adv. Math. 306, 89--122 (2017; Zbl 1356.14039)]. In this paper we determine which fundamental slide polynomial refinements of key polynomials, indexed by strong compositions, are multiplicity free. We also give a recursive algorithm to determine all terms in the fundamental slide polynomial refinement of a key polynomial indexed by a strong composition. From here, we apply our results to begin to classify which fundamental slide polynomial refinements, indexed by weak compositions, are multiplicity free. We completely resolve the cases when the weak composition has at most two nonzero parts or the sum has at most two nonzero terms. Schubert polynomials; Lascoux-Schützenberger key polynomials Symmetric functions and generalizations, Combinatorial aspects of representation theory, Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds Slide multiplicity free key polynomials | 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 For \(s_0,\dots,s_{n+1}\in\mathbb{C}\) and a topological path \(\gamma (t)(0\leq t\leq 1)\) with \(\gamma(0)=s_0\), \(\gamma(1)=s_{n+1}\), and \(\{\gamma(t)\}\cap\{s_1,\dots, s_n\}=\varphi\), the iterated integral is defined to be \(\int_{0\leq t_1\leq \cdots\leq 1_n\leq 1}\frac{d\gamma(t_1)} {\gamma(t_1)-s_1}\wedge\cdots\wedge\frac {d\gamma(t_n)}{\gamma (t_n)-s_n}\). In this paper the authors construct an element \(I(s_0;s_1, \dots,s_n;s_{n+1})\) in the motivic Hopf algebra \(\chi_F(n)\) for any field \(F\). Here \(\chi_F(n)\) denotes a fundamental object in [\textit{S. Bloch} and \textit{I. Kriz}, Ann. Math. 140, 557--605 (1994; Zbl 0935.14014)], which is used to construct the category MTM\((F)\) of mixed Tate motives over \(F\). Actually MTM\((F)\) is defined to be the category of finite dimensional graded comodules over the Hopf algebra \(\chi_F(n)\), which is the 0-th cohomology of a certain commutative differential Hopf algebra constructed through cubical cycle complex. The main theorem of this paper shows that if \(F\) is a subfield of \(C\), the Hodge realization of the motivic integral \((-1)^nI(s_0;s_1,\dots,s_n;s_{n+1})\) agrees with the framed mixed Hodge Tate structure corresponding to the iterated integral above. algebraic cycles; motives; iterated integrals Furusho, Math. Res. Lett. 14 pp 923-- (2007) Applications of methods of algebraic \(K\)-theory in algebraic geometry, Rational points, Algebraic cycles, Derived categories, triangulated categories Algebraic cycles and motivic generic iterated integrals | 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 was published in the book announced in this Zbl 0342.00005.] This paper is concerned with certain combinatorial problems arising from the Schubert calculus. The two main topics treated are: (1) a combinatorial description of the cohomology ring \(H^*(G_{dn};\mathbb Z)\) of the Grassmann manifold \(G_{dn}\) over \(\mathbb C\) in terms of Schur functions, and (ii) the determination of the Hilbert function of the homogeneous and affine coordinate rings of the Schubert varieties \(\Omega(a_0a_1\dots a_d)\). These are in general well-known results, but the explicit combinatorial approach given here should make the subject more accessible to combinatorialists. A few new observations are made, e.g., if \(H(m)\) is the Hilbert function of a polynomial ring over \(\mathbb C\) in the entries of a generic \(\alpha\times\beta\) matrix modulo the \(\gamma\times\gamma\) minors, then a combinatorial Interpretation is stated for the numbers \(f_i\) defined by \(H(m)=\Sigma f_i\tbinom {m-1} {i-1}\). There also is given a definition and analysis of a class of varieties called skew Schubert varieties which generalize the ordinary Schubert varieties. Their multiplication rule in the ring \(H^*(G_{dn};\mathbb Z)\) is described, and the ``postulational formula'' is extended to these varieties. Itshould be pointed out that Theorem 4.5 is false for skew Schubert varieties, although it is true for ordinary Schubert varieties. It is an interesting open question to find the correct form of Theorem 4.5, or equivalently to find the number of points on a skew Schubert variety over a finite field. Stanley, R.; Foata, D., Some combinatorial aspects of the Schubert calculus, Combinatoire et représentation du groupe symétrique (Actes Table Ronde CNRS, Univ. Louis-Pasteur Strasbourg, Strasbourg, 1976), (1977), Springer, Berlin Exact enumeration problems, generating functions, Enumerative problems (combinatorial problems) in algebraic geometry Some combinatorial aspects of the Schubert calculus | 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 perfect field of characteristic \(p\). Let \(X\) be an integral \(k\)-scheme. There is a pro-finite, unipotent, local, fundamental group scheme \(\pi_1^{\circ}(X)\) associated to \(X\), defined in [\textit{M. V. Nori}, Proc. Indian Acad. Sci., Math. Sci. 91, 73--122 (1982; Zbl 0586.14006)]. Let \(X:=\mathrm{Spec}(A)\) and \(Y:=\mathrm{Spec}(B)\) be two integral affine schemes over \(k\). Then the author proves that a map \(f:X \to Y\) induces an isomorphism \(\pi_1^{\circ}(X)\) to \(\pi_1^{\circ}(Y)\) if and only if it induces an isomorphism \(B/B^p \to A/A^p\).
The result is an analog of a result of \textit{K. Kedlaya} proved in [Can. J. Math. 60, No. 1, 140--163 (2008; Zbl 1144.14013)]. fundamental group scheme; schemes in positive characteristic Ünver, S., \textit{on the local unipotent fundamental group scheme}, Canad. Math. Bull., 53, 187-191, (2010) Group schemes, Varieties over finite and local fields, Homotopy theory and fundamental groups in algebraic geometry On the local unipotent fundamental group scheme | 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 consider graded Gorenstein quotients of \(R=k[X_1,X_2, \dots,X_s]\) of codimension \(r\). If \(s=r\), we let \(PGor(H)\) be the space parametrizing all graded Gorenstein \(R\)-algebras \(R/I\) with Hilbert function \(H\) \((H(i)=\dim(R/I)_i)\), with a scheme structure induced by the vanishing of the relevant catalecticant minors [as explained by \textit{S. J. Diesel}, Pac. J. Math. 172, No. 2, 365-397 (1996; Zbl 0882.13021)]. If \(s>r\) we define \(PGor(H)\) similarly and endow it with an apparently different scheme structure (remark 1.6). The main theorems of this note are concerned with \(PGor(H)\) for \(s=r=3\) in which case we prove that \(PGor (H)\) is a smooth irreducible scheme and we compute its dimension. For \(s>r=3\) a corresponding theorem is proved by \textit{J. O. Kleppe} and \textit{R. M. Miró-Roig} [J. Pure Appl. Algebra 127, No. 1, 73-82 (1998)]. As a corollary we prove a conjecture of Geramita, Pucci, and Shin on the Hilbert function of \(R/I^2\). If \(s=r>3\), we find conditions for \(PGor(H)\) to be smooth and we prove a useful linkage result for computing its dimension. We also prove some results on the codimension of \(PGor(H)\) embedded in some natural strata of the punctual Hilbert scheme. In particular if \(s=r=3\), we compute the dimension of \(ZGor(H)\) of all (not necessarily graded) Gorenstein quotients of \(k[[X_1,X_2, \dots, X_s]]\) with symmetric Hilbert function \(H\) at a graded quotient \(R\to A\), leading to a criterion for \(A\) to be non-alignable. Our method of proof applies the theorem of \textit{A. Iarrobino} and \textit{V. Kanev} [``The length of a homogeneous form, determinantal loci of catalecticants and Gorenstein algebras'' (preprint)] where they determine the tangent space of \(PGor(H)\). punctual Hilbert scheme; graded Gorenstein quotients; Hilbert function; dimension Kleppe, JO, The smoothness and the dimension of \({\mathrm PGor}(H)\) and of other strata of the punctual Hilbert scheme, J. Algebra, 200, 606-628, (1998) Parametrization (Chow and Hilbert schemes), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) The smoothness and the dimension of \(PGor(H)\) and of other strata of the punctual Hilbert scheme | 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 interesting paper the authors study homomorphisms from the abelianization \(\Pi(X,x_0)^{ab}\) of the \(F\)-divided fundamental group scheme of a proper smooth variety \(X\) over algebraically closed field \(k\) of characteristic \(p > 0\) to the \(F\)-divided fundamental group \(\Pi(A, \alpha(x_0))\) of the Albanese variety \(A\) of \(X\). The second author [J. Algebra 317, No. 2, 691--713 (2007; Zbl 1130.14032)] of the paper under review gives a description of \(F\)-divided fundamental group scheme in the case of abelian varieties over algebraically closed field of characteristic \(p > 0\). He proves that it is correspond to a representation of the fundamental group scheme by \textit{M. V. Nori} [Compos. Math. 33, 29--41 (1976; Zbl 0337.14016)]. The authors generalize this analysis to proper and smooth varieties over \( k\). The main result (Theorem 4.1) of the paper under review is as followa. The homomorphism \(\alpha_{\#}: \Pi(X,x_0)^{ab} \to \Pi(A, \alpha(x_0))\) is surjective and its kernel is \(\mathrm{Diag}(\mathrm{NS}(X))^{`} \times K\). Here \(\alpha: X \to A\) is the Albanese morphism, \(\mathrm{NS}(X)^{`} \) is the subgroup of elements whose order is finite and prime to \(p\) of the Néron-Severi group \( NS(X) \) and \(K\) is the group of \( k\)-points of the Cartier dual of the local affine group scheme \(\mathbf{ Pic}^{0}(X)/ \mathbf{ Pic}^{0}_{\mathrm{red}}(X)\). The proof is given by means of a number of propositions together with applications of results by \textit{W. C. Waterhouse} [Introduction to affine group schemes. New York, Heidelberg, Berlin: Springer-Verlag (1979; Zbl 0442.14017)] and results by the second author [loc. cit.]. The authors of the paper under review indicate that analogous considerations for the essentially finite fundamental group scheme were undertaken by \textit{M. Antei} [Isr. J. Math. 186, 427--446 (2011; Zbl 1263.14047)]. \(F\)-divided fundamental group scheme; Frobenius; Picard scheme; Albanese Positive characteristic ground fields in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry, Picard schemes, higher Jacobians Abelianization of the \(F\)-divided fundamental group scheme | 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 that for any constants \(a<1/4\), and \(b,c\), there are at least \(y^{by^a+c}\) many irreducible components of the moduli space of regular surfaces of general type with \(K^2= c^2_1=y\) and fixed \(c_2\), where \(c_1\) and \(c_2\) are the Chern classes of the tangent bundle of the surface, and \(K\) is the canonical class. The same result holds true for the Hilbert scheme of surfaces in \(\mathbb{P}^4\) with \(K^2=y\) and fixed Hilbert polynomials. Similar results are given for threefolds. The idea of the proof is to look at projectively normal subvarieties of codimension two in the projective space with some very special resolution of the ideal sheaf. components of the moduli space; regular surfaces of general type; Chern classes; Hilbert scheme; surfaces; threefolds; codimension two M.-C. Chang, The number of components of Hilbert schemes. \textit{Internat. J. Math}. 7 (1996), 301-306. MR1395932 Zbl 0892.14006 Parametrization (Chow and Hilbert schemes), Low codimension problems in algebraic geometry, Enumerative problems (combinatorial problems) in algebraic geometry, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) The number of components 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 Let \(T\) be a torus over a local field \(K\). Let \(\varphi\) be the scheme of connected components of the Néron model of \(T\); it is an étale group scheme of finite type. Hence \(\varphi\) is totally determined by a finitely generated \(G_ k\)-module, where \(G_ k\) is the absolute Galois group of \(k\). In this paper we give an explicit description of \(\varphi\) as \(G_ k\)-module in terms of the complex of \(G_ k\)-modules \(R\Gamma(I,X)\), where \(I\) is the inertia subgroup of the absolute Galois group of \(K\), and \(X\) is the character group of \(T\). --- The main result is that there exists an isomorphism respecting the \(G_ k\)-action: \(R\text{Hom}_ \mathbb{Z}(\varphi,\mathbb{Z})\cong\tau_{\leq 1} R\Gamma(I,X)\). In particular, we can compute \(\varphi\) as the zero cohomology of the complex \(R\text{Hom}_ \mathbb{Z}(\tau_{\leq 1} R\Gamma(I,X),\mathbb{Z})\) using any \(I\)-acyclic resolution of \(X\). The functor \(\tau_{\leq 1}\) truncates the complex by degree 1, but maintains the 0 and 1 cohomology. torus over a local field; Néron model; absolute Galois group Xarles, X., The scheme of connected components of the Néron model of an algebraic torus, J. reine angew. math., 167-179, (1993) Minimal model program (Mori theory, extremal rays), Group actions on varieties or schemes (quotients) The scheme of connected components of the Néron model of an algebraic torus | 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 devoted to a finer analysis of a certain class of algebraic schemes generalizing quasi-projective schemes. More precisely, the authors reconsider the class of divisorial schemes, which was introduced by \textit{M. Borelli} exactly forty years ago [Pac. J. Math. 13, 375--388 (1963; Zbl 0123.38102)]. Recall that a quasi-compact and quasi-separated scheme \(X\) is called divisorial if there is a collection \(L_1, \dots, L_r\) of invertible sheaves on \(X\) satisfying the following condition: The open sets \(X_F\) for any \(f\in \Gamma(X, L_1^{d_1}\otimes\cdots\otimes L^{d_r}_r)\) and any multiindex \(d:=(d_1, \dots, d_r)\in\mathbb Z^r\) form a base of the Zariski topology of the scheme \(X\). Such a collection \(L_1, \dots, L_r\) is then called an ample family on \(X\).
For instance, all quasi-projective schemes, all smooth varieties, and all locally \(\mathbb Q\)-factorial varieties are known to be divisorial in this sense.
In the paper under review, the authors first generalize Grothendieck's construction of homogeneous spectra for \(\mathbb N\)-graded rings [\textit{A. Grothendieck}, EGA III, premiére partie, Publ. Math., Inst. Hautes Étud. Sci. 11, 349--511 (1962; Zbl 0118.36206)] to so-called multihomogeneous spectra of multigraded rings, that is to rings graded by an arbitrary, finitely generated abelian group \(D\). For such a ring \(S=\bigoplus_{d\in D}S_d\), the multihomogeneous spectrum \(\text{Proj}(S)\) is obtained by patching certain affine open pieces \(D_+(f)=\text{Spec}(S_{(f)})\), where \(f\in S\) are special homogeneous elements in \(S\). In the particular case of \(\mathbb N^r\)-graded rings, a similar construction has been carried out by \textit{P. Roberts} [``Multiplicities and Chern classes in local algebra'', Camb. Tracts Math. 133 (1998; Zbl 0917.13007)]
As the authors point out, their multihomogeneous spectra share many properties of the classical homogeneous spectra, but are possibly non-separated. In the sequel, the authors relate multihomogeneous spectra to divisorial schemes and simplicial toric varieties. In this context, one of their main results states that a scheme is divisorial if and only if it admits an embedding into a suitable multihomogeneous spectrum of a multigraded ring. This follows from a characterization of ample families on a scheme \(X\) in terms of the multihomogeneous spectrum \(\text{Proj}(S)\) for the multigraded ring \(S:=\bigoplus_{d\in\mathbb N}^r\Gamma(X,L^{d_1}_1\dots L^{d_r}_r)\). Moreover, generalizing \textit{H. Grauert}'s classical criterion of ampleness [Math. Ann. 146, 331--368 (1962; Zbl 0173.33004)] to families of invertible sheaves, the authors also characterize ample familiea, and therefore divisorial schemes, in terms of affine hulls and contraction maps. In addition, they provide a cohomological characterization of divisoriality, which may be regarded as an analogue of Serre's criterion of ampleness. Finally, as an application of the foregoing results, Grothendieck's algebraization theorem for formal schemes [\textit{A. Grothendieck}, ``Techniques de construction et théorèmes d'existence en géométrie algébrique. III: Préschémas quotients.'' Sem. Bourbaki 13 (1960/61), No.212, 20 p. (1961; Zbl 0235.14007)] is generalized as follows:
A proper formal scheme \({\mathcal X}\to\text{Spf}(R)\) is algebraizable if there is a finite collection of invertible formal sheaves restricting to an ample family on the closed fibre and satisfying an additional technical condition.
All these very substantial results are comprehensively derived in the six sections of the present paper, and that in an utmost lucid, detailed and rigorous manner, with numerous clarifying remarks and bibliographical references. schemes and morphisms; divisorial schemes; ample sheaves; graded rings; formal schemes; group actions; geometric invariant theory; ampleness criteria; algebraization Brenner, H.; Schröer, S.: Ample families, multihomogeneous spectra, and algebraization of formal schemes. Pacific J. Math. (2001) Schemes and morphisms, Divisors, linear systems, invertible sheaves, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Geometric invariant theory, Formal methods and deformations in algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies Ample families, multihomogeneous spectra, and algebraization of formal 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 a zero-dimensional scheme \(Z\subset \mathbb{P}^2\) and a positive integer \(a\), the curves of degree \(a\) containing \(Z\) form a linear system \(|aL-Z|\) of dimension at least \(v(Z, a) = a(a + 3)/2-\mathrm{length Z}\). \(|aL-Z|\) is said to have maximal rank if it is empty or \(\dim |aL- Z| = v(Z, a)\). \(Z\) itself has maximal rank (equivalently, its graded ideal has maximal Hilbert function) if \(|aL- Z|\) has maximal rank for all \(a > 0\). Given positive integers \(n, e\), denote \(Z(e^n) = \bigcup ^n_{i=1} p^e_i\) as the scheme formed by \(n\) general points \(p_1,\dots , p_n \in\mathbb{P}^2\) taken with multiplicity \(e\). Theorem 1 and Theorem 2 are evidence for the homogeneous Segre-Harbourne-Gimigliano-Hirschowitz conjecture, which says that such \(Z\) have maximal rank for every \(e\) if \(n\geq 9\). Theorem 3 is evidence for the Greuel-Lossen-Shustin conjecture, which says that general singularity schemes have maximal rank in every degree larger than the sum of the three biggest multiplicities (which in the case of ADE singularities is 9). This reduces the proof of the Greuel-Lossen-Shustin conjecture for ADE singularities to checking a finite number of cases.
Theorems 1, 2 and 3 are proved by a generalization of the differential Horace method developed by J. Alexander, A. Hirschowitz and L. Evain, applied to suitable specializations. This differential method consists in studying a family \(Z_t\) of zero-dimensional schemes whose special member \(Z_0\) is tractable but gives dimension larger than \(Z_t\), by taking \(t^{p_1} = 0, t^{p_2} = 0,\dots\) for suitable exponents \(p_i\). Like Evain's, the families \(Z_t\) are essentially monomial and can be described by combinatorial objects called staircases, but in this paper the reduction of algebraic computations to combinatorics is only valid under additional hypotheses on the \(p_i\)'s, which we check for each \(Z_t\) involved. zero-dimensional scheme; maximal rank; differential Horace method Divisors, linear systems, invertible sheaves, Singularities of curves, local rings, Plane and space curves, Fibrations, degenerations in algebraic geometry Maximal rank for schemes of small multiplicity by Évain's differential Horace method | 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 finite group over a field \(k\) of positive characteristic. A full triangulated subcategory \(\mathcal{C}\) of the stable module category \(StMod \, G\) of possibly infinite-dimensional \(G\)-modules is called a \textit{colocalising subcategory} if it is closed under set-indexed products. It is \(\mathrm{Hom}\) closed if whenever \(M\) is in \(\mathcal{C}\), so is \(\mathrm{Hom}_k(L,M)\) for any \(G\)-module \(L\). The main result of the paper gives a bijection between the Hom closed colocalising subcategories of \(StMod \, G\) and the subsets of \(\mathrm{Proj} \, H^*(G,k)\) where the latter is the set of homogeneous prime ideals not containing \(H^{\geq 1}(G,k)\). This bijection is given by sending \(\mathcal{C}\) to its \(\pi\)-cosupport.
In earlier work [the first author et al., J. Am. Math. Soc. 31, No. 1, 265--302 (2018; Zbl 1486.16011)] the authors had classified the tensor closed localising subcategories of \(StMod \;G\). Combined with the present work the assignment \(\mathcal{C} \mapsto \mathcal{C}^{\bot}\) gives a bijection between the tensor closed localising subcategories of \(StMod \, G\) and the Hom closed colocalising subcategories of \(StMod \, G\). cosupport; stable module category; finite group scheme; colocalising subcategory Benson, Dave; Iyengar, Srikanth B.; Krause, Henning; Pevtsova, Julia, Colocalising subcategories of modules over finite group schemes, Ann. K-Theory, 2379-1683, 2, 3, 387\textendash408 pp., (2017) Modular representations and characters, Derived categories, triangulated categories, Group schemes, Cohomology theory for linear algebraic groups, Cohomology of groups Colocalising subcategories of modules over finite 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 We present a detailed and elementary construction of the inverse perfection of a scheme and discuss some of its main properties. We also establish a number of auxiliary results (for example, on inverse limits of schemes) which do not seem to appear in the literature. perfect closure; perfect scheme Schemes and morphisms, Perfectoid spaces and mixed characteristic, Finite ground fields in algebraic geometry, Group schemes On the perfection of 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 \textit{E. Bierstone} and \textit{P. D. Milman} [Invent. Math. 101, No. 2, 411--424 (1990; Zbl 0723.32005)] and \textit{A. Parusinski} [Trans. Am. Math. Soc. 344, No. 2, 583--595 (1994; Zbl 0819.32006)] proved the following important rectilinearization result for continuous subanalytic functions:
Let \(U\) be a real analytic manifold and \(f:U\to\mathbb{R}\) a continuous subanalytic function. Then there is a locally finite covering \((\pi_j:U_j\to U)_j\) such that
(i) each \(\pi_j\) is a composite of finitely many mappings each of which is either a local blowing-up with smooth center or a local power substitution;
(ii) each \(f\circ \pi_j\) is analytic and identically \(0\) or a normal crossing or the inverse of a normal crossing.
In the paper under review, the author proves a rectilinearization theorem for functions definable in an o-minimal structure generated by a convergent Weierstrass system. Convergent Weierstrass systems were introduced in [\textit{L. Van den Dries}, J. Symb. Log. 53, No. 3, 796--808 (1988; Zbl 0698.03023)]. They are induced by certain subrings of the ring of all restricted real analytic functions satisfying similar properties, notably being closed under Weierstrass division. Taking all restricted analytic functions, one obtains the o-minimal structure \(\mathbb{R}_{\mathrm{an}}\) and the above result for globally subanalytic functions in a global form. The proofs rely on the fact that functions definable in an o-minimal structure generated by a convergent Weierstrass system are piecewise given by terms in a reasonable language. rectilinearization; subanalytic functions; Weierstrass systems DOI: 10.4064/ap99-2-2 Semi-analytic sets, subanalytic sets, and generalizations, Modifications; resolution of singularities (complex-analytic aspects), Real-analytic and semi-analytic sets, Quantifier elimination, model completeness, and related topics Rectilinearization of functions definable by a Weierstrass system and its applications | 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 spirit of Alain Lascoux, the authors propose the use of Schubert polynomials for (certain) computations with polynomials in several variables. The idea comes from situations like doing computations with symmetric functions: There, computations are (usually) not done with monomials, but with a basis adapted to the specific problem that we are dealing with, such as Schur functions, for example.
Most of the paper is devoted to survey the background and basic facts about Schubert polynomials. When we regard the complete ring of polynomials in \(x_1,x_2,\dots,x_n\) as a ring over the ring of symmetric polynomials in \(x_1,x_2,\dots,x_n\), then the Schubert polynomials indexed by permutations in \(S_n\) (the symmetric group on \(n\) elements) constitute a linear basis. Similarly, the ring of polynomials in \(x_1,x_2,\dots,x_n\) with coefficients that are polynomials in \(y_1,y_2,\dots,y_n\) has as a linear basis the double Schubert polynomials. In order to use Schubert polynomials efficiently for computations in such rings, one of the first things we need is a rule for multiplying Schubert polynomials. No general formula for multiplying Schubert polynomials has been found yet (in contrast to Schur functions, where we have the Littlewood-Richardson rule). At least, at the very basic level, there is Monk's formula for the multiplication of a Schubert polynomial in \(x_1,x_2,\dots,x_n\) by one of the variables. However, this formula (possibly) involves Schubert polynomials which are indexed by permutations in \(S_{n+1}\) (and not just \(S_n\)). The authors show how to modify the formula so that one obtains, within the ring of polynomials in \(x_1,x_2,\dots,x_n\), regarded as a ring over the symmetric polynomials, expansions consisting of Schubert polynomials indexed by permutations in \(S_n\). A ``Monk's formula'' for double Schubert polynomials is proved as well. Schubert polynomials; symmetric functions; Monk's formula; divided differences Kohnert, A.; Veigneau, S.: Using Schubert basis to compute with multivariate polynomials. Adv. appl. Math. 19, 45-60 (1997) Symmetric functions and generalizations, Polynomials, factorization in commutative rings, Determinantal varieties, Grassmannians, Schubert varieties, flag manifolds Using Schubert basis to compute with multivariate polynomials | 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 \(X\subset \mathbb{P}^{m-1},\, \, X'\subset \mathbb{P}^{r-1},\, r<m\),\, denote two algebraic toric sets defined over the finite field \(\mathbb{F}_q\),\, with \(X'\)\, embedded in \(X\). The aim of the paper is to study the relations of the parameterized codes associated to \(X\)\, and \(X'\).
The parameterized codes of order \(d\), \(\mathcal{C}_X(d)\), associated to the toric set \(X\),\, were defined by \textit{C. Rentería, A. Simis} and \textit{R. H. Villarreal} [Finite Fields Appl. 17, No. 1, 81--104 (2011; Zbl 1209.13037)]. They are the image of the evaluation map from the homogeneous polynomials of degree \(d\)\, of \(S=\mathbb{F}_q[X_1,\dots , X_m]\)\, in the points of \(X\). Section 2 relates the parameters of \(\mathcal{C}_X(d)\)\, (length, dimension and minimum distance) with the algebraic invariants of \(S/I_X,\, I_X\)\, vanishing ideal of \(X\).
Section 3 considers the case \(X'\)\, embedded in \(X\)\, and recalls and generalizes previous results of \textit{M. Vaz Pinto} and \textit{R. H. Villarreal} [Commun. Algebra 41, No. 9, 3376--3396 (2013; Zbl 1286.13022)] relating the basic invariants of \(\mathcal{C}_X(d)\)\, and \(\mathcal{C}_X'(d)\). Section 4 provides formulas for the regularity index of \(S/I_X\)\, when \(X\)\, is the toric set parameterized by the edges of a complete graph with an even number of vertices (Corollary 5) or an odd number (Corollary 7). Finally Section 5 finds lower and upper bounds for the minimum distance of \(\mathcal{C}_X(d)\),\, \(X\)\, associated with the edges of a complete graph, both in the case of even number of vertices (Corollary 8) and odd number of vertices (Corollary 9). algebraic toric sets; embedded sets; regularity index; parameterized codes; minimum distance; complete graphs. González-Sarabia, M.; Rentería, C.; Sarmiento, E., Parameterized codes over some embedded sets and their applications to complete graphs, Math. Commun., 18, 377-391, (2003) Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.), Algebraic coding theory; cryptography (number-theoretic aspects), Applications to coding theory and cryptography of arithmetic geometry, Geometric methods (including applications of algebraic geometry) applied to coding theory, Linear codes (general theory) Parameterized codes over some embedded sets and their applications to complete graphs | 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 we consider the concept of the multiplicity of intersection points of plane algebraic curves \(p,q=0,\) based on partial differential operators. We evaluate the exact number of maximal linearly independent differential conditions of degree \(k\) for all \(k\ge 0.\) On the other hand, this gives the exact number of maximal linearly independent polynomial and polynomial-exponential solutions, of a given degree \(k,\) for homogeneous PDE system \(p(D)f=0,q(D)f=0.\) intersection point; multiplicity; PDE system Interpolation in approximation theory, Multidimensional problems, Plane and space curves On constant coefficient PDE systems and intersection multiplicities | 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 subspace arrangement in a vector space, one can associate to it a variety, called its wonderful model in the sense of De Concini and Procesi. These varieties appear in many fields of research: toric geometry, discrete geometry, theory of moduli spaces of curves, to name just a few. For each arrangement one can construct several wonderful models; they are indexed by certain combinatorial data (``building sets''). Among these, there always exist a minimal and a maximal building set with respect to inclusion; the corresponding wonderful models are also called a minimal and a maximal one.
This paper is devoted to a particularly important class of arrangements: the root arrangements. Cohomology bases for maximal complex models of root arrangements were described by \textit{S. Yuzvinski } [Invent. Math. 127, No. 2, 319--335 (1997; Zbl 0989.14006)]. The authors use his results to compute inductive formulas for Poincaré series of maximal complex models associated to root arrangements of types A, B, C, and D. models; arrangements; Poincaré series; roots G. Gaiffi and M. Serventi, ``Poincarè series for maximal De Concini-Procesi models of root arrangements,'' Rendiconti Lincei-Matematica e Applicazioni. In press. Configurations and arrangements of linear subspaces, Root systems, Homogeneous spaces and generalizations Poincaré series for maximal de Concini-Procesi models of root arrangements | 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 No review copy delivered. [7]I. Cascudo, R. Cramer and C. Xing, Torsion limits and Riemann--Roch systems for function fields and applications, IEEE Trans. Information Theory 60 (2014), 3871--3888. Algebraic functions and function fields in algebraic geometry, Applications to coding theory and cryptography of arithmetic geometry, Combinatorial codes Torsion limits and Riemann-Roch systems for function fields and applications | 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 It is shown that if a mapping from the \(n\)-dimensional hypercube to itself has the property that all the Boolean eigenvalues of the discrete Jacobian matrix of each element of the hypercube are zero, then it has a unique fixed point. This answers to the ``Combinatorial fixed point conjecture'', a combinatorial version of the Jacobian conjecture. Automata network; Jacobian conjecture; Fixed point; 01-strings; Hypercube; \(k\)-subcube; Discrete Jacobian matrix; Boolean eigenvalue; Hamming metric Shih, M.-H., \& Dong, J.-L. (2005). A combinatorial analogue of the Jacobian problem in automata networks. Adv. Appl. Math., 34, 30--46. Combinatorial aspects of matrices (incidence, Hadamard, etc.), Cellular automata (computational aspects), Jacobian problem, Boolean functions A combinatorial analogue of the Jacobian problem in automata networks | 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 \(\Gamma \subset \mathrm{SL}_2(\mathbb{C})\) be a nontrivial finite subgroup and the surface \(S = \widehat{\mathbb{C}^2/\Gamma}\) be the minimal resolution of \(\mathbb{C}/\Gamma\). Associated to \(\Gamma\) is a Heisenberg algebra of affine type, \(\mathfrak{h}_\Gamma\), and the Hilbert schemes of points \(\mathrm{Hilb}^n(S)\). \textit{I. Grojnowski} [Math. Res. Lett. 3, No. 2, 275--291 (1996; Zbl 0879.17011)] and \textit{H. Nakajima} [Ann. Math. (2) 145, No. 2, 379--388 (1997; Zbl 0915.14001)] construct a representation of the Heisenberg algebra (actually a slightly different version from the one considered in this paper) on the cohomology of the Hilbert schemes. Algebraically, \textit{I. Frenkel}, \textit{N. Jing} and \textit{W. Wang} [Int. Math. Res. Not. 2000, No. 4, 195--222 (2000; Zbl 1011.17020)] construct the basic representation of \(\mathfrak{h}_\Gamma\) on the Grothendieck group of the category of \(\mathbb{C}[\Gamma^n \rtimes S_n]\)-modules.
In this paper, the authors define a 2-category \(\mathcal{H}_\Gamma\) and their first main result (3.4) states that \(\mathcal{H}_\Gamma\) categorifies the Heisenberg algebra \(\mathfrak{h}_\Gamma\).
The second main result of the paper (4.4) is a categorical action of \(\mathcal{H}_\Gamma\) on a 2-category \(\bigoplus_{n\geq 0} D(A_n^\Gamma -\mathrm{gmod})\). Here \(D(A_n^\Gamma -\mathrm{gmod})\) denotes the bounded derived category of finite-dimensional, graded \(A_n^\Gamma\)-modules, where
\[
A_n^\Gamma = [(\mathrm{Sym}^*((\mathbb{C}^2)^\vee) \rtimes \Gamma) \otimes \ldots \otimes (\mathrm{Sym}^*((\mathbb{C}^2)^\vee) \rtimes \Gamma) ] \rtimes S_n.
\]
As explained in Section 8, \(D(A_n^\Gamma -\mathrm{gmod})\) is known to be equivalent to \(D\mathrm{Coh}(\mathrm{Hilb}^n(S))\) and thus the second main theorem categorifies a representation similar to that of Grojnowski [Zbl 0879.17011] and Nakajima [Zbl 0915.14001].
In Section 9, another 2-representation of \(\mathcal{H}_\Gamma\) is introduced that is related to the first by Koszul duality. In section 9.6, it is shown that this 2-representation categorifies an action similar to that constructed by Frenkel, Jing and Wang [Zbl 1011.17020].
For the most part geometry appears only in Section 8. The main definitions are algebraic and a number of the proofs are based on graphical calculus. Section 10 contains a description of various connections to other categorical actions and some open problems.
As the case \(\Gamma = \mathbb{Z}/2\) differs slightly, the necessary modifications are addressed separately in a short appendix. categorification; Heisenberg algebra; McKay correspondence; Hilbert scheme S. Cautis & A. Licata, ``Heisenberg categorification and Hilbert schemes'', Duke Math. J.161 (2012) no. 13, p. 2469-2547 Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Quantum groups and related algebraic methods applied to problems in quantum theory, Frobenius induction, Burnside and representation rings, Lie algebras and Lie superalgebras, Parametrization (Chow and Hilbert schemes) Heisenberg categorification 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 This paper studies the Hilbert scheme of \(n\) points in affine \(d\) space, for \(n\) at most 8. In particular the authors focus on when this scheme is reducible. The main theorem classifies precisely when this happens, namely for \(n=8\) and \(d \geq 4\).
In general the Hilbert scheme is a difficult and complicated object to study and it is well known that it may contain arbitrarily ``bad'' singularities [\textit{R.~Vakil}, Invent. Math. 164, No. 3, 569--590 (2006; Zbl 1095.14006)] (however Hartshorne did prove that it is always connected). The authors do a wonderful and thorough study of these particular Hilbert schemes. The paper has many explicit calculations and examples. The study is comprehensive and includes a characterization of which points are smoothable (i.e. belongs to the smoothable component of the Hilbert scheme). The authors finish with some interesting open questions which would nicely extend their work in this paper. Hilbert scheme; zero-dimensional ideal; smoothable D. A. Cartwright, D. Erman, M. Velasco, B. Viray, Hilbert schemes of 8 points. \textit{Algebra Number Theory}\textbf{3} (2009), 763-795. MR2579394 Zbl 1187.14005 Parametrization (Chow and Hilbert schemes), Commutative Artinian rings and modules, finite-dimensional algebras Hilbert schemes of 8 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 We prove the following theorem.
Fix integers \(s>0\) and \(m_1\geq \cdots \geq m_s>0\). Let \(Z\subset \mathbb{P}^3\) be a general union of \(s\) fat points of multiplicity \(m_1,\dots,m_s\). Let \(k\) be the critical value of \(Z\). Assume \(m_i=1\) for at least \((m_1-1)k^2/2+k(2m_1+5)^3/12\) integers \(i\in\{1,\dots,s\}\). Then \(Z\) has the expected minimal free resolution.
Here, the critical value of \(Z\) is the minimal integer \(k>0\) such that \(z\leq \binom{k+3}{3}\). We will also say that \(k\) is the critical value of the integer \(z>0\). \(Z\) is said to have maximal rank if for every integer \(t\) the restriction map \(\rho_{Z,t}: H^0(\mathcal O(t))\rightarrow H^0(Z,\mathcal O_Z(t))\) has maximal rank. If \(z\) has critical value \(k\), then \(Z\) has maximal rank if and only if \(h^0(\mathcal I_Z(k-1))=h^1(\mathcal I_Z(k))=0\). \(Z\) has the expected minimal free resolution if it has maximal rank and no line bundle appears in two different steps of the minimal free resolution of \(Z\). Projective techniques in algebraic geometry On the minimal free solution for fat points schemes in \(\mathbb P^3\) | 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 Motivated by definitions in mixed Hodge theory, we define the weight filtration and the monodromy weight filtration on the combinatorial intersection cohomology of a fan. These filtrations give a natural definition of the multivariable invariants of subdivisions of polytopes, lattice polytopes and fans, namely the mixed \(h\)-polynomial, the refined limit mixed \(h^\ast\)-polynomial, and the mixed \textit{cd}-index, defined by Katz-Stapledon and Dornian-Katz-Tsang. Previously, only the refined limit mixed \(h^\ast\)-polynomial had a geometric interpretation, which came from filtrations on the cohomology of a schön hypersurface. Consequently, we generalize a positivity result on the mixed \(h\)-polynomial by Katz and Stapledon using the relative hard Lefschetz theorem of Karu. fan subdivisions; toric \(h\)-polynomial; \(h^\ast \)-polynomial; \textit{cd}-index; filtrations Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.) Filtrations on combinatorial intersection cohomology and invariants of subdivisions | 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 Liaison theory concerns an equivalence relation among equidimensional projective subschemes of the same dimension. This equivalence relation is generated by {direct links}. Roughly, schemes \(V\) and \(V'\) are directly linked if their union is arithmetically Gorenstein. Liaison theory has seen a surge of activity since it was placed in a modern context by \textit{C. Peskine} and \textit{L. Szpiro} [Invent. Math. 26, 271--302 (1974; Zbl 0298.14022)] in 1974. The majority of results for the first two decades dealt with the special case where the links were in fact via complete intersections (CI-liaison), but in the last 7 years or so much of the focus has changed to the more general case (G-liaison), beginning with the memoir by \textit{J. O. Kleppe}, \textit{J. C. Migliore}, \textit{R. Miró-Roig}, \textit{U. Nagel} and \textit{C. Peterson} [``Gorenstein liaison, complete intersection liaison invariants and unobstructedness'', Mem. Am. Math. Soc. 732 (2001; Zbl 1006.14018)], although the groundwork had been laid by \textit{P. Schenzel} [J. Math. Kyoto Univ. 22, 485--498 (1982; Zbl 0506.13012)] in 1982/83.
The most complete picture is for codimension two, where complete intersections and Gorenstein ideals coincide. However, recent work gives some hope that changing the focus to Gorenstein liaison will allow us to prove results for higher codimension that are analogous those in codimension two, even though the same statements are false for CI-liaison in higher codimension. One particularly nice codimension two result is a classical result due independently to Apery and to Gaeta (which now is a special case of a much bigger theory): in modern language it says that for zero-dimensional subschemes of \({\mathbb P}^2\), there is only one liaison class. This is known to be false in higher codimension, if we restrict to CI-liaison. One of the most intriguing questions at the moment is whether there is just one G-liaison class for zero-dimensional subschemes of higher projective space. (This is in turn a special case of a more general open question which we omit here.)
The author, together with \textit{R. Miró-Roig}, has proved many striking results for G-liaison of subschemes of rational normal scrolls [J. Pure Appl. Algebra 164, 325--343 (2001; Zbl 1070.14522)]. In this paper, the author takes a step toward answering the above question. She proves that a {general} set of points on a smooth rational normal scroll surface is in the G-liaison class of a complete intersection (i.e. {glicci}). This generalizes a result of \textit{R. Hartshorne} [Collect. Math. 53, No. 1, 21--48 (2002; Zbl 1076.14065)] for points on a quadric surface. As a consequence, the author proves that a general set of \(\leq N+4\) points in \({\mathbb P}^N\) is glicci. The Gorenstein links are produced using the theory developed in the memoir of Kleppe et al. mentioned above, which develops G-liaison as a theory of divisors. Gorenstein liaison; zero-dimensional schemes; rational normal scroll; arithmetically Cohen-Macaulay Casanellas, M.: Gorenstein liaison of 0-dimensional schemes. Manuscripta math. 11, 265-275 (2003) Linkage, Divisors, linear systems, invertible sheaves, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Gorenstein liaison of 0-dimensional 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 present algorithms to compute approximations to the size of curves over finite fields, and to the number of points on one of the coordinate axis with projection fibre on the curve of the same cardinality. The authors refer to this problem as the ``Computational Weil estimate''. The approach is based on the ``Strip-counting'' method introduced in [\textit{J. von zur Gathen, M. Karpinski} and \textit{I. E. Shparlinski}, Comput. Complexity 6, 64-99 (1997)], that uses the principle that the behaviour of the curve can be deduced from its behaviour over a wide enough vertical strip. The idea in this paper is that, for some properties, the strip can be taken small.
Related papers to the problem are: \textit{R. J. Schoof}, Math. Comput. 44, 483-494 (1985; Zbl 0579.14025) and \textit{M. D. Huang} and \textit{Y. C. Wong}, Solving systems of polynomial congruences modulo a large prime. Proc. 1996 IEEE Symp. on Foundations of Computer Science, 115-124 (1996). curves over finite fields; projections; computational Weil estimate; strip counting; computational algebraic geometry von zur Gathen J., Shparlinski I.E.: Computing components and projections of curves over finite fields. SIAM J. Comput. 28, 822--840 (1998) Curves over finite and local fields, Arithmetic ground fields for curves, Computational aspects of algebraic curves, Symbolic computation and algebraic computation Computing components and projections of curves over finite fields | 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 field of characteristic zero. For \(1\leq j\leq p\) \quad let \(X_j\) be a smooth variety over \(k\) and \( f_j: X_j\rightarrow \mathbb A_k^{1}\) be a function. Denote by \(X_0( f)\) the set of common zeroes on \(X={\prod}_{j}X_j\) of the compositions of the appropriate projections with the functions \(f_j\). Let \(P\in k[y_1,\dots,y_p]\) be a polynomial which is nondegenerate with respect to its Newton polyhedron and \(P(f)\) be the corresponding composition. The authors show that the motivic nearby cycles \({\mathcal S}_{P(f)}\) on \(X_0( f)\) of the function \(P(f)\) on \(X\) can be expressed as a sum over the set of compact faces \(\delta\) of the Newton polyhedron of \(P\). Let \(P_{\delta}\) denote the quasihomogoneous polynomial corresponding to \(\delta\). The authors define a convolution operator \({\Psi}_{P_{\delta}}\). For a compact face \(\delta\) they define generalized nearby cycles \({\mathcal S}_{f}^{{\sigma}(\delta)}\). The main result of the paper is the following formula:
\[
i^{*}{\mathcal S}_{P(f)}={\Sigma}_{J\subset\{1,\dots,p\}}{\Sigma}_{\delta\in {\Gamma}^J} {\Psi}_{P_{\delta}}({\mathcal S} _{f_J}^{{\sigma}(\delta), l_{\Gamma}}).
\]
G. Guibert, F. Loeser and M. Merle, Nearby cycles and composition with a non-degenerate polynomial, Int. Math. Res. Not. IMRN 31 (2005), 1873-1888. Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Algebraic cycles, Singularities in algebraic geometry, Zeta functions and \(L\)-functions, Toric varieties, Newton polyhedra, Okounkov bodies Nearby cycles and composition with a nondegenerate polynomial | 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 theory of motives is still mostly hypothetical. In this paper, a triangular category of mixed motives is constructed instead of an abelian category required by the standard conjectures. homology of schemes; triangular category of mixed motives Voevodsky, V., \textit{homology of schemes}, Selecta Math. (N.S.), 2, 111-153, (1996) (Co)homology theory in algebraic geometry, Generalizations (algebraic spaces, stacks), Applications of methods of algebraic \(K\)-theory in algebraic geometry Homology of 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 paper under review gives an explicit construction of a functor from the category of finite flat commutative group schemes of period \(2\) defined over a valuation ring of a \(2\)-adic field with algebraically closed residue field to a category of filtered modules satisfying some properties, and shows that the functor is an anti-equivalence.
More specifically, let \(k = \bar{k}\) be a field of characteristic \(2\). Let \(W(k)\) be the ring of Witt vectors and let \(K_{00}\) be the field of fractions of \(W(k)\). Let \(K_0\) be a field extension of \(K_{00}\) of degree \(e\), and let \(O_0\) be the valuation ring of \(K_0\). Let \(\text{Gr}_{O_0}\) be the category of finite flat commutative group schemes \(G\) over \(O_0\) such that \(2 \, \text{id}_G = 0\). For any \(G_1, G_2 \in \text{Gr}_{O_0}\), let \(R(G_1,G_2)\) be the set of morphisms that are of the form \(G_1 \to G_1^{\text{et}} \to G_2^{\text{mult}} \to G_2\) where the first is the natural quotient morphism \(j^{\text{et}} : G_1 \to G_1^{\text{et}}\) and the third morphism is the natural monomorphism \(i^{\text{mult}} : G_2^{\text{mult}} \to G_2\). Let \(\text{Gr}_{O_0}^*\) be the category that has the same objects set as \(\text{Gr}_{O_0}\) and morphism set \(\text{Hom}_{\text{Gr}_{O_0}^*}(G_1,G_2) = \text{Hom}_{\text{Gr}_{O_0}}(G_1,G_2)/R(G_1,G_2)\).
Set \(S = k[[t]]\) where \(t\) is a variable and let \(\sigma : S \to S\) be such that \(\sigma(s) = s^2\). Let \(\text{MF}_S^e\) be the category consisting of the triples \((M^0, M^1, \varphi_1)\) such that \(t^eM^0 \subset M^1 \subset M^0\) are \(S\)-modules, \(M^0\) is a free \(S\)-module of finite rank, \(\varphi_1 : M^1 \to M^0\) is a \(\sigma\)-linear morphism of \(S\)-modules satisfying \(\varphi_1(M^1)S = M^0\). In the category \(\text{MF}_S^e\), each element \({\mathcal M} = (M^0, M^1, \varphi_1)\) has a unique maximal etale subobject \(i^{\text{et}} : {\mathcal M}^{\text{et}} := (M^{0,\text{et}}, t^eM^{1, \text{et}}, \varphi_1) \to {\mathcal M}\), and has a unique maximal multiplicative quotient \(j^{\text{mult}}:{\mathcal M} \to {\mathcal M}^{\text{mult}} := (M^{0,\text{mult}}, M^{1,\text{mult}}, \varphi_1)\). Let \(\text{MF}_S^{e*}\) be the category that has the same objects set as \(\text{MF}_S^e\) and morphism set \(\text{Hom}_{\text{MF}_S^{e*}}({\mathcal M}_1, {\mathcal M}_2) = \text{Hom}_{\text{MF}_S^{e}}({\mathcal M}_1, {\mathcal M}_2)/R({\mathcal M}_1, {\mathcal M}_2)\) where \(R({\mathcal M}_1, {\mathcal M}_2)\) consists of the morphisms of \(\text{MF}_S^e\) of the form \({\mathcal M}_1 \to {\mathcal M}_1^{\text{mult}} \to {\mathcal M}_2^{\text{et}} \to {\mathcal M}_2\). The main theorem of the paper is the following:
Theorem. There is an antiequivalence of categories
\[
{\mathcal F}_{O_0}^{O*} : \text{Gr}_{O_0}^* \to \text{MF}_S^{e*}.
\]
finite flat commutative group schemes; characteristic \(2\); filtered module Group schemes, Galois theory, Algebraic moduli of abelian varieties, classification Group schemes of period 2 | 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 \(M\) be a smooth algebraic variety over a field \(k\), algebraically closed and of characteristic zero. Let \(X\subset M\) be a hypersurface, i.e., a closed subscheme locally defined by one equation. Then the singular locus \(Y\) of \(X\) has a natural scheme structure defined by the jacobian ideal.
The author defines a class \(\mu_{\mathcal L}(Y)\) in the Chow group of \(Y\), depending on the data \((M,X)\): he then shows that the class only depends on \(Y\) and on the line bundle \({\mathcal L}:={\mathcal O}_M(X)|_Y\), and gives explicit methods for computing it in the case where \(Y\) is itself smooth.
In \S 2 it is proven that \(\deg(\mu_{\mathcal L}(Y))\) is Parusiński's generalized Milnor number; the class is then used to study local properties near \([X]\) of the dual variety of \((M,{\mathcal O}_M(X))\), i.e., the variety of sections of \({\mathcal O}_M(X)\) defining singular hypersurfaces, recovering and partially extending previous results of Ein, Holme, Landman, Parusiński and Zak.
In \S 3 it is shown that the \(\mu\) class imposes strong restrictions on which schemes \(Y\) can appear as singular subschemes of a hypersurface. hypersurface singularity; Milnor number; jacobian ideal; Chow group Paolo Aluffi, ``On the singular schemes of hypersurfaces'', Duke Math. J.80 (1995) no. 2, p. 325-351 Hypersurfaces and algebraic geometry, Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes) Singular schemes of 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 Given \(c \in \mathbb Q\) and a positive integer \(N\), let \(f_c\) be the endomorphism of the affine line \(\mathbb A^1_{\mathbb Q}\) defined by \(f_c (x) = x^2 +c\), and let \(f^N_c\) be the \(N\)-fold composition of \(f_c\). If \(f^{-N}_c\) denotes the \(N\)-fold preimage, the set of rational preimage of \(a \in \mathbb A^1 (\mathbb Q)\) is given by
\[
\bigcup_{N \geq 1} f^{-N}_c (a) (\mathbb Q) = \{ x_0 \in \mathbb A^1 (\mathbb Q) \mid f^N_c (x_0) = a \; \text{ for some } \; a \in \mathbb A^1 (\mathbb Q) \} .
\]
This paper is concerned with the problem of bounding the number of rational points that eventually landing at the origin after iteration. Subject to the validity of the Birch-Swinnerton-Dyer conjecture and some other related conjectures for the \(L\)-series of a special abelian variety and using a number of modern tools for locating rational points on higher genus curves, the authors prove that the maximum number of rational iterated preimages is six. They also provide further insight into the geometry of the preimage curves. quadratic dynamical systems; arithmetic geometry; rational points Faber, Xander; Hutz, Benjamin: On the number of rational iterated pre-images of the origin under quadratic dynamical systems, (2008) Rational points, Arithmetic aspects of modular and Shimura varieties, Computer solution of Diophantine equations, Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets On the number of rational iterated preimages of the origin under quadratic dynamical 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 In the paper under review, the authors study schemes supported on the singular loci of hyperplane arrangements in \(\mathbb{P}^{n}_{\Bbbk}\), where \(\Bbbk\) is a field of characteristic zero. Let \(\mathcal{A}\) be a hyperplane arrangement in \(\mathbb{P}^{n} := \mathbb{P}^{n}_{\Bbbk}\) containing \(d\) hyperplanes and let \(F\) be a product of linear forms defining hyperplanes in \(\mathcal{A}\), and \(J\) be the Jacobian ideal of \(F\). The ideal \(J\) has height two. Let \(E\) be the syzygy module of \(J\), so we have an exact sequence \[0\rightarrow E \rightarrow \bigoplus_{i=1}^{n+1}R(1-d) \rightarrow J \rightarrow 0,\] where \(R = \Bbbk[x_{0},\dots,x_{n}]\), and let \[J = \mathfrak{q}_{1} \cap \dots \cap \mathfrak{q}_{r}\] be a primary decomposition of \(J\). Denote by \(\mathfrak{p}_{i}\) the associated prime of \(\mathfrak{q}_{i}\). When all the \(\mathfrak{p}_{i}\) have the same height, we say that \(J\) is unmixed. In the present paper, the authors are interested in three unmixed ideals that arise somehow naturally from \(J\). First of all, the authors consider the intersection of the codimension two ideals in a primary decomposition of \(J\), which will be denote by \(J^{\mathrm{un}}\). The second ideal that the authors study is the radical of \(J\), denoted by \(\sqrt{J}\), which is just the intersection of the associated primes of \(J\). The third ideal is less natural, i.e., the authors consider fatten components of \(\sqrt{J}\). The first main result of the paper under review can be formulated as follows.
Theorem A. Let \(\mathcal{A}\) be a hyperplane arrangement in \(\mathbb{P}^{n}\) defined by a product of linear forms \(F\). Let \(J, \sqrt{J}, J^{\mathrm{un}}\) be the ideals defined above. Assume that no linear factor of \(F\) is in the associated prime for any two non-reduced components of \(J^{\mathrm{un}}\). Then both \(R/J^{\mathrm{un}}\) and \(R/\sqrt{J}\) are Cohen-Macaulay.
As an application, the authors translate this result to graphic arrangements, namely let \(G\) be a graph and we assume that no two \(3\)-cycles of \(G\) share an edge, then for the associated graphic arrangement \(\mathcal{A}_{G}\) and the Jacobian ideal \(J\), \(R/\sqrt{J}\) and \(R/J^{\mathrm{un}}\) are Cohen-Macaulay.
From now on we assume that \(n=3\). In the case of a curve \(C \in \mathbb{P}^{3}\) there is a natural way to measure the failure of \(R/I_{C}\) to be Cohen-Macaulay degree by degree, and this is the so-called Hartshorne-Rao module of \(C\), denote here by \(M(C)\). We denote by \(C_{F}^{\mathrm{un}}\) the scheme defined by \(J^{\mathrm{un}}\), invoking the form \(F\) defining the arrangement of \(\mathcal{A}\). This scheme \(C_{F}^{\mathrm{un}}\) is the equidimensional top dimensional part of \(C_{F}\), removing components of higher codimension. Similarly, one denotes by \(C_{F}^{\mathrm{red}}\) the support of \(C_{F}^{\mathrm{un}}\) which is defined by \(\sqrt{J}\). In this setting, the authors show the following theorem.
Theorem B. Let \(r\geq 1\) be a positive integer. Then
1) there exists a positive integer \(N\) and a product of linear forms \(F\), defining an arrangement \(\mathcal{A}_{F}\) in \(\mathbb{P}^{3}\) such that \[\dim \, M(C_{F}^{\mathrm{un}})_{N} = r\] and all other components of \(M(C)_{F}^{\mathrm{un}}\) are zero;
1) there exists a positive integer \(N'\) and a product of linear forms \(F'\), defining an arrangement \(\mathcal{A}_{F'}\) in \(\mathbb{P}^{3}\) such that \[\dim \, M(C_{F'}^{\mathrm{red}})_{N'} = r\] and all other components of \(M(C_{F'}^{\mathrm{red}})\) are zero;
3) for each \(h\geq 1\) we can replace \(N\) by \(N+h\) and find a polynomial \(G\) so that \[\dim \, M(C_{G}^{\mathrm{un}})_{N+h} = r\] and all other components of \(M(C_{G}^{\mathrm{un}})\) are zero. The curve \(C_{G}^{\mathrm{un}}\) is in the same even liaison class as \(C_{F}^{\mathrm{un}}\). The analogous result for \(C^{\mathrm{red}}\) also holds. hyperplane arrangements; ideals; freeness; Cohen-Macaulay algebras Configurations and arrangements of linear subspaces, Polynomial rings and ideals; rings of integer-valued polynomials Schemes supported on the singular locus of a hyperplane arrangement 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 We study interpolation by Grassmannian Schubert polynomials (Schur functions). We prove versions of the Sturmfels-Zelevinsky formula for the product of the maximal minors of rectangular matrices corresponding to elementary symmetric functions and Schur functions, and deduce from them generalizations of formulae for the Cauchy-Vandermonde determinant and Cauchy's formula for Schur functions. We define generalizations of higher Bruhat orders whose elements encode connected components of configuration spaces, and also generalizations of discriminantal Manin-Schechtman arrangements. Schur functions; symmetric functions; Cauchy-Vandermonde determinant; Cauchy's formula; Manin-Schechtman arrangements. Symmetric functions and generalizations, Configurations and arrangements of linear subspaces, Determinants, permanents, traces, other special matrix functions, Numerical interpolation Symmetric function interpolation and alternating higher Bruhat orders. | 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 provide here an infinite family of finite subgroups \(\{G_n\subset\text{SL}(\mathbb{C}\}_{n\geq 2}\) for which the \(G\)-Hilbert scheme \(G_n\)-Hilb \(\mathbb{A}^n\) is a crepant resolution of \(\mathbb{A}^n/G_n\), via the Hilbert-Chow morphism. The proof is based on an explicit description of the toric structure of \(G_n\)-Hilb \(\mathbb{A}^n\), \(n\geq 2\), in terms of Nakamura's \(G_n\)-graphs. DOI: 10.1016/j.crma.2006.11.033 Parametrization (Chow and Hilbert schemes), Group actions on varieties or schemes (quotients), Global theory and resolution of singularities (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies Smooth toric \(G\)-Hilbert schemes via \(G\)-graphs | 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 first part of the paper, we present two criteria to characterise lexicographic sets among Borel sets: one criterion by two combinatorial invariants of a Borel set, the other by an extremal property of a packing problem. In the second part, we apply these results to prove the simple-connectedness of certain Hilbert schemes by Gröbner basis theory. lexicographic Borel set; Hilbert schemes; Gröbner basis [Mal97] D. Mall, Characterizations of lexicographic sets and simply-connected Hilbert schemes, In: Proceedings of AAECC-12, Lecture Notes in Computer Science, Vol. 1255, Springer Verlag, 1997, pp. 221--136. Parametrization (Chow and Hilbert schemes), Combinatorics of partially ordered sets, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) Characterisations of lexicographic sets and simply-connected 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 \(E\) be an elliptic curve and let \(V\) be the vector bundle of rank \(2\) obtained as a non-trivial extension \(0\to O_E\to
V\to O_E\to 0\). Consider the ruled surface \(S=\mathbb P(E)\). The author studies the interpolation problem with multiplicities,
with respect to the line bundle \(L=F+\ell D\) on \(S\), where \(F\) is the fiber over a fixed point and \(D\) is the section induced by the embedding
\(O_E\to V\) in the
exact sequence above. The behavior of general divisors on ruled surfaces with respect to the interpolation problem is well known to be
often quite special, even when one imposes just one single point of multiplicity \(m\geq 3\). The author proves that sets
\(m_1P_1+\dots+m_nP_n\), with \(P_1,\dots, P_n\in S\) general points, impose the expected number of conditions to the divisor \(L\) if and only if the
same is true for \(n=1\) (i.e. for a single multiple point). Then, the author gives examples, in positive characteristic,
of ruled surfaces \(S\) as above for which a single multiple point fails to impose the expected number of conditions to
the divisor \(L=F+\ell D\). ruled surfaces; interpolation Divisors, linear systems, invertible sheaves, Elliptic curves Elliptic surfaces and linear systems with fat 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 a previous paper by the first author and \textit{M. Manaresi} [cf. Rev. Roum. Math. Pures Appl. 38, No. 7-8, 569-578 (1993; Zbl 0807.14003)] the so called \(K\)-rational components of the Stückrad-Vogel intersection cycle [see e.g. \textit{W. Vogel}, ``Lectures on results on Bezouts's theorem'' (Tata Lect. Math. Phys., Math. 74 (1984; Zbl 0553.14022)] were characterized by the maximality of the analytic spread. In the present paper explicit examples of intersections of subvarieties of a projective space are computed using the above characterization, the computation being based on MACAULAY and CoCoA programs. These examples are related to open questions in the intersection theory. reduction ideal; Stückrad-Vogel intersection cycle; analytic spread; MACAULAY; CoCoA R. Achilles and D. Aliffi. On the computation of the Stfickrad-Vogel intersection cycle by analytic spread and minimal reductions.Bull. Soc. Math. Belg. (1993), to appear Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics, Computational aspects of higher-dimensional varieties On the computation of the Stückrad-Vogel cycle by analytic spread and minimal reductions | 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{R. Hartshorne} [Publ. Math., Inst. Hautes Étud. Sci. 29, 5-48 (1966; Zbl 0171.41502)] showed that the full Hilbert scheme for projective subschemes with a fixed Hilbert polynomial is connected. Often one studies certain subsets of the Hilbert scheme which parametrize subschemes satisfying a certain property. For example, one can consider the Hilbert scheme \(H(d,g)\) of locally Cohen-Macaulay curves in the projective space \(\mathbb{P}^3\) of fixed degree \(d\) and genus \(g\). In this case, it is not known whether \(H(d,g)\) is connected. In the present paper the author shows that \(H(3,g)\) is connected. This is achieved by giving a classification of the locally Cohen-Macaulay space cubics of genus \(g\), determining the irreducible components of \(H(3,g)\), and giving certain specializations to show connectedness. Curiously, there are curves which lie in the closure of each irreducible component of \(H(3,g)\). locally Cohen-Macaulay space curve; Rao module; Koszul module; Hilbert scheme; connectedness [MDP1]\textsc{M. Martin-Deschamps--D. Perrin},\textit{Sur la classification des courbes gauches}, Astérisque, Vol.\textbf{184-185}, 1990. Parametrization (Chow and Hilbert schemes), Plane and space curves, Topological properties in algebraic geometry The Hilbert schemes of degree three 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 Computing elliptic-curve scalar multiplication is the most time consuming operation in any elliptic-curve cryptosystem. In the last decades, it has been shown that pre-computations of elliptic-curve points improve the performance of scalar multiplication especially in cases where the elliptic-curve point \(P\) is fixed. In this paper, we present an improved fixed-base comb method for scalar multiplication. In contrast to existing comb methods such as proposed by \textit{C. H. Lim} and \textit{P. J. Lee} [Advances in cryptology -- CRYPTO 1994, Lect. Notes Comput. Sci. 839, 95--107 (1994; Zbl 0939.94537)] or \textit{W.-J. Tsaur} and \textit{C.-H. Chou} [Appl. Math. Comput. 168, No. 2, 1045--1064 (2005; Zbl 1076.94017)], we make use of a width-\(\omega \) non-adjacent form representation and restrict the number of rows of the comb to be greater or equal \(\omega \). The proposed method shows a significant reduction in the number of required elliptic-curve point addition operation. The computational complexity is reduced by 33 to 38,\% compared to Tsaur and Chou method even for devices that have limited resources. Furthermore, we propose a constant-time variation of the method to thwart simple-power analysis attacks. elliptic-curve cryptosystem; scalar multiplication; Lim-Lee method; Tsaur-Chou method; non-adjacent form; width-\(\omega \) NAF Mohamed, N. A. F.; Hashim, M. H. A.; Hutter, M.: Improved fixed-base comb method for fast scalar multiplication. Africacrypt 2012 7374, 342-359 (2012) Cryptography, Applications to coding theory and cryptography of arithmetic geometry, Mathematical problems of computer architecture Improved fixed-base comb method for fast scalar multiplication | 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 \(X\) and \(Y\) are closed subschemes of \(\mathbb{P}^n\) with respective ideal sheaves \(\mathcal{I}_X\) and \(\mathcal{I}_Y\), then one can define the residual scheme of \(X\) with respect to \(Y\). This subscheme is denoted by \(\mathrm{res}_Y X\) and is determined by the condition that its ideal sheaf is given by the colon ideal sheaf \((\mathcal{I}_X : \mathcal{I}_Y)\).
In the article under review, the authors fix closed subschemes \(X,Y \subseteq \mathbb{P}^n\) and are interested in the schemes \(\text{res}_Y X\) and \(\text{res}_{Y \cap V}(X \cap V)\) for \(V \subseteq \mathbb{P}^n\) a linear subspace. They prove two theorems.
To describe the first theorem, let \(H\subset \mathbb{P}^n\) denote a hyperplane and let \(I_X\), \(I_Y\), and \(I_H\) denote the saturated homogeneous ideals of \(X\), \(Y\), and \(H\) respectively. The authors prove that if \(H\) is general and if the ideals \(I_X+I_H\) and \((I_X : I_Y)+I_H\) are saturated, then the Castelnuovo-Mumford regularity of \(\text{res}_{Y \cap H}(X\cap H)\) and \(\text{res}_Y X\) are equal.
The second theorem concerns the concept of uniform position for a set of points in \(\mathbb{P}^n\). To state it, suppose that \(V \subseteq \mathbb{P}^n\) is a linear subspace of codimension \(r\) determined by hyperplanes in general position with respect to \(X\) and \(Y\). The authors prove that if \(\text{res}_Y X\) is irreducible of dimension \(r\), then the closed subscheme \(\text{res}_{Y\cap V}(X\cap V)\) is a set of points in uniform position. graded modules; schemes; regularity; uniform position principle Schemes and morphisms, Syzygies, resolutions, complexes and commutative rings, Graded rings, Divisors, linear systems, invertible sheaves On the regularity of the residual scheme | 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 \(p\) be an odd prime number, \(K\) a complete, discrete valuation field of mixed characteristic \((0,p)\) with perfect residue class field \(k\), such that \(K\) is finite and totally ramified over the quotient field of the Witt ring of \(k\). Let \(K_\infty\) be a maximal totally ramified \(p\)-extension of \(K\) inside its algebraic closure \(\overline K\) and \(G_{K_\infty}\) its absolute Galois group. Then one has a field \(\mathcal X \simeq k((u))\) with valuation ring \(\mathcal O_{\mathcal X} \simeq k[[u]]\), and with absolute Galois group \(G_{\mathcal X} \simeq G_{K_\infty}\), where this latter isomorphism is compatible with the upper ramification subgroups.
Now let \(\mathcal G\) and \(\mathcal H\) be finite flat commutative group schemes over \(\mathcal O_K\) and \(\mathcal O_{\mathcal X}\), resp., which are killed by \(p\) and which correspond to each other under the correspondence developed by C. Breuil and M. Kisin. For any Breuil-Kisin module \(\mathfrak M\) (i.e., a free \(k[[u]]\)-module with certain properties) one then obtains an isomorphism of \(G_{K_\infty}\)-modules
\[
\mathcal G (\mathfrak M) (\mathcal O_{\overline K}) \big | _{G_{K_\infty}} \to \mathcal H (\mathfrak M) (\mathcal O_{\mathcal X^{\text{sep}}}).
\]
The main result of this paper shows that this isomorphism induces isomorphisms of the upper and the lower ramification subgroups of these modules. So studying ramification of the group scheme \(\mathcal G\) over \(\mathcal O_K\) can be translated to studying the problem in the equal characteristic case. The proof uses Cartier duality to connect the upper ramification filtration with the lower one. Cartier duality; Breuil-Kisin modules; Abbes-Saito ramification Hattori, S, Ramification correspondence of finite flat group schemes over equal and mixed characteristic local fields, J. Number Theory, 132, 2084-2102, (2012) Group schemes, Ramification and extension theory, Witt vectors and related rings Ramification correspondence of finite flat group schemes over equal and mixed characteristic local fields | 0 |
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