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The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let W be an open irreducible subset of the Hilbert scheme of \({\mathbb{P}}^ r\) parametrizing smooth irreducible curves of degree n and genus g and \(\pi\) : \(W\to {\mathcal M}_ g\) the natural map to the moduli space of curves of genus g. W is said to have the expected number of moduli if \(\dim(\pi (W))=\min \{3g-3,3g-3+\rho \}\), where \(\rho =\rho (g,n,r)\) is the Brill-Noether number.
The purpose of the paper is the construction of such families of curves for negative \(\rho\) 's, extending the previous existence range shown by \textit{E. Sernesi} [Invent. Math. 75, 25-57 (1984; Zbl 0541.14024)]. For \(r\geq 12\) or \(r=10\), it is shown the existence of families having the expected number of moduli in the range \(-(5r+\epsilon)g/(4r+9- \epsilon)+f(r)\leq \rho \leq 0\) where \(\epsilon\) is defined by \((r- \epsilon)/3=\lceil (r-1)/3\rceil\) and f(r) is a rational function of r asymptotically like \(5/4r^ 2\) (for the explicit expression see section 1 of the paper). - Note that the range where the existence problem makes sense is \(\rho \geq -3g+3\) and that the above result gives an affirmative answer roughly for \(\rho \geq -5(g-r^ 2)/4.\)
The proof has two parts:
(a) A careful study of the normal bundle of general nonspecial curves in \({\mathbb{P}}^ r\). This is accomplished by studying the special case of nonspecial curves lying in suitable rational normal surface scrolls;
(b) an inductive proof starting from curves \(C\subset {\mathbb{P}}^ r\) whose existence was proved by E. Sernesi [loc. cit.] and attaching general nonspecial curves \(\Gamma\) at \(r+4\) general points. The results in (a) are then used to show that this construction is possible and that the curves \(C\cup \Gamma\) are flatly smoothable (the latter by standard deformation theory techniques developed by E. Sernesi [loc. cit.]). Hilbert scheme; expected number of moduli; Brill-Noether number Lopez, AF, On the existence of components of the Hilbert scheme with the expected number of moduli, Math. Ann., 289, 517-528, (1991) Families, moduli of curves (algebraic), Parametrization (Chow and Hilbert schemes) On the existence of components of the Hilbert scheme with the expected number of moduli | 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 observe that there exists a Białynicki-Birula decomposition of the Hilbert scheme \(\text{Hilb}^P_n\) such that the cells are homeomorphic to Gröbner strata of homogeneous ideals with fixed initial ideal. Using such a decomposition, we show that \(\text{Hilb}^P_n\) is singular at a monomial scheme if the corresponding Gröbner stratum is singular at \(J\). Hilbert scheme; Białynicki-Birula decomposition; Gröbner bases Parametrization (Chow and Hilbert schemes), Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Polynomial rings and ideals; rings of integer-valued polynomials Computable Białynicki-Birula decomposition 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 We construct a public-key cryptosystem based on an NP-complete problem in algebraic geometry. It is a problem of finding sections on fibered algebraic surfaces; in other words, we use a solution to a system of multivariate equations of high degrees. Our cryptosystem is a revised version of the algebraic surface cryptosystem (ASC) we constructed earlier. We revise its encryption algorithm to avoid known attacks. Further, we show that the key size of our cryptosystem is one of the shortest among those of post-quantum public-key cryptosystems known at present. public-key cryptosystem; algebraic surface; section finding problem Akiyama, Koichiro; Goto, Yasuhiro; Miyake, Hideyuki, An algebraic surface cryptosystem, 425-442, (2009), Berlin, Heidelberg Cryptography, Computational aspects of algebraic surfaces, Effectivity, complexity and computational aspects of algebraic geometry An algebraic surface cryptosystem | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A tropical curve \(\Gamma \) is a metric graph with possibly unbounded edges, and tropical rational functions are continuous piecewise linear functions with integer slopes. We define the complete linear system \(|D|\) of a divisor \(D\) on a tropical curve \(\Gamma \) analogously to the classical counterpart. We investigate the structure of \(|D|\) as a cell complex and show that linear systems are quotients of tropical modules, finitely generated by vertices of the cell complex. Using a finite set of generators, \(|D|\) defines a map from \(\Gamma \) to a tropical projective space, and the image can be modified to a tropical curve of degree equal to deg\((D)\) when \(|D|\) is base point free. The tropical convex hull of the image realizes the linear system \(|D|\) as a polyhedral complex. We show that curves for which the canonical divisor is not very ample are hyperelliptic. We also show that the Picard group of a \({\mathbb{Q}}\)-tropical curve is a direct limit of critical groups of finite graphs converging to the curve. tropical curves; divisors; linear systems; canonical embedding; chip-firing games; tropical convexity , Curves in algebraic geometry, Divisors, linear systems, invertible sheaves, Games on graphs (graph-theoretic aspects) Linear systems on 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 A tropical curve \(\Gamma \) is a metric graph with possibly unbounded edges, and tropical rational functions are continuous piecewise linear functions with integer slopes. We define the complete linear system \(|D|\) of a divisor \(D\) on a tropical curve \(\Gamma \) analogously to the classical counterpart. We investigate the structure of \(|D|\) as a cell complex and show that linear systems are quotients of tropical modules, finitely generated by vertices of the cell complex. Using a finite set of generators, \(|D|\) defines a map from \(\Gamma \) to a tropical projective space, and the image can be modified to a tropical curve of degree equal to \(\deg(D)\). The tropical convex hull of the image realizes the linear system \(|D|\) as a polyhedral complex. tropical curves; divisors; linear systems; canonical embedding; chip-firing games; tropical convexity Haase, C.; Musiker, G.; Yu, J., Linear systems on tropical curves, Math. Z., 270, 3-4, 1111-1140, (2012) Linear systems on 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 Given a polytope \(\sigma\subset \mathbb{R}^{m}\), its characteristic distribution \(\delta_{\sigma}\) generates a \(D\)-module which we call the characteristic \(D\)-module of \(\sigma\) and denote by \(M_\sigma\). More generally, the characteristic distributions of a cell complex \(K\) with polyhedral cells generate a \(D\)-module \(M_K\), which we call the characteristic \(D\)-module of the cell complex. We prove various basic properties of \(M_K\), and show that under mild topological conditions on \(K\), the \(D\)-module theoretic direct image of \(M_{K}\) coincides with the module generated by the \(B\)-splines associated to the cells of \(K\) (considered as distributions). We also give techniques for computing \(D\)-annihilator ideals of polytopes. \(B\)-splines; \(D\)-modules Ketil Tveiten, Period integrals and mutation, arXiv:1501.05095 [math.AG], 2015. Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Sheaves of differential operators and their modules, \(D\)-modules, \(n\)-dimensional polytopes \(B\)-splines, polytopes, and their characteristic \(D\)-modules | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \((R,m)\) be a Noetherian local ring, and let \(M\) denote a finitely generated \(R\)-module. A Buchsbaum \(R\)-module \(M\) is maximal if \(M\) has the same dimension as \(R\), and \(R\) has finite Buchsbaum representation type if there are only finitely many isomorphism classes of indecomposable maximal Buchsbaum \(R\)-modules.
In this paper after a short overview on the subject the author studies the case of curves; he proves under the assumptions that \(R\) is complete and \(R/m\) is infinite that the following conditions are equivalent:
(i) \(R\) has finite Buchsbaum representation type.
(ii) \(e(R) \leq 2\), \(\nu (R) \leq 2\) and the ring \(R/H^ 0_ m (R)\) is reduced \((\nu (R)\) is the embedding dimension).
The author obtains this theorem after the proof of the following: (same hypothesis) \(R\) is a Cohen-Macaulay ring of finite Buchsbaum representation type if and only if \(R\) is a reduced ring of \(e(R) \leq 2\). maximal Buchsbaum modules; Buchsbaum type; Noetherian local ring Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Singularities of curves, local rings, Multiplicity theory and related topics, Commutative Noetherian rings and modules, Algebraic functions and function fields in algebraic geometry Curve singularities of finite Buchsbaum-representation type | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The subject of the paper under review is resolution of singularities in algebraic geometry. The main result is a stacky resolution theorem in log geometry. To state it, let \(S\) be the spectrum of a field, endowed with the trivial log structure. Let \(X\) be a fine saturated (fs) log scheme which is log smooth over \(S\). Then the result is that there exists a smooth log smooth Artin stack \(\mathfrak X\) over \(S\) and a morphism \(f: \mathfrak X \to X\) over \(S\) such that \(f\) is a good moduli space morphism in the sense of \textit{J. Alper} [Ann. Inst. Fourier 63, No. 6, 2349--2402 (2013; Zbl 1314.14095)] and the base change of \(f\) to the smooth locus of \(X\) is an isomorphism. Furthermore, \(\mathfrak X\) admits a moduli description in terms of log geometry. All of this generalizes the case when the charts of \(X\) are all simplicial toric varieties, which was established by \textit{I. Iwanari} [Publ. Res. Inst. Math. Sci. 45, No. 4, 1095--1140 (2009; Zbl 1203.14058)]. The paper gives two applications of the main result: a generalization of the Chevalley-Shephard-Todd theorem to the case of diagonalizable group schemes; and a generalization of work of \textit{L. Borisov, L. Chen} and \textit{G. Smith} [J. Am. Math. Soc. 18, No. 1, 193--215 (2005; Zbl 1178.14057)], \textit{B. Fantechi} et al. [J. Reine Angew. Math. 648, 201--244 (2010; Zbl 1211.14009)], and \textit{I. Iwanari} [Compos. Math. 145, No. 3, 718--746 (2009; Zbl 1177.14024)] on toric Deligne-Mumford stacks and stacky fans to the case of certain toric Artin stacks produced in this paper. log structure; Chevalley-Shephard-Todd; toric stack; stacky fan; resolution of singuarities Matthew Satriano, Canonical Artin stacks over log smooth schemes, Math. Z. 274 (2013), no. 3-4, 779 -- 804. Stacks and moduli problems, Toric varieties, Newton polyhedra, Okounkov bodies Canonical Artin stacks over log smooth 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 JFM 06.0358.06 rational curves Questions of classical algebraic geometry, Plane and space curves On systems of points on rational 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 \(f:(\mathbb{K}^{n},0)\rightarrow (\mathbb{K},0)\) be an analytic function germ, where \(\mathbb{K}=\mathbb{R}\) or \(\mathbb{K=C}\). The Fukui numerical set \(A(f)\) of \(f\) is defined by
\[
A(f)=\{\text{ord}f(\gamma (t))\in \mathbb{N\cup }\{\infty \}:\text{ } \gamma :(\mathbb{K},0)\rightarrow (\mathbb{K}^{n},0)\text{ is an analytic arc }\}.
\]
\textit{T. Fukui} [Compos. Math. 105, 95--108 (1997; Zbl 0873.32008)] proved in real case (\(\mathbb{K}=\mathbb{R}\)) that \(A(f)\) is an invariant under the blow-analytic equivalence. The authors:
1. discuss invariance of \(A(f)\) in complex case under topological equivalence (it is a counterpart of the blow-analytic equivalence of real case to the complex one). They prove that \(A(f)\) is such an invariant for \(n=2,\)
2. find \(A(f)\) for various families of function germs,
3. discuss relations between real blow-analiticity and complex topological triviality for families of function germs. They show that if the complexification of a family of two variable real analytic function germs is topological trivial then the original real family is blow-analytically trivial,
4. pose several questions concerning related problems. Fukui numerical set; blow-analytic equivalence; topological equivalence; Kuo-Lu tree model; Seifert form Local complex singularities, Singularities in algebraic geometry, Equisingularity (topological and analytic), Real algebraic and real-analytic geometry Some questions on the Fukui numerical set for complex function germs | 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 There have been several generalizations of the concept of a fundamental group for algebraic varieties. Grothendieck defined the étale fundamental group by considering finite étale covers of the variety. In the complex case, under suitable hypotheses, this turns out to be the profinite completion of the topological fundamental group. Later, Nori defined the fundamental group scheme of a scheme \(X\) over a perfect field \(k\) as the affine group scheme associated to the neutral Tannaka category of essentially finite vector bundles over \(X\). If \(k\) is algebraically closed, this has the étale fundamental group as its quotient, and is in fact equal to it if \(k\) has characteristic 0. In addition to these two fundamental groups, Simpson defined the universal complex pro-algebraic group as the inverse limit of the directed system of representations \(\rho: \pi_1(X,x) \to G\) for complex algebraic groups \(G\), such that the image of \(\rho\) is Zariski-dense in \(G\). This group can also be described as being associated to the neutral Tannaka category of semistable Higgs bundles with vanishing rational Chern classes. \textit{I. Biswas}, \textit{A. J. Parameswaran} and \textit{S. Subramanian} [Duke Math. J. 132, No. 1, 1--48 (2006; Zbl 1106.14032)] defined the S-fundamental group scheme of a smooth, projective curve \(X\) over an algebraically closed field as the affine group scheme associated to the neutral Tannaka category of strongly semistable vector bundles of degree zero over \(X\).
In the paper under review, the author seeks to study the S-fundamental group scheme for a general complete, connected, reduced \(k\)-scheme \(X\). This is defined as the affine group scheme associated to the neutral Tannaka category of strongly semistable vector bundles with vanishing Chern classes over \(X\), directly generalizing the definition given in Biswas. We note that, for a smooth and projective variety \(X\), this category can also be described as the category of numerically flat vector bundles. Here, a vector bundle \(E\) on \(X\) is called numerically flat if both \(E\) and \(E^*\) are nef. For \(G\) a connected reductive \(k\)-group and \(E_G\) a numerically flat principal \(G\)-bundle on \(X\), the monodromy group scheme of \(E_G\) is defined and it is shown that this is the smallest subgroup scheme of \(G\) to which \(E_G\) has a reduction of structure group. Next, the author discusses basic properties of the S-fundamental group scheme. Among these, we have that the S-fundamental group scheme of the projective space is trivial, and the S-fundamental group scheme is well-behaved under blow-ups and base change. The author also conjectures that the S-fundamental group scheme of a product of complete \(k\)-varieties is the product of the corresponding S-fundamental group schemes. (The author proves this result in a sequel to this paper.) After two vanishing theorems for the cohomology of strongly semistable sheaves with vanishing Chern classes, the author proves Lefschetz-type theorems for the S-fundamental group scheme, with similar statements for Nori's and étale fundamental groups deriving as a corollary. fundamental group; positive characteristic; numerically flat bundles; Lefschetz type theorems Langer, A., On the \(S\)-fundamental group scheme, Ann. Inst. Fourier, 61, 2077-2119, (2011) Homotopy theory and fundamental groups in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Group schemes On the S-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 A locally recoverable (LRC) code is a code over a finite field \(\mathbb{F}_q\) such that any erased coordinate of a codeword can be recovered from a small number of other coordinates in that codeword. As an answer to multiple device failures case the concept of localities \((r, \delta)\) of an LRC code is considered. It means the minimum size of a set \(\overline{R}\) of positions so that any \(\delta-1\) erasures in \(\overline{R}\) can be recovered from the remaining \(r\) coordinates in this set to be determined.
The authors construct LRC codes, which are subfield-subcodes of some \(J\)-affine variety codes and compute their localities \((r, \delta)\). They also show that some of the constructed LRC codes have lengths \(n \gg q\) and are \((\delta-1)\)-optimal. LRC codes; subfield subcodes; \(J\)-affine variety codes Geometric methods (including applications of algebraic geometry) applied to coding theory, Linear codes (general theory), Affine geometry, Algebraic coding theory; cryptography (number-theoretic aspects), Applications to coding theory and cryptography of arithmetic geometry Locally recoverable \(J\)-affine variety codes | 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 \(m,n\) be integers and \(\mathcal{Z}_{2}^{m,n}\) be the determinantal variety given by the vanishing of all the \(2\times2\) minors of a generic \(m\times n\) matrix. Let \(\mathcal{Z}_{2,2}^{m,n}\) be the affine variety of first-order jets over \(\mathcal{Z}_{2}^{m,n}\). If \(2<m\leq n\), then \(\mathcal{Z}_{2,2}^{m,n}\) has two irreducible components. One of them is isomorphic to \(\mathbb{A}^{mn}\). The other one is called the principal component, and is denoted by \(Z_0\). Let \(R=\mathbb{F}[x_{i,j},y_{i,j}:1\leq i\leq m, 1\leq j\leq n]\) be the polynomial ring over an algebraically closed field \(\mathbb{F}\) and \(\mathcal{I}=\mathcal{I}_{2,2}^{m,n}\) be the ideal which corresponds to \(\mathcal{Z}_{2,2}^{m,n}\). Let \(\mathcal{I}_0\) be the ideal which corresponds to \(Z_0\). By using combinatorial techniques, the authors compute the Hilbert series of \(R/\mathcal{I}_0\) and prove that this is the square of the Hilbert series of the base variety \(\mathcal{Z}_{2}^{m,n}\). As a consequence, they determine the \(a\)-invariant of \(R/\mathcal{I}_0\) and the Hilbert series of its graded canonical module. They characterize also the property of \(Z_0\) of being Gorenstein. jet schemes; Hilbert series; determinantal varieties Determinantal varieties, Combinatorial aspects of commutative algebra, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series Hilbert series of certain jet schemes of determinantal 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 Der Verfasser beweist analytisch folgenden Satz:
``Zieht man von einem Punkte \(A\) an eine Plancurve dritter Ordnung vierter Klasse die 4 Tangenten, und verbindet die ihnen angehörigen 4 Berührungspunkte mit dem Doppelpunkte, so ist der Ort derjenigen Punkte \(A\), für welche die 4 Verbindungsstrahlen harmonisch sind, deine Curve dritter Ordnung, welche die ursprüngliche Curve in den drei Wendepunkten, also sonst noch in 6 auf einem Kegelschnitte liegenden Punkten schneidet.'' An diesen Satz werden einige verwandte angeschlossen. third order curves; fourth order curves; harmonic points; rational curves; conics Plane and space curves, Projective techniques in algebraic geometry, Projective analytic geometry, Rational and birational maps Harmonic point systems of third and fourth order rational 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 \(X\) be a variety over a field \(k\) and let \(X[T]:=X[t_1,\dots , t_r]\) denote its polynomial extension. \(X\) is called \(K_i\)-regular if the natural map \(K_i(X)\to K_i(X[T])\) is an isomorphism for all \(r\geq 1\), where \(K_i(X)\) is the \(i\)th-stable homotopy group of the non-connective spectrum \(K(X)\) of perfect complexes on \(X\). Given an abelian group \(A\), denote by \(A\{p\}\) its \(p\)-primary torsion group. The author proves the following Theorem: Let \(k\) be a field of positive characteristic \(p\), assume that resolution of singularities holds over \(k\) and let \(X\) be a variety of dimension \(d\) over \(k\). Then
{\parindent5mm
\begin{itemize}\item[1)] \(K_i(X,\mathbb{Z}/n)=0\) for \(i<-d-1\) and for all \(n\geq 1\). \item[2)] \(K_{-d-2}(X[T])\) is a divisible group and \(K_{-d-2}(X[T])=K_{-d-2}(X[T])\{ p\}\).
\item[3)] \(K_i(X)=0\) and \(X\) is \(K_i\)-regular for \(i<-d-2\).
\end{itemize}} singular varieties; topological cyclic homology Krishna, A, On the negative \(K\)-theory of schemes in finite characteristic, J. Algebra, 322, 2118-2130, (2009) \(K\)-theory of schemes, Negative \(K\)-theory, NK and Nil, Applications of methods of algebraic \(K\)-theory in algebraic geometry On the negative \(K\)-theory of schemes in finite characteristic | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X,Y\) be reduced and irreducible compact complex spaces and \(S\) be the set of all isomorphism classes of reduced and irreducible compact complex spaces \(W\) such that \(X\times Y\simeq X\times W\). It is proved that \(S\) is at most countable.
This result is then applied to show that \(S(X):=\) the set of all \(W\) such that \(X\times X^\sigma\simeq W\times W^\sigma\) is at most countable, where \(A^\sigma\) denotes the complex conjugate of any variety \(A\). Compact complex \(n\)-folds, Surfaces and higher-dimensional varieties, Real algebraic and real-analytic geometry Discreteness for the set of complex structures on a real variety | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Efficient construction of long algebraic geometric-codes resulting from optimal towers of function fields is known to be difficult. In the following a tower which is both optimal and unramified after its third level, is investigated in the hope that its simple ramification structure can be exploited in the construction of algebraic geometric-codes. Results are mostly negative, but help clarifying the difficulties in computing bases of Riemann-Roch spaces. Arithmetic theory of algebraic function fields, Applications to coding theory and cryptography of arithmetic geometry, Geometric methods (including applications of algebraic geometry) applied to coding theory An optimal unramified tower of function 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 We introduce and study the Hermitian matrix model with potential \(V_{s,t}(x)=x^{2}/2 - stx/(1 - tx)\), which enumerates the number of linear chord diagrams with no isolated vertices of fixed genus with specified numbers of backbones generated by \(s\) and chords generated by \(t\). For the one-cut solution, the partition function, correlators and free energies are convergent for small \(t\) and all \(s\) as a perturbation of the Gaussian potential, which arises for \(st=0\). This perturbation is computed using the formalism of the topological recursion. The corresponding enumeration of chord diagrams gives at once the number of RNA complexes of a given topology as well as the number of cells in Riemann's modulispaces for bordered surfaces. The free energies are computed here in principle for all genera and explicitly in genus less than four. chord diagrams; matrix model; Riemann's moduli space; RNA; topological recursion Jørgen E. Andersen, Leonid O. Chekhov, R. C. Penner, Christian M. Reidys, and Piotr Sułkowski, Topological recursion for chord diagrams, RNA complexes, and cells in moduli spaces, Nuclear Phys. B 866 (2013), no. 3, 414 -- 443. String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Perturbative methods of renormalization applied to problems in quantum field theory, Statistical thermodynamics Topological recursion for chord diagrams, RNA complexes, and cells in moduli spaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The quest for regular models of arithmetic surfaces allows different viewpoints and approaches: using valuations or a covering by charts. In this article, we sketch both approaches and then show in a concrete example, how surprisingly beneficial it can be to exploit properties and techniques from both worlds simultaneously. arithmetic surfaces; regular models; valuations; desingularization Global theory and resolution of singularities (algebro-geometric aspects), Special surfaces, Software, source code, etc. for problems pertaining to algebraic geometry Desingularization of arithmetic surfaces: algorithmic aspects | 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 gives a fairly self-contained introduction to Kato's Euler systems. These Euler systems may be applied to prove finiteness results for elliptic curves, in particular over \(\mathbb Q\). The paper consists of five sections (the second with an appendix on higher \(K\)-theory of modular varieties) and a list of references.
To fix the notations and introduce some basic notions, let \(f:E\rightarrow S\) be an elliptic curve over a (suitable) base scheme \(S\). Write \(e:S\rightarrow E\) for the zero section. Then there is a standard invertible sheaf \(\omega= \omega_{E/S}=f_*\Omega^1_{E/S}=e^*\Omega^1_{E/S}\). This \(\omega\) can be considered as a sheaf on the moduli stack \(\mathcal M\) of elliptic curves. A (meromorphic) modular form of weight \(k\) is then a section of \(\omega^ {\otimes k}_{E/S}\), i.e.\ an element of \(\Gamma(\mathcal M,\omega^{\otimes k})\). Of special importance is the discriminant \(\Delta(E/S)\) which gives a trivialisation of \(\omega^{\otimes 12}_{E/S}\). This fact may be held responsable for the existence of so called \textit{Kato-Siegel} functions:
For an integer \(D\) with \((D,6)=1\) there is exactly one rule \(\vartheta_D\) which associates to each elliptic curve \(E\rightarrow S\) a section \(\vartheta_D^{(E/S)}\in\mathcal O^*(E-\text{ker}[\times D])\) such that \(\vartheta_D^{(E/S)}\) has divisor \(D^2(e)-\text{ker}[\times D]\), where \(\times D\) means multiplication by \(D\). The \(\vartheta_D^{(E/S)}\) are compatible with base change and natural under isogenies. Furthermore, \(\vartheta_{-D}= \vartheta_D\), \(\vartheta_1=1\) and \([\times D]_*\vartheta_D=1\). Over \(\mathbb C\) with \(E/{\mathbb C}\) given by the lattice \({\mathbb Z}+\tau{\mathbb Z}\), \(\text{Im}(\tau)>0\), one has
\[
\vartheta_D^{(E/{\mathbb C})}=(-1)^{{D-1}\over 2}\Theta(u,\tau)^{D^2} \Theta(Du,\tau)^{-1},
\]
where \({\Theta(u,\tau)=q^{1\over 12} (t^{1\over 2}-t^{-1\over 2})\prod_{n>0}(1-q^nt)(1-q^nt^{-1})}\) with \(q=e^ {2\pi i\tau}\), \(t=e^{2\pi i u}\).
Let \(x\in E/S\) be a torsion section of order \(N\) with \((N,D)=1\). The vertical logarithmic derivative \(\text{d}\log_v\vartheta_D\in\Gamma(E-\text{ker}[\times D],\Omega^1_{E/S})\) gives rise to a weight one modular form (Eisenstein series)
\[
{}_D\text{Eis}(x) ={}_D\text{Eis}(E/S,x):=x^*\text{d}\log_v\vartheta_D\in\Gamma(S, x^*\Omega^1_{E/S})=\Gamma(S,\omega_{E/S}).
\]
Then, for \(D\equiv 1\bmod N\), one gets
\[
\text{Eis}(E/S,x):{1\over D^2-D}\cdot{}_D\text{Eis} (E/S,x)\in\Gamma \Biggl(S\otimes{\mathbb Z} \biggl[{1\over D(D-!)} \biggr],\omega\Biggr)
\]
independent of \(D\). Over \(\mathbb C\) one recovers the classical Eisenstein series
\[
\text{Eis}(E/{\mathbb C},x)=\sum_{m_i\in{a_i\over N}+{\mathbb Z}}\left. {1\over(m_1\omega_1+m_2\omega_2)|(m_1\omega_1+m_2\omega_2)|^s}\right|_{s=0} du,
\]
where \(u\) is the variable in the complex plane and where \(x\in E({\mathbb C})- \{e\}\) is the torsion point \((a_1\omega_1+a_2\omega_2)/N\in N^{-1}\Lambda/ \Lambda\), with \(\Lambda\) the lattice defining \(E/{\mathbb C}\).
The second section deals with norm relations in algebraic \(K\)-theory. The Euler system in \(K_2\) of modular curves is constructed. For an elliptic curve \(E/S\) one denotes by \(S(N)(T)\) the level \(N\) structures on \(E\times_ST\), i.e.\ isomorphisms \(\alpha:({\mathbb Z}/N)^2_T{\buildrel\sim\over\longrightarrow}\text{ker}[\times N]_T\) (in a functorial way). For integers \(D,D''\) and a prime \(\ell\) such that \((6\ell,DD'')=1\) one writes \(\vartheta= \vartheta^{(E/S)}_D\) and \(\vartheta''=\vartheta^{(E/S)}_{D''}\). Also one writes \(N_{?/?}\) for the push-forward maps \(\pi_{?/?*}\) on \(K_2\), i.e.\ for a proper map of schemes \(\pi_{?/?}:X''\rightarrow X\) one has \(\pi_{?/?*}:K_2X''\rightarrow K_2X\). If \(S=Y(N)/H\), \(H\subset GL_2 ({\mathbb Z}/N)\), is a modular curve of level prime to \(\ell\), \(E=\mathcal E^ {\text{univ}}\rightarrow S\) the universal elliptic curve, and \(z\), \(z^ {\prime}\) are torsion sections of \(E/S(\ell)\) whose projections onto \(\text{ker}[\times\ell]\) are linearly independent, then
\[
N_{S(\ell)/S}\{\vartheta(z),\vartheta''(z'')\}=(1-T_{\ell} \circ\langle\ell\rangle_*+\ell\langle\ell\rangle_*)\{\vartheta(\ell z), \vartheta''(\ell z''\},
\]
where \(\{.,.\}\) denotes a symbol in \(K_2\), and where \(T_{\ell}\) is a Hecke operator. One may generalize to norm relations for \(\Gamma(\ell^n)\). Also, one may deduce norm relations for products of the Eisenstein series \({}_D\text{Eis}(E/S,x)\) and \({}_D\text{Eis}(E/S,x'')\). One should pass these facts to Galois cohomology. The problem becomes to show that the corresponding cohomology classes are non-trivial (if the corresponding value of the \(L\)-function do not vanish). To this end Kato formulated a general reciprocity theory. This is the subject of the third section.
In the third section the dual exponential map is introduced. Let \(K\) be a finite extension of \({\mathbb Q}_p\) with ring of integers \(\mathfrak o\) and fixed algebraic closure \(\overline{K}\). Denote by \(G_K\) the Galois group \(\text{Gal} (\overline{K}/K)\). and let \(\zeta_{p^n}\) be a primitive \(p^n\)-th root of unity in \(\overline{K}\). Write \(K_n=K(\zeta_{p^n})\) with valuation ring \(\mathfrak o_n\). For a continuous finite-dimensional representation \(V\) of \(G_K\) which is supposed to be \textit{de Rham} in Fontaine's terminology, let \(DR(V)=(B_{\text{dR}}\otimes_{{\mathbb Q}_p} V)^{G_K}\) be the associated filtered \(K\)-vector space with decreasing filtration \(DR^i(V)\) coming from the filtration on \(B_{\text{dR}}\)). Then Kato's dual exponential map is defined as the composite
\[
H^1(K,V)\rightarrow H^1(K,B^0_{\text{dR}}\otimes_{{\mathbb Q}_p}V) =H^1(K,\text{Fil}^0(B_{\text{dR}}\otimes KDR(V)))\simeq DR^0(V).
\]
Apply this to \(V=H^1(Y_{\overline{K}},{\mathbb Q}(p))(1)\) for a smooth \(\mathfrak o\)-scheme \(Y\), which is the complement in a smooth proper \(\mathfrak o\)-scheme \(X\) of a divisor \(Z\) with relatively normal crossings. Write \(H^i_{\text{dR}}(Y/\mathfrak o)=H^i(X,\Omega^{\bullet}_{X/\mathfrak o}(Z))\). One calculates \(DR^0(V)=\text{Fil}^1H^1_{\text{dR}}(Y/\mathfrak o) \otimes_{\mathfrak o}K\). The explicit reciprocity theorem now becomes the commutativity of a diagram starting with \(\displaystyle{\lim_{\buildrel\longleftarrow\over n}(K_2(Y\otimes\mathfrak o_n)\otimes\mu_{p^n} ^{\otimes -1})}\) and going (via the dual exponential) to \(K_m\otimes_{\mathfrak o} \text{Fil}^1H^1_{\text{dR}}(Y/\mathfrak o)\). The proof of this result (in fact, of a theorem of which the reciprocity theorem is a corollary) consumes a lot of pages and uses the Faltings-Fontaine-Hyoda approach to \(p\)-adic Hodge theory.
In the fourth section a sketch is given of Kato's Rankin-Selberg integral method to compute the projection of the image of the dual exponential into a Hecke eigenspace. The presentation is adelic. One transplants the Eisenstein series into the adelic setting to obtain \(E_{k,s}\) and \(E_k:=E_{k,0}\), where \(E_{k,s}\) is absolutely convergent for \(k+2\) \(\text{Re}(s)>2\). The \(E_{k,s}\) depend on several arguments: a locally constant function \(\phi:{\mathbb A}^2_f\rightarrow {\mathbb C}\) of compact support and with an action of the group \(G_f=GL_{2, {\mathbb A}_f}\), and the 'lattice' parameter \(\tau\). One can then write \(E_{k,s}(\phi)\) as a function \(E_{k,s,f}\) where \(f:G_f\rightarrow{\mathbb C}\) is a suitable locally constant function such that \(E_{k,s,f}\) can be expressed in terms of \(f\). Then, for adelic modular forms \(F\), \(G\) of weights \(k+\ell\), \(k\) respectively, at least one is a cusp form, define
\[
\Omega=E_{\ell,s,f}G \overline{F}y^{k+\ell+s-2}|\det(g)|^{-k-\ell-s} d\tau\wedge d\overline{\tau}.
\]
This is a left \(G^+_{{\mathbb Q}}\)-invariant form on \(\mathfrak H\times G_f\). One is interested in computing \({\int_{G^+_{{\mathbb Q}}\mathfrak H\times G_f}\Omega dg}\), which is by definition the inner product \(\langle E_{\ell,s,f}G,F \rangle\). In particular, one takes for \(F\) a cusp form, belonging to an irreducible representation \(\pi=\otimes''\pi_p\) of \(G_f\), satisfying some more suitable conditions, and let \(G=E_k(\phi'')\) a suitable Eisenstein series. Then the inner product \(\langle E_{\ell,s}(\phi)E_k( \phi''),F\rangle\) can be decomposed explicitly into local integrals. The upshot is that one obtains a formula for the inner product in terms of values of \(L\)-functions related to \(\pi\). A supplementary character may also be included.
Let \(D,D''\) be integers prime to \(6Np\) and define \(\mathcal R''_p\) as the set of squarefree integers prime to \(NpDD''\). Also, let \(\mathcal R_p=\{r=r_0p^m\mid r_0\in\mathcal R''_p\), \(m\geq 1\}\). Suppose that for any \(r\in\mathcal R_p\) one has points \(z_r,z_r''\in\mathcal E^{\text{univ}} (Y(Nr)) \simeq({\mathbb Z}/Nr)^2\) having suitable properties, in particular, for \(r=p^m\), the orders of \(z_r,z_r''\) are divisible by a prime other than \(p\). One defines for any \(r\in\mathcal R_p\):
\[
\overline{\sigma}_r=\{\vartheta_D(z_r),\vartheta _{D''}(z_r'')\}\in K_2(Y(Nr)),
\]
and also
\[
\sigma_r=N_{Y(Nr)/Y(N) \otimes{\mathbb Q}(\mu_r)}\overline{\sigma}_r\in K_2(Y(N)\otimes{\mathbb Q}(\mu_r)).
\]
The \(\sigma\)'s define an almost Euler system for \(K_2\) in the sense that:
(i) \(N_ {{\mathbb Q}(\mu_{rp})/{\mathbb Q}(\mu_r)}\sigma_{rp}=\sigma_r\);
(ii) If \(\ell\) is prime and \((\ell,NDD''r)=1\) then
\[
N_{{\mathbb Q}(\mu_{\ell r})/{\mathbb Q} (\mu_r)}\sigma_{\ell r}=(1-T_{\ell}\langle\ell\rangle_*\otimes\text{Frob}_ {\ell}+\ell\langle\ell\rangle_*\otimes\text{Frob}^2_{\ell})\sigma_r.
\]
Define \({\mathbb T}_{p,N}=H^1(Y(N)\otimes_{\mathbb Q}\overline{\mathbb Q},{\mathbb Z}_p(1))\). Then, by Abel-Jacobi and corestriction, the family \(\sigma_{r_0p^n}\otimes [\zeta_{p^n}]^{-1}\) maps to the Galois cohomology \(H^1({\mathbb Q}(\mu_r), {\mathbb T}_{p,N})\). Write \(\xi_r=\xi_r(Y(N))\in H^1({\mathbb Q}(\mu_r), {\mathbb T}_{p,N})\) for its image. Then it follows that the \(\xi\)'s also satisfy Euler system-like identities. Looking at an elliptic curve \(E/{\mathbb Q}\) with conductor \(N_E\) and Weil parametrisation \(\varphi_E:X_0(N_E)\rightarrow E\), considering the composite morphism \(\varphi_{E,N}:X(N)\rightarrow X_0(N_E) {\buildrel\varphi_E\over\longrightarrow}E\), one gets Galois-equivariant maps
\[
\varphi_{E,N*}:H^1(X(N)\otimes_{\mathbb Q}\overline{\mathbb Q},{\mathbb Z}_p(1))\rightarrow H^1(E\otimes_{\mathbb Q}\overline{\mathbb Q},{\mathbb Z}_p(1))=T_p(E),
\]
the Tate module, and also a restriction to \({\mathbb T}_{p.N}\). The (proof of the) Manin-Drinfeld theorem now implies that one can eventually define elements \(\xi_r(E)\in H^1 ({\mathbb Q}(\mu_r),T_p(E))\) from the \(\xi_r\) via \(\varphi_{E,N*}\).
The main result of the fifth section is now that \textsl{the family \(\{\xi_r(E)\}\) is an Euler system for \(T_p(E)\).} In the reasoning behind these facts the dual exponential plays an important role. Finally, one may relate the Euler system to the \(L\)-function by using the Rankin-Selberg integral described above. For a character \(\lambda:\text{Gal}({\mathbb Q}(\mu_r)/{\mathbb Q})\rightarrow {\mathbb C}^{\times}\) such that \(\lambda(-1)=\pm 1\), one gets an interesting formula relating the sum (over the Galois group) of the dual exponentials of the conjugates of the Euler system elements times the character values in terms of \(L(E,\lambda,1)\), the periods of \(E\) and \(\omega_E\). Euler system; Eisenstein series; Galois cohomology; \(p\)-adic Hodge theory; \(L\)-function; \(K\)-theory of modular varieties; Kato-Siegel functions; Kato's dual exponential map A. J. Scholl, ''An introduction to Kato's Euler systems,'' in Galois Representations in Arithmetic Algebraic Geometry, Cambridge: Cambridge Univ. Press, 1998, vol. 254, pp. 379-460. Elliptic curves over global fields, Holomorphic modular forms of integral weight, \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture, Rational points, Elliptic curves An introduction to Kato's Euler 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 Often there is an interest in solutions of polynomial systems of equations. Enumerative geometry is concerned with the counting of the number of solutions for those systems arising in geometric situations. The intersection theory provides methods to accomplish the enumeration. The author investigates some problems from enumerative geometry, illustrating some applications of symbolic computation to this important part of solving polynomial equations. The considered enumerative problems include those which are overdetermined respectively improper. There are also considerations of real solutions of these geometric properties. A particular investigation is devoted to the study of lines tangent to 4 spheres respectively lines tangent to real quadrics sharing a real conic in three space respectively in higher dimensions. Macaulay2; enumerative geometry; systems of polynomial equations; tangent to 4 spheres; tangent to real quadrics F. Sottile, \textit{From enumerative geometry to solving systems of polynomial equations}, in Computations in Algebraic Geometry with Macaulay 2, ACM, New York, 2002, pp. 101--128. Enumerative problems (combinatorial problems) in algebraic geometry, Numerical computation of solutions to systems of equations, Polynomial rings and ideals; rings of integer-valued polynomials, Symbolic computation and algebraic computation, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) From enumerative geometry to solving systems of polynomial equations | 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 with curved geometric objects forms the basis for many algorithms in areas such as geometric modeling, computer aided design and robot motion planning. In general, such computations cannot be carried out reliably with standard machine precision arithmetic.
Slightly more than a decade ago robustly and efficiently dealing with conics and Bézier curves in 2D and quadrics and splines in 3D was considered an enormous challenge. This picture has changed. Our first successes were achieved for conics and quadrics, mainly relying on properties of the involved low-degree polynomials. In a second step, to tackle general algebraic curves and surfaces, we exploited more involved mathematical tools such as subresultants. In addition with clever filtering techniques, these methods already beat the previous specialized solutions. The most recent drastical success in performance gain for algebraic curves is due to several ingredients: The central one consists of a cylindrical algebraic decomposition with a revised lifting step. Using results from algebraic geometry we avoid any change of coordinates and replace the costly symbolic operations by numerical tools. The new algorithms for curve topology computation only need to compute the resultant and the gcd of bivariate polynomials and to perform numerical root finding. For the symbolic operations, we can rely on implementations exploiting graphics hardware, which is several magnitudes faster than corresponding CPU implementations.
All algorithms have been implemented as contributions to the C++! project Cgal. Excellent practical behavior of our algorithms has been shown in exhaustive sets of experiments, where we compared them with our previous and recent competing approaches. Beyond, the algorithms are also proven to be efficient in theory. Recent work shows that our implemented and practical algorithm needs \(\tilde{O}(d^6 + d^5\tau)\) bit operations (\(d\) degree, \(\tau \) bitsize of coefficients) to compute the topology of an algebraic curve and for solving bivariate systems.
Joint work with Pavel Emeliyanenko, Michael Kerber, Kurt Mehlhorn, Michael Sagraloff, Alexander Kobel, and Pengming Wang. algebraic curves; algebraic surfaces; geometric computing; symbolic operations; arrangements Software, source code, etc. for problems pertaining to algebraic geometry, Geometric aspects of numerical algebraic geometry, Computer-aided design (modeling of curves and surfaces), Numerical algebraic geometry Robustly and efficiently computing algebraic curves and 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 This work studies transversal morphisms and their effect in the set points of highest multiplicity of a variety, inspired by their role in Constructive Resolution of Singularities. Arc spaces and invariants of singularities induced by the Nash multiplicity sequence [\textit{M. Lejeune-Jalabert}, Am. J. Math. 112, No. 4, 525--568 (1990; Zbl 0743.14002)] turn out to give a characterization of the desired property of transversal morphisms in the main result, Theorem 1.1.
Given two algebraic varieties over a perfect field, a transversal morphism between them is a finite, dominant morphism that preserves the maximum value of the multiplicity in the following sense: if the generic rank of the morphism \(\beta: X' \longrightarrow X\) is \(r\), and the maximum value that the multiplicity achieves in \(X\) is \(m\), then the maximum value that it achieves in \(X'\) is precisely \(rm\). For non-transversal morphisms \(rm\) is only an upper bound for the multiplicity in \(X'\). Transversality of morphisms is stable under blow ups with regular centers contained in the set of points of maximum multiplicity, as well as under open restrictions and multiplication by an affine line, which is one of the properties of interest here. These types of transformations do not allow the multiplicity to increase, and any sequence of them, ending (at latest) when the multiplicity drops for the first time, is what is called an \(F_m\)-local sequence over \(X\).
Strongly transversal morphisms are transversal morphisms \(\beta :X' \longrightarrow X\) that induce a homeomorphism between the (closed) set of points of \(X'\) with multiplicity \(rm\), where \(r\) is again the generic rank of \(\beta \), and the (also closed) set of points of \(X\) with multiplicity \(m\), and for which this property is preserved under \(F_m\)-local sequences.
It is possible to represent the set of points of maximum multiplicity of a variety \(X\) by means of a Rees algebra \(\mathcal{G}_X\), in a way such that \(F_m\)-local sequences are also represented by algebraic transformations of this algebra [\textit{O. E. Villamayor}, Adv. Math. 262, 313--369 (2014; Zbl 1295.14015)]. This is highly useful in the context of Resolution of Singularities, where it has been proven that a simplification of the multiplicity can be the central building step for constructing a Resolution in the characteristic zero case: in this approach, \(F_m\)-local sequences are constructed to simplify the multiplicity several times, until a resolution of the singularities is complete.
In this context, strongly transversal morphisms of varieties guarantee a strong relation between \(F_m\)-local sequences over \(X\) and \(F_{rm}\)-local sequences over \(X'\): via a strongly transversal morphism, a simplification of the multiplicity of \(X'\) induces a simplification of the multiplicity of \(X\) and vice versa. There is also a relation between the corresponding Rees algebras: a transversal morphism \(\beta: X' \longrightarrow X\) guarantees an extension \(\mathcal{G}_X\subset \mathcal{G}_{X'}\) of the corresponding Rees algebras, and if \(\beta \) is strongly transversal, this extension is finite. This relation is even stronger when the characteristic of the field is zero, as in that case the finiteness of extensions \(\mathcal{G}_X\subset \mathcal{G}_{X'}\) characterizes strong transversality of morphisms [\textit{C. Abad} et al., Math. Nachr. 293, No. 1, 8--38 (2020; Zbl 07197931)].
This work is focused in understanding what the condition of \(\mathcal{G}_X\subset \mathcal{G}_{X'}\) being finite means by itself, as it is not equivalent to \(\beta \) being strongly transversal if the characteristic of the field is positive. A characterization of this finiteness, valid for any perfect field, is found in terms of the persistance of arcs [\textit{A. Bravo} et al., Indiana Univ. Math. J. 69, No. 6, 1933--1973 (2020; Zbl 1475.14026)], centered in the set of higher multiplicity of \(X\) and \(X'\). In particular, the main result is stated for transversal morphisms \(\beta \) of generic rank \(r\), inducing a homeomorphism between the set of points of \(X\) with multiplicity \(m\) and the set of points of \(X'\) of multiplicity \(rm\) (although maybe not strongly transversal, so this condition might not be preserved by \(F_m\)-local sequences). For any such morphism, \(\mathcal{G}_X\subset \mathcal{G}_{X'}\) will be finite if and only if any arc in \(X'\) whose center has multiplicity \(rm\) shares the same persistence as the corresponding arc in \(X\) via the induced morphism of arc spaces \(\beta _{\infty}:\mathcal{L}(X')\longrightarrow \mathcal{L}(X)\). The persistance of an arc gives a measure of how strong is the contact of this arc with the variety at its center, so this result indicates that the finiteness of the extension of Rees algebras induced by a transversal morphism is linked to whether such morphism preserves the contact of arcs centered at points of highest multiplicity. singularities; finite morphisms; multiplicity; arc spaces; constructive resolution of singularities Arcs and motivic integration, Singularities in algebraic geometry Finite morphisms and Nash multiplicity sequences | 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 \(Y\subset X\) be integral projective varieties and \(V\) a linear system on \(X\). Here we study the rank of the restriction map \(V\to V|Z\) where \(Z\) is a general zero-dimensional connected scheme supported by a general \(P\in Y\) and for which we prescribe the integers length \((Z\cap Y^{(t-1)}), t>0\), where \(Y^{(t-1)}\) is the infinitesimal neighborhood of order \(t\) of \(Y\) in \(X\). Divisors, linear systems, invertible sheaves Surfilinear zero-dimensional schemes: Osculation and postulation. | 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 computational study of bottlenecks on algebraic varieties. The bottlenecks of a smooth variety \(X \subseteq \mathbb{C}^n\) are the lines in \(\mathbb{C}^n\) which are normal to \(X\) at two distinct points. The main result is a numerical homotopy that can be used to approximate all isolated bottlenecks. This homotopy has the optimal number of paths under certain genericity assumptions. In the process we prove bounds on the number of bottlenecks in terms of the Euclidean distance degree. Applications include the optimization problem of computing the distance between two real varieties. Also, computing bottlenecks may be seen as part of the problem of computing the reach of a smooth real variety and efficient methods to compute the reach are still to be developed. Relations to triangulation of real varieties and meshing algorithms used in computer graphics are discussed in the paper. The resulting algorithms have been implemented with Bertini [\textit{D. J. Bates} et al., ``Bertini: software for numerical algebraic geometry'', \url{doi:10.7274/R0H41PB5}] and Macaulay2 [\textit{D. R. Grayson} and \textit{M. E. Stillman}, ``Macaulay2, a software system for research in algebraic geometry'', \url{http://www.math.uiuc.edu/Macaulay2}]. numerical algebraic geometry; systems of polynomials; triangulation of manifolds; reach of manifolds Geometric aspects of numerical algebraic geometry, Effectivity, complexity and computational aspects of algebraic geometry, Numerical aspects of computer graphics, image analysis, and computational geometry, Numerical algebraic geometry The numerical algebraic geometry of bottlenecks | 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 Most of the authors who proposed algorithms dealing with curved objects used a set of oracles allowing to perform basic geometric operations on curves (as computing the intersection of two curves). If the handled curves are algebraic, the oracles involve the resolution of algebraic equations, and in a geomeric computing framework no method of solving algebraic equations is considered available.
In this paper, we address the problem of the incidence graph of an arrangement of curves and we propose a method that completely avoids algebraic equations, all the computations to be done concerning linear objects. This will be done via suitable polynomial approximations of the given curves; we start by presenting a ``polygonal'' method in a case where the required polygonal approximations exist by definition, the case of composite Bézier curves, and then we show how we can construct these polygonal approximations in the general case of Jordan arcs. algebraic curves; computational geometry; oracles; incidence graph; polynomial approximations; Bézier curves; Jordan arcs Numerical aspects of computer graphics, image analysis, and computational geometry, Plane and space curves, Computational aspects of algebraic curves Polygonal approximations for curved problems: An application to 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 article generalizes several results to the level of motivic Milnor fiber and motivic zeta function, introduced by \textit{J. Denef} and \textit{F. Loeser} [J. Algebr. Geom. 7, No. 3, 505--537 (1998; Zbl 0943.14010)], previously proved for topological or Hodge theoretical invariants. For composed singularities, \textit{A. Némethi} [Compos. Math. 79, No. 1, 63--97 (1991; Zbl 0724.32020)] determined their Milnor fiber and monodromy zeta function. For such a computation one needs some good properties for the discriminant of the pair \((h,g)\). One such particular situation is when this discriminant is a normal crossing divisor. E.g., in such a case, the Hodge spectrum of the composed map also can be determined (Némethi-Steenbrink).
The present article targets the analogous results at the level of the motivic zeta function: at this level of technicality already the case when the discriminant is a normal crossing divisor needs essentially new ideas. The article clarifies completely the situation when \(h\) and \(g\) have no variables in common. The proof involves a reinterpretation of \textit{G. Guibert}'s theorem [Comment. Math. Helv. 77, No. 4, 783--820 (2002; Zbl 1046.14008)]. Milnor fiber; monodromy; monodromy zeta function; Hodge spectrum; motivic Milnor fiber; motivic zeta function; composed singularities; plane curve singularities G. Guibert, F. Loeser and M. Merle, Composition with a two variable function, Math. Res. Lett. 16 (2009), 439-448. Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry, Germs of analytic sets, local parametrization, Local complex singularities, Invariants of analytic local rings, Milnor fibration; relations with knot theory Composition with a two variable function | 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 focus our attention on a particular way of describing algebraic varieties (a geometric resolution of an equidimensional variety) that was already known to \textit{L. Kronecker} [Berlin. Reimer. Kronecker J. XCII. 1--123 (1882; JFM 14.0038.02)] and that has since proved to be very useful for solving system of polynomial equations algorithmically. We briefly show how this description can be attained and state some of its applications. Computational aspects in algebraic geometry A parametric description of algebraic varieties for algorithmic purposes | 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 commutative domain of characteristic 0 which is finitely generated over \(\mathbb{Z}\) as a \(\mathbb{Z}\)-algebra. Denote by \(A^*\) the unit group of \(A\) and by \(\overline{K}\) the algebraic closure of the quotient field \(K\) of \(A\). We shall prove effective finiteness results for the elements of the set
\[
\mathcal{C}:=\{ (x,y)\in (A^*)^2 \mid F(x,y)=0 \}
\]
where \(F(X,Y)\) is a non-constant polynomial with coefficients in \(A\) which is not divisible over \(\overline{K}\) by any polynomial of the form \(X^{m}Y^{n}-\alpha\) or \(X^{m}-\alpha Y^{n}\), with \(m, n\in\mathbb{Z}_{\geq0}\), \(\max(m,n)>0\), \(\alpha\in\overline{K}^*\). This result is a common generalisation of effective results of \textit{J.-H. Evertse} and \textit{K. Győry} [Math. Proc. Camb. Philos. Soc. 154, No. 2, 351--380 (2013; Zbl 1332.11040)] on \(S\)-unit equations over finitely generated domains, of \textit{E. Bombieri} and \textit{W. Gubler} [Heights in Diophantine geometry. Cambridge: Cambridge University Press (2006; Zbl 1115.11034)] on the equation \(F(x, y) = 0\) over \(S\)-units of number fields, and it is an effective version of \textit{S. Lang}'s general but ineffective theorem [Publ. Math., Inst. Hautes Étud. Sci. 6, 319--335 (1960; Zbl 0112.13402)] on this equation over finitely generated domains. The conditions that \(A\) is finitely generated and \(F\) is not divisible by any polynomial of the above type are essentially necessary. [2] A. Bérczes, \(Effective results for unit points on curves over finitely generated domains\), Math. Proc. Cambridge Phil. Soc., 158 (2015), 331-353. &MR 33 Curves over finite and local fields, Multiplicative and norm form equations, Polynomials over commutative rings, Solving polynomial systems; resultants, Heights, Rational points, Arithmetic ground fields for curves Effective results for unit points on curves over finitely generated domains | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(S\) be a connected Dedekind scheme, \(X\) a connected scheme and \(f:X \to S\) a faithfully flat morphism of finite type with a section. The author proves that \(X\) has a fundamental group scheme. This is defined to be the \(S\)-group scheme in the initial object for the category \(\mathcal{P}(X)\) whose objects are triples \((Y,G,y)\) where \(Y \to X\) is an fpqc-torsor over a finite and flat \(S\)-group scheme \(G\) and \(y\) is an \(S\)-valued point of \(Y\). He also proves that for an \textit{affine} scheme \(X\), the fundamental group of \(X_{\text{red}}\) is a closed subgroup of the fundamental group of \(X\). These results generalize former results of the author, and also of \textit{M. A. Garuti} [Proc. Am. Math. Soc. 137, No. 11, 3575--3583 (2009; Zbl 1181.14053)]. torsor; fundamental group scheme Antei M.: The fundamental group scheme of a non reduced scheme. Bull. Sci. Math. 135(5), 531--539 (2011) Group schemes, Positive characteristic ground fields in algebraic geometry The fundamental group scheme of a non-reduced 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 authors develop a computational model for algebraic formal power series based on a symbolic codification of the series by means of the implicit function theorem. They then reduce the problem of handling a finite set of algebraic series to some corresponding problem involving suitable polynomial rings. In this model they show that most of the usual local commutative algebra can be effectively performed on algebraic series. --- The main result of the paper is an effective version of the Weierstrass preparation theorem: the authors are able to prepare a distinguished polynomial and contemporaneously reduce the involved locally smooth system to one with one less variable. This theorem will allow them to have an effective version of the Weierstrass division theorem, to handle an effective elimination theory for algebraic series and to give an effective version of the Noether normalization lemma.
In section 1 the authors recall the basic theory of standard bases in rings of formal power series. The second section is devoted to the presentation of the proposed computational model for algebraic series, based on the concept of locally smooth systems. In section 3 they show how to modify a locally smooth system to compute effectively algebraic series. In section 4 they then give an algorithm to compute a standard basis for the ring of algebraic series. In section 5 they give effective versions of the Weierstrass preparation and division theorems, which are used in section 6 to present algorithms for computing the elimination of variables and the Noether normal position of an ideal of algebraic formal power series. By means of the Artin-Mazur theorem the authors show in the appendix how to reduce classically defined algebraic series to their model and conversely. symbolic codification of formal power series; Weierstrass preparation theorem; Weierstrass division theorem; Noether normalization lemma; standard bases Alonso, María Emilia; Mora, Teo; Raimondo, Mario, A computational model for algebraic power series, J. Pure Appl. Algebra, 0022-4049, 77, 1, 1-38, (1992) Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Formal power series rings, Geometric methods (including applications of algebraic geometry) applied to coding theory, Relevant commutative algebra A computational model for algebraic power series | 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 a class of local systems on the complement of a germ of irreducible plane curves. We exhibit local systems which were studied by \textit{O. Neto} [Compos. Math. 127, 229--241 (2001; Zbl 0995.58018)]. These local systems give rise to regular holonomic \(D\)-modules whose characteristic variety is the union of the zero section with the conormal of the curve. local systems; monodromy; D-modules Silva, P. C.: On a class of local systems associated to plane curves. C. R. Acad. sci. Paris, ser. I. 335, 421-426 (2002) Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Singularities in algebraic geometry, Singularities of curves, local rings On class of local systems associated to plane 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 polynomial \(F:\mathbb C^n\to\mathbb C^n\) one defines \(\mu(F)\) to be the dimension of the algebra obtained by quotienting out polynomials on \(\mathbb C^n\) by the components of \(F\). This number can also be interpreted as the expected number of solutions of the system \(F=0\). It is known that \(\mu(F)\) is bounded above by \(n! \mathrm{Vol}(\widetilde{\Gamma}_+(F))\), where \(\widetilde{\Gamma}_+(F)\) is a convex polyhedron obtained when studying which monomials appear with non-0 coefficients in \(F\). The main result of this paper is that in this estimate we have equality if an only if \(F\) is Newton non-degenerate at infinity -- an important condition on polynomials, involving the convex geometry of the faces of \(\widetilde{\Gamma}_+(F)\). In the last section the authors introduce and study the notion of \(F\) being non-degenerate with respect to a ``global Newton polyhedron''. In Theorem 4.9 these maps are proved to be the ones for which a certain \(\mu\)-vs-Volume estimate is sharp. complex polynomial maps; Milnor number; multiplicity; Newton polyhedron Complex singularities, Singularities in algebraic geometry, Computational aspects and applications of commutative rings Polynomial maps with maximal multiplicity and the special closure | 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 investigate a version of Viro's method for constructing polynomial systems with many positive solutions, based on regular triangulations of the Newton polytope of the system. The number of positive solutions obtained with our method is governed by the size of the largest \textit{positively decorable} subcomplex of the triangulation. Here, positive decorability is a property that we introduce and which is dual to being a subcomplex of some regular triangulation. Using this duality, we produce large positively decorable subcomplexes of the boundary complexes of cyclic polytopes. As a byproduct, we get new lower bounds, some of them being the best currently known, for the maximal number of positive solutions of polynomial systems with prescribed numbers of monomials and variables. We also study the asymptotics of these numbers and observe a log-concavity property. polynomial systems; triangulations; cyclic polytopes Topology of real algebraic varieties, Solving polynomial systems; resultants, Computational aspects of higher-dimensional varieties, Special polytopes (linear programming, centrally symmetric, etc.), Polynomials in real and complex fields: location of zeros (algebraic theorems), Complete intersections A polyhedral method for sparse systems with many positive solutions | 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 bielliptic curve. Here we give an existence theorem for \(\alpha\)-stable coherent systems on \(X\), using as the main tool the study of the coherent systems on elliptic curves made by \textit{H. Lange} and \textit{P. Newstead} [Int. J. Math. 16, No.~7, 787--805 (2005; Zbl 1078.14045)]. coherent systems on curves; vector bundles on curves; stable vector bundles T. Graber and E. Zaslow, ''Open-string Gromov-Witten invariants: Calculations and a mirror ``theorem'','' in Orbifolds in Mathematics and Physics, Providence, RI: Amer. Math. Soc., 2002, vol. 310, pp. 107-121. Vector bundles on curves and their moduli Stable coherent systems on bielliptic 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 In this booklet algebraic geometric codes are discussed. In the first 6 chapters, the author explains the basic theory of algebraic curves over finite fields, their function fields and the Goppa codes associated to certain divisors. For the proofs of most key results the reader is referred to the literature. In Chapter 7 the author exposes some result of \textit{D. V. Chudnovky} and \textit{G. V. Chudnovsky} [Proc. Natl. Acad. Sci. USA 84, 1739-1743 (1987; Zbl 0644.68066)] on the related complexity of multiplication in extension fields. Finally, in chapters 8 and 9, Goppa codes associated to divisors of curves of genus 1 are discussed. The author expresses the weight distributions of these codes in terms of the number of words of minimal weight. algebraic geometric codes; algebraic curves over finite fields; function fields; Goppa codes; complexity of multiplication in extension fields; divisors of curves of genus 1; weight distributions; minimal weight Other types of codes, Geometric methods (including applications of algebraic geometry) applied to coding theory, Finite ground fields in algebraic geometry, Algebraic functions and function fields in algebraic geometry Contributions to the theory of coding and complexity using algebraic function 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 This article is a continuation of the author's previous article [\textit{B.~Mesablishvili}, Appl. Categ. Struct. 12, No. 5--6, 485--512 (2004; Zbl 1084.14003)]. In that article it is shown that a quasi-compact morphism of schemes is a stable effective descent morphism for the scheme-indexed category of quasi-coherent modules if and only if it is pure, a notion also developed in that article. In this paper, the authors shows that this characterisation of (quasi-compact) effective descent morphisms by purity is also true for the scheme-indexed categories of (i) quasi-coherent modules of finite type, (ii) flat quasi-coherent modules, (iii) flat quasi-coherent modules of finite type and (iv) locally projective quasi-coherent modules of finite type [compare also \textit{B.~Mesablishvili}, Theory Appl. Categ. 10, 180--186, electronic only (2002; Zbl 1044.18006)].
Further results of this text are alternative descriptions of (quasi-compact) pure morphisms. The author proves the following two theorems:
A quasi-compact morphism of schemes is pure if and only if it is a stable regular epimorphism (i.e. a morphisms such that each of its pullbacks is an epimorphism that is a coequiliser).
A stable schematically dominant morphism of schemes [\textit{A.~Grothendieck} and \textit{J.~A.~Dieudonné}, Éléments de géométrie algébrique.~I (Die Grundlehren der mathematischen Wissenschaften. 166, Springer-Verlag, Berlin-Heidelberg-New York) (1971; Zbl 0203.23301)] is pure. If, in addition, the morphism is quasi-compact, the converse also holds.
In order to derive the characterisation for stable effective descent morphisms of the four examples (i)--(iii) of scheme-indexed category given above, the author also introduces the notion of a pure stack, which is a stack \(\mathcal F\) on the Zariski site of schemes (and in particular a scheme-indexed category) such that every pure morphism of affine schemes is an effective \(\mathcal F\)-descent morphism. The author then derives general results for pure stacks, in particular comparison results between two pure stacks. scheme; pure morphism; descent theory B. Mesablishvili, ''More on descent theory for schemes,'' Georgian Math. J., 11, No. 4, 783--800 (2004). Schemes and morphisms, Epimorphisms, monomorphisms, special classes of morphisms, null morphisms, Factorization systems, substructures, quotient structures, congruences, amalgams More on descent theory for 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 In this paper we outline some algorithms for working with real numerical varieties with an application to parameterizing quadratic surface intersection curves (QSIC). Computational aspects of algebraic curves, Real algebraic sets, Numerical aspects of computer graphics, image analysis, and computational geometry, Computer graphics; computational geometry (digital and algorithmic aspects) Algorithms for real numerical varieties with application to parameterizing quadratic surface intersection 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 We answer a question in [\textit{J. M. Landsberg} and \textit{N. Ressayre}, Differ. Geom. Appl. 55, 146--166 (2017; Zbl 1380.68202)], showing the regular determinantal complexity of the determinant \(\det_m\) is \(O(m^3)\). We answer questions in, and generalize results of \textit{N. R. Aravind} and \textit{P. S. Joglekar} [Lect. Notes Comput. Sci. 9210, 95--105 (2015; Zbl 1380.68455)], showing there is no rank one determinantal expression for \(\operatorname{perm}_m\) or \(\det_m\) when \(m \geq 3\). Finally we state and prove several ``folklore'' results relating different models of computation. Ikenmeyer, Christian; Landsberg, J.M., On the complexity of the permanent in various computational models, (2016) Symbolic computation and algebraic computation, Effectivity, complexity and computational aspects of algebraic geometry, Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.) On the complexity of the permanent in various computational models | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(w=(w_ 1,w_ 2,\dots,w_ n)\) be a permutation in the symmetric group \(S_ n\). An explicit combinatorial construction of the Schubert polynomial \({\mathcal S}_ w\) is given as a sum of weights \(x^ D\) of diagrams \(D\) (finite nonempty sets of lattice points \((i,j)\) in the positive quadrant), where the sum is over the set \(\Omega(w)\) of diagrams \(D\) which are obtainable from an initial diagram \(D(w)\) by repeated application of allowable \(B\)-moves. If we view a diagram as a set of checkers (or black stones for those who prefer) distributed over the positive quadrant, then a \(B\)-move is just a move of one checker from a position to an adjacent unoccupied position subject to a set of rules, thereby obtaining a new diagram. Since the classical Schur functions can be viewed as special cases of the Schubert polynomials for appropriate choices of the permutation \(w\), it has been hoped that there was a combinatorial algorithm for constructing Schubert polynomials which extends (or is at least in analogy to) the construction of the Schur function \(S_ \lambda(x_ 1,\dots,x_ r)\) as a sum of weights of column strict tableaux of shape \(\lambda\), where \(\lambda\) is a partition. The algorithm given here is not quite an extension of the usual construction for Schur functions, however a simple bijection between \(\Omega(w)\) and the set of column strict tableaux of shape \(\lambda\) is given in the particular case where the Schubert polynomial \({\mathcal S}_ w\) is the Schur function \(S_ \lambda(x_ 1,\dots,x_ r)\). A different algorithm for constructing the general Schubert polynomial \({\mathcal S}_ w\) was conjectured (and proved when \(w\) is a vexillary permutation) by \textit{A. Kohnert} [Weintrauben, Polynome, Tableaux, Bayreuther Math. Schr. 38, 1-97 (1991; Zbl 0755.05095)]. A sketch is given at the end of the paper here of how one would show the equivalence of the rule given here and that of Kohnert. symmetric group; Schubert polynomial; Schur functions; tableaux; algorithm Bergeron N., J. Comb. Theory 60 pp 168-- (1992) Symmetric functions and generalizations, Permutations, words, matrices, Grassmannians, Schubert varieties, flag manifolds A combinatorial construction of the Schubert 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 We construct five families of 2D moduli spaces of parabolic Higgs bundles (respectively, local systems) by taking the equivariant Hilbert scheme of a certain finite group acting on the cotangent bundle of an elliptic curve (respectively, twisted cotangent bundle). We show that the Hilbert scheme of \(m\) points of these surfaces is again a moduli space of parabolic Higgs bundles (respectively, local systems), confirming a conjecture of \textit{P. Boalch} [Publ. Math., Inst. Hautes Étud. Sci. 116, 1--68 (2012; Zbl 1270.34204)] in these cases and extending a result of \textit{A. Gorsky} et al. [Commun. Math. Phys. 222, No. 2, 299--318 (2001; Zbl 0985.81107)]. Using the McKay correspondence, we establish the autoduality conjecture for the derived categories of the moduli spaces of Higgs bundles under consideration. Gröchenig, M., Hilbert schemes as moduli of Higgs bundles and local systems, Int. math. res. not. IMRN, 2014, 23, 6523-6575, (2014) Relationships between algebraic curves and integrable systems, Parametrization (Chow and Hilbert schemes), Algebraic moduli problems, moduli of vector bundles, McKay correspondence Hilbert schemes as moduli of Higgs bundles and local 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 numerical algebraic geometry witness sets are numerical representations of positive dimensional solution sets of polynomial systems. Considering the asymptotics of witness sets we propose certificates for algebraic curves. These certificates are the leading terms of a Puiseux series expansion of the curve starting at infinity. The vector of powers of the first term in the series is a tropism.
For proper algebraic curves, we relate the computation of tropisms to the calculation of mixed volumes. With this relationship, the computation of tropisms and Puiseux series expansions could be used as a preprocessing stage prior to a more expensive witness set computation. Systems with few monomials have fewer isolated solutions and fewer data are needed to represent their positive dimensional solution sets. numerical examples; algebraic curves Verschelde, J., Polyhedral methods in numerical algebraic geometry, No. 496, 243-263, (2009), Providence Numerical computation of solutions to systems of equations, Numerical computation of roots of polynomial equations, Computational aspects of algebraic curves Polyhedral methods in numerical algebraic geometry | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Not reviewed Algebraic geometry Remarque sur la décomposition des courbes de diramation des plans multiples | 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 Henselian and Japanese discrete valuation ring \(A\) and a flat and projective \(A\)-scheme \(X\), we follow the approach of \textit{I. Biswas} and the second author [J. Inst. Math. Jussieu 10, No. 2, 225--234 (2011; Zbl 1214.14037)] to introduce a \textit{full subcategory} of coherent modules on \(X\) which is then shown to be Tannakian. We then prove that, under normality of the generic fibre, the associated affine and flat group is pro-finite in a strong sense (so that its ring of functions is a Mittag-Leffler \(A\)-module) and that it classifies finite torsors \(Q\to X\). This establishes an analogy to Nori's theory of the essentially finite fundamental group. In addition, we compare our theory with the ones recently developed by Mehta-Subramanian and Antei-Emsalem-Gasbarri. Using the comparison with the former, we show that any quasi-finite torsor \(Q\to X\) has a reduction of the structure
group to a finite one. discrete valuation rings; group schemes; Tannakian categories; coherent sheaves Sheaves in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry, Group schemes, Local ground fields in algebraic geometry, Group actions on varieties or schemes (quotients) Finite torsors on projective schemes defined over a discrete valuation ring | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author discusses the postulation of general unions of zero-dimensional schemes in a projective space, proving that is has maximal rank in two cases: for curvilinear zero-dimensional schemes and ``curvilinear prolongations of fat points''. fat point; Hilbert function; curvilinear Projective techniques in algebraic geometry, Syzygies, resolutions, complexes and commutative rings, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series Postulation of general unions of a large number zero-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 We consider the multicanonical systems \(|mK_S|\) of quasi-elliptic surfaces with Kodaira dimension 1 in characteristic 2. We show that for any \(m \geq 6 |mK_S|\) gives the structure of quasi-elliptic fiber space, and 6 is the best possible number to give the structure for any such surfaces. characteristic \(p\); multicanonical system; quasi-elliptic surface Elliptic surfaces, elliptic or Calabi-Yau fibrations, Divisors, linear systems, invertible sheaves, Arithmetic ground fields (finite, local, global) and families or fibrations, Special surfaces On the multicanonical systems of quasi-elliptic surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(C\) be a smooth curve of genus \(g \geq 2\). A coherent system of type \((n,d,k)\) on \(C\) is a pair \((E,V)\), where \(E\) is a rank \(n\) vector bundle on \(C\) with degree \(d\) and \(V\) is a \(k\)-dimensional linear subspace of \(H^0(C,E)\). For any \(\alpha \in \mathbb {R}\) set \(\mu _\alpha (E,V):= d/n + \alpha k/n\) (the \(\alpha\)-slope). \((E,V)\) is \(\alpha\)-stable if \(\mu _\alpha (F,W) < \mu _\alpha (E,V)\) for all proper coherent subsystems \((F,W)\) of \((E,V)\). For each \(\alpha\) there is a moduli space of \(\alpha\)-stable coherent systems of type \((n,d,k)\). Varying \(\alpha\) these moduli spaces vary only at certain critical \(\alpha\)'s and in a controlled way (flips) [\textit{S. B. Bradlow}, \textit{O. García-Prada}, \textit{V. Munõz} and \textit{P. E. Newstead}, Int. J. Math. 14, No. 7, 683--733 (2003; Zbl 1057.14041)]. Here (case \(k \leq n\)) the authors for large \(\alpha\) compute the Picard groups and the first and second homotopy groups of these moduli spaces. For \(k=n-2\) they give their Poicaré polynomial. For \(k=n-1\) they give an explicit description of them as a fibration over the Jacobian \(J^d(C)\) with a Grassmannian as fibers. coherent system; moduli of vector bundles; algebraic curves; Brill-Noether loci Bradlow S.B., García-Prada O., Mercat V., Muñoz V., Newstead P.E., On the geometry of moduli spaces of coherent systems on algebraic curve Vector bundles on curves and their moduli, Algebraic moduli problems, moduli of vector bundles, Special divisors on curves (gonality, Brill-Noether theory) On the geometry of moduli spaces of coherent systems on 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 a smooth, projective algebraic surface; for any integral divisor \(H\) on \(S\), the problem of determining the dimension of the (non-complete) linear system given by the curves in \(|H|\) which are singular at \(r+1\) given points of \(S\) is quite a classical one in Algebraic Geometry. Such a problem is strongly related (via Terracini's Lemma.when \(H\) is very ample and the points are generic in \(S\)) to the one of determining the dimension of the \(r\)-secant variety to the surface which is the image of \(S\) via the embedding defined by \(|H|\). Such a subsystem of \(|H|\) is said to be \(special\) when its dimension is bigger than expected (the corresponding secant variety is said to be ``defective'').
The case which is studied here is given by the surfaces \(S\) for which \(H^i(S,B)=0\), \(\forall i>0\) and for all integral curves \(B\subset S\). As a result, a classification of these special subsystem is given for simple abelian surfaces, K3 surfaces with no elliptic curves on them and anticanonical rational surfaces.
In particular, in this last case, it is shown a nice case of the Harbourne-Hirschowitz Conjecture about the postulation of a generic set of fat points in the plane, namely that the conjecture is true for a set of nine points with arbitrary multiplicities and any number of double points. linear systems; double points; secant varieties; postulation Divisors, linear systems, invertible sheaves Special systems through double points on an algebraic 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 \(X\) be a smooth irreducible projective curve of genus \(g\geq 1\) and let \(J(X)\) be the jacobian of \(X\). Let \(I(d):X^{(d)}\to J(X)\) be the natural morphism and consider \(W^ r_ d=\{x\in J(X):\dim(I(d)^{- 1}(x))\geq r\}\). Let \(\rho^ r_ d(g)\) be the Brill-Noether number. The author proves some results concerning the schemes \(W^ r_ d\). For example: ``Suppose \(\dim(W^ r_{d-1})=\rho^ r_{d-1}(g)\geq 0\) and \(\rho^ r_ d(g)<g\). If \(W^ r_{d-1}\) is a reduced (respectively irreducible) scheme, then \(W^ r_ d\) is a reduced (respectively irreducible) scheme''. As an application one proves some dimension theorems for the schemes \(W^ 1_ d\). jacobian; Brill-Noether number; dimension \textsc{M. Coppens,} Some remarks on the schemes \(W^{r}_{d}\), Ann. Math. Pura Appl. (4) \textbf{97} (1990), 183-197. Jacobians, Prym varieties Some remarks on the schemes \(W_ d^ r\) | 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 limiting case of the Krichever construction of orthogonal curvilinear coordinate systems when the spectral curve becomes singular. We show that when the curve is reducible and all its irreducible components are rational curves, the construction procedure reduces to solving systems of linear equations and to simple computations with elementary functions. We also demonstrate how well-known coordinate systems, such as polar coordinates, cylindrical coordinates, and spherical coordinates in Euclidean spaces, fit in this scheme. Миронов, А. Е.; Тайманов, И. А., Ортогональные криволинейные системы координат, отвечающие сингулярным спектральным кривым, Proc. Steklov Inst. Math., 255, 180-196, (2006) Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions, Relationships between algebraic curves and integrable systems, Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry Orthogonal curvilinear coordinate systems corresponding to singular spectral 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 We survey the theory of local models of Shimura varieties. In particular, we discuss their definition and illustrate it by examples. We give an overview of the results on their geometry and combinatorics obtained in the last 15 years. We also exhibit their connections to other classes of algebraic varieties such as nilpotent orbit closures, affine Schubert varieties, quiver Grassmannians and wonderful completions of symmetric spaces. local model; Shimura variety; affine flag varity Pappas, G.; Rapoport, M.; Smithling, B., \textit{local models of Shimura varieties, I. geometry and combinatorics}, Handbook of moduli, Vol. III, 135-217, (2013), International Press, Somerville, MA Research exposition (monographs, survey articles) pertaining to algebraic geometry, Modular and Shimura varieties, Arithmetic aspects of modular and Shimura varieties, Grassmannians, Schubert varieties, flag manifolds Local models of Shimura varieties, I. Geometry and combinatorics | 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 compute the intersection number between two cycles \(A\) and \(B\) of complementary dimensions in the Hilbert scheme \(H\) parameterizing subschemes of given finite length \(n\) of a smooth projective surface \(S\). The \((n+1)\)-cycle \(A\) corresponds to the set of finite closed subschemes the support of which has cardinality 1. The \((n-1)\)-cycle \(B\) consists of the closed subschemes the support of which is one given point of the surface. Since \(B\) is contained in \(A\), indirect methods are needed. The intersection number is \(A.B=(-1)^{n-1}n\), answering a question by H. Nakajima. punctual Hilbert scheme; intersection numbers Ellingsrud G., Strømme S.A.: An intersection number for the punctual Hilbert scheme of a surface. Trans. Amer. Math. Soc. 350, 2547--2552 (1999) Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Parametrization (Chow and Hilbert schemes) An intersection number for the punctual Hilbert scheme of 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 We construct a spectral sequence associated to a stratified space, which computes the compactly supported cohomology groups of an open stratum in terms of the compactly supported cohomology groups of closed strata and the reduced cohomology groups of the poset of strata. Several familiar spectral sequences arise as special cases. The construction is sheaf-theoretic and works both for topological spaces and for the étale cohomology of algebraic varieties. As an application we prove a very general representation stability theorem for configuration spaces of points. configuration spaces; representation stability; homological stability; stratified spaces; spectral sequence Discriminantal varieties and configuration spaces in algebraic topology, General theory of spectral sequences in algebraic topology, Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects), Classical real and complex (co)homology in algebraic geometry, Sheaf cohomology in algebraic topology A spectral sequence for stratified spaces and configuration spaces of 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 Let \(S\to C\) be a smooth quasi-projective surface properly fibered onto a smooth curve. We prove that the multiplicativity of the perverse filtration on \(H^*(S^{[n]},\mathbb{Q})\) associated with the natural map \(S^{[n]}\to C^{(n)}\) implies that \(S\to C\) is an elliptic fibration. The converse is also true when \(S\to C\) is a Hitchin-type elliptic fibration. perverse filtration; \(P = W\) conjecture; Hilbert schemes of points; Nakajima Heisenberg operators Topological properties in algebraic geometry Multiplicativity of perverse filtration for Hilbert schemes of fibered surfaces. II | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\mathcal C\) be a real plane algebraic curve defined by the resultant of two polynomials (resp., by the discriminant of a polynomial). Geometrically such a curve is the projection of the intersection of the surfaces defined implicitly by the equations \(P(x,y,z)=Q(x,y,z)=0\) (resp. \(P(x,y,z)=\frac{\partial P}{\partial z}(x,y,z)=0\)), and generically its singularities are nodes (resp. nodes and ordinary cusps). In the previous literature, one may find numerical algorithms that compute the topology of smooth curves but usually fail to certify the topology of singular ones. The main challenge is to find practical numerical criteria that guarantee the existence and the uniqueness of a singularity inside a given box \(B\), while ensuring that \(B\) does not contain any closed loop of \(\mathcal C\). The authors solve this problem by first providing a square deflation system, based on subresultants, that can be used to certify numerically whether \(B\) contains a unique singularity \(p\) or not. Then, it is introduced a numeric adaptive separation criterion based on interval arithmetic to ensure that the topology of \(\mathcal C\) in \(B\) is homeomorphic to the local topology at \(p\). The algorithms presented are implemented and experiments show their efficiency compared to previous symbolic or homotopic methods. topology of algebraic curves; resultant; discriminant; subresultant; numerical algorithm; singularities; interval arithmetic; node and cusp singularities Plane and space curves, Singularities of curves, local rings, Real algebraic sets, Computational aspects of algebraic curves, Numerical algorithms for computer arithmetic, etc. A certified numerical algorithm for the topology of resultant and discriminant 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 probabilistic analysis of condition numbers is of central importance to turn condition-based complexity analyses of numerical algorithsm into probabilistic complexity analyses. While the former explain how the algorithm behaves at a particular input, the former allows us to see how the algorithm behaves in general.
Unfortunately, passing from a condition-based complexity analysis to a probabilistic complexity analysis requires to choose a probabilistic distribution of the input. With the notable exception of numerical linear algebra and the theory of random matrices, probabilistic analyses of condition numbers in numerical algebraic geometry rely exclusively on some form of Gaussian assumption of the input.
In their previous paper [\textit{A. A. Ergür} et al., Found. Comput. Math. 19, No. 1, 131--157 (2019; Zbl 1409.65033)], they performed an average complexity analysis in which they consider the condition number, as introduced in [\textit{F. Cucker} et al., J. Complexity 24, No. 5--6, 582--605 (2008; Zbl 1166.65021)], of random real polynomial systems under robust assumptions. In this paper, they extend their results to the smoothed analysis framework of \textit{D. Spielman} and \textit{S.-H. Teng} [in: Proceedings of the thirty-third annual ACM symposium on theory of computing, STOC 2001. Hersonissos, Crete, Greece, July 6--8, 2001. New York, NY: ACM Press. 296--305 (2001; Zbl 1323.68636)]. In these framework, we don't just consider a random input, but an arbitrary input perturbed by random noise. As numerical algorithms work with inputs submitted to errors, this is a more realistic framwork.
Moreover, the authors do not only provide an smoothed complexity analysis, but they also do so for a class of estructured random polynomial systems. The structure is chosen by taking the random polynomials out of subspaces of the space of polynomials, for which one can guarantee nice evaluation bounds. This is the first probabilistic analysis for random structured polynomials.
Finally, let us note that to achive this, Ergür, Paouris and Rojas rely on results coming from geometric functional analysis for subgaussian and anti-concentrated random variables such as those in [\textit{R. Vershynin}, High-dimensional probability. An introduction with applications in data science. Cambridge: Cambridge University Press (2018; Zbl 1430.60005); \textit{M. Rudelson} and \textit{R. Vershynin}, Int. Math. Res. Not. 2015, No. 19, 9594--9617 (2015; Zbl 1330.60029)]. condition number; random polynomials; grid method; smoothed analysis; structured polynomials Geometric aspects of numerical algebraic geometry, Numerical computation of matrix norms, conditioning, scaling, Real algebraic and real-analytic geometry, Complexity and performance of numerical algorithms Smoothed analysis for the condition number of structured real 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 The main topic of this paper is to give characterizations of geometric properties of 0-dimensional subschemes \(X \subseteq \mathbb{P}^d\) in terms of the algebraic structure of the canonical module of their projective coordinate ring. The author characterizes Cayley-Bacharach, (higher order) uniform position, linearly and higher order general position properties, and derives inequalities for the Hilbert functions of such schemes. Finally the structure of the canonical module is related to properties of the minimal free resolution of \(X\).
Applications of this work are contained in a paper by the author [in: Zero-dimensional Schemes, Proc. Int. Conf. Ravello/Italy 1992, 243-252 (1994; see the following review)] and by \textit{K. Yanagawa} [J. Algebra 170, No. 2, 429-439 (1994)]. Cayley-Bacharach scheme; uniform position property; 0-dimensional scheme; Hilbert functions M. Kreuzer, On the canonical module of a \(0\)-dimensional scheme , Canad. J. Math. 46 (1994), 357-379. Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Projective techniques in algebraic geometry, Cycles and subschemes, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series On the canonical module of a 0-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 This paper answers affirmatively the question in the survey article by \textit{P. A. Griffiths} [``An introduction to the theory of special divisors on algebraic curves'', Regional Conf. Ser. Math. 44 (1980; Zbl 0446.14010)]: Does the variety \(W_d^r\) of linear systems on a general curve of genus \(g\) with degree \(d\) and dimension at least \(r\) have the Brill-Noether dimension \(g - (r +1)(g - d +r)\)? Moreover the authors determine the class of this variety in the cohomology of the Jacobian and show that it is without multiple components.
The proof is by a detailed geometrical analysis of a classical degeneration idea of Castelnuovo's. It is formalized as the Castelnuovo-Severi-Kleiman conjecture (CSK): the family of \(P^k\)'s in \(P^d\) meeting the chords of a rational normal curve in \(P^d\) has the same dimension as if the chords were lines in general position, and the family has no multiple components.
The paper has three parts: (I) The reduction to CSK -- except for the absence of multiple components to \(W_d^r\); (II) The proof of CSK; (III) Proofs of the absence of multiple components.
(I) has been previously achieved by \textit{S. Kleiman} [Adv. Math. 22, 1--31 (1976; Zbl 0342.14012)]. The proof in this paper is by a geometrical argument based on duality of special divisors. (II) uses a degeneration of the chords to span an osculating flag. (III) is again by degeneration. The degenerate case is chosen so that number of intersections of two varieties as a set equals the algebraic intersection number. The techniques are those of classical algebraic geometry and Schubert calculus. variety of linear systems on a general curve; cohomology of Jacobian; Castelnuovo-Severi-Kleiman conjecture; algebraic intersection number; Schubert calculus Griffiths, P. \& Harris, J.,On the variety of special linear systems on a general algebraic curves, Duke Math. J.,47(1980), 233--272. Jacobians, Prym varieties, Grassmannians, Schubert varieties, flag manifolds, Divisors, linear systems, invertible sheaves, Special algebraic curves and curves of low genus, Enumerative problems (combinatorial problems) in algebraic geometry On the variety of special linear systems on a general algebraic 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 This is a more complete version, including proofs and examples, of a conference proceedings article by the authors [\textit{R. Matsumoto} and \textit{S. Miura} in: Applied algebra, algebraic algorithms and error correcting codes, 13th Int. Symp., AAECC-13, Honolulu 1999, Proc. Lect. Notes Comput. Sci. 1719, 271-281 (1999; Zbl 0958.14041)]. affine algebraic curve; place at infinity; ideal quotient; Gröbner basis; algebraic geometry code; Weierstraß semigroup; Riemann-Roch space 11. Matsumoto, R., Miura, S.: Finding a basis of a linear system with pairwise distinct discrete valuations on an algebraic curve. J. Symb. Comput. 30 (3), 309-323 (2000). Computational aspects of algebraic curves, Geometric methods (including applications of algebraic geometry) applied to coding theory, Valuations and their generalizations for commutative rings, Divisors, linear systems, invertible sheaves, Riemann surfaces; Weierstrass points; gap sequences Finding a basis of a linear system with pairwise distinct discrete valuations on an algebraic 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 give a simple and effective method for the construction of algebraic curves over finite fields with many rational points. The curves are given as Kummer covers of the projective line. algebraic curves; finite fields; rational points; Kummer extensions Rational points, Plane and space curves Division algorithm and construction of curves with many 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 \textit{A. Joseph} invented multidegrees in his article [J. Algebra 88, 238--278 (1984; Zbl 0539.17006)] to study \textit{orbital varieties}, which are the components of an \textit{orbital scheme}, itself constructed by intersecting a nilpotent orbit with a Borel subalgebra. Their multidegrees are known as \textit{Joseph polynomials}, and these polynomials give a basis of a (Springer) representation of the Weyl group. In the case of the nilpotent orbit \(\{M^2 = 0\}\), the orbital varieties can be indexed by noncrossing chord diagrams in the disk. {
}
In this paper we study the normal cone to the orbital scheme inside this nilpotent orbit \(\{M^2 = 0\}\). This gives a better-motivated construction of the Brauer loop scheme we introduced in [Adv. Math. 214, No. 1, 40--77 (2007; Zbl 1193.14068)], whose components are indexed by all chord diagrams (now possibly with crossings) in the disk.
The multidegrees of its components, the \textit{Brauer loop varieties}, were shown to reproduce the ground state of the \textit{Brauer loop model} in statistical mechanics [\textit{P. Di Francesco} and the second author, Commun. Math. Phys. 262, No. 2, 459--487 (2006; Zbl 1113.82026)]. Here, we reformulate and slightly generalize these multidegrees in order to express them as solutions of the rational quantum Knizhnik-Zamolodchikov equation associated to the Brauer algebra. In particular, the vector of the multidegrees satisfies two sets of equations, corresponding to the \(e_i\) and \(f_i\) generators of the Brauer algebra. The proof of the analogous statement in Knutson and Zinn-Justin (loc. cit.) was slightly roundabout; we verified the \(f_i\) equation using the geometry of multidegrees, and used algebraic results of Di Francesco and Zinn-Justin (loc. cit.) to show that it implied the \(e_i\) equation. We describe here the geometric meaning of both \(e_i\) and \(f_i\) equations in our slightly extended setting. {
}
We also describe the corresponding actions at the level of orbital varieties: while only the \(e_i\) equations make sense directly on the Joseph polynomials, the \(f_i\) equations also appear if one introduces a broader class of varieties. We explain the connection of the latter with matrix Schubert varieties. quantum Knizhnik-Zamolodchikov equation; orbital varieties; Brauer loops Knutson, Allen and Zinn-Justin, Paul, {G}rassmann--{G}rassmann conormal varieties, integrability and plane partitions, Université de Grenoble. Annales de l'Institut Fourier, (None) Operator algebra methods applied to problems in quantum theory, Group actions and symmetry properties, Finite-dimensional groups and algebras motivated by physics and their representations, Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations, Groups and algebras in quantum theory and relations with integrable systems, Coadjoint orbits; nilpotent varieties, Grassmannians, Schubert varieties, flag manifolds The Brauer loop scheme and orbital varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The partition of projective geometry over the field \(\mathbb F_q\) into Schubert sets allows to convert an incidence graph to symbolic Grassmann automaton. Special symbolic computations of these automata produce bijective transformation of the largest Schubert cell. Some of them are chosen as maps which are used in new cryptosystems. symbolic Grassmann automaton; Schubert cell Cryptography, Grassmannians, Schubert varieties, flag manifolds On Schubert cells in Grassmannians and new algorithms of multivariate cryptography | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper, the notion of the Gröbner cell for the Hilbert scheme of points in the plane, as well as that of the punctual Hilbert scheme is comprehensively defined. An explicit parametrization of the Gröbner cells in terms of minors of a matrix is recalled. The main core of this paper shows that the decomposition of the Punctual Hilbert scheme into Grönber cells induces that of the compactified Jacobians of plane curve singularities. As an important application of this decomposition, the topological invariance of an analog of the compactified Jacobian and the corresponding motivic superpolynomial for families of singularities is concluded. Hilbert schemes; affine plane; Grothendieck-Deligne map; Gröbner cells; zeta functions; plane curve singularities Parametrization (Chow and Hilbert schemes), Singularities of curves, local rings, Plane and space curves, Exact enumeration problems, generating functions, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Singularities in algebraic geometry, Jacobians, Prym varieties, Hecke algebras and their representations, Combinatorial aspects of representation theory, Braid groups; Artin groups, Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.) Gröbner cells of punctual Hilbert schemes in dimension two | 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 concerns the conjectures of Kobayashi, of Green, Griffiths and Lang and also \textit{X. Chen} et al. [Adv. Math. 384, Article ID 107735, 64 p. (2021; Zbl 1479.14016)] as well as Seshadri constants. The Kobayashi Conjecture says that a very general hypersurface \(X\) of degree \(d\) in \(\mathbb{P}^n\) is Brody hyperbolic if \(d\) is sufficiently large. Moreover, the complement \(\mathbb{P}^n \setminus X\) is also Brody hyperbolic for large enough \(d\). The suspected optimal bound for \(d\) is approximately \(d \geq 2n-1\). However, the best current bound is for \(d\) greater than about \((en)^{2n+2}\). Green, Griffiths, Lang conjecture states that if \(X\) is a variety of general type, then there is a proper subvariety \(Y \subset X\) containing all the entire curves of \(X\). The authors give a new proof of the Kobayashi conjecture using results on the Green, Griffiths and Lang conjecture; namely they prove that a general hypersurface in \(\mathbb{P}^n\) of degree \(d\) admits no nonconstant holomorphic maps from \(\mathbb{C}\) for \(d \geq d_{2n-3}\), where \(d_2 = 286, d_3 = 7316\) and
\[
d_n = \left\lfloor \frac{n^4}{3} (n \log(n \log(24n)))^n \right\rfloor.
\]
They also give a short proof that if \(X\) is a general hypersurface in \(\mathbb{P}^n\) of degree at least \(d_{2n}\), where \(d_n = (5n)^2 n^n\), then \(\mathbb{P}^n \setminus X\) is Brody hyperbolic.
Chen, Lewis, and Sheng [loc. cit.] conjectured the following:
Let \(X \subset \mathbb{P}^n\) be a very general complete intersection of multidegree \((d_1, \dots, d_k)\). Then for every \(p \in X\), the space of points of \(X\) rationally equivalent to \(p\) has dimension at most \(2n-k-\sum_{i=1}^k d_i\) (no such points if \(2n-k-\sum_{i=1}^k d_i < 0\)). The authors prove the conjecture for all the cases, except \(2n-k-\sum_{i=1}^k d_i = -1\), when they prove the result holds with the exception of possibly countably many points.
Then the authors consider the Seshadri constants. By \(\epsilon(p,X)\) they denote the Seshadri constant of \(X\) at the point \(p\), ie. the infimum of \(\frac{\deg C}{\text{mult}_p C}\) over all curves \(C\) in \(X\) passing through \(p\), and by \(\epsilon(X)\) the Seshadri constant of \(X\), ie. the infimum (over all \(p\)) of \(\epsilon(p,X)\). The authors prove: If for a very general hypersurface \(X_0 \subset \mathbb{P}^{2n-1}\) of degree \(d\), the Seshadri constant at a general point \(p\) we have \(\epsilon(p,X_0)\geq r\), then for a very general \(X \subset \mathbb{P}^n\) of degree \(d\), we have \(\epsilon(X) \geq r\). Grassmannian technique; hyperbolicity; Kobayashi conjecture; Seshadri constants Divisors, linear systems, invertible sheaves Applications of a Grassmannian technique to hyperbolicity, Chow equivalency, and Seshadri constants | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Here we solve a polynomial interpolation problem for double points in \( \mathbb{P}^n\), some of the points being in a fixed hyperplane. Projective techniques in algebraic geometry Polynomial interpolation with restricted partial support | 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 simply connected Lie group with Lie algebra \(\mathfrak{g}\), and \(\Gamma\) a cocompact discrete subgroup of \(G\). If \(G\) is nilpotent, then Chen's (closed) iterated integrals induced from \(\bigwedge \mathfrak{g}_\mathbb{C}^\ast\) represent the coordinate ring of the Malcev completion of \(\Gamma\). In this paper, a generalized case is discussed where \(G\) is a solvable Lie group and (hence) \(\Gamma\) is a torsion-free polycyclic group. The author applies the theory of \textit{C. Miller} [Topology 44, No. 2, 351--373 (2005; Zbl 1149.57315)] and shows that the coordinate ring of the algebraic hull of \(\Gamma\) is represented by Miller's (closed) exponential iterated integrals induced from a \(\mathbb{Z}\)-lattice of \(\mathfrak{g}_\mathbb{C}^\ast\) and \(\bigwedge \mathfrak{g}_\mathbb{C}^\ast\). exponential iterated integral; algebraic hull; solvmanifold Nilpotent and solvable Lie groups, Coverings of curves, fundamental group, Rational homotopy theory, Special aspects of infinite or finite groups Algebraic hulls of solvable groups and exponential iterated integrals on solvmanifolds | 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 To any smooth projective curve one can associate an abelian variety, its Jacobian given by a period matrix. However, except for very special curves an explicit relation between a curve and its periodic matrix is unknown. This paper gives a brief informal description of a method to recover from a \(2 \times 2\) period matrix \(Z\) a numerical approximation of the polynomial of the corresponding curve \(C\). This introductory paper deals only with the special case of real curves with 3 real components. A more general treatment will appear in the thesis of \textit{Y. Chekouri}. The method is associating a standard basis of the homology of \(C\) to the double covering of \(P_ 1\) defined by the polynomial giving \(C\) and then numerically integrating the usual basis of holomorphic differentials of \(C\) along this standard basis. Jacobian; period matrix; real curves with 3 real components; standard basis Jacobians, Prym varieties, Period matrices, variation of Hodge structure; degenerations, Special algebraic curves and curves of low genus, Real algebraic and real-analytic geometry A numerical approach to defining a Legendre map in genus 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 Recently, \textit{A.~Knutson} and \textit{P.~Zinn-Justin} introduced a non-standard multiplication in \(M_n(\mathbb C)\), the space of all \(n\) by \(n\) matrices, denoted \(\bullet\) [see Adv. Math. 214, No. 1, 40--77 (2007; Zbl 1193.14068)]. By definition, the Brauer loop scheme is \(E:=\{M\in M_n({\mathbb C})\mid M\bullet M=0\}\). Knutson and Zinn-Justin have classified all the top dimensional irreducible components of \(E\). The main result of the article under review is that \(E\) is equidimensional, i.e., there are no other irreducible components. This was already conjectured in [loc. cit]. upper-triangular matrices; Brauer loop model; orbit; link diagram B. Rothbach, Equidimensionality of the Brauer loop scheme. Electron. J. Combin. 17 (2010), no. 1, Research Paper 75, 11 pp.MR 2651728 Zbl 1196.14045 Special varieties Equidimensionality of the Brauer loop 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 [in: Can. J. Math. 46, No. 2, 357-379 (1994; see the preceding review)] defined several kinds of uniformities for a 0-dimensional subscheme \(Z\) of \(\mathbb{P}^d\) generalizing the classical Cayley-Bacharach property of general hyperplane sections of an integral curve of \(\mathbb{P}^{d + 1}\) (in characteristic 0). Here, as well in the author's joint paper with \textit{L. Robbiano}, ``On maximal Cayley-Bacharach schemes'' [Commun. Algebra 23, No. 9, 3357-3378 (1995)] and the references quoted there the author studies \(Z\) with algebraic tools (local duality and the canonical module). His results are applied here also to the study of combinatorial properties (``purity'', ``flawless'') of the Hilbert function of \(Z\). 0-dimensional subscheme; hyperplane sections; Cayley-Bacharach schemes Kreuzer, M.: Some applications of the canonical module of a 0-dimensional scheme. Zero-dimensional schemes, proc. Conf. ravello 1992, 243-252 (1994) Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Projective techniques in algebraic geometry, Plane and space curves, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Some applications of the canonical module of a \(0\)-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 Abstract: We present a generalization of the functional equation for the Weierstrass \(\mathcal{P}\)-function for hyperelliptic surfaces of infinite genus arising from iteration of the horseshoe and baker maps. The ramified cover of these infinite genus surfaces over the complex plane are associated to a quadratic differential of finite norm with simple poles accumulating to infinity. We study the geometry of its critical trajectories emanating from these poles and their rate of accumulation. complex dynamics; hyperelliptic surface R. Chamanara, F. Gardiner and N. Lakic, A hyperelliptic realization of the horseshoe and baker maps, Ergodic Theory and Dynamical Systems 26 (2006), 1749--1768. Dynamical systems over complex numbers, Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable, Elliptic curves, Dynamical systems involving maps of the circle, Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics) A hyperelliptic realization of the horseshoe and baker maps | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The aim of the present paper is to study the structure of the punctual Hilbert schemes for the curve singularities of types \(E_6\) and \(E_8\). Our analysis uses computational methods to decompose a punctual Hilbert scheme into affine cells. We also use known results about the compactified Jacobians of singular curves. punctual Hilbert schemes; curve singularities of types \(E_6\) and \(E_8\); compactified Jacobians of singular curves Y. S\={}oma, M. Watari: Punctual Hilbert schemes for irreducible curve singularities of types E6and E8. J. Singularites. 8, 135-145, (2014). Parametrization (Chow and Hilbert schemes), Singularities of curves, local rings The punctual Hilbert schemes for the curve singularities of types \(E_6\) and \(E_8\) | 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 Plane valuations at infinity are classified in five types. Valuations in one of them determine weight functions which take values on semigroups of \(\mathbb Z^2\). These semigroups are generated by \(\delta \)-sequences in \(\mathbb Z^2\). We introduce simple \(\delta \)-sequences in \(\mathbb Z^2\) and study the evaluation codes of maximal length that they define. These codes are geometric and come from order domains. We give a bound on their minimum distance which improves the Andersen-Geil one. We also give coset bounds for the involved codes. evaluation code; simple \(\delta \)-sequence; valuation at infinity; order domain; coset bound Galindo, C; Pérez-Casales, R, On the evaluation codes given by simple \(\delta \)-sequences, AAECC, 27, 59-90, (2016) Geometric methods (including applications of algebraic geometry) applied to coding theory, Applications to coding theory and cryptography of arithmetic geometry On the evaluation codes given by simple \(\delta \)-sequences | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In order to investigate the intersection of two algebraic surfaces in \(R^3\), the authors use the Groebner bases. They especially consider one application of the reduced Groebner basis to plane sections of a special conoid, obtained by a motion of the system of generatrices along three directrices, where a cubic egg curve is one of them. An egg curve based conoid is analyzed from the aspect of project geometry, while its plane sections are analyzed from the aspect of analytic geometry. Using this example, a method based on the technique of reduced Groebner bases is developed. This method enables investigation of the existence of the assigned type of planar section (non-degenerated conic). The computer algebra system Maple and Maple Package ``Groebner'' are used to compute the reduced Groebner bases. The authors say that in some cases it is possible, using the presented method, to determine if the other algebraic surfaces have a planar section that consists of non-degenerate conics. reduced Groebner basis; egg curve; conoid; planar intersection Projective techniques in algebraic geometry, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) An application of Groebner bases to planarity of intersection of 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 The authors of this paper concerns about the problem of computing the Lebesgue volume of compact basic semialgebraic sets, using the Moment-SOS methodology. This involves solving an infinite-dimensional linear program (LP) and obtaining the volume by taking the limit of a sequence of solutions as it converges.
However, the convergence of the sequence can be slow due to the Gibbs phenomenon, which can impede the accuracy of the approximations. This issue can be resolved by introducing additional linear moment constraints obtained from an application of Stokes' theorem for integration on the set, which greatly improves convergence.
While this approach has shown significant promise, the rationale behind its efficacy was unclear so far. The authors provide a refined version of the LP formulation, demonstrating that when the set is a smooth super-level set of a single polynomial, the dual of the refined LP has an optimal solution that is a continuous function. As a result, the dual approximates a continuous function with a polynomial, eliminating the Gibbs phenomenon and further accelerating convergence.
The authors utilize recent results on Poisson's partial differential equation (PDE) in their proof of this technique. Overall, this paper presents a valuable contribution to the field of computational mathematics, providing a deeper understanding of the effective computation of Lebesgue volumes of semialgebraic sets. numerical methods for multivariate integration; real algebraic geometry; convex optimization; Stokes' theorem; Gibbs phenomenon Semialgebraic sets and related spaces, Semidefinite programming, Fields related with sums of squares (formally real fields, Pythagorean fields, etc.), Effectivity, complexity and computational aspects of algebraic geometry, Length, area, volume, other geometric measure theory, Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation, Integral geometry, Numerical integration, Approximation methods and heuristics in mathematical programming Stokes, Gibbs, and volume computation of semi-algebraic sets | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors analyze an existing method [\textit{J. Zhang} and \textit{X. Xin}, Pure Appl. Math. 24, No. 1, 133--135, 208 (2008; Zbl 1174.94386)] and get an improved implementation method of Koblitz curve cryptography for computing \(kP\). The analysis shows that the efficiency is increased by a factor of over 55\% and this method can be used to improve the efficiency of elliptic curve cryptography (ECC). elliptic curve cryptography; multiplying point; Koblitz curve Cryptography, Applications to coding theory and cryptography of arithmetic geometry An improved method for computing multiplying points on Koblitz curve cryptography | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\mathbb{P}^ n\) be the projective space of dimension \(n\) over an algebraically closed field of characteristic zero. A finite subscheme \(X\) in \(\mathbb{P}^ n\) is said to be in linearly general position if for every proper linear subspace \(L \subset \mathbb{P}^ n\) one has \(\deg (X \cap L) \leq 1 + \dim (L)\). If one cuts a reduced, irreducible, nondegenerate projective variety \(V\) of codimension \(n\) in \(\mathbb{P}^ r\) with a general linear subspace of dimension \(n\), one gets a set \(X\) of distinct points on this \(\mathbb{P}^ n\) in linear general position. This explains why this notion has a central role in many problems of algebraic geometry, especially in the so-called Castelnuovo theory as developed by Castelnuovo and more recently by \textit{J. Harris} (with the collaboration of \textit{D. Eisenbud}) [``Curves in projective space'', Sém. Math. Supér. 85 (1982; Zbl 0511.14014)] and \textit{M. L. Green} [J. Differ. Geom. 19, 125-171 (1984; Zbl 0559.14008)]. A classical result in this field is the so-called Castelnuovo lemma which states that any set \(X \subset \mathbb{P}^ n\) of \(d \geq 2n + 3\) points in linearly general position, which imposes a most \(2n + 1\) conditions on the system of quadrics in \(\mathbb{P}^ n\), lies on a rational normal curve. In the quoted paper, M. Green showed a more subtle result, the so-called strong Castelnuovo lemma, for any set \(X \subset \mathbb{P}^ n\) of points in linearly general position.
Continuing recent works of D. Bayer, D. Eisenbud and J. Harris, leading to extend Castelnuovo theory to more general finite subschemes of the projective space, in the present paper the authors extend the proof of the strong Castelnuovo lemma for zero-dimensional schemes in linearly general position from the reduced to the nonreduced case. linearly general position; strong Castelnuovo lemma; zero-dimensional schemes Cavaliere, M. P.; Rossi, M. E.; Valla, G.: The strong Castelnuovo lemma for zerodimensional schemes. (1994) Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Divisors, linear systems, invertible sheaves, Schemes and morphisms, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series The strong Castelnuovo lemma for zerodimensional 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 \(\mathcal{A}\) be a collection of \(n+1\) points in a lattice and let \(X \subseteq \mathbb{P}^n\) be the corresponding (possibly non-normal) projective toric variety. For each integer \(k \geq 0\), let \(\mathbf{F}_k(X)\) denote the Fano scheme of \(k\)-planes in \(X\), i.e. the fine moduli space that parametrises \(k\)-dimensional linear subspaces of \(\mathbb{P}^n\) which are contained in \(X\).
The authors of the article under review construct subvarieties of \(\mathbf{F}_k(X)\), i.e. families of \(k\)-planes contained in \(X\), by considering the faces of the convex hull of \(\mathcal{A}\) which are Cayley polytopes. Being Cayley is a well studied property of lattice polytopes, which was applied to the study of linear subspaces contained in polarised toric varieties in [\textit{C. Casagrande} and \textit{S. di Rocco}, Commun. Contemp. Math. 10, No. 3, 363--389 (2008; Zbl 1165.14036)] and [\textit{A. Ito}, Adv. Math. 270, 598--608 (2015; Zbl 1333.14048)].
The main result of the article under review is a combinatorial description of the irreducible components and of the connected components of \(\mathbf{F}_k(X)\). Moreover, in the case \(k = \mathrm{dim} (X) -1\), the authors are able to describe the schematic (i.e. non-reduced) structure of \(\mathbf{F}_k(X)\). toric varieties; Fano schemes; Cayley polytopes Toric varieties, Newton polyhedra, Okounkov bodies, Parametrization (Chow and Hilbert schemes), Fano varieties On Fano schemes of toric 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 We give a construction of linear error-correcting codes over an arbitrary algebraic surface, and then we focus on linear codes over ruled surfaces. At the end we discuss another approach to getting codes over algebraic surfaces using sections of rank two bundles. The new codes are not linear but do have a group structure. linear codes; algebraic surface; ruled surfaces Bouganis T.: Error correcting codes over algebraic surfaces. In: Lecture Notes in Computer Science, vol. 2643, pp. 169--179. Springer Verlag, Berlin, (2003). Geometric methods (including applications of algebraic geometry) applied to coding theory, Rational and ruled surfaces, Surfaces of general type Error-correcting codes over algebraic 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 Bertini\_real is a command line program for numerically decomposing the real portion of a one- or two-dimensional complex irreducible algebraic set in any reasonable number of variables. Using numerical homotopy continuation to solve a series of polynomial systems via regeneration from a witness set, a set of real vertices is computed, along with connection information and associated homotopy functions. The challenge of embedded singular curves is overcome using isosingular deflation. This decomposition captures the topological information and can be used for further computation and refinement. numerical algebraic geometry; cell decomposition; algebraic surface; algebraic curve; homotopy continuation; deflation Software, source code, etc. for problems pertaining to algebraic geometry, Real algebraic sets, Geometric aspects of numerical algebraic geometry, Symbolic computation and algebraic computation Bertini\_real: software for one- and two-dimensional real algebraic 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 Denote by \(\Gamma_g\) the mapping class group of the closed oriented surface \(\Sigma_g\) of genus \(g\geq 1\). \textit{W. Meyer} [Math. Ann. 201, 239--264 (1973; Zbl 0241.55019)] introduced the signature cocycle \(\tau_g\colon {\Gamma_g\times \Gamma_g\to \mathbb{Z}}\), defined by \(\tau_g(f_1,f_2)=\text{Sign}(E(f_1,f_2))\), where \(E(f_1,f_2)\) is the \(\Sigma_g\)-bundle over the pair of pants having \(f_1\) and \(f_2\) as the monodromies as one travels around two of the boundary circles, and \(\text{Sign}(E(f_1,f_2))\) is its usual signature invariant. Meyer also gave a linear algebraic formulation of \(\tau_g\), and proved that, for \(g=1\) or \(g=2\), there is a unique \(\mathbb{Q}\)-valued \(1\)-chain \(\phi_g\) such that \(\delta\phi_g=\tau_g\).
Various analogues of the Meyer functions \(\phi_g\) have been given for \(g>2\). The version in this paper is defined as follows. Let \(X\) be a smooth complex projective variety of dimension \(n\) embedded in the \(N\)-dimensional complex projective space \(\mathbb{P}_N\). For \(k=N-n+1\) let \(G_k(\mathbb{P}_N)\) be the Grassmannian of \(k\)-planes in \(\mathbb{P_N}\). Denote by \(D_X\) the set of \(k\)-planes in \(\mathbb{P}_N\) meeting \(X\) non-transversely (for \(n=2\) this is the classical dual variety of \(X\)), and let \(U^X=G_k(\mathbb{P}_N)-D_X\). For \(W\in U^X\), \(W\) and \(X\) meet transversely so their intersection is a compact Riemann surface. Putting \(C^X=\{(x,W)\in \mathbb{P}_N\times U^X \colon x\in X\cap W\}\) and \(p_X=p_2|_{C^X}\colon C^X\to U^X\) defines a \(\Sigma_g\)-bundle for some \(g\). Let \(\rho_X\colon \pi_1(U^X)\to \Gamma_g\) be its topological monodromy, and let \(\rho_X^*(\tau_g)\) be the pullback of \(\tau_g\). The main theorem of the paper is that there is a unique \(\mathbb{Q}\)-valued \(1\)-cochain \({\phi_X\colon \pi_1(U^X)\to \mathbb{Q}}\) such that \(\delta\phi_X=\rho_X^*(\tau_g)\). This \(\phi_X\), which depends on the projective embedding of \(X\), is called the Meyer function.
The second section of the paper gives extensive background material. The third contains the proof of the main theorem, and uses the theory of Lefschetz pencils to calculate \(M\) having \(\Sigma_g\)-fiberings \(p\colon M\to B\) with singular fibers over a finite set of points \(\{b_i\}\). For \(g\leq 2\), \textit{Y. Matsumoto} [Proc. Japan Acad., Ser. A 58, 298--301 (1982; Zbl 0515.55012) and in: Kojima, Sadayoshi (ed.) et al., Topology and Teichmüller spaces. Proceedings of the 37th Taniguchi symposium, Katinkulta, Finland, July, 24--28, 1995. Singapore: World Scientific. 123--148 (1996; Zbl 0921.57006)] gave a formula for \(\text{Sign}(M)\) in terms of local signatures near the singular fibers. Let \(F_i\) be the germ of neighborhoods of \(p^{-1}(b_i)\), represented by \(p^{-1}(\Delta)\) where \(\Delta\) is a small closed disk containing \(b_i\). The local monodromy around \(\partial \Delta\) gives an element \(x_{F_{i}}\in \Gamma_g\). For \(g\leq 2\), the local signature is defined by \(\sigma(F_i)=\phi_g(x_{F_i})+\text{Sign}(p^{-1}(\Delta))\), and the formula says that \(\text{Sign}(M)=\sum \sigma(F_i)\).
Matsumoto's formula has been extended to higher \(g\) in various ways. \textit{H. Endo} [Math. Ann. 316, No. 2, 237--257 (2000; Zbl 0948.57013)] established it for all \(g\), provided that the fibers are hyperelliptic, and the author [ibid. 342, No. 4, 923--949 (2008; Zbl 1162.57018)] obtained a version when the fibers are nonhyperelliptic of genus 3. \textit{T. Ashikaga} and \textit{K. Konno} [in: Usui, Sampei (ed.) et al., Algebraic geometry 2000, Azumino. Proceedings of the symposium, Nagano, Japan, July 20--30, 2000. Tokyo: Mathematical Society of Japan. Adv. Stud. Pure Math. 36, 1--49 (2002; Zbl 1088.14010)] and \textit{T. Ashikaga} and \textit{K. Yoshikawa} [in: Brasselet, Jean-Paul (ed.) et al., Singularities, Niigata-Toyama 2007. Proceedings of the 4th Franco-Japanese symposium, Niigata, Toyama, Japan, August 27--31, 2007. Tokyo: Mathematical Society of Japan. Advanced Studies in Pure Mathematics 56, 1--34 (2009; Zbl 1196.14024); appendix, \textit{K. Konno}, ibid. 35--38 (2009; Zbl 1196.14014)] gave a general version, but requiring a definition of local signature that uses algebro-geometric conditions, and allows fiber germs that are topologically trivial but have nontrivial local signature.
The author gives a different formulation of the local signature based on the Meyer functions \(\phi_X\), and using only the smoothness of \(M\) and \(f\). The idea is to define, under certain circumstances, a universal nonsingular \(\Sigma_g\)-fibering having fibers in a subset \(A\) of all Riemann surfaces \(\Sigma_g\). For such a universal fibering \((C_u,p_u,B_u)\), a fibering \(p\colon M\to B\) whose nonsingular fibers are in \(A\) admits classifying maps \(\rho\colon B-\{p_i\}\to B_u\) for the bundle \({M-\bigcup p^{-1}(b_i)\to B-\{p_i\}}\). Provided that \((C_u,p_u,B_u)\) admits a Meyer function -- that is, a \(1\)-cochain \(\phi_A\colon \pi_1(B_u)\to \mathbb{Q}\) satisfying \(\delta\phi_A=\rho_u^*(\tau_g)\) -- the local signature \(\sigma_A\) is defined by \(\sigma_A(F)=\phi_A(x_F) + \text{Sign}(p^{-1}(\Delta))\). This vanishes on locally trivial singular fibers, and the author proves that it satisfies the generalized Matsumoto formula \(\text{Sign}(M)=\sum \sigma_A(F_i)\). The remainder of the paper gives extensive calculations of \(\sigma_A\), focusing on the cases when \(A\) consists of the genus-\(4\) surfaces of rank \(4\), and when it consists of the genus-\(5\) nontrigonal surfaces. mapping class group; Meyer function; bounded cohomology; local signature Y. Kuno, The Meyer functions for projective varieties and their application to local signatures for fibered 4-manifolds, Algebr. Geom. Topol. 11 (2011), 145-195. Topology of Euclidean 4-space, 4-manifolds, Structure of families (Picard-Lefschetz, monodromy, etc.), Fibrations, degenerations in algebraic geometry The Meyer functions for projective varieties and their application to local signatures for fibered 4-manifolds | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors use the theory of Thom polynomials of right-left singularities to prove enumerative results on the points of a variety $X$ in projective space that have prescribed contact with a line. \par Right-left singularities are obtained by considering map germs from $m$-dimension to $n$-dimension up to holomorphic re-parametrization of the source and the target. Consider a family of such germs. Thom polynomials are universal formulas for the number of (or cohomology class represented by) points in the family where the germ belongs to a given singularity. Thom polynomials are rather well known for so-called K-singularities, but not much is known for right-left singularities. Hence, the authors use the interpolation method to calculate the Thom polynomials of certain right-left singularities for maps between low dimensional spaces. \par The second part of the paper is devoted to geometric applications, as follows. Consider a variety $X^m$ in $\mathbb{P}^{n+1}$. The notion ``a line $l$ having prescribed local contact type with $X$'' is re-phrased as a certain germ from $\mathbb{C}^m$ to $\mathbb{C}^n$ having a particular right-left singularity. Then, the Thom polynomial formulas developed in the first half of the paper are used to calculate the number or rather the degree of such loci. For smooth surfaces in $\mathbb{P}^2$ classical formulas are rediscovered, and are generalized to singular surfaces. New formulas are proved for surfaces in $\mathbb{P}^3$ and 3-folds in $\mathbb{P}^4$, in similar spirit. Thom polynomial; right-left singularity; enumeration of contacts; singular projections Classical problems, Schubert calculus, Characteristic classes and numbers in differential topology, Singularities of differentiable mappings in differential topology, Global theory of complex singularities; cohomological properties Thom polynomials in \(\mathcal A\)-classification I: counting singular projections of 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 In this sequel to a previous paper [J. Lond. Math. Soc. 65, 575--590 (2002; Zbl 1016.16018)] the authors study the scheme of line modules for several classes of quantum \(\mathbb{P}^3\), including Clifford algebras, homogenized \({\mathfrak {sl}}(2)\) and algebras associated to smooth quadrics in \(\mathbb{P}^3\). The authors also prove that a quantum \(\mathbb{P}^3\) with enough symmetry in its defining relations has a line scheme of dimension at least two, with infinitely many line modules incident to any point module. Clifford algebras; quantum \(\mathbb{P}^3\) Shelton B., Schemes of Line Modules Noncommutative algebraic geometry, Rings arising from noncommutative algebraic geometry, Schemes of line modules. II. | 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 smooth projective threefold of general type. Assume \(q ( X ) > 0\). We show that the \(m\)-canonical map of \(X\) is birational for all \(m \geqslant 5\). Rational and birational maps, \(3\)-folds On quint-canonical birationality of irregular 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 Here we study the stability of coherent systems defined over \(\mathbb{F}_q\) on genus 0 curves over \(\mathbb{F}_q\). Vector bundles on curves and their moduli, Finite ground fields in algebraic geometry Coherent systems on \(P^1_{{\mathbb F}_q}\) | 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 his fundamental paper [``Techniques de construction et théorèmes d'existence en géométrie algébrique. IV: Les schemes de
Hilbert'', Sémin. Bourbaki, exp. 221 (1961; Zbl 0236.14003)], \textit{A. Grothendieck} introduced the so called Hilbert scheme, which parametrizes all projective subschemes of the projective space with fixed Hilbert polynomial. Problems naturally arising in the study of the Hilbert scheme are irreducibility and number of components, dimension and smoothness. For instance, one knows that if \(X\subset \mathbb{P}^r\) is a local complete intersection projective subscheme and \(h^1(X,\mathcal N_{X,\mathbb{P}^r})=0\) (\(\mathcal N_{X,\mathbb{P}^r}=\) normal bundle of \(X\) in \(\mathbb{P}^r\)), then \(X\) is unobstructed, i.e. the corresponding point \([X]\) in the Hilbert scheme is smooth, and in such case the local dimension at \([X]\) is \(h^0(X,\mathcal N_{X,\mathbb{P}^r})\). But, in general, a necessary and sufficient condition for a subscheme to be unobstructed is not known [see also \textit{D. Mumford}, Am. J. Math. 84, 642--648 (1962; Zbl 0114.13106)] and \textit{E. Sernesi} [``Topics on families of projective schemes'', Queen's Pap. Pure Appl. Math. 73 (1986)].
Continuing previous works by \textit{J. O. Kleppe} [``The Hilbert-flag scheme, its properties and its connection with the Hilbert scheme. Applications to curves in \(3-\)space'', Preprint (part of thesis), Univ. of Oslo, March (1981)], \textit{G. Bolondi} [Arch. Math. 53, No. 3, 300--305 (1989; Zbl 0658.14005)], and \textit{M. Martin-Deschamps} and \textit{D. Perrin} [``Sur la classification des courbes gauches'', Astérisque 184--185 (1990; Zbl 0717.14017)], in the paper under review the author exhibits sufficient conditions and necessary conditions for unobstructedness of space curves \(C\subset \mathbb{P}^3\) which satisfy \(_{0}{\text{Ext}}^2_R(M,M)=0\) (e.g. of diameter\((M)\leq 2\)), and computes the dimension of the Hilbert scheme \(H(d,g)\) at \([C]\) under the sufficient conditions. Here \(C\subset \mathbb{P}^3\) denotes an equidimensional, locally Cohen-Macaulay subscheme of dimension one, \(d\) and \(g\) the degree and the arithmetic genus of \(C\subset \mathbb{P}^3\), \(M=\bigoplus_{v}H^1(\mathbb{P}^3, \mathcal I_C(v))\) denotes the Hartshorne-Rao module of \(C\), \(R=k[x_0,x_1,x_2,x_3]\) the polynomial ring over an algebraically closed field \(k\) of characteristic zero, and diameter\((M):=\max\{v\,| \,H^1(\mathbb{P}^3, \mathcal I_C(v))\neq 0\}-\min\{v\,| \,H^1(\mathbb{P}^3, \mathcal I_C(v))\neq 0\}+1\) (when \(H^1(\mathbb{P}^3, \mathcal I_C(v))=0\) for all \(v\), i.e. when \(C\) is arithmetically Cohen-Macaulay, then by \textit{G. Ellingsrud} [Ann. Sci. Éc. Norm. Supér. (4) 8, 423--431 (1975; Zbl 0325.14002)] one already knows that \(C\) is unobstructed).
In the diameter one case, the necessary and sufficient conditions coincide, and the unobstructedness of \(C\) turns out to be equivalent to the vanishing of certain graded Betti numbers of the free minimal resolution of the ideal \(I=\bigoplus_{v}H^0(\mathbb{P}^3, \mathcal I_C(v))\subset R\) of \(C\). The author also gives a description of the number of irreducible components of \(H(d,g)\) which contain an obstructed diameter one curve, and shows that in the diameter one case every irreducible component is reduced. Hilbert scheme; space curve; Buchsbaum curve; unobstructedness; cup-product; graded Betti numbers; ghost terms; linkage; normal module; postulation Hilbert scheme Dan, A.: Non-reduced components of the Noether-Lefschetz locus. Preprint arXiv:1407.8491v2 Parametrization (Chow and Hilbert schemes), Plane and space curves, Linkage, complete intersections and determinantal ideals, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series The Hilbert scheme of space curves of small diameter | 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 non-Archimedean field, and let \({\mathfrak X}\) be a formal scheme locally finitely presented over the ring of integers \(k^ 0\). In this work one constructs and studies the vanishing cycles functor from the category of étale sheaves on the generic fibres \({\mathfrak X}_ \eta\) of \({\mathfrak X}\) (which is a \(k\)-analytic space) to the category of étale sheaves on the closed fibre \({\mathfrak X}_{\overline s}\) of \({\mathfrak X}\) (which is a scheme over the residue field of the separable closure of \(k)\). One proves that if \({\mathfrak X}\) is the formal completion \(\hat {\mathcal X}\) of a scheme \({\mathcal X}\) finitely presented over \(k^ 0\) along the closed fibre, then the vanishing cycles sheaves of \(\hat {\mathcal X}\) are canonically isomorphic to those of \({\mathcal X}\) [as defined by \textit{P. Deligne} in Sémin. Géométrie algébrique, 1967-1969, SGA7 II, Lect. Notes Math. 340, Exposé XIII, 82-115 (1973; Zbl 0266.14008)]. In particular, the vanishing cycles sheaves of \({\mathcal X}\) depend only on \(\hat {\mathcal X}\), and any morphism \(\varphi:\hat {\mathcal Y} \to \hat {\mathcal X}\) induces a homomorphism from the pullback of the vanishing cycles sheaves of \({\mathcal X}\) under \(\varphi_{\overline s}:{\mathcal Y}_{\overline s} \to {\mathcal X}_{\overline s}\) to those of \({\mathcal Y}\). Furthermore, one proves that, for each \(\hat {\mathcal X}\), there exists a nontrivial ideal of \(k^ 0\) such that if two morphisms \(\varphi,\psi:\hat {\mathcal Y} \to \hat {\mathcal X}\) coincide modulo this ideal, then the homomorphisms between the vanishing cycles sheaves induced by \(\varphi\) and \(\psi\) coincide. These facts were conjectured by P. Deligne.
The second fact is deduced from a theorem on the continuity of the action of the set of morphisms between two analytic spaces on their étale cohomology groups. Its particular case states the following. Let \(X={\mathcal M} ({\mathcal A})\) be a \(k\)-affinoid space, and let \(f_ 1,\dots,f_ n\) be a \(k\)-affinoid generating system of elements of \({\mathcal A}\). Then for any discrete \(\text{Gal} (k^ s/k)\)-module \(\Lambda\) and any element of \(\alpha \in H^ q (X,\Lambda)\) there exist \(t_ 1, \dots,t_ n>0\) such that, for any pair of morphisms \(\varphi,\psi:Y \to X\) over \(k\) with \(\max_{y \in Y} | (\varphi^* f_ i-\psi^*f_ i)(y) | \leq t_ i\), \(1 \leq i \leq n\), one has \(\varphi^*(\alpha)=\psi^*(\alpha)\) in \(H^ q(Y,\Lambda)\). The essential ingredient of the proof is a generalization of the classical Krasner lemma. This result implies, in particular, the following fact. If a \(k\)-analytic group \(G\) acts on a \(k\)-analytic space \(X\), then the étale cohomology groups of \(X\) with compact support are discrete \(G(k)\)-modules. The present paper is based on the previous works of the author [``Spectral theory and analytic geometry over non-Archimedean fields'', Math. Surveys Monographs 33 (1990; Zbl 0715.14013) and ``Étale cohomology for non-Archimedean analytic spaces'', Publ. Math., Inst. Hautes Étud. Sci. 78, 5-171 (1993)]. analytic group; non-Archimedean field; formal scheme; vanishing cycles functor; étale sheaves V.\ G. Berkovich, Vanishing cycles for formal schemes, Invent. Math. 115 (1994), no. 3, 539-571. Étale and other Grothendieck topologies and (co)homologies, (Co)homology theory in algebraic geometry, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects), Local ground fields in algebraic geometry, Algebraic cycles Vanishing cycles for 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 On a smooth complex projective threefold \(X\) there are two curve counting theories, which are conjecturally equivalent: Donaldson-Thomas (DT) invariants, studied by \textit{D. Maulik}, \textit{N. Nekrasov}, \textit{A. Okounkov} and \textit{R. Pandharipande} in [Compos. Math. 142, No. 5, 1263--1285 (2006; Zbl 1108.14046)], and Pandharipande-Thomas (PT) invariants, studied by \textit{R. Pandharipande} and \textit{R. P. Thomas} in [Invent. Math. 178, No. 2, 407--447 (2009; Zbl 1204.14026)].
If \(\beta\in H_{2}(X,\mathbb{Z})\) and \(n\in\mathbb{Z}\), let \(I_{n}(X,\beta)\) be the Hilbert scheme of subschemes \(Z\) of \(X\) in the class \([Z]=\beta\) with holomorphic Euler characteristic \(\chi(\mathcal{O}_{Z})=n\), and \(I_{n,\beta}:=e(I_{n}(X,\beta))\) be its Euler characteristic. Moreover, let \(P_{n}(X,\beta)\) be the moduli space of stable pairs \((F,s)\), where \(F\) is a pure sheaf on \(X\) with Chern character \((0,0,\beta,-n+\beta\cdot c_{1}(X)/2)\) and \(s\) is a section of \(F\) with \(0\)-dimensional cokernel, and let \(P_{n,\beta}:=e(P_{n}(X,\beta))\). Letting \(Z^{I}_{\beta}(X)\) and \(Z^{P}_{\beta}(X)\) be the generating series of the \(I_{n,\beta}\) and of the \(P_{n,\beta}\) respectively, the author prove that \(Z^{P}_{\beta}(X)=Z^{I}_{\beta}(X)/Z^{I}_{0}(X)\), which is a topological version of the DT/PT-correspondence. For Calabi-Yau threefolds, this was obtained by \textit{Y. Toda} in [J. Am. Math. Soc. 23, No. 4, 1119--1157 (2010; Zbl 1207.14020)] and by \textit{T. Bridgeland} in [J. Am. Math. Soc. 24, No. 4, 969--998 (2011; Zbl 1234.14039)] with different methods. The strategy of the proof is the following: let \(C\) be any Cohen-Macaulay curve in \(X\) and \(I_{n,C}\) (resp. \(P_{n,C}\)) the Euler characteristic of the subset of \(I_{n}(X,\beta)\) (resp. \(P_{n}(X,\beta)\)) of subschemes whose underlying Cohen-Macaulay curve is \(C\) (resp. of pairs supported at \(C\)), and \(Z^{I}_{C}(X)\) (resp. \(Z^{P}_{C}(X)\)) their generating series. The authors show that \(Z^{I}_{C}(X)=Z^{P}_{C}(X)\cdot Z^{I}_{0}(X)\): integrating over all \(C\), one gets the previous statement involving \(Z^{I}_{\beta}(X)\) and \(Z^{P}_{\beta}(X)\). In order to relate \(Z^{I}_{C}(X)\) and \(Z^{P}_{C}(X)\), the authors provide a GIT wall-crossing between \(I_{n}(X,\beta)\) and \(P_{n}(X,\beta)\), and they study the relation between the Euler characteristic of the fibers of this wall-crossing using the Ringel-Hall algebra machinery of Joyce.
In section 2, the authors provide a GIT construction of \(P_{n}(X,\beta)\): Le Potier's construction of the moduli space of stable coherent systems presents \(P_{n}(X,\beta)\) as a quotient of a Quot scheme \(Q:=\mathrm{Quot}(\mathcal{H},P_{\beta})\), where \(\mathcal{H}=H^{0}(F(m))\otimes\mathcal{O}_{X}(-m)\). Here \(m\) is chosen so that for every pair \((F,s)\) the sheaf \(F(m)\) is globally generated, and we let \(V:=H^{0}(F(m))\).
Let \(R_{m}:=H^{0}(\mathcal{O}_{X}(m))\): for any pair \((F,s)\) there is an inclusion \(H^{0}(F)\subseteq V\otimes R_{m}^{*}\). As \(s\) spans a one-dimensional subspace of \(H^{0}(F)\), the pair \((F,s)\) corresponds to a point of the projective space \(\mathbb{P}(V\otimes R_{m}^{*})\). The authors construct a suitable closed subscheme \(\mathcal{N}\) of \(\mathbb{P}(V\otimes R^{*}_{m})\times Q\) parameterizing stable pairs, together with a natural action of \(\mathrm{SL}(V)\) and two ample \(\mathbb{Q}\)-linearizations \(\mathcal{L}_{0}\) and \(\mathcal{L}_{1}\) for the action of \(\mathrm{SL}(V)\). Using the Hilbert-Mumford criterion, the authors show that \(P_{n}(X,\beta)=\mathcal{N}^{s}//_{\mathcal{L}_{1}}\mathrm{SL}(V)\) and \(I_{n}(X,\beta)=\mathcal{N}^{s}//_{\mathcal{L}_{0}}\mathrm{SL}(V)\). Letting \(\mathcal{L}_{t}:=(1-t)\mathcal{L}_{0}+t\mathcal{L}_{1}\), the authors show that the quotient \(SS_{n}(X,\beta)=\mathcal{N}^{ss}//_{\mathcal{L}_{t*}}\mathrm{SL}(V)\) (for some \(0<t^{*}<1\)) has a stratification \(SS_{n}(X,\beta)=\coprod_{k=0}^{n}I^{\mathrm{pur}}_{n-k}(X,\beta)\times S^{k}(X)\), where \(I^{\mathrm{pur}}_{n-k}(X,\beta)\) is the locus of the Cohen-Macaulay closed subschemes of \(I_{n-k}(X,\beta)\), and \(S^{k}(X)\) is the \(k\)-th symmetric product of \(X\). Moreover, there are two morphisms \(\varphi_{P}:P_{n}(X,\beta)\longrightarrow SS_{n}(X,\beta)\) and \(\varphi_{I}:I_{n}(X,\beta)\longrightarrow SS_{n}(X,\beta)\), which are isomorphism on the subschemes of pairs with surjective sections and pure support respectively. This is the GIT wall-crossing between \(I_{n}(X,\beta)\) and \(P_{n}(X,\beta)\).
Now, letting \(I_{n}(X,C)=\varphi_{I}^{-1}(C,S^{n}(X))\), \(P_{n}(X,C)=\varphi^{-1}_{P}(C,S^{n}(X))\) for any Cohen-Macaulay curve \(C\) in \(X\), and \(I_{n,C}=e(I_{n}(X,C))\), \(P_{n,C}=e(P_{n}(X,C))\), the authors show that \(I_{n,C}=P_{n,C}+e(X)P_{n-1,C}+e(\mathrm{Hilb}^{2}(X))P_{n-2,C}+\dots+e(\mathrm{Hilb}^{n}(X))P_{0,C}\), which implies the relation between the generating series \(Z^{I}_{C}(X)\) and \(Z^{P}_{C}(X)\). This is obtained by using the Ringel-Hall algebra machinery of Joyce: if \(\mathcal{T}\) is the stack of \(0\)-dimensional sheaves, Joyce provides a Ringel-Hall algebra \(H(\mathcal{T})\) together with an integration map \(P_{q}:H(\mathcal{T})\longrightarrow\mathbb{Q}(q^{1/2})[t]\). The generating series \(Z^{I}_{C}(X)\) and \(Z^{P}_{C}(X)\) are the limit (for \(q\rightarrow 1\)) of the integration map \(P_{q}\) computed over some stacks mapping to \(\mathcal{T}\) (namely: for \(Z^{I}_{C}(X)\) the stack \(\Hom(\mathcal{I}_{C},-)\), whose fiber over \(T\) is \(\Hom(\mathcal{I}_{C},T)\), and for \(Z^{P}_{C}(X)\) the stack \(Ext^{1}(-,\mathcal{O}_{C})\), whose fiber over \(T\) is \(Ext^{1}(T,\mathcal{O}_{C})\)). Using convolutions and relations between these stacks, one concludes with the relation between \(Z^{I}_{C}(X)\) and \(Z^{P}_{C}(X)\). Turaev, V.G.: The Conway and Kauffman modules of a solid torus. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) \textbf{167}(Issled. Topol. 6), 7989 (1988) (Russian) [English translation: J. Soviet Math. \textbf{52}, 27992805 (1990)] Parametrization (Chow and Hilbert schemes), Algebraic moduli problems, moduli of vector bundles, Geometric invariant theory Hilbert schemes and stable pairs: GIT and derived category wall crossings | 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 purpose of this paper is twofold. The first is to apply the method introduced in the works of \textit{A. Nakayashiki} and \textit{F. A. Smirnov} [in: The Kowalevski property. Providence, RI: American Mathematical Society (AMS). CRM Proc. Lect. Notes 32, 239--246 (2002; Zbl 1064.14027)] on the Mumford system to its variants. The other is to establish a relation between the Mumford system and the isospectral limit \(\mathcal{Q}_g^{(I)}\) and \(\mathcal{Q}_g^{(II)}\) of the Noumi-Yamada system [\textit{M. Noumi} and \textit{Y. Yamada}, Funkc. Ekvacioj, Ser. Int. 41, No. 3, 483--503 (1998; Zbl 1140.34303)]. As a consequence, we prove the algebraically completely integrability of the systems \(\mathcal{Q}_g^{(I)}\) and \(\mathcal{Q}_g^{(II)}\), and get explicit descriptions of their solutions. Inoue, R.; Yamazaki, T.: Cohomological study on variants of the Mumford system, and integrability of the Noumi--Yamada system. Comm. math. Phys. 265, 699-719 (2006) Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions, Jacobians, Prym varieties, Relationships between algebraic curves and integrable systems Cohomological study on variants of the Mumford system, and integrability of the Noumi-Yamada system | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a scheme and denote by \({\mathcal Q}coh\, X\) the category of quasi-coherent sheaves on \(X.\) The purpose of this paper under review is to study the obstructions to a \(k\)-linear right exact functor \(F:{\mathcal Q}coh\, X \rightarrow {\mathcal Q}coh\,Y\) which commutes with direct limits being isomorphic to tensoring with a bimodule. This can be viewed as a generalization of the Eilenberg-Watts theorem given independently by \textit{S. Eilenburg} [J. Indian Math. Soc., n. Ser. 24, 231--234 (1961; Zbl 0100.26103)] and \textit{C. E. Watts} [Proc. Am. Math. Soc. 11, 5--8 (1960; Zbl 0093.04101)].
Let \(k\) denote a commutative ring, \(Z=\text{Spec}\,k\) and assume all schemes and products of schemes are over \(Z.\) Assume further that \(X\) is a quasi-compact and separated scheme and \(Y\) is a separated scheme. Let \(\text{Funck}_k({\mathcal Q}coh\, X, {\mathcal Q}coh\,Y)\) denote the category of \(k\)-linear functors. Its full subcategory consisting of right exact functors commuting with direct limits is denoted by \({\mathcal B}imod_k(X-Y).\) Call the functor \(W:{\mathcal B}imod_k(X-Y) \rightarrow{\mathcal Q}coh(X \times Y)\) defined by \textit{M. Van Den Bergh} [Non-Commutative \(\mathbb{P}^1\)-bundles over commutative schemes, preprint \url{arXiv:math/0102005v3} (2010)] the Eilenberg-Watts functor. Here is the main result of the paper.
Theorem: If \(F \in {\mathcal B}imod_k(X-Y),\) then there exists a natural transformation
\[
\Gamma_F:F \longrightarrow - \bigotimes_{\mathcal{O}_X} W(F)
\]
such that \(\ker \Gamma_F\) and \(\text{coker} \Gamma_F\) are totally global. Furthermore, \(\Gamma_F\) is an isomorphism if
1) \(X\) is affine or
2) \(F\) is exact or
3) \(F \cong - \otimes_{\mathcal{O}_X} \mathcal{F}\) in \({\mathcal Q}coh(X \times Y)\). schemes; quasi-coherent sheaves; Eilengerg-Watts theorem Nyman, A, The Eilenberg-Watts theorem over schemes, J. Pure Appl. Algebra, 214, 1922-1954, (2010) Categories in geometry and topology, Noncommutative algebraic geometry, Module categories in associative algebras, Functor categories, comma categories The Eilenberg-Watts theorem over 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 consider families of parameterizations of reduced curve singularities over a Noetherian base scheme and prove that the delta invariant is semicontinuous. In our setting, each curve singularity in the family is the image of a parameterization and not the fiber of a morphism. The problem came up in connection with the right-left classification of parameterizations of curve singularities defined over a field of positive characteristic. We prove a bound for right-left determinacy of a parameterization in terms of delta, and the semicontinuity theorem provides a simultaneous bound for the determinacy in a family. The fact that the base space can be an arbitrary Noetherian scheme causes some difficulties but is (not only) of interest for computational purposes. curve singularity; parameterization; delta invariant; completed tensor product; semicontinuity; determinacy Integral closure of commutative rings and ideals, Étale and flat extensions; Henselization; Artin approximation, Singularities in algebraic geometry, Deformations of singularities On delta for parameterized curve singularities | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\mathbb Z_{\geq 0}\) be the set of nonnegative integers. A monoid of \(\mathbb Z_{\geq 0}\) such that its complement is a finite set is called a numerical semigroup. The cardinality of the complement of a numerical semigroup is called genus of the numerical semigroup. A numerical semigroup is called sparse semigroup if every two subsequent gaps are spaced by at most 2. Let \(g\) be the genus and \(\ell_g\) the largest gap value.
In this paper, the structures of the sparse semigroups are considered. In particular, under a condition on \(g\) and \(\ell_g\), the sparse semigroup is completely determined. These results seem to have many applications to the theory of sigma function and Abelian function. Recently, the sigma function of algebraic curves are extensively studied in connection with the problem of mathematical physics. The sigma function is directly related with the defining equations of algebraic curves. The algebraic property of sigma function is shown for certain algebraic curves constructed from a semigroup with \(\ell_g=2g-1\). Since this paper gives the structure of a semigroup of \(\ell_g<2g-1\) completely, it seems to give a powerful tool to study the sigma function of algebraic curves constructed from a semigroup with \(\ell_g<2g-1\). numerical semigroups; sparse semigroups; Weierstrass points; algebraic curves Contiero, A.; Moreira, C. G. T. A.; Veloso, P. M., On the structure of numerical sparse semigroups and applications to Weierstrass points, J. Pure Appl. Algebra, 219, 3946-3957, (2015) Commutative semigroups, Riemann surfaces; Weierstrass points; gap sequences, Computational aspects of algebraic curves On the structure of numerical sparse semigroups and applications to Weierstrass 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 It is known that the Schur expansion of a skew Schur function runs over the interval of partitions, equipped with dominance order, defined by the least and the most dominant Littlewood-Richardson filling of the skew shape. We characterise skew Schur functions (and therefore the product of two Schur functions) which are multiplicity-free and the resulting Schur expansion runs over the whole interval of partitions, i.e., skew Schur functions having Littlewood-Richardson coefficients always equal to 1 over the full interval. skew Schur functions; ribbons; positive Littlewood-Richardson coefficients; multiplicity-free; dominance order; interval support Symmetric functions and generalizations, Combinatorial aspects of representation theory, Combinatorial aspects of partitions of integers, Classical problems, Schubert calculus, Representations of finite symmetric groups Multiplicity-free skew Schur functions with full interval support | 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 concerned with real and complex \((\mu,S)\)-frames, finite frames having a given frame operator \(S\) and consisting of vectors \(\{f_1, f_2, \dots, f_N\}\) having specified lengths \(\{\mu_1, \mu_2, \dots, \mu_N\}\), respectively. In terms of the synthesis operator \(F=[f_1 f_2 \cdots f_N]\), a \((\mu,S)\)-frame satisfies the quadratic equations \(FF^*=S\) and \(\mathrm{diag}(F^* F)=\mu\). From a geometric viewpoint, the condition involving \(S\) means that \(F\) is in an ellipsoidal, deformed Stiefel manifold, while the norm condition means that it is in a product of spheres. The intersection of these two manifolds forms a variety with possible singular points. The intersection of these two matrix manifolds forms a real algebraic variety with possible singular points.
The paper characterizes the singular points in this variety as orthodecomposable frames. This result is the generalization of earlier work with Dykema on spherical tight frames [\textit{K. Dykema} and \textit{N. Strawn}, ``Manifold structure of spaces of spherical tight frames'', Int. J. Pure Appl. Math. 28, No. 2, 217--256 (2006; Zbl 1134.42019)]. A finite frame is orthodecomposable if it can be partitioned into subsequences which are spanning for mutually orthogonal subspaces. If a \((\mu,S)\)-frame is not orthodecomposable, then it is at a non-singular point in the variety. An explicit, real-analytic parameterization for the neighborhood of any non-singular point is constructed. The method of construction begins with a suitable permutation of the frame vectors, assigning the order inductively so that the last \(d\) vectors form a basis \(B\) for the Hilbert space. The parameterization is realized by perturbing the first \(N-d\) vectors in a norm-preserving way and by compensating the change in the frame operator with the remaining \(d\) vectors in \(B\). There is some residual freedom in the choice of the remaining vectors which amounts to choosing the entries below the subdiagonal of the synthesis operator of \(B\). The construction is explained in detail for \((\mu, S)\)-frames in real Hilbert spaces. The appendix of the paper explains a similar strategy for complex Hilbert spaces. finite frames; real and complex Hilbert space; Stiefel manifold; tangent space; nonsingular points; local coordinates Strawn, N.: Finite frame varieties: nonsingular points, tangent spaces, and explicit local parameterizations. J. Fourier Anal. Appl. 17, 821--853 (2011) General harmonic expansions, frames, Computational aspects of higher-dimensional varieties, Special varieties, Special matrices, Differentiable manifolds, foundations Finite frame varieties: Nonsingular points, tangent spaces, and explicit local parameterizations | 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 irreducibility and the smoothness of the generic plane curve of degree \(d\) passing through \(r\) generic points with prescribed multiplicities \(m_i\). We prove that, under the assumptions that the multiplicities are at most 3 and the degree is high enough, these curves are irreducible and smooth away from the prescribed singularities. plane curve; prescribed multiplicities; prescribed singularities Mignon, T., Systèmes linéaires de courbes planes à singularités ordinaires imposées, C. R. Acad. Sci. Paris Ser. I Math., 327, 7, 651-654, (1998) Singularities of curves, local rings, Plane and space curves Linear systems of plane curves with prescribed ordinary singularities | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author gives some theorems on the subject of group scheme actions on a field via inner automorphisms coming from a larger algebra. Chase, S. U.: Group scheme actions by inner automorphisms. Comm. algebra 4, 403-434 (1976) Group schemes, Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.) Group scheme actions by inner automorphisms | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems From the existence of smooth Castelnuovo curves and the existence of integral non-degenerate curves of all geometric genera from 0 to the Castelnuovo bound [shown by the author in Compos. Math. 41, 107-126 (1980; Zbl 0399.14018)] the author deduces that for any integer \(p\geq 0\), \(d>1\), \(n>1\), there exists a smooth irreducible curve of genus p which contains a linear system \(g_ d^{N(p,d)}\) and a smooth irreducible curve of genus p containing a \(g^ n_{D(p,n)}\). Here N(p,d) is the maximal possible dimension and D(p,n) is the minimal possible degree with respect to p, d, n for such linear systems, the bounds coming from inverting the Castelnuovo bound. Castelnuovo curves; Castelnuovo bound; linear system Special algebraic curves and curves of low genus, Divisors, linear systems, invertible sheaves A note about 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 \(H(d,g)_{\mathrm{sc}}\) be the Hilbert scheme of smooth connected space curves. In the literature, many questions have been considered including smoothness of components, the existence of non-reduced components, the dimension of components, etc. In this paper the authors consider maximal irreducible closed subsets \(W\) of \(H(d,g)_{\mathrm{sc}}\) whose general element corresponds to a curve \(C\) lying on a surface \(S\) of degree \(s\), especially when \(s=4\) and \(s=5\). The authors ask when \(W\) is a non-reduced, or generically smooth, component. They also determine \(\dim W\), and they obtain connections with the Picard group of \(S\). In an appendix, the first author finds new classes of non-reduced components of \(H(d,g)_{\mathrm{sc}}\), making progress on a conjecture about non-reduced components for maximal families \(W \subset H(d,g)_{\mathrm{sc}}\) of linearly normal curves on a smooth cubic surface; he significantly extends the known range where the conjecture holds. space curves; quartic surfaces; cubic surfaces; Hilbert scheme; Hilbert-flag scheme Kleppe, J.O., Ottem, John C.: Components of the Hilbert scheme of space curves on low-degree smooth surfaces. Int. J. Math. \textbf{26}(2) (2015) Parametrization (Chow and Hilbert schemes), Divisors, linear systems, invertible sheaves, Picard schemes, higher Jacobians, \(K3\) surfaces and Enriques surfaces, Plane and space curves Components of the Hilbert scheme of space curves on low-degree smooth 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 Classically, liaison theory has studied the equivalence relation on subschemes of projective space generated by identifying two subschemes if their union is a complete intersection. Such schemes are said to be CI-linked. In order to extend this to allow common components, there is an algebraic version involving ideal quotients of the corresponding homogeneous ideals. In the last decade, there has been a great deal of activity around the idea of replacing ``complete intersection'' by ``arithmetically Gorenstein scheme''. The resulting equivalence relation is called \(G\)-liaison. If we insist on even numbers of links, we obtain even \(G\)-liaison, or the closely related \(G\)-biliaison (which puts it in the context of divisors on arithmetically Cohen-Macaulay subschemes). Background material on liaison theory can be found in the reviewer's book [``Introduction to liaison theory and deficiency modules''. Boston, MA: Birkhäuser (1998; Zbl 0921.14033)].
A long-standing open question is whether every arithmetically Cohen-Macaulay subscheme of projective space is in the even \(G\)-liaison class of a complete intersection (so-called glicci schemes). The analogous statement for even CI-liaison has long been known to be false, although it is true in codimension two. Because this question is not known even in codimension three, any large class for which it is true gives an interesting result. (Of course, even one subscheme for which it is false would be immensely interesting!) The first large class for which it was shown to be true is that of standard determinantal schemes [due to \textit{J. O. Kleppe}, the reviewer, \textit{R. Miró-Roig, U. Nagel} and \textit{C. Peterson}, Gorenstein liaison, complete intersection liaison invariants and unobstructedness, Mem. Am. Math. Soc. 732 (2001; Zbl 1006.14018)].
In this paper, the author considers the slightly stronger question of \(G\)-biliaison. She proves it for another large class, namely that of symmetric determinantal schemes. These are defined by the minors of a homogeneous symmetric matrix with polynomial entries, under the assumption that the codimension is the largest possible given the size of the matrix and of the minors defining the ideal. The author explicitly describes the biliaisons used to arrive at a complete intersection (which in fact is shown to be a linear variety). All divisors involved in the \(G\)-biliaisons are symmetric determinantal, and the \(G\)-biliaisons are performed on almost-symmetric determinantal schemes, in analogy to the proof given by Kleppe et al. mentioned above. liaison; \(G\)-biliaison; glicci scheme; arithmetically Cohen-Macaulay scheme; arithmetically Gorenstein scheme; minor; symmetric matrix; determinantal scheme Gorla, E., The G-biliaison class of symmetric determinantal schemes, J. Algebra, 310, 2, 880-902, (2007) Linkage, complete intersections and determinantal ideals, Determinantal varieties, Linkage, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) The \(G\)-biliaison class of symmetric determinantal 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 prove that, for \(d \leq 9\), the Gorenstein locus of the Hilbert scheme parametrizing \(d\) points in projective space is irreducible. Moreover they give a complete description of the singular sub-locus. This description leads the authors to conjecture on the nature on this singular locus for any \(d\).
The paper begins with an introduction on the history of the study of Grothendieck's Hilbert scheme, quickly bringing the reader up to date on the most important advancements. The picture of interest -- the locus of Gorenstein subschemes -- is introduced along with the relevant state of knowledge, including the important fact that it is an open subset of the Hilbert scheme.
The paper proceeds with thorough and explicit calculations of rings with particular Hilbert functions. This information is then put together to establish the main theorems of the paper. Along the way some useful examples are given. punctual Hilbert scheme; Gorenstein Casnati, G.; Notari, R., On the Gorenstein locus of some punctual Hilbert schemes, \textit{J. Pure Appl. Algebra}, 213, 2055-2074, (2009) Parametrization (Chow and Hilbert schemes), Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) On the Gorenstein locus of some punctual 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 It is well-known that a smooth variety has a reduced (moreover smooth) arc space. But there are many examples that schemes are reduced and the corresponding arc spaces are not reduced. In this paper, the author proves that a plane curve over a subfield \(k\) of \(\mathbb{C}\) has a reduced arc space if and only if it is smooth over \(k\). arc space; plane curve Julien Sebag, ''Arcs schemes, derivations and Lipman's theorem'', J. Algebra347 (2011), p. 173-183 Plane and space curves, Arcs and motivic integration Arcs schemes, derivations and Lipman's theorem | 0 |
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