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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 g be a fixed integer. Let \(X_ g=Spec(R_ g)\) (where \(R_ g=S.(S_ gV)\), V being a two-dimensional vector space) be the space of binary forms of degree \( g.\) Let \(X_{p,g}\subset X_ g\) be the subset of binary forms having a root of multiplicity \(\geq p\) and \(J_ p\) be the ideal of polynomial functions vanishing on \(X_{p,g}\). The main result of the paper is the following theorem 3:
The generators of the ideal \(J_ p\) have degrees \(\leq 4\) for \(p\geq [g/2]+1.\)
The generators of \(J_ p\) are explicitly described for \(p>[g/2]+1\). The Hilbert function of \(R_ g/J_ p\) and the decomposition of \(R_ g/J_ p\) into representations of SL(V) are also explicitly described. generators of ideal of algebraic set; binary forms; polynomial functions; Hilbert function Weyman, J., The equations of strata for binary forms, J. Algebra, 122, 1, 244-249, (1989) Schemes and morphisms, Rational and unirational varieties The equations of strata for binary forms | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 study, we consider a graph surface associated to Cobb-Douglas production function in economics on product time scales. We classify this surface based on the flatness and minimality properties for several product time scales. Then, we interpret the obtained results from the perspective of production theory in economics. Therefore, we extend the known results in Euclidean geometry by considering time scale calculus. time scales; surfaces; delta derivatives; production function Surfaces of general type, Differentiation theory (Gateaux, Fréchet, etc.) on manifolds, Production theory, theory of the firm A certain class of surfaces on product time scales with interpretations from economics | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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{J. Bochnak} and \textit{W. Kucharz} [Invent. Math. 97, 585-611 (1989; Zbl 0687.14023)] proved that for every \(C^{\infty}\) m-dimensional compact manifold M there exist countably many algebraic manifolds \(X_ i\) diffeomorphic to M, where \(X_ i\) are irreducible \({\mathbb{R}}\)-algebraic subsets of \({\mathbb{R}}^{2m+1}\) and for \(i\neq j\), \(X_ i\) is not birationally equivalent to \(X_ j\). They conjectured that in fact there exist uncountably many such birationally inequivalent algebraic models of M.
The author of the present paper proves this conjecture roughly in the following way. For a compact m-dimensional \(C^{\infty}\) submanifold M of \({\mathbb{R}}^{2m+1}\) he constructs a holomorphic map between complex spaces p: \(X\to T\) defined over \({\mathbb{R}}\), where locally for \(U\subset T\), \(p^{-1}(U)\hookrightarrow U\times {\mathbb{P}}^ N({\mathbb{C}})\) for some N (and p commutes with the Cartesian projection) such that for \(t\in T\), the real part of \(X_ t=p^{-1}(t)\) is diffeomorphic to M, \(X_ t\) is not birationally equivalent to \(X_ u\) if \(t\neq u\), moreover there exists a family of \(C^{\infty}\) embeddings \(g_ t: M\hookrightarrow {\mathbb{R}}^{2m+1}\) (arbitrarily close to the identity) such that for every \(t\in T\), \(X_ t\) is a complexification of \(g_ t(M)\). In the proof the author uses the proofs of the paper by J. Bochnak and W. Kucharz, and standard methods of complex algebraic geometry, e.g. the theory of deformation and moduli.
The discussed conjecture has been independently proved by J. Bochnak in another way. algebraic model of a \(C^{\infty }\) manifold; real algebraic; coarse moduli scheme; Kuranishi family; Hilbert scheme; polarized manifold; algebraic space; Hilbert polynomial Ballico, E., ?An addendum on algebraic models of smooth models?,Geom Dedicata 38 (1991), 343-346. Real-analytic and Nash manifolds, Real-analytic and semi-analytic sets, Parametrization (Chow and Hilbert schemes), Moduli, classification: analytic theory; relations with modular forms, Minimal model program (Mori theory, extremal rays), Deformations of complex structures An addendum on algebraic models of smooth manifolds | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(F=F/k\) be an algebraic function field of genus \(g=g(F)\geq 2\) defined over the finite field \(k={\mathbb F}_q\) of order \(q\), let \(h=h(F)\) denote the class number of \(F\). The goal of this paper is twofold: (i) To improve bounds on \(h\); (ii) Estimate the class number of all steps of certain towers of algebraic function fields over finite fields. For (i) the starting point is the exact formula \(S(F)=hR(F)\, (*)\) stated in [\textit{G. Lachaud} and \textit{M. Martin-Deschamps}, Acta Arith. 56, No. 4, 329--340 (1990; Zbl 0727.14019)], where \(S(F)\) depends on the numbers \(A_n(F)\) of effective divisors of degree \(n\) of \(F\) with \(0\leq n\leq g-1\) while \(R(F)\) depends on the reciprocal roots \((\pi,\bar\pi_i)\) of the enumerator of the zeta-function of \(F\). In general, one does not expect to compute all the data involved in \((*)\) and in fact one is only able to deduce accurate estimates for \(h\) such as \(h\geq q^{g-1}(q-1)^2/(q+1)(g+1)\) (loc. cit.). In this paper quite involved formulas give bounds on \(h\) which in some cases improve on among all the known bounds.
A tower \(\mathcal F\) over \(k\) is a sequence, \((F_i)_{i\geq 0}\) of function fields over \(k\) such that \(k\) is algebraically closed in \(F_i\) for each \(i\), \(F_i\subsetneq F_{i+1}\) with \(F_{i+1}|F_i\) finite and separable, and \(g(F_i)\to \infty\) as \(i\to\infty\). Under the hypothesis \(\beta_m(\mathcal F):=\lim{i\to\infty} B_m(F_i)/g(F_i)>0\), for some \(m\), where \(B_m(F_i)\) is the number of places of degree \(m\) of \(F_i|k\), the authors study the following parameters of each step \(F_i|k\) of \(\mathcal F\): the genus, the number of places of certain degree, and the class number. The results are illustrated by taking as a building block the tower in [\textit{A. Garcia} and \textit{H. Stichtenoth}, Invent. Math. 121, No. 1, 211--222 (1995; Zbl 0822.11078)], or the so called recursively defined towers in the sense that \(F_{i+1}=F_i(x_{i+1})\) with \(f\in k[X,Y]\), \(f(x_i,x_{i+1}=0\). finite field; Jacobian; algebraic function field; class number; tower Algebraic functions and function fields in algebraic geometry, Finite fields (field-theoretic aspects) Effective bounds on class number and estimation for any step of towers of algebraic function fields over finite fields | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This book is a remarkable introduction from various points of view (differential, topological, analytical and algebro-geometric) of a historical, but nowadays in top of research, theory, namely that of completely integrable Hamiltonian systems. Although, the most recent framework of these systems is provided by Poisson geometry, the author restricts to the case of symplectic manifolds in order to handle a lot of examples with a mechanical or geometrical flavor such as Hénon-Heiles, the simple and spherical pendulum, the anharmonic oscillator, the rigid body with a fixed point and so on.
The organization of the book is excellent. Its first chapter contains all the background material in symplectic geometry which is needed. The technical part of the book covers the next three chapters and is presented in a highly professional way. More precisely, the second chapter provides an introduction to the main tool of the theory, namely action-angle variables, through the celebrated Liouville-Arnold theorem. The next chapter is devoted to the algebraic approach based on differential Galois theory. More specifically, the author emphasizes the role of a theorem by \textit{J. J. Moralez-Ruiz} and \textit{J. P. Ramis} [Methods Appl. Anal. 8, No. 1, 33--95 (2001; Zbl 1140.37352); ibid., 97--111 (2001; Zbl 1140.37354)] in order to show that certain Hamiltonian systems are not completely integrable. The last chapter turns to algebraic geometry which in some cases is able to give a more tractable version of the Liouville-Arnold theorem. Finally, two appendices summarize useful definitions and properties from differential Galois theory and the theory of algebraic curves, respectively.
As a reference work, this monograph is invaluable while as a work of pedagogy, through its several examples and exercises, could be used in a graduate course being an extremely helpful textbook. Complemented sometimes with the author's previous work [Spinning tops. A course on integrable systems, Cambridge Studies in Advanced Mathematics, 51, Cambridge: Cambridge Univ. Press (1996; Zbl 0867.58034)] this book will become a standard reference in this field. Hamiltonian system; symplectic geometry; action-angle variables; integrable systems; not completely integrable systems; Galois theory; Liouville-Arnold theorem; algebraic curves Audin, M., Hamiltonian systems and their integrability, (2008), American Mathematical Society (AMS)/Société Mathématique de France (SMF) Providence, RI/Paris Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory, Poisson manifolds; Poisson groupoids and algebroids, Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics, , Differential algebra, Relationships between algebraic curves and integrable systems Hamiltonian systems and their integrability. Transl. from the French by Anna Pierrehumbert. Translation edited by Donald Babbitt | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The main result of the paper is the following theorem.
Let \(k\) be an infinite field and \(G\) a smooth reductive group-scheme over \(k\). Let \(\mathbb{X}\) be an irreducible smooth scheme of finite type over \(k\). Let \((P, \pi, m)\) be a principal \(G\)-bundle on \(\mathbb{X}\) which is trivial over a nonempty Zariski open subset of \(\mathbb{X}\). Then for each \(x \in \mathbb{X}\) there is a Zariski open neighbourhood \(\mathbb{X} (x)\) of \(x\) in \(\mathbb{X}\) such that \((P, \pi, m)\) is trivial over \(\mathbb{X} (x)\).
This result is an affirmation of the conjecture of Grothendieck and Serre in the case of an infinite field and \(\mathbb{X}\) is of finite type over \(k\). It is new only when \(k\) is not perfect and \(G\) has an anisotropic factor over \(k\); when \(k\) is infinite perfect or \(G\) has no anisotropic factors, the theorem is due to \textit{J.-L. Colliot-Thélène} and \textit{M. Ojanguren} [see Publ. Math., Inst. Hautes Étud. Sci. 75, 97-122 (1992; Zbl 0795.14029)]. principal \(G\)-bundle; group-scheme Raghunathan, M.S.: Erratum: ''Principal bundles admitting a rational section''. Invent. Math. 116(1--3), 409--423 (1994) (Invent. Math. 121(1), 223 (1995)) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Group actions on varieties or schemes (quotients) Principal bundles admitting a rational section | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The aim of this paper is the study of the Artin-Schreier tower \({\mathcal H}\) recursively defined by the equation \(y^2+y=\frac{ x}{x^2+x+1}\) over \({\mathbb F}_2\). In [Finite Fields Appl. 12, No. 1, 56--77 (2006; Zbl 1104.11052)], \textit{P. Beelen} et al. studied the classification, according to their asymptotic behavior, of recursive towers of function fields over a finite field \({\mathbb F}_q\). They studied recursive towers defined by equations of the form \(f(y)=g(x)\) where \(f\) and \(g\) are rational functions over \({\mathbb F}_q\) and obtained classification results for towers of Kummer and of Artin-Schreier types and they gave a complete list of all \((f,g)\)-towers of Artin-Schreier type with \(\deg f=\deg g=2\) over \({\mathbb F}_2\). It turns out that all cases had been considered before except for the Artin-Schreier tower defined by \(y^2+y=\frac{x}{x^2+x+1}\).
For a tower \({\mathcal F}=\{F_i\}_i\) of function fields over \({\mathbb F}_q\), it is defined \(\lambda({\mathcal F})= \lim_{i\to\infty}\frac{N(F_i)}{g(F_i)}\), where \(N(F_i)\) is the number of places of degree \(1\) in \(F_i\) and \(g(F_i)\) denotes the genus of \(F_i\). We have \(\lambda({\mathcal F})\leq A(q):= \limsup_{g\to \infty}\frac{N_q(g)}{g}\leq \sqrt{g}-1\), where \(N_q(g)\) is the maximum possible number of places of degree \(1\) in a function field over \({\mathbb F}_q\) of genus \(g\) and the last inequality is the Hasse-Weil bound.
The genus \(\gamma({\mathcal F})\) of \({\mathcal F}\) over \(F_0\) is defined as \(\gamma({\mathcal F}):=\lim_{i\to\infty}\frac{g(F_i)}{[F_i:F_0]}\). The tower is \textit{asymptotically good} if \(\lambda({\mathcal F})>0\) and \(\gamma({\mathcal F})<\infty\). The authors first show that \(\gamma({\mathcal H}) \leq 4\) over \({\mathbb F}_{2^s}\) for every positive integer \(s\). The main result is that \({\mathcal H}\) is asymptotically good over \({\mathbb F}_4\) with \(\lambda({\mathcal H})\geq \frac{1}{8}\). As a corollary it follows that \({\mathcal H}\) is asymptotically good over \({\mathbb F}_{2^s}\) for any even integer \(s\) with \(\lambda({\mathcal H})\geq \frac{1}{8}\). (4 Refs.) global function fields; Artin-Schreier extensions; genus; rational places; towers; limit of towers; asymptotically good towers Arithmetic theory of algebraic function fields, Curves over finite and local fields, Algebraic functions and function fields in algebraic geometry A problem of Beelen, Garcia and Stichtenoth on an Artin-Schreier tower in characteristic 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 aim of this paper is to study the extension of Painlevé-gauge theory correspondence to circular quivers by focusing on the special case of \(\mathrm{SU}(2)\; \mathcal{N}=2^*\) theory.
The authors show that the Nekrasov-Okounkov partition function of this gauge theory provides an explicit combinatorial expression and a Fredholm determinant formula for the tau-function describing isomonodromic deformations of \(\mathrm{SL}_2\) flat connections on the one-punctured torus. This is achieved by reformulating the Riemann-Hilbert problem associated to the latter in terms of chiral conformal blocks of a free-fermionic algebra. This viewpoint provides the exact solution of the renormalization group flow of the \(\mathrm{SU}(2)\; \mathcal{N}=2^*\) theory on a self-dual-background and, in the Seiberg-Witten limit, an elegant relation between the IR and UV gauge couplings.
The paper is organized as follows. In Section 2 the authors recall known results on the two-sphere. In Section 3 they illustrate the new results obtained for the one-punctured torus case, namely they discuss the free fermion
realization of the Riemann-Hilbert problem kernel and the corresponding isomonodromic tau function. In Section 4 they provide a Fredholm determinant formula for the latter and discuss gauge theory/topological string implications of their results. The conclusions are reported in Section 5. The remaining third of the paper consists of technical appendices. Yang-Mills; gauge theory; tau-function; isomonodromic deformations; Riemann-Hilbert problem Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Vector bundles on curves and their moduli, Riemann-Hilbert problems in context of PDEs, Algebraic theory of abelian varieties, Yang-Mills and other gauge theories in quantum field theory \(\mathcal{N} = 2^*\) gauge theory, free fermions on the torus and Painlevé VI | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(K={\mathbb{F}}_ r(T)\) \(({\mathbb{F}}_ r\) the finite field with \(r=p^ m\) elements), one has notions analogous with the following data over the field \(K={\mathbb{Q}}\) of the rationals: cyclotomic extensions, Teichmüller character, Bernoulli numbers.
In the note, a summary of some new results on divisibility properties of p-class groups of such extensions is given. global function fields; Kummer criterion; divisibility; p-class groups Goss, D, Units and class groups in the arithmetic of function fields, Bull. Am. Math. Soc., 13, 131-132, (1985) Arithmetic theory of algebraic function fields, Finite ground fields in algebraic geometry Units and class-groups in the arithmetic theory 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 Algebraic combinatorics lies at the junction of not only algebra and combinatorics, as the name suggests, but also areas such as number theory, algebraic geometry and topology. In this paper, a leader in the field surveys some of the recent progress that has been made in relation to the latter three areas. It is a tale that will appeal to both the experts and the novices, with its clarity, illustrative examples and comprehensive bibliography.
First we are given an informal taste of cluster algebras and the work of Fomin and Zelevinsky proving the integrability of the sequences Somos-4 through -7. We also get to glimpse at the relation between Somos-4 and perfect matchings. Second, after connecting symmetric functions to Schubert calculus we are introduced to the recent work of Postnikov on toric Schur functions, whose expansion into Schur functions yield Gromov-Witten invariants as the coefficients of the expansion. Lastly we are taken on a tour of increasingly complex polytopes. Our first stop is simplicial polytopes whose h-vectors are completely characterized by the g-theorem. We then meet a combinatorial description of the toric h-vector for rational polytopes. Moving to non-rational polytopes we are introduced to the recent work of Karu on the hard Lefschetz theorem that a certain bijection exists. Finally, we are told of the half hard Lefschetz theorem that a certain injection exists for matroid complexes, recently proved by Swartz. cluster algebra; toric Schur function; hard Lefschetz theorem Stanley, R. P.: Recent developments in algebraic combinatorics. Israel J. Math. 143, 317-339 (2004) Symmetric functions and generalizations, Combinatorial aspects of representation theory, Group actions on posets and homology groups of posets [See also 06A09], Special sequences and polynomials, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Polyhedra and polytopes; regular figures, division of spaces Recent developments in algebraic 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 The author considers the Lagrangian field theory for some physical fields. It is known that in many cases there exists Poisson brackets defined on the space of local functionals of a fiber bundle. Three different cases of interest are investigated. These cases depend on the dimensions of the vector bundles characterizing the theories and the corresponding Hamiltonian differential operators. It is known that the evolutionary systems of Hamiltonian type have the form \(\partial u/\partial t = {\mathcal{D}}\delta\mathcal{H}\), where \(\mathcal{H}\) is a local functional called the Hamiltonian, \(\mathcal{D}\) is a Hamiltonian differential operator, \(\delta \) is the variational derivative. The solutions of the Hamiltonian system are sections of the fiber bundle. Recall that the variational derivative of a local functional \({\mathcal{P}}=\int_{M}Pv\) is given by \({\mathbf{E}}_a=(-D)_I\partial P/\partial u_I^a\), where \(D_i=\displaystyle\frac{\partial }{\partial x^i}+u_{iJ}^a\displaystyle\frac{\partial } {\partial u_{J}^a}\), \(I=\{i_1,i_2,\ldots , i_k\}\), \(k>0\), \(D_I=(D_{i_1}\circ\cdots\circ D_{i_k})\), and \(P\) is some density, that is, a local function representing \(\mathcal{P}\), and \(v=fdx^1\wedge\cdots \wedge dx^n\) is a volume element on the base manifold \(M\). An interesting result for the first case where \(\dim M=1\), \(\dim E=2\), \({\mathcal{D}}=\omega D_x\) with \(\omega\in \mathrm{Loc}_E\) -- the algebra of local functions on \(J^{\infty }E\) -- is that the induced transformation \(\Psi \) on functionals is canonical, i.e., \(\{\Psi ({\mathcal{P}}), \Psi ({\mathcal{Q}})\}=\Psi (\{{\mathcal{P}},{\mathcal{Q}}\})\) for \({\mathcal{P}},{\mathcal{Q}}\in {\mathcal{F}}\) if and only if \(D_x(\text{det}\psi_M\partial \psi_E/\partial u)=0\), and \(\omega\circ j\psi = \omega (\text{det}\psi_M)^2(\partial \psi_E/\partial u)^2\). Here, \({\mathcal{F}}\) is the space of functionals, and \(j\psi :J^{\infty }E\to J^{\infty }E\) is the jet prolongation of the automorphism \(\psi :E\to E\). Furthermore, the author studies other two cases with different dimensions: dim\(M=n\), dim\(E=m+n\), \({\mathcal{D}}=\omega^{abi}D_i\), and the case dim\(M=1\), dim\(E=2\), \({\mathcal{D}}=\omega^{i}D_i\). The necessary and sufficient conditions for a transformation on the space of local functionals to be canonical are established. The conservation laws are also discussed. Some examples illustrating the theory are given as well. canonical transformation; Hamiltonian; Poisson brackets; Lagrangian; Euler-Lagrange operator; fiber bundle; variational derivative; evolutionary systems , Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry, , Lagrangian formalism and Hamiltonian formalism in mechanics of particles and systems, Canonical and symplectic transformations for problems in Hamiltonian and Lagrangian mechanics Canonical transformations and Hamiltonian evolutionary 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 abstract elliptic function fields Arithmetic theory of algebraic function fields, Algebraic functions and function fields in algebraic geometry On the theory of abstract elliptic 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 Let \(F = F/\mathbb{F}_q\) be an algebraic function field in one variable over finite field \(\mathbb{F}_q\), briefly function field. Let \(g(F)\) be the genus of \(F\), and \(N(F)\) of \(\mathbb{F}_q\)-rational places. The author studies the towers \(\mathcal{F} = (F_0 , F_1, F_2, \ldots, F_m, \ldots)\) of function fields where \(F_{m+1}/F_m\) is separable extension of positive degree, and for some \(m \geq 0\), \(F_m/\mathbb{F}_q\) is non-rational and non-elliptic.
\textit{A. Garcia} and \textit{H. Stichtenoth} [J. Number Theory 61, No. 2, 248--273 (1996; Zbl 0893.11047)] proved that the limit \(\lambda(\mathcal{F}) = \lambda(\mathcal{F}/\mathbb{F}_q) = \lim_{m \to \infty} N(F_m)/g(F_m)\) exists; it is called Garcia-Stichtenoth number. The author obtains the upper bound for Garcia-Stichtenoth numbers, computes this number for several towers of function fields, and examines several examples, among them the examples of optimal towers. algebraic function fields over finite fields; Kummer extensions; coding theory T. Hasegawa, An upper bound for the Garcia-Stichtenoth numbers of towers, Tokyo J. Math., 28 (2005), 471-481. Arithmetic theory of algebraic function fields, Algebraic functions and function fields in algebraic geometry, Finite ground fields in algebraic geometry, Algebraic coding theory; cryptography (number-theoretic aspects), Arithmetic codes An upper bound for the Garcia-Stichtenoth numbers of towers | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors give a computation of the homotopy groups of the \(\eta\)-periodic motivic sphere spectrum over certain fields. Let \(S\) be the motivic sphere spectrum. Let \(\eta:\mathbb{G}_m\to S\) be the motivic Hopf map deduced from the canonical morphism \(\mathbb{A}^2\setminus0\to\mathbb{P}^1\), and let \(\eta^{-1}S\) be the stabilization of \(S\) with respect to \(\eta\). Its homotopy groups are denoted as \(\pi_{m+n\alpha}\eta^{-1}S=[(S^1)^{\wedge m}\wedge(\mathbb{G}_m^{\wedge n}),\eta^{-1}S]_{\mathbf{SH}}\). The main result (Theorem 4.8) states that if \(k\) is a field of characteristic different from \(2\) of finite cohomological dimension in which \(-1\) is the sum of four squares, then there is an isomorphism of bigraded algebras
\[
\pi_{*}\eta^{-1}S\simeq W(k)[\eta^\pm,\sigma,\mu]/(\sigma^2)
\]
where \(|\sigma|=3+4\alpha\) and \(|\mu|=4+5\alpha\). In particular, \(\pi_{m+n\alpha}\eta^{-1}S\simeq W(k)\) if \(m\geq0\) is congruent to \(0\) or \(3\) mod \(4\), and vanishes in other cases. The main tool used is the \(\alpha_1\)-periodic slice spectral sequence introduced in Section 2, which computes the homotopy groups of the slice completion of the \(\eta\)-periodic sphere spectrum. motivic homotopy theory; stable motivic homotopy sheaves; slice spectral sequence Motivic cohomology; motivic homotopy theory, Stable homotopy of spheres The homotopy groups of the \(\eta \)-periodic motivic sphere spectrum | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 the function field of a curve \(C\) over a field \(\mathbb{F}\) of either odd or zero characteristic. Following the work by Serre and Mason on \(\operatorname{SL}_2\), we study the action of arithmetic subgroups of \(\operatorname{SU}(3)\) on its corresponding Bruhat-Tits tree associated to a suitable completion of \(K\). More precisely, we prove that the quotient graph ``looks like a spider'', in the sense that it is the union of a set of cuspidal rays (the ``legs''), parametrized by an explicit Picard group, that are attached to a connected graph (the ``body''). We use this description in order to describe these arithmetic subgroups as amalgamated products and study their homology. In the case where \(\mathbb{F}\) is a finite field, we use a result by Bux, Köhl and Witzel in order to prove that the ``body'' is a finite graph, which allows us to get even more precise applications. algebraic function fields; arithmetic subgroups; Bruhat-Tits trees; quotient graphs; special unitary groups Groups acting on trees, Other matrix groups over rings, Algebraic functions and function fields in algebraic geometry, Cohomology of groups, Linear algebraic groups over global fields and their integers, Cohomology of arithmetic groups Quotients of the Bruhat-Tits tree by arithmetic subgroups of special unitary groups | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(f\in \mathbb C[x_1,\ldots,x_n]\) be a polynomial which determines a central hyperplane arrangement (not necessarily reduced) and let \(A_n(\mathbb C) = \mathbb C[x_1,\ldots,x_n, \partial_1, \ldots, \partial_n]\) be the Weyl algebra. Given a factorization \(f = f_1 \cdots f_r\) (not necessarily into linear terms) and two divisors \(f^\prime\) and \(g\) of \(f\), the author considers an analogue of Bernstein-Sato ideal in \(\mathbb C[s_1,\ldots,s_r]\) consisting of the polynomials \(B(s)\) satisfying the functional equation \(B(s)f^\prime f_1^{s_1} \cdots f_r^{s_r} \in A_n(\mathbb C)[s_1, \cdots, s_r]gf^\prime f_1^{s_1+1} f_r^{s_r+1}\) (cf. [\textit{P. Maisonobe}, ``Idéal de Bernstein d'un arrangement central générique d'hyperplans'', Preprint, \url{arXiv:1610.03357}]). First he computes the zero locus of the usual (when \(f^\prime = 1\)) Bernstein-Sato ideal in the sense of \textit{N. Budur} [Ann. Inst. Fourier 65, No. 2, 549--603 (2015; Zbl 1332.32038)] for any factorization of a free and reduced arrangement and for certain factorizations of a non-reduced \(f\). Then he gives an explicite expression for the roots of the Bernstein-Sato polynomial for any power of a free and reduced arrangement, generalizes a duality formula from [\textit{L. Narváez Macarro}, Adv. Math. 281, 1242--1273 (2015; Zbl 1327.14090)], and so on. If \(f\) is tame, the author obtains a combinatorial formula for the roots containing in \([-1, 0)\). For \(f^\prime\neq 1\) and any factorization of a line arrangement, he computes the zero locus of the Bernstein-Sato ideal, and so on. As an interesting application he also shows that ``small roots of the Bernstein-Sato polynomial can force lower bounds for the minimal number of hyperplanes one must add to a tame \(f\) so that the resulting arrangement is free''; this question is closely related to the paper [\textit{D. Mond} and \textit{M. Schulze}, J. Singul. 7, 253--274 (2013; Zbl 1292.32014)]. b-function; D-modules; Weyl algebra; Bernstein-Sato polynomials; hyperplane arrangements; tame arrangements; free divisors; Euler-homogeneous divisors; Saito-holonomic divisors; logarithmic differential forms; logarithmic vector fields; Spencer complex Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Local complex singularities, Relations with arrangements of hyperplanes, Sheaves of differential operators and their modules, \(D\)-modules Combinatorially determined zeroes of Bernstein-Sato ideals for tame and free 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 Let \(\mathbb{F}_{2^r}\) be the finite field of \(2^r\) elements. Let \(f \in \mathbb{F}_{2^r}[T]\) be a monic irreducible polynomial of degree \(n\). \(f\) is said to be an \(A\)--polynomial if \(\operatorname{Tr}_{\mathbb{F}_{2^r}/\mathbb{F}_2}(a_{n-1})=1\) and \(\operatorname{Tr}_{\mathbb{F}_{2^r}/\mathbb{F}_2}(a_1/a_0)=1\).
In the paper under review, the authors obtain an explicit counting formula for the number of \(A\)-polynomials in \(\mathbb{F}_{2^r}[T]\) of degree \(n=2^km\), with \(m\) odd. To achieve this, they relate the number of \(A\)--polynomials of degree \(n\) with the number of inert places of degree \(n\) in a suitable extension of elliptic function fields. From this formula it can be deduced that if \((r, n)\neq (1,3)\) this number is always positive. Therefore, except for an isolated exception, such polynomials exist for each degree over every finite field of characteristic \(2\). These polynomials play an important role in the construction of high degree self-reciprocal irreducible polynomials over the finite field \(\mathbb{F}_2\). The result presented in this article is a generalization of the result of \textit{H. Niederreiter}, for the case \(r=1\) [Appl. Algebra Eng. Commun. Comput. 1, No. 2, 119--124 (1990; Zbl 0726.11077)]. \(A\)-polynomials; elliptic function fields; Kloosterman sums; \(Q\)-transform Algebraic functions and function fields in algebraic geometry, Computational aspects of algebraic curves, Polynomials over finite fields, Arithmetic theory of polynomial rings over finite fields, Arithmetic theory of algebraic function fields Enumeration of a special class of irreducible polynomials in characteristic 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 We give a geometric criterion for Dirichlet \(L\)-functions associated to cyclic characters over the rational function field \(\mathbb{F}_q (t)\) to vanish at the central point \(s = \frac{1}{2}\). The idea is based on the observation that vanishing at the central point can be interpreted as the existence of a map from the projective curve associated to the character to some abelian variety over \(\mathbb{F}_q\). Using this geometric criterion, we obtain a lower bound on the number of cubic characters over \(\mathbb{F}_q (t)\) whose \(L\)-functions vanish at the central point where \(q = p^{4n}\) for any rational prime \(p \equiv 2 \mod 3\). We also use recent results about the existence of supersingular superelliptic curves to deduce consequences for the \(L\)-functions of Dirichlet characters of other orders. Chowla's conjecture; \(L\)-functions; zeta functions of curves; Carlitz extensions; cyclotomic function fields; abelian varieties over finite fields Zeta and \(L\)-functions in characteristic \(p\), Zeta functions and \(L\)-functions, Cyclotomy, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Vanishing of Dirichlet \(L\)-functions at the central point over 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 discuss the use of elliptic curves in cryptography. In particular, we propose an analogue of the Diffie-Hellman key exchange protocol which appears to be immune from attacks of the style of Western, Miller and Adleman. With the current bounds for infeasible attack, it appears to be about 20\% faster than the Diffie-Hellmann scheme over \(\mathrm{GF}(p)\). As computational power grows, this disparity should get rapidly bigger.
[For the entire collection see Zbl 0583.00049.] bounds for infeasible attack; Diffie-Hellman scheme Miller, V.: Use of elliptic curves in cryptography. Lecture notes in comput. Sci. 218, 47-426 (1986) Cryptography, Applications to coding theory and cryptography of arithmetic geometry, Elliptic curves over global fields Use of elliptic curves in 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 \(K\) be a complete discrete valuation field of mixed characteristic with residue field \(F\) of characteristic \(p\) and not necessarily perfect. Consider \({\mathcal G}:= \text{Spf}(B)\), a connected finite flat group scheme of \(p\)--power order over \({\mathcal O}_ K\), the ring of integers of \(K\). Let \(\big\{{\mathcal G}^ j\big\}_ {j>0}\) be the \(j\)--ramification filtration of \({\mathcal G}\) in the sense of Abbes and Saito, and let \({\mathcal G}^{j^ +}\) be the schematic closure of \({\mathcal G}^ {j^ +}(\bar{K}):=\bigcup_{j'>j}{\mathcal G}^ {j'}(\bar{K})\) in \({\mathcal G}\). Let \(G_ K:= \text{Gal\;} (\bar{K}/K)\) be the absolute Galois group of \(K\). The author proves that the \(G_ K\)-module \({\mathcal G}^ {j}(\bar{K})/{\mathcal G}^ {j^ +}(\bar{K})\) is tame and killed by \(p\). Also, if \(I_ K\) denotes the inertia subgroup of \(G_ K\), then the \(I_ K\)-module \({\mathcal G}^ {j}(\bar{K})/{\mathcal G}^ {j^ +}(\bar{K}) \otimes_{{\mathbb F}_ p}\bar{{\mathbb F}}_ p\) is the direct sum of fundamental characters of level \(j\).
As corollaries of these results, it is obtained that the order of the image of the homomorphism \(I_ K\to \Aut({\mathcal G}(\bar{K}))\) is a power of \(p\) if and only if every jump \(j\) of the ramification filtration \(\big\{{\mathcal G}^ j\big\}_ {j>0}\) is an element of \({\mathbb Z}[1/p]\).
In the last section, the author computes the conductor of a Raynaud \({\mathbb F}\)-vector space scheme over \({\mathcal O}_ K\), where \({\mathbb F}\) is a finite extension of the finite field \({\mathbb F}_ p\) of \(p\)--elements. For integers \(0\leq s_ 1,\ldots,s_ r\leq e\) where \(e\) is the absolute ramification index, let \({\mathcal G}(s_ 1,\ldots, s_ r)\) be the Raynaud \({\mathbb F}\)-vector space scheme over \({\mathcal O}_ K\) defined by \(T_ 1^ p=\pi ^ {s_ 1} T_ 2, T_ 2^ p=\pi^ {s_ 2}T_ 3,\ldots, T_ r^ p= \pi^{s_ r}T_ 1\) where \(\pi\) is a uniformizing parameter of \(K\). Set \(j_ k:=(p s_ k+ p^ 2 s_ {k-1}+\cdots+p^ k s_ 1+ p^{k+1} s_ r+p^ {k+2}s_{r-1}+\cdots+p^ r s_ {k+1})/(p^ r-1)\). Then the conductor \(c({\mathcal G}(s_ 1,\ldots,s_ r))\) is equal to \(\sup_ k j_ k\). Galois representations; characters; group scheme; ramification; complete discrete fields doi:10.1016/j.jnt.2007.05.006 Group schemes, Ramification and extension theory, Formal groups, \(p\)-divisible groups, Group actions on varieties or schemes (quotients) Tame characters and ramification of finite flat group schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Generalized algebraic-geometry codes (in short GAG-codes) have recently been introduced by \textit{C. Xing, H. Niederreiter} and \textit{K. Y. Lam} [IEEE Trans. Inf. Theory 45, No. 7, 2498--2501 (1999; Zbl 0956.94023)] and they have allowed us to obtain codes with better parameters compared with Brouwer's table. They are a generalization of the well-known geometric Goppa codes since, for their construction, we can use not only rational places but also place of higher degree as well. For our purposes we are interested in GAG-codes constructed with places which are all of the same degree. For such GAG-code, \textit{A. G. Spera} [J. Pure Appl. Algebra 210, No. 3, 837--845 (2007; Zbl 1162.94008)] proved that its automorphism group admits a subgroup which is isomorphic to an automorphism group of the underlying function field. This result is similar to a result by \textit{H. Stichtenoth} [J. Algebra 130, No. 1, 113--121 (1990; Zbl 0696.94012)] on geometric Goppa codes.
In the present paper, following \textit{S. Wesemeyer} [IEEE Trans. Inf. Theory 44, No. 2, 630--643 (1998; Zbl 0913.94022)] (see also \textit{D. Joyner} and \textit{A. E. Ksir}, IEEE Trans. Inf. Theory 52, No. 7, 3325--3329 (2006; Zbl 1225.94029)]) and introducing the notion of \(n\)-automorphism for GAG-codes, we are able to invert Spera's afore-mentioned result in the rational, elliptic and hyperelliptic cases. We show that, under certain suitable conditions, the \(n\)-automorphism group of a GAG-code can be embedded in the automorphism group of the underlying function field. geometric Goppa codes; generalized algebraic-geometry codes; algebraic function fields; automorphisms; finite fields Picone, A.; Spera, A. G.: Automorphisms of hyperelliptic GAG-codes. Electron. notes discrete math. 26, 123-130 (2006) Geometric methods (including applications of algebraic geometry) applied to coding theory, Special algebraic curves and curves of low genus, Applications to coding theory and cryptography of arithmetic geometry Automorphisms of hyperelliptic GAG-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 \(X\) be a scheme over a finite field \(K=\mathbb F_ q\) and let \(G\) be a finite group of automorphisms of \(X\) defined over \(k\). Let \(\chi\) be a complex-valued character of \(G\). One defines the Artin \(L\)-function \(L(t;X,\chi)\) by \(L(t;X,\chi)=\exp (\sum v_ n(\chi)t^ n/n)\), where \(v_ n(\chi)=| G|^{-1}\sum_{g\in G}\chi(g)\Lambda(gF^ n),\) \(F\) is the Frobenius endomorphism, and, \(\Lambda(\lambda)=\) number of fixed points of \(\lambda: X\to X\) over the algebraic closure of \(k\). One knows by Grothendieck that \(L(t;X,\chi)\) is a rational function of \(t\). In this note the authors complement the result of Grothendieck, and they note that one can reduce to assuming that \(X\) is \textit{affine} and \(G\) is a \textit{linear action}. The ``total degree'' of \(L\) is the number of zeros and poles of \(L\) counted with multiplicity; it is denoted ``\(\text{totdeg}(L(t;X,\chi))\)''.
Let \(j: X\to \mathbb A^ N\) be the affine embedding. The authors define the pair \((X,G)\) to be in \textit{standard form} iff: (1) \(j\) is an affine embedding over \(k\); (2) \(G\) acts linearly on \(X\).
The authors then establish the following theorem: If \((X,G)\) is in standard form and \(X\) has degree \(d\), then
\[
\text{totdeg}(L(t;X,\chi)\leq \chi(1)^ 2(4d+9)^{4N}.
\]
The proof depends upon the fact that one can find twistings \(X_ g\) of \(X\) so that the number of fixed points of \(g^ nF^ n\) on \(X =\) number of fixed points of \(F^ n\) on \(X_ g\) for all \(n\). finite field; finite group of automorphisms; Artin L-function; total degree Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Finite ground fields in algebraic geometry On the degree of Artin \(L\)-functions in characteristic \(p\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 note contains a résumé of a seminar given by the author at the ''Seminario Matematico'', March 1982, at the University of Torino. Here the author gives a preliminary version of work subsequently written up more fully in Compos. Math. 52, 99-113 (1984; Zbl 0544.14025). arithmetically effective canonical divisor; threefolds of general type; base curves of multicanonical systems; normal bundles; configurations of curves \(3\)-folds, Divisors, linear systems, invertible sheaves, Families, moduli, classification: algebraic theory Threefolds with arithmetically effective canonical divisor | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems \((X,0) \subseteq ({\mathbb C}^n , 0)\) denotes a germ of a reduced space curve. For a finite function germ \(f:(X,0) \to ({\mathbb C} , 0)\), the Milnor number \(\mu (f)\) (in the sense of \textit{V. Goryunov} [J. Lond. Math. Soc., II. Ser. 61, 807--822 (2000; Zbl 0965.58030)], \textit{D. Mond} and \textit{D. van Straten} [J. Lond. Math. Soc., II. Ser. 63, No. 1, 177--187 (2001; Zbl 1017.32022)]) is an invariant with the following properties:
(1) It is preserved under simultaneous deformation of \(f\) and \((X,0)\).
(2) For smooth germs \((X,0) \), the number \(\mu (f)\) coincides with the usual Milnor number.
If \((X,0) \) is smoothable, then \(\mu (f)\) is determined by the above conditions.
The authors show: \(\mu (f) = \mu (X,0) + \text{deg}(f) -1\).
This generalizes a result of the first author and Jorge Pérez. An algebraic proof of the general formula is given.
The result is applied to study Whitney-equisingularity for a family \((X_t)\) of space curves. \((X_t)\) is shown to be Whitney-equisingular iff the first polar multiplicity of \((X_t)\) is constant. Furthermore, for a family \(f_t:X_t \to {\mathbb C}\) of functions on space curves, the following is obtained:
(a) \(f_t\) is topologically trivial iff \(\mu (f_t)\) is constant.
(b) \(f_t\) is Whitney-equisingular iff \(\mu (f_t)\) and the multiplicity \(m_0(X_t,0)\) are both constant. space curve; Milnor number; finite function germ; Whitney-equisingular Nuño-Ballesteros, JJ; Tomazella, JN, The Milnor number of a function on a space curve germ, Bull. Lond. Math. Soc., 40, 129-138, (2008) Deformations of complex singularities; vanishing cycles, Critical points of functions and mappings on manifolds, Equisingularity (topological and analytic), Singularities in algebraic geometry The Milnor number of a function on a space curve germ | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Fix a rational base point \(b\) and a rational number \(c\). For the quadratic dynamical system \(f_c(x)=x^{2}+c\), it has been shown that the number of rational points in the backward orbit of \(b\) is bounded independent of the choice of rational parameter \(c\). In this short note we investigate the dependence of the bound on the base point \(b\), assuming a strong form of the Mordell conjecture. quadratic dynamical systems; rational pre-images; Mordell conjecture Faber X.: A remark on the effective Mordell conjecture and rational pre-images under quadratic dynamical systems. C. R. Math. Acad. Sci. Paris 348(7--8), 355--358 (2010) Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps, Height functions; Green functions; invariant measures in arithmetic and non-Archimedean dynamical systems, Heights, Rational points A remark on the effective Mordell conjecture and rational pre-images under quadratic dynamical systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We give a simple proof of a criterion for the existence of global sections for line bundles on Schubert varieties. Unlike previous proofs, which use the nontrivial fact that Schubert varieties are normal, our proof is based on an elementary geometric property of root systems and their Weyl groups. global sections; line bundles; Schubert varieties; root systems; Weyl groups R. Dabrowski, A simple proof of a necessary and sufficient condition for the existence of nontrivial global sections of a line bundle on a Schubert variety, in: Kazhdan-Lusztig Theory and Related Topics (Chicago, IL, 1989), Contemp. Math., Vol. 139, Amer. Math. Soc., Providence, RI, 1992, pp. 113-120. Linear algebraic groups over arbitrary fields, Grassmannians, Schubert varieties, flag manifolds, Simple, semisimple, reductive (super)algebras A simple proof of a necessary and sufficient condition for the existence of nontrivial global sections of a line bundle on a Schubert variety | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The present account constitutes a further elaboration of the idea that ``geometry/space'' is always the result of functions that characterize it, which is also in accordance with the point of view that ``physical geometry'' is the outcome of the physical laws. The same lies also at the root of a ``relativistic quantization'', where functions are transformed into sections of appropriate (topological) algebra sheaves, corresponding to a ``relativistic localization'', while the indispensable, in that context, ``dynamics'' is further accomplished, by employing an abstract form, à la Leibniz, viz. no ``space'', at all(!), of the classical differential geometric machinery, that is, in terms of the so-called ``abstract (alias, `modern') differential geometry'' (ADG). This enables one to exercise herewith, even a quite general context, as, for instance, topos theory, or what we may call topological algebra schemes, an ``Einstein (topological) algebra space'', being, in effect, a particular instance, thereat, in terms thus of which one can formulate, for example, Einstein's quantized equation (in vacuo). topological algebras; geometric topological algebras; Gel'fand presheaf; topological algebra scheme; locally affine; local information; Yoneda functor; Yoneda's lemma; functor of points, spectrum functor; Šilov's problem; dynamical algebra; dynamical relativistic localization; Einstein topological algebra space; extension of scalars functor; abstract/modern differential geometry; ADG; Heisenberg's incompatibility; principle of locality Mallios, A.: On algebra spaces. Contemp. Math. 427, 263--283 (2007) Representations of commutative topological algebras, Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.), Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects), Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Generalizations (algebraic spaces, stacks), Differential geometric methods, including holonomy, Berry and Hannay phases, Aharonov-Bohm effect, etc. in quantum theory, Axiomatic quantum field theory; operator algebras, Einstein's equations (general structure, canonical formalism, Cauchy problems), Quantization of the gravitational field On algebra 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 article generalizes results and methods of \textit{T. Ohsawa} and \textit{K. Takegoshi} theorem from [Math. Z. 195, 197--204 (1987; Zbl 0625.32011)] to the case when the subvariety \(Y\) is not necessarily reduced, by using a multiplier ideal sheaf and jumping numbers. For a holomorphic vector bundle \(E\) on a complex manifold \(X\) one can discuss the existence of global holomorphic extensions \(F\in H^0(X,E)\) of a section \(f\in H^0(Y,E|_{Y})\) together with \(L^2\) approximations. The article considers this problem when \(X\) is a weakly pseudoconvex Kähler manifold with Kähler metric \(\omega\) and when the holomorphic vector bundle \(E\) is equipped with a (possibly singular) hermitian metric \(h=e^{-\varphi}\). Let \(\psi\) denote a quasi-psh function on \(X\) with neat analytic singularities and with log canonical singularities along a analytic subvariety \(Y=V(\mathcal{I}(\psi))\) (so that \(Y\) is reduced). If the Chern curvature tensor \(\Theta_{E,h}\) has the property that \(i\Theta_{E,h}+\alpha i\partial\overline{\partial}\otimes Id_{E}\) is Nakano semipositive for all \(\alpha\in [1,1+\delta]\) and some \(\delta>0\), then for every section \(f\in H^0(Y^0,(K_X\otimes E)|_{Y^0})\) on \(Y^{0}=Y_{\mathrm{reg}}\) such that
\[
\int_{Y_{0}}|f|^2_{\omega,h}dV_{Y^0,\omega}[\psi]<+\infty
\]
there exists an extension \(F\in H^{0}(X,K_{X}\otimes E)\) whose restriction to \(Y^{0}\) is equal to \(f\), such that
\[
\int_{X}\gamma(\delta \psi)|F|^2_{\omega,h}e^{-\psi}dV_{X,\omega}<\frac{34}{\delta}\int_{Y_{0}}|f|^2_{\omega,h}dV_{Y^0,\omega}[\psi]
\]
The remark states that if \(F\) is a \((n,0)\)-form then the product \(|F|^2_{\omega,h}dV_{X,\omega}\) does not depend on \(\omega\). The author claims that the constant \(\frac{34}{\delta}\) in the inequality is not optimal.
The concept of the multiplier ideal sheaf used in the proof is parallel yet more general than the one presented by \textit{D. Popovici} [Nagoya Math. J. 180, 1--34 (2005; Zbl 1116.32017)]. holomorphic function; plurisubharmonic function; multiplier ideal sheaf; \(L^2\) extension theorem; Ohsawa-Takegoshi theorem; log canonical singularities; non reduced subvariety Kähler metric; multiplier ideal sheaf; jumping numbers Demailly, J. P., Extension of holomorphic functions defined on non reduced analytic subvarieties, The Legacy of Bernhard Riemann after One Hundred and Fifty Years, I, 191-222, (2016) Transcendental methods, Hodge theory (algebro-geometric aspects), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Analytic sheaves and cohomology groups Extension of holomorphic functions defined on non reduced analytic subvarieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 book is the first volume of a modern introduction to transcendental algebraic geometry, with a special emphasis on the underlying aspects of Kählerian geometry and Hodge theory. Accordingly, this first volume focuses on the merely complex-analytic foundations of the subject without particularly mentioning complex projective manifolds, whilst the forthcoming second volume will be devoted to the systematic application of the analytic framework in different algebro-geometric directions relating Hodge theory, topology and the study of algebraic cycles on smooth projective manifolds. The ultimate goal of the text, as a whole, is to provide a comprehensive, self-contained and up-to-date account of Hodge theory and the theory of algebraic cycles as developed, during the past thirty years, by P. Griffiths and his school, P. Deligne, S. Bloch, and many others.
With regard to this program and its methodical realization, the present first volume is perfectly suitable for seasoned students either as a profound general introduction to the analytic foundations of complex algebraic geometry, or as an excellent preparation for the second volume, which apparently will lead them to the forefront of contemporary research in complex algebraic geometry.
Now, as to this first volume under review, its main purpose is to explain the existence of special structures on the cohomology of Kähler manifolds, in particular the Hodge decomposition and the Lefschetz decomposition, and to analyze their basic properties and fundamental consequences. The entire text is subdivided into twelve chapters which form, by their arrangement, the four principal parts of the book. Part I is entitled ``Preliminaries'' and comprises the first four chapters. Basically, this introductory part provides the reader with the essential background material from the theory of complex manifolds. Chapter 1 recalls the main results of the theory of analytic functions of several variables, including the Riemann and Hartogs extension theorems. Chapter 2 introduces complex manifolds and holomorphic vector bundles over them. This chapter also treats the integrability of almost complex structures on differential manifolds, inclusively the Newlander-Nirenberg theorem, and the Dolbeault complex of a holomorphic bundle. Chapter 3 is devoted to Kählerian geometry. The author gives various characterizations of the Kähler metrics, introduces Chern connections of Hermitean vector bundles, and discusses some constructions and examples of Kähler manifolds. Chapter 4 concludes the introductory part I with an introduction to the theory of sheaves and their cohomology, together with the underlying categorical framework and, as a first important application, the de Rham theorems.
Part II, the main body of the book, turns to Hodge theory and comes with the title ``The Hodge decomposition''. This part encompasses the next four chapters and is mainly devoted to proving the Hodge and Lefschetz decomposition theorems. Chapter 5 briefly presents the basic ideas and fundamental concepts of classical Hodge theory, omitting, however, the intricate proofs of the necessary results from pure analysis. This includes sufficiently elaborated accounts of Laplacian operators, general elliptic differential operators, harmonic differential forms and their cohomology, and the related cohomological duality theorems. Chapter 6 is centered around the application of Hodge theory to the special case of Kähler manifolds. After proving the Kähler identities and studying the Lefschetz operator, this chapter culminates in the proofs of the three fundamental results in this context: the Hodge decomposition, the Lefschetz decomposition, and the Hodge index theorem. The following two chapters deal with conceptual applications of these central results. Chapter 7 explains Hodge structures, polarised Hodge structures, weighted Hodge structures, and their functorial behavior. The discussion submitted here also touches upon the highly important Kodaira embedding theorem. Finally, the many examples given in this chapter are enriched by exploring the relation between Hodge structures of weight one and abelian varieties. Chapter 8, which concludes part II of the book, is thematically devoted to the holomorphic de Rham complex and the interpretation of the Hodge theorem in terms of degeneracy of the Frölicher spectral sequence. To this end, the necessary framework, including hyperhomology, the logarithmic complex, and the technique of spectral sequences, is efficiently developed, and a brief digression into the Hodge theory of open manifolds (with mentioning Deligne's theorem on Hodge structures on compact Kähler manifolds, 1971) completes this particularly profuse second part of the text.
Part III consists of chapters 9 and 10, which thematically turn toward the study of variations of Hodge structures. The author first introduces the notion of a family of complex manifolds and treats the construction of the Kodaira-Spencer map associated to such a family. Then the Gauss-Manin connection of the local system of the cohomology of the fibres of a family is expounded, followed by a special treatment of the Kähler case, and the stability theorem for Kähler manifolds concludes chapter 9. The topic of variations of Hodge structures is dealt with in the following chapter 10. The author gives a concise account of P. Griffiths's approach to the moduli theory of Kähler manifolds via the period domain and the period map for families of them. Apart from the basics of the theory of variations of Hodge structures, Hodge bundles and their transversality property, and the structure of the differential of the period map, the author presents some applications to the special case of moduli of curves and Calabi-Yau manifolds, including the related Torelli-type theorems.
Part IV is then even more advanced and directly leads over to one of the major themes of the forthcoming second volume of the book: the interaction between algebraic cycles and the Hodge theory of a projective complex manifold. Remaining in the framework of Kählerian geometry, the author discusses here the analytic aspects of the theory of cycles, i.e., analytic cycles and cycle classes on compact Kähler manifolds. Chapter 11, entitled ``Hodge classes'', exhibits the basic properties of analytic subsets and their cohomology classes for general complex manifolds. Cycle classes, Chern classes and Hodge classes are the main subjects of the discussion, whereat the special case of Kähler manifolds undergoes a respective specific inspection. In the course of the discussion, the author also touches upon the celebrated Hodge conjecture, which perfectly fits into (and actually motivates) the treatment of Hodge classes. The concluding chapter 12 is devoted to Deligne cohomology and the Abel-Jacobi map. These objects will be studied much more deeply in the second volume of this book, but here the first basics of them are briefly introduced. This includes the defintion of intermediate Jacobians, the Abel-Jacobi map, Picard and Albanese varieties, the Deligne complex, and Deligne cohomology, together with the first fundamental properties of these objects.
Each of the twelve chapters comes with a bunch of carefully selected exercises, however with any hints to their solution. These exercises, in regard of their thematic content and methodical significance, perfectly serve the purpose of illustrating and enhancing the material covered by the text, and the reader will profit a great deal from seriously working on them. The bibliography, at the end of the book, is up-to-date and reasonably sweeping, however mainly focusing on related textbooks and research monographs in the field.
Indeed, this introductory text to Hodge theory and Kählerian geometry is essentially self-contained for readers with a basic knowledge of complex analysis, differential geometry, topology, and algebra, that is for a seasoned graduate student. It represents, altogether, an excellent and modern introduction to the subject, shining with comprehensiveness, strictness, clarity, rigor, thematic steadfastness of purpose, and catching enthusiasm for this fascinating field of contemporary mathematical research. This book is exceedingly instructive, inspiring, challenging and user-friendly, which makes it truly outstanding and extremely valuable for students, teachers, and researchers in complex geometry. transcendental algebraic geometry; Kählerian geometry; Hodge theory; Hodge decomposition; Lefschetz decomposition; Hodge index theorem; de Rham complex; Frölicher spectral sequence; variations of Hodge structures; period domain; algebraic cycles; Deligne cohomology; Abel-Jacobi map C. Voisin, \textit{Hodge theory and complex algebraic geometry. I}. Translated from the French original by Leila Schneps, Cambridge Studies in Advanced Mathematics, 76, Cambridge University Press, Cambridge, 2002.Zbl 1005.14002 MR 1967689 Transcendental methods, Hodge theory (algebro-geometric aspects), Research exposition (monographs, survey articles) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces, Algebraic cycles, Transcendental methods of algebraic geometry (complex-analytic aspects), Variation of Hodge structures (algebro-geometric aspects), Period matrices, variation of Hodge structure; degenerations Hodge theory and complex algebraic geometry. I. Translated from the French by Leila Schneps | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 triangulated category \(D\) whose objects have bounded cohomology, a \(t\)-structure on it, an autoequivalence \(K \to K (1)\) of \(D\) respecting the \(t\)-structure and a morphism
\[
{\eta : K \to K (1)}\quad [2],
\]
the condition (LV) states that for all \(i \geq 0\), the \(i\)-th iterate of \(\eta\) induces an isomorphism \(H^{-i} (K) \to H^ i (K) (i)\). An earlier paper [``Théorème de Lefschetz et critères de dégénérescence des suites spectrales'', Publ. Math., Inst. Hautes Étud. Sci. 35(1968), 107-126 (1969; Zbl 0159.225)] of the author showed that this implies that \(K\) is isomorphic to the direct sum of the \(H^ i (K)\), each placed in an appropriate degree.
The first objective of this paper is to establish the existence of a canonical such isomorphism. This is achieved by imposing an extra condition: the construction is by induction on a filtration obtained using the \(t\)-structure, and the condition (too technical to give here) yields uniqueness at the inductive step.
The hypothesis on \(D\) passes to the opposite category, but the isomorphism first attained is not self-dual in this sense. If \(D\) is defined over \(\mathbb{Q}\), a self-dual isomorphism is constructed by a more delicate argument involving an action by the Lie algebra \(s \ell (2)\).
The concepts introduced in the proofs are applied to \(\ell\)-adic cohomology, where they lead to an attractive reformulation of conjectures relating algebraic cycles and the motivic derived category.
An appendix gives a treatment (due to \textit{J.-L. Verdier}) of spectral objects in the derived category, and of the induced spectral sequences. \(t\)-structure; \(\ell\)-adic cohomology; triangulated category; bounded cohomology; algebraic cycles; motivic derived category; spectral sequences Deligne, P., Décompositions dans la catégorie dérivée, (Décompositions dans la catégorie dérivée, motives, Seattle, WA, 1991, Proc. Symp. Pure Math., vol. 55, (1994), Amer. Math. Soc. Providence, RI), 115-128, Part 1 Derived categories, triangulated categories, Algebraic cycles, \(p\)-adic cohomology, crystalline cohomology, Spectral sequences, hypercohomology Decomposition in the derived category | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 rational prime \(p\), denote by \(K\) a finite extension of \({\mathbb Q}_p\) and by \(m\) the maximal ideal of the ring of integers \({\mathfrak O}_K\).
Let \(A\) be an elliptic curve over \(K\), with identity element \(e\), and, for any positive integer \(n\), denote by \(n\iota:A\to A\) the multiplication by \(n\) of the points of \(A\). Let \(D\) be a \(K\)-rational divisor on \(A\) with degree \(0\), disjoint from \(e\) modulo \(m\), and for any subgroup \(G\) of points of \(A\), denote by \(I=I_G\) the ideal in the group-ring \({\mathbb Z}[G]\) of the zero-cycles of degree \(0\) and denote by \([D,{\mathfrak a}]\) the pairing of \(D\) with a cycle \({\mathfrak a}\in I^2\).
For a suitable subgroup \(G\) of \(A(K)\) and \({\mathfrak a}\in I_G\), Néron proved that the limit \(\vartheta_D^\nu({\mathfrak a}) = \lim_{r\to\infty}[D, p^r{\mathfrak a}-(p^r\iota){\mathfrak a}]^{1/p^r}\) exists and defined a function, \(a\mapsto \vartheta_D^\nu((a)-(e))\), analytic on \(G\), with values in \(1+m\). This is the Néron theta function associated to \(D\) [cf. \textit{A. Néron}, Sémin. Delange-Pisot-Poitou, Prog. Math. 22, 149-174 (1982; Zbl 0492.14035)].
Given a uniformization \(1\to q^{\mathbb Z}\to K^\times \buildrel{\pi}\over{\to} A\to 0\) of \(A\), denote by \(\theta_D\) a theta function (in the sense of Tate) associated to \(D\), normalized by putting \(\theta_D(e)=1\). If \(G\) is the subgroup of definition of the Néron theta function and \({\mathfrak a}=(\pi(a))-(e)\in I_G\), the author states that \([D, p^r{\mathfrak a}-(p^r\iota){\mathfrak a}]\theta_D(a^{p^r})=\theta_D(a)^{p^r}\) and that, for \(|a|<1\), the limit \(\lim_{r\to\infty}\theta_D(a^{p^r})^{1/p^r}\) exists in \(1+m\). As a consequence the author states that, in a suitable neighborhood of the origin of \(A\), the quotient of the two theta functions is analytic and its values are units. \(p\)-adic theta functions; \(p\)-adic elliptic curves; Néron theta function; Tate theta function Theta functions and abelian varieties, Elliptic curves, Arithmetic ground fields for abelian varieties, Local ground fields in algebraic geometry Comparison of \(p\)-adic theta functions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper is concerned with the classification of (connected) branched coverings of the 2-sphere, where two such coverings, \(f_1: X_1\to S^2\) and \(f_2: X_2\to S^2\), are considered equivalent if there exist homeomorphisms \(g: S^2\to S^2\) and \(h: X_1\to X_2\) such that \(g f_1=f_2 h\). In 1891, A. Hurwitz studied the case when all branch points are simple, i.e., have fibers of cardinality \(d-1\) where \(d\) is the degree (number of sheets) of the covering. He proved that two such coverings are equivalent if and only if they have the same degree and the same number of branch points. \textit{P. Kluitmann} [Contemp. Math. 78, 299-325 (1988; Zbl 0701.20019)] extended the classification to coverings with exactly one special, i.e. not simple, branch point. He proved that the two Hurwitz invariants together with a further obvious invariant, the branching data (i.e. the list of multiplicities of the points in the fiber) of the special branch point, are sufficient to distinguish among such coverings. The case of two special branch points was first considered by \textit{E. Looijenga} [Invent. Math. 121, No. 2, 411-419 (1995; Zbl 0851.14017)] under the restriction that both special branch points have singleton fibers. In the present paper the author uses Looijenga's method to obtain a complete classification of coverings with exactly two special branch points. He proves that two such coverings are equivalent if and only if they have the same degree, number of branch points (or genus), branching data for the two special branch points, and monodromy group. He shows that these invariants are not sufficient to distinguish among coverings with three special branch points. branched covering of the Riemann sphere; Hurwitz action of the braid group; simple branch point; special branch point; branching data; monodromy group; Hurwitz scheme; Hurwitz invariants B.Wajnryb, Orbits of Hurwitz action for coverings of a sphere with two special fibers, Indag. Math. (N. S.), 7 (no. 4) (1996), 549--558. Low-dimensional topology of special (e.g., branched) coverings, Riemann surfaces; Weierstrass points; gap sequences Orbits of Hurwitz action for coverings of a sphere with two special fibers | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(p_1,\dots,p_n\) be general points in the projective plane \(\mathbb P^2\), with corresponding homogeneous ideals \(P_1,\dots,P_n\). An ideal of the form \(I({\mathbf m},n) = P_1^{m_1} \cap \dots \cap P_n^{m_n}\) (where \(\mathbf m = (m_1,\dots,m_n)\) is an \(n\)-tuple of non-negative integers) defines a fat point subscheme \(Z \subset \mathbb P^2\). If \(m_1 = \dots = m_n = m\) (say), we say that \textbf{m} (or \(Z\)) is uniform and just write \(I(m,n)\). There are several conjectures about the Hilbert function and minimal free resolution of the ideal \(I(\mathbf m,n)\), and the answers are known in some cases (mostly uniform cases) and counterexamples are known in a few cases. However, all known failures involve \(n < 9\). The conjectures all involve some sort of maximal rank property.
In this paper the authors introduce the notion of quasiuniformity, saying that \(\mathbf m\) is quasiuniform if \({\mathbf m} = (m_1,m_2,\dots,m_n)\) with \(n \geq 9\) and \(m_1 = \dots = m_9 \geq m_{10} \geq \dots \geq m_n \geq 0\). They give conjectures for the Hilbert function and minimal free resolution of quasiuniform fat point schemes. Since quasiuniformity is an extension of the notion of uniformity, they note that their conjectures contain as special cases the existing conjectures (or in a few cases, theorems) for the uniform case. They also prove a number of solid results that give substantial evidence for their conjectures, especially for the uniform case. In particular, they prove the conjectures for infinitely many \(m\) for each of infinitely many \(n\), and for infinitely many \(n\) for every \(m>2\). They also show that in many cases the Hilbert function conjecture implies the resolution conjecture. As a by-product of their work, they get a strong bound on the regularity of \(I(m,n)\). ideal generation conjecture; symbolic powers; minimal free resolution; fat points; maximal rank; Hilbert function; regularity; quasiuniformity Harbourne, B.; Holay, S.; Fitchett, S., Resolutions of ideals of quasiuniform fat point subschemes of \(\mathbb P^2,\), Trans. Amer. Math. Soc., 355, 2, 593-608, (2003) Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Syzygies, resolutions, complexes and commutative rings, Multiplicity theory and related topics Resolutions of ideals of quasiuniform fat point subschemes of \(\mathbb{P}^2\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(A\) be a henselian discrete valuation ring and let \(l\) be a prime number invertible in \(A\). We prove:
Theorem 1. For a proper semistable family over \(A\) whose special fiber has smooth irreducible components, the Steenbrink-Rapoport-Zink \(l\)-adic weight spectral sequence degenerates in \(E_2\)-terms.
The degeneracy was proved in cases where
(i) \(A=\mathbb{C}\{t\}\) by \textit{J. H. M. Steenbrink} [Invent. Math. 31, 229--257 (1976; Zbl 0303.14002)]; and
(ii) The residue field of \(A\) is finite by a result by \textit{M. Rapoport} and \textit{Th. Zink} [Invent. Math. 68, 21-101 (1982; Zbl 0498.14010)].
The second case was proved by the Weil conjecture on Frobenius weights. \textit{L. Illusie} [Astérisque 223, 9-57 (1994; Zbl 0837.14013)] conjectured that the degeneracy holds over an arbitrary A. Theorem 1 gives an affirmative answer to it. log geometry; discrete valuation ring; weight spectral sequence Nakayama, C., \textit{degeneration of \textit{\(\mathcal{l}\)}-adic weight spectral sequences}, Amer. J. Math., 122, 721-733, (2000) Local ground fields in algebraic geometry, \(p\)-adic cohomology, crystalline cohomology, Fibrations, degenerations in algebraic geometry Degeneration of \(l\)-adic weight spectral 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 The author proves two theorems on the canonical basis in Lubin-Tate formal modules in the case of local field with perfect residue field (Theorem 1) and in the case of imperfect residue field (Theorem 2) .
These canonical bases are obtained by applying a variant of the Artin-Hasse function.
The chief tools used in the paper are the construction of a basis in \({\mathbb Z}_p\)-module of principle units for complete discrete valuation field with an arbitrary residue field by \textit{S. V. Vostokov} [J. Math. Sci., New York 188, No. 5, 570--581 (2013; Zbl 1300.12009); translation from Zap. Nauchn. Semin. POMI 394, 174--193 (2011)], the construction of primary elements in formal modules by \textit{S. V. Vostokov} and \textit{I. L. Klimovitskii} [Proc. Steklov Inst. Math. 282, S140--S149 (2013; Zbl 1318.11156); translation from Sovrem. Probl. Mat. 17, 153--163 (2013)] and the construction of the Shafarevich basis in higher-dimensional local fields by \textit{E. V. Ikonnikova} and \textit{E. V. Shaverdova} [J. Math. Sci., New York 202, No. 3, 410--421 (2014; Zbl 1319.11088); translation from Zap. Nauchn. Semin. POMI 413, 115--133 (2013)]. Lubin-Tate formal module; local field; canonical basis; Artin-Hasse function; perfect residue field; imperfect residue field; isogeny Formal groups, \(p\)-divisible groups, Class field theory; \(p\)-adic formal groups The Hensel-Shafarevich canonical basis in Lubin-Tate formal 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 This paper describes the group of line bundles, or the Picard group, of the moduli space of line bundles on smooth curves as well as some closely related moduli spaces. The Picard group had been studied before, notably in [\textit{J. Ebert} and \textit{O. Randal-Williams}, Doc. Math., J. DMV 17, 417--450 (2012; Zbl 1273.14058); \textit{C. Fontanari}, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 16, No. 1, 45--59 (2005; Zbl 1222.14055)], and [\textit{A. Kouvidakis}, J. Differ. Geom. 34, No. 3, 839--850 (1991; Zbl 0780.14004)], but this is the first paper to carefully treat stack-theoretic issues.
Recall the moduli space of degree \(d\) line bundles on smooth curves of genus \(g\), denoted \(\mathcal{J}ac_{d,g}\) in the paper, exists as an algebraic stack. This stack has the property that every stabilizer group contains the multiplicative group \(\mathbb{G}_{m}\) (acting by scalar multiplication on line bundles), and a natural rigidification construction removes the \(\mathbb{G}_{m}\)s to produce a new stack \(\mathcal{J}_{d,g}\). The stabilizer groups of the new stack \(\mathcal{J}_{d,g}\) are finite groups, in fact \(\mathcal{J}_{d,g}\) is a Deligne-Mumford stack, but the stabilizers can be nontrivial, so \(\mathcal{J}_{d,g}\) is not an algebraic variety. The nontrivial stabilizer groups can, however, be removed to produce a third object, the coarse moduli space of \(\mathcal{J}_{d,g}\) which is a quasi-projective variety. In the literature, all of these spaces are often called the universal Jacobian, although in the present paper the term is reserved for \(\mathcal{J}ac_{d,g}\).
The universal Jacobian is studied alongside certain moduli spaces parameterizing line bundles on (possible unstable) nodal curves that satisfy the balancedness condition (a numerical condition on the multidegree). Similar to the previous situation, one considers the moduli stack \(\overline{\mathcal{J}ac}_{d,g}\) of these objects as well as a Deligne-Mumford stack \(\overline{\mathcal{J}}_{d,g}\) obtained by removing \(\mathbb{G}_{m}\)s from stabilizers and a projective variety \(\overline{J}_{d,g}\) obtained by removing all stabilizers. All of these spaces are often called the compactified universal Jacobian as \(\overline{J}_{d,g}\) is proper.
The main results of this paper compute the Picard groups of these stacks when \(g \geq 3\). The results are easiest to state for \(\mathcal{J}ac_{d,g}\) and \(\overline{\mathcal{J}ac}_{d,g}\). The universal family of curves over \(\mathcal{J}ac_{d,g}\) admits two natural line bundles: the universal line bundle \(\mathcal{L}\) and the relative dualizing sheaf \(\omega\). Theorem A states that the Picard group of \(\mathcal{J}ac_{d,g}\) is freely generated by the determinants of cohomology of \(\mathcal{L}\), \(\omega\), and \(\mathcal{L} \otimes \omega\). Furthermore, the Picard group of \(\overline{\mathcal{J}ac}_{d,g}\) is freely generated by these line bundles together with the line bundles associated to the irreducible components of the boundary (i.e. the complement of \(\mathcal{J}ac_{d,g}\) in \(\overline{\mathcal{J}ac}_{d,g}\)).
The Picard groups of \(\mathcal{J}_{d,g}\) and \(\overline{\mathcal{J}}_{d,g}\) are described in Theorem B. This result is more complicated to state because the universal family of curves over \(\mathcal{J}_{d,g}\) does not admit a universal family of line bundles. The result states that the Picard group of \(\mathcal{J}_{d,g}\) is freely generated by the determinant of cohomology of \(\omega\) and a line bundle that is more complicated to describe but is similar to a certain explicit linear combination of the determinants of cohomology of \(\mathcal{L}\) and \(\mathcal{L} \otimes \omega\). These line bundles together with the line bundles associated to the irreducible components of the boundary freely generate the Picard group of \(\overline{\mathcal{J}}_{d,g}\).
The analogous results for \(J_{d,g}\) and \(\overline{J}_{d,g}\) were known by [Zbl 1222.14055], and the authors also relate the descriptions in that work to their Theorems A and B in Theorem C.
The main theorems are proven by using Kouvidakis's work [Zbl 0780.14004] to compute the Picard group of \(\mathcal{J}_{d,g}\) and then relating the other Picard groups of interest to \(\text{Pic}(\mathcal{J}_{d,g})\). Results similar to the results about \(\mathcal{J}ac_{d,g}\) and \(\mathcal{J}_{d,g}\) were proven in [Zbl 1273.14058], which appeared shortly after a preliminary of the present paper was made publically available. The authors of that paper compute the Picard groups of topological stacks that are expected to be topological models of \(\mathcal{J}ac_{d,g}\) and \(\mathcal{J}_{d,g}\). The proofs there are very different from the proofs in the present paper. In [Zbl 1273.14058] the main results are proven using algebraic topology, especially ideas from homotopy theory, while the present paper uses algebraic geometry. The relation between the results of the two papers is carefully described in Section 1.1 of the present paper and Section 4.5 of [Zbl 1273.14058]. Brauer group; Picard group; gm-gerbe; universal Jacobian stack and scheme; compactified universal Jacobian Melo, M.; Viviani, F., The Picard group of the compactified universal Jacobian, Doc. Math., 19, 457-507, (2014) Families, moduli of curves (algebraic), Jacobians, Prym varieties, Picard groups, Generalizations (algebraic spaces, stacks), Geometric invariant theory The Picard group of the compactified universal Jacobian | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Gauss' class number problem for imaginary quadratic fields \(K_ D:=\mathbb Q(\sqrt{D})\), where \(D\in\mathbb Z\), \(D<0\), denotes the discriminant, consists in finding an effective algorithm for determining all such fields \(K_ D\) with given class number \(h_ D=h\). Of course, this task presupposes the truth of Gauss' conjecture stating that there is only a finite number of those fields \(K_ D\) with fixed class number \(h_ D\).
The author first describes the history of the problem, starting out with the observation of Euler and Legendre that \(x^ 2-x+41\) and \(x^ 2+x+41\) is a prime for \(x=1, 2,\dots, 40\) and \(x=0,1,\dots,39\) respectively, and winding up with the recent solution of the problem due to Goldfeld--Gross--Zagier. In fact, Gauss' problem is settled by the following theorem of Goldfeld--Gross--Zagier: For every \(\varepsilon >0\), there exists an effectively computable constant \(c>0\) such that \(h_ D>c (\log | D|)^{1-\varepsilon}.\)
Oesterlé (1984) computed the constant in this theorem and Mestre, Oesterlé and Serre, by showing that a certain elliptic curve is modular, made it possible to establish a complete list of all imaginary quadratic fields \(K_ D\) with class number \(h_ D=3\). The corresponding lists for the cases of \(h_ D=1\) and \(h_ D=2\) had been obtained earlier by Heegner-Baker-Stark-Deuring-Siegel and by Baker-Stark, respectively. The author points out that, when combined with the solution of the case of \(h_ D=4\), these results would yield the complete finite list of all integers n admitting a unique representation as a sum of three squares: \(n=x^ 2+y^ 2+z^ 2\) \((x\geq y\geq z\geq 0).\)
Gauss' conjecture itself turned out to be true already as a consequence of the theorem of Hecke--Deuring--Heilbronn stating that \(h_ D\to \infty\) as \(D\to -\infty\). A theorem of the author (1976) then reduces Gauss' class number problem via the famous conjecture of Birch and Swinnerton-Dyer to showing that there exists a modular elliptic curve \(E_ 0\) over \(\mathbb Q\) whose Hasse-Weil \(L\)-function \(L(E_ 0,s)\) has a triple zero at \(s=1\). The amazing theorem of Gross-Zagier (1983) implies that this is true, e.g., for the curve \(E_ 0: -139y^ 2=x^ 3+4x^ 2-48x+80\) of conductor \(N=37\cdot (139)^ 2\). The combination of these latter two theorems finally led to the above-cited decisive theorem of Goldfeld--Gross--Zagier and hence to a complete solution of Gauss' class number problem.
We should like to mention that in the list of contributors to the class-number-one problem given on pp. 32--33, an important paper of \textit{C. Meyer} [J. Reine Angew. Math. 242, 179--214 (1970; Zbl 0218.12007)] is missing, in which, aside from a detailed description of the history of the problem, a conjecture of Weber is proved. This conjecture played a controversial role in the interpretation of Heegner's original reasoning in his solution of the class-number-one case. Birch-Swinnerton-Dyer conjecture; sums of squares; class number problem; imaginary quadratic fields; Gauss' conjecture; modular elliptic curve; Hasse-Weil L-function; class-number-one problem \BibAuthorsD. Goldfeld, Gauss' class number problem for imaginary quadratic fields, Bull. Amer. Math. Soc. 13 (1) (1985), 23--37. Class numbers, class groups, discriminants, Quadratic extensions, Algebraic number theory computations, Elliptic curves over global fields, History of mathematics in the 18th century, History of number theory, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Holomorphic modular forms of integral weight Gauss' class number problem for imaginary quadratic 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 paper provides new results on a conjecture on exceptional almost perfect nonlinear (APN) functions. A (polynomial) function \(f: \mathbb F_{2^n}\rightarrow \mathbb F_{2^n} \) is an APN function if \(\forall a, b\in \mathbb F_{2^n},\, a\neq 0\), the equation \(f(x + a) - f(x) = b\)\, has at most 2 solutions. An APN function is called exceptional if it is APN on infinitely many extensions of \(\mathbb F_{2^n}\). \textit{Y. Aubry} et al. [Contemp. Math. 518, 23--31 (2010; Zbl 1206.94025)] conjectured that the only exceptional APN functions are the Gold monomials \(f(x)= x^{2^k+1}\)\, and the Kasami-Welch monomials \(f(x)=x^{2^{2k}-2^k+1}\). They also proved that a polynomial function \(f(x)\)\, of odd degree is not exceptional APN function when \(\deg(f)\)\, is not a Gold number \(2^k+1\)\, or a Kasami-Welch number \(2^{2k}-2^k+1\).
Section 1 of the present paper points out the relationships of APN functions with cyclic error-correcting codes and formulates the conjecture of Aubry, McGuire and Rodier. Section 2 recalls previous results of the authors concerning Gold polynomials \(f(x)= x^{2^k+1}+h(x)\),\, providing families of polynomials which are not exceptional APN.
The conjecture is true for polynomials \(f(x)= x^{2^k+1}+h(x)\)\, if \(d=\deg(h)\) is odd and \(d\)\, is not a Gold number or is a Gold number with \(d=2^l+1\)\, and \((l,k)=1\) (see Theorem 6). Section 3 deals with the case \((l,k)\neq 1\) (Theorems 7 and 8). Finally Section 4 discusses the case \(d\)\, even (Theorem 9). almost perfect nonlinear (APN); cyclic codes; Deligne estimate; Lang-Weil estimate; absolutely irreducible polynomial; CCZ-equivalence; EA-equivalence; Gold function, Kasami-Welch function Delgado M., Janwa H.: Further results on exceptional APN functions. AGCT-India (2013) http://www.math.iitb.ac.in/~srg/AGCT-India-2013/Slides/. Applications to coding theory and cryptography of arithmetic geometry, Algebraic coding theory; cryptography (number-theoretic aspects), Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Cryptography Some new results on the conjecture on exceptional APN functions and absolutely irreducible polynomials: the Gold case | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We study the connection between periodic finite-difference Intermediate Long Wave (\(\Delta\)ILW) hydrodynamical systems and integrable many-body models of Calogero and Ruijsenaars-type. The former describe quantum cohomology and quantum \(K\)-theory of the ADHM moduli space of Abelian instantons, while the latter arise in the instanton counting of four- and five-dimensional supersymmetric gauge theories with eight supercharges in the presence of defects. Using string theory dualities, we provide correspondences between hydrodynamical and many-body integrable systems. In particular, we match the energy spectra on both sides. integrable systems; supersymmetric gauge theories; quantum \(K\)-theory; quantum hydrodynamics; geometric representation theory P. Koroteev and A. Sciarappa, \textit{Quantum Hydrodynamics from Large-N Supersymmetric Gauge Theories}, arXiv:1510.00972 [INSPIRE]. Groups and algebras in quantum theory and relations with integrable systems, String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Supersymmetric field theories in quantum mechanics, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Quantum groups (quantized enveloping algebras) and related deformations, Quantum groups (quantized function algebras) and their representations, Quantum hydrodynamics and relativistic hydrodynamics, Quantum groups and related algebraic methods applied to problems in quantum theory Quantum hydrodynamics from large-\(n\) supersymmetric gauge theories | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 will investigate the relationship between Ihara's zeta functions of Ramanujan graphs and Hasse-Weil's congruent zeta functions of modular curves. As an application we will describe the limit value of Hasse-Weil's congruent zeta functions in terms of the corresponding Ramanujan graphs. Moreover we will show a congruence relation of the Fourier coefficients of a normalized Hecke eigenform of weight 2. Ramanujan graph; Ihara zeta function; a modular form; Hasse-Weil zeta function Enumeration in graph theory, Paths and cycles, Graphs and linear algebra (matrices, eigenvalues, etc.), Arithmetic aspects of modular and Shimura varieties, Curves over finite and local fields, Zeta and \(L\)-functions in characteristic \(p\), Zeta and \(L\)-functions: analytic theory, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Zeta functions of Ramanujan graphs and modular forms | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 nonsingular projective variety over the algebraically closed field \(k\), \({\mathcal L}\) a line bundle on \(X\) and \(h^ 0 ({\mathcal L}^{\otimes n}) = \dim_ k H^ 0(X, {\mathcal L}^{\otimes n})\). The Riemann-Roch problem consists in studying the function \(h^ 0({\mathcal L}^{\otimes n})\) for large \(n\). The problem was solved for curves and was first studied by Italian geometers for surfaces. \textit{O. Zariski} [Ann. Math., II. Ser. 76, 560-615 (1962; Zbl 0124.370)] proved that if \(S\) is a nonsingular surface and \(D\) an effective divisor on \(S\) then \(h^ 0 ({\mathcal O}_ S (nD)) = P(n) + \Lambda (n)\) for \(n \gg 0\), where \(P(x)\) is a quadratic polynomial and \(\Lambda (n)\) is a bounded function. In this fundamental paper Zariski also stated that ``it is an open question whether \(\Lambda (n)\) is always a periodic function of \(n\)''. Zariski himself obtained some partial results in this area. He proved that if \(D\) is an effective divisor on \(S\) such that the Kodaira dimension \(K(D) = \text{tr deg} (\bigoplus_{n \geq 0} H^ 0({\mathcal O}_ S (nD))) - 1\) is not greater than 1 then \(\bigoplus_{n \geq 0} H^ 0 ({\mathcal O}_ S (nD))\) is a finitely generated \(k\)-algebra and hence \(\Lambda (n)\) is periodic. Because \(K(D) \leq 2\) on a surface, the remaining case is \(K(D) = 2\).
The authors of the present paper solve the Zariski problem completely. The main results are the following:
Theorem 2: If \(\text{char} k = 0\), \(S\) is a normal surface proper over \(k\) and \(D\) is an effective Cartier divisor on \(S\), then for \(n \gg 0\), \(h^ 0 ({\mathcal O}_ S (nD)) = P(n) + \Lambda (n)\) where \(P(n)\) is a quadratic polynomial and \(\Lambda (n)\) is a periodic function.
Theorem 3: If \(k\) is a finite field, under the same hypothesis on \(S\), then \(\bigoplus_{n \geq 0} H^ 0 (S, {\mathcal O}_ S (nD))\) is a finitely generated \(k\)-algebra and \(h^ 0 ({\mathcal O}_ S (nD)) = P(n) + \Lambda (n)\), where \(P(n)\) is a quadratic polynomial and \(\Lambda (n)\) is periodic for \(n \gg 0\).
Theorem 3 is the stronger result because Zariski showed that \(\bigoplus_{n \geq 0} H^ 0({\mathcal O}_ s (nD))\) can be a nonfinitely generated \(k\)-algebra. The paper also contains many interesting other results. It is furnished a very nice but sophisticated example of a ruled surface \(X = \mathbb{P} ({\mathcal E})\), \(X @>\pi>>C\), \({\mathcal E} = {\mathcal O}_ \mathbb{C} + {\mathcal O}_ \mathbb{C} (P)\), over a field of positive characteristic \(p\) on which the Zariski problem has a negative answer for a line bundle \({\mathcal L} = {\mathcal O}_ X(1) \otimes \pi^* {\mathcal M}\). The Riemann-Roch problem in higher dimensions is also considered and it is given an example of an effective divisor \(D\) on a 3-fold \(X\) (projective and nonsingular) such that \(h^ 0 ({\mathcal O}_ X (nD))\) is a polynomial of degree 3 in \([n(2 - \sqrt 3/3)]\). In such way Zariski's problem has a negative answer even for effective divisors in dimension greater than 2. dimensions of linear systems; line bundle; Riemann-Roch problem; effective divisor S. D. Cutkosky and V. Srinivas, On a problem of Zariski on dimensions of linear systems , Ann. of Math. (2) 137 (1993), 531-559. JSTOR: Divisors, linear systems, invertible sheaves, Riemann-Roch theorems On a problem of Zariski on dimensions of linear systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors construct bases of non vanishing cohomology and homology groups, provide an interpretation as pairing of a cohomology class and a homology class to the Riemann-Wirtinger integral, and lastly describe the Gauss-Manin connection on the cohomology groups. theta function; integral representation Theta functions and abelian varieties, Classical hypergeometric functions, \({}_2F_1\), Homology with local coefficients, equivariant cohomology, de Rham cohomology and algebraic geometry, Analytic sheaves and cohomology groups Twisted cohomology and homology groups associated to the Riemann-Wirtinger integral | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 local zeta functions over \(p\)-adic fields are defined as integrals over open and compact subsets with respect to the Haar measure.
In this article, the author introduces new integrals defined over submanifolds as follows. Let \(K\) be a \(p\)-adic field, \(f_1,\ldots,f_l\) polynomials in \(K[x_1,\ldots,x_n]\) or, more generally, \(K\)-analytic functions on an open and compact subset \(U\subset K^n\) and let \(V^{(j)}=\left\{x\in U| f_i(x)=0,1\leq i\leq j\right\}\), for any \(1\leq j\leq l\) such that \(V^{(l-1)}\) is a closed submanifold of \(U\) and such that \(f_l:V^{(l-1)}\rightarrow K\) is an analytic function on \(V^{(l-1)}\). Let \(\Phi:K^n\rightarrow \mathbb{C}\) be a Bruhat-Schwartz function and let \(\omega\) be a quasi-character of \(K^{\times}\), i.e., a continuous homomorphism from \(K^{\times}\) into \(\mathbb{C}^{\times}\). To the above data, one can associate the following local zeta function:
\[
Z_{\Phi}\left(\omega,f_1,\ldots,f_l,K\right):=Z_{\Phi}(\omega,V^{(l-1)},f_l)=\int_{V^{(l-1)}(K)}\Phi(x)\omega\left(f_l(x)|\gamma_{GL}(x)|\right),
\]
where \(|\gamma_{GL}(x)|\) is the measure induced on \(V^{(l-1)}\) by a Gel'fand-Leray differential form, i.e., by a form satisfying \(\gamma_{GL}\wedge \bigwedge_{i=1}^{l-1}df_i=\bigwedge_{i=1}^{n}dx_i\).
In the present paper, the author provides a geometric description of the poles of the meromorphic continuation of \(Z_{\Phi}\left(\omega,f_1,\ldots,f_l,K\right)\) when \(f_1,\ldots,f_l\) are nondegenerate with respect to their Newton polyhedra.
Also, the connections between the above defined integral and a similar integral defined over nondegenerate complete intersection varieties and some arithmetical problems such as estimation of exponential sums \(\mod p^m\) are studied. In particular, Igusa's method for estimating exponential sums \(\bmod p^m\) is extended to the case of exponential sums \(\bmod p^m\) along nondegenerate smooth varieties.
Moreover, the author shows the existence of an asymptotic expansion for oscillatory integrals of type
\[
E_{\Phi}\left(z, V^{(l-1)},f_l\right)=\int_{V^{(l-1)}}\Phi(x)\Psi(zf_l(x))|\gamma_{GL}(x)|,
\]
for \(|z|_K \gg 0\), which is controlled by the poles of \(Z_{\Phi}(\omega,V^{(l-1)},f_l)\). Also, an estimation of the number of solutions of a polynomial congruence over a smooth algebraic variety is given.
In addition, several conjectures, problems and open questions connected with the local zeta functions are pointed out in this article. Igusa zeta function; p-adic fields; exponential sums; congruences in many variables; Newton polyhedra; complete intersection varieties W. A. Zúñiga-Galindo, ''Local zeta functions supported on analytic sets and Newton polyhedra,'' Intern. Math. Res. Notices (2009); doi: 10.1093/imrn/rnp035. Zeta functions and \(L\)-functions, Zeta and \(L\)-functions in characteristic \(p\), Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Local zeta functions supported on analytic submanifolds and Newton polyhedra | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This article determines an effective family of spectral curves appearing in Hitchin fibrations. By embedding the effective family of Jacobian varieties of the spectral curves into the Sato Grassmannian, the authors showed that the KP flows are tangent to each fiber of the Hitchin fibration. Since the moduli space of Higgs bundles of rank \(n\) and degree \(n(g-1)\) on an algebraic curve \(C\) of genus \(g\geq 2\) is embedded into the relative Grassmannian, this article gives the fact that the Hitchin integrable system, the natural algebraically completely integrable Hamiltonian system defined on the Higgs moduli space, coincides with the KP equations. The authors proved that the Krichever construction transforms the Serre duality of the geometric data consisting of algebraic curves and vector bundles on them to the formal adjoint of pseudo-differential operators acting on the Grassmannian. By identifying the fixed-point-set of the Serre duality and the formal adjoint operation, the Hitchin integrable system on the moduli space of \(Sp_{2m}\)-Higgs bundles is expressed in terms of a reduction of the KP equations. Since there are two ways to reduce an algebraically completely integrable Hamiltonian system, one by restriction and the other by taking a quotient of a Lie algebra action that is similar to the symplectic quotient, the authors showed the fact that both constructions yield SYZ-mirror pairs [\textit{A. Strominger, S.-T. Yau} and \textit{E. Zaslow}, Nucl. Phys., B 479, No. 1--2, 243--259 (1996; Zbl 0896.14024)] by applying them to the moduli spaces of Higgs bundles. The \(SL-PGL\) and \(Sp_{2m}-SO_{2m+1}\) dualities are also interpreted. Hitchin integrable system; spectral curve; Higgs bundle; Sato Grassmannian; symplectic KP equation Hodge, A.; Mulase, M., Hitchin integrable systems, deformations of spectral curves, and KP-type equations, Adv. Stud. Pure Math., 59, 31-77, (2010) Vector bundles on curves and their moduli, Relationships between algebraic curves and integrable systems, KdV equations (Korteweg-de Vries equations), Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests Hitchin integrable systems, deformations of spectral curves, and KP-type 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 Let \(X\subset\mathbb P^n\) be an integral non-degenerate subvariety. Fix \(P\in\mathbb P^n\) and an integer \(k>0\). Let \({\mathcal Z}(X,P,k)\) (resp. \({\mathcal S}(X,P,k)\)) be the set of all zero-dimensional subscheme (resp., zero-dimensional and reduced) \(Z\subset X\) such that \(\deg(Z)=k\), \(P\) is in the linear \(\text{span}\langle Z\rangle\) of \(Z\) but \(P\not\in\langle Z'\rangle\) for all \(Z'\subsetneqq Z\). We study these sets when \(X\) is a linearly normal curve with low genus with respect to \(n\). ranks; \(X\)-rank; zero-dimensional scheme; linearly normal curve Projective techniques in algebraic geometry Subschemes of a projective subvariety \(X\subset\mathbb P^n\) minimally spanning a given \(P\in\mathbb P^n\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For a system of (complex, multivariate) polynomial equations \(f_1=\cdots=f_s=0\), with \(f_1,\ldots,f_s\) generating the ideal \(I=\langle f_1,\ldots,f_s\rangle\subseteq R={\mathbb C}[x_1,\ldots,x_n]\), and with finitely many solutions \(V(I)\subseteq{\mathbb C}^n\), numerical eigenvalue methods for root finding rely on different variants of the eigenvalue theorem. Classically, for a given \(g\in{\mathbb C}[x_1,\ldots,x_n]\) one considers the multiplication map \(m_g:R/I\rightarrow R/I\) given by \(f+I\mapsto gf+I\) whose eigenvalues are \(g(z)\in {\mathbb C}\) for \(z\in V(I)\), and if one has a vector space basis \(b_1,\ldots, b_d\in R/I\) and the ideal \(I\) is radical, then the eigenvectors of \(m_g\) are the tuples \((b_1(z),\ldots,b_d(z))\in{\mathbb C}^d\) for each zero \(z\in V(I)\), and the coordinates of the solutions can be recovered from these eigenvectors.
In previous work, \textit{S. Telen} [J. Pure Appl. Algebra 224, No. 9, Article ID 106367, 26 p. (2020; Zbl 1442.14187)] studied numerically robust algorithm for sparse homogeneous polynomials systems taken from the Cox ring of a compact toric variety \(X\). The corresponding algorithm computes homogeneous coordinates of the solutions from the eigenvalues of a multiplication map in certain degrees of the Cox ring, requiring that the solutions of the system have multiplicity one and lie in the simplicial part of the toric variety \(X\).
The main contribution of the present paper extends the use of eigenvalue computations to solve polynomial systems on a toric variety \(X\) allowing that the equations may have isolated singularities which need not belong to the simplicial part of \(X\). In order to do that, they introduce a notion of regularity for these zero-dimensional systems, similar to the classical Castelnuovo--Mumford regularity for projective space and let the degrees to vary in the class group of \(X\). The first main result shows that for a homogeneous ideal \(I\subseteq \text{Cox}(X)\) in the Cox ring of a toric variety \(X\) defining a zero-dimensional subscheme \(V_X(I)\subseteq X\) of degree \(\delta^+\) and for any regularity pairs \(\alpha,\alpha_0\in\text{Cl}(X)\), if \(\phi\) is a regular rational function on \(V_X(I)\), the multiplication by \(\phi\) map on \((\text{Cox}(X)/I)_{\alpha+\alpha_0}\) has eigenvalues \(\phi(\zeta_i)\) of algebraic multiplicity \(\mu_i\), for \(\zeta\in V_X(I)\) and where \(\delta^+=\mu_1+\cdots+\mu_{\delta}\). Using this result and its proof which characterize the eigenvectors of the multiplication map, the authors obtain a numerical algorithm for computing homogeneous coordinates of the points in \(V_X(I)\). This algorithm depends on the regularity of the pair \((\alpha+\alpha_0)\), and the second main result gives a criterion for extending a regular degree \(\alpha\) to a regularity pair when \(X\) is a complete intersection. solving polynomial systems; sparse polynomial systems; toric varieties; Cox rings; eigenvalue theorem; symbolic-numeric algorithm Toric varieties, Newton polyhedra, Okounkov bodies, Numerical computation of roots of polynomial equations, Numerical computation of solutions to systems of equations Toric eigenvalue methods for solving sparse 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 author studies Hamiltonian systems $dF=0$ and their perturbations $dF+\varepsilon \omega =0$ by polynomial 1-forms $\omega$. The displacement function
$\Delta (t,\varepsilon )=\sum _{j=\mu}^{\infty}\varepsilon ^jM_j(t)$ along a cycle $\gamma (t)$ on a given level set $F^{-1}(t)$ is considered, where $M_{\mu}$ is the first nonzero Melnikov function.
\textit{Y. S. Ilyashenko} [Mat. Sb. (N.S.), 78, 360--373, (1969)] and \textit{J. P. Francoise} [Ergodic Theory Dyn. Syst. 16, No. 1, 87--96 (1996; Zbl 0852.34008)] have shown that $M_{\mu}$ is an abelian integral.
\textit{L. Gavrilov} [Ann. Fac. Sci. Toulouse, Math. (6) 14, No. 4, 663--682 (2005; Zbl 1104.34024)]
has shown that in general it is an iterated integral of length at most $\mu$.
The author continues the study of linear deformations of a family of non-generic Hamiltonian systems fulfilling certain geometric conditions. It turns out that the first nonzero Melnikov function is an iterated integral of length at most two. displacement function; Melnikov function; limit cycles; Hamiltonian system General theory of finite-dimensional Hamiltonian and Lagrangian systems, Hamiltonian and Lagrangian structures, symmetries, invariants, Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms, Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications) for ordinary differential equations, Structure of families (Picard-Lefschetz, monodromy, etc.) The first nonzero Melnikov function for a family of good divides | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 the moment and the distribution of \(L(1,\chi_P)\), where \(\chi_P\) varies over quadratic characters associated to irreducible polynomials \(P\) of degree \(2g+1\) over \(\mathbb{F}_q[T]\) as \(g\to\infty\). In the first part of the paper, we compute the integral moments of the class number \(h_P\) associated to quadratic function fields with prime discriminants \(P\), and this is done by adapting to the function field setting some of the previous results carried out by Nagoshi in the number field setting. In the second part of the paper, we compute the complex moments of \(L(1,\chi_P)\) in large uniform range and investigate the statistical distribution of the class numbers by introducing a certain random Euler product. The second part of the paper is based on recent results carried out by Lumley when dealing with square-free polynomials. mean values of \(L\)-functions; finite fields; function fields Zeta and \(L\)-functions in characteristic \(p\), \(\zeta (s)\) and \(L(s, \chi)\), Curves over finite and local fields, Relations with random matrices, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) The moments and statistical distribution of class number of primes over 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 Let \(F\) be an algebraically closed field with \(\text{char}(F)=p>0\) and let \(C \subset \mathbb{P}^ n\) \((n \geq 4)\) be an integral, nondegenerate curve with degree \(d\). Let \(\Gamma \subset C \times \mathbb{P}^{n^*}\) be the incidence correspondence. \((\mathbb{P}^{n^*}=\) hyperplanes in \(\mathbb{P}^ n)\). The author proves a result about the Galois group \(G\) of the normal extension of the function field \(F(\mathbb{P}^{n^*})\) generated by \(F(\Gamma)\). Assume that the linear general position property does not hold, i.e.: for a general hyperplane \(H\) there is a subset \(S \subset H \cap C\) with \(\text{card} (S) \leq n\) such that \(S\) spans a linear space of dimension \(<\text{card}(S)-1\). Assume that the trisecant lemma holds for \(C\) and that \(d>22\). Then \(d=2^ k\) for some \(k \geq n-1\) and \(G \cong \text{AGL} (k,2)\). hyperplane section of curve; curves on hyperplanes; Galois group of function field; incidence; trisecant lemma E. Ballico, On the general hyperplane section of a curve in char. p, Rendiconti Istit. Mat. Univ. Trieste 22 (1990), 117--125. Arithmetic ground fields for curves, Finite ground fields in algebraic geometry, Plane and space curves, Hypersurfaces and algebraic geometry On the general hyperplane section of a curve in char\( p\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems After reviewing the main properties of the Brieskorn lattice in the framework of tame regular functions on smooth affine complex varieties, we prove a conjecture of Katzarkov-Kontsevich-Pantev in the toric case. Brieskorn lattice; irregular Hodge filtration; irregular Hodge numbers; tame function Transcendental methods, Hodge theory (algebro-geometric aspects), Singularities in algebraic geometry, Local complex singularities, de Rham cohomology and algebraic geometry, Mixed Hodge theory of singular varieties (complex-analytic aspects), Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects) Some properties and applications of Brieskorn lattices | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 present paper provides an extension of the theory of perverse sheaves to algebraic stacks and therefore to moduli problems, \(\mathbb{Q}\)-varieties, algebraic spaces, etc. We also include a detailed study of the intersection cohomology of algebraic stacks and their associated moduli spaces. Smooth group scheme actions on singular varieties and the associated derived category turn up as special cases of the more general results on algebraic stacks. -- The main goals of the present paper are as follows:
(i) generalize much of the basic theory of perverse sheaves as done by \textit{A. A. Beilinson}, \textit{J. Bernstein} and \textit{P. Deligne} [``Faisceaux pervers'', Astérisque 100 (1982; Zbl 0536.14011)] to algebraic stacks; as a result the main results on perverse sheaves (for example the decomposition theorems for direct images of perverse sheaves by a proper map) are shown to hold in much more generality and apply in much wider contexts, for example moduli problems, \(\mathbb{Q}\)-varieties, algebraic spaces etc.
(ii) define and study the intersection cohomology of algebraic stacks (and their associated moduli spaces) in arbitrary characteristics. Recall that the only previous study of the intersection cohomology of moduli spaces is by \textit{F. C. Kirwan} [see ``Cohomology of quotients in symplectic and algebraic geometry'', Math. Notes 31 (1984; Zbl 0553.14020) and Invent. Math. 90, 153-167 (1987; Zbl 0631.14012)]; however her study is from an entirely different point of view and is only valid for complex varieties.
(iii) using the observation [see \textit{M. Artin}, Invent. Math. 27, 165-189 (1974; Zbl 0317.14001); p. 180] that algebraic stacks may be viewed as groupoid objects in the category of algebraic spaces, we are able to include the study of smooth group-scheme actions and the associated `equivariant' derived category as a special case of our general study of algebraic stacks. This provides an alternate construction of the equivariant derived category along with all the relevant machinery; the equivariant derived category becomes the natural home of the equivariant intersection cohomology complexes introduced by the author [see \textit{R. Joshua}, Math. Z. 195, 239-253 (1987; Zbl 0637.14014)] and has further applications [see \textit{R. Joshua}, ``Construction of simple (Deligne-Langlands) quotients for representations of \(p\)-adic groups'' (to appear)]. perverse sheaves; algebraic stacks; moduli problems; algebraic spaces; intersection cohomology; group-scheme actions; equivariant derived category Joshua, R.: The intersection cohomology and the derived category of algebraic stacks. In: Algebraic KTheory and Algebraic Topology, NATO ASI Series C, Vol. 407, pp. 91--145, Kluwer Academic, Dordrecht, 1993 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Group actions on varieties or schemes (quotients), Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Derived categories, triangulated categories The intersection cohomology and derived category of algebraic stacks | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors present some results related to plateaued functions and mappings whose derivatives are \(2^s\)-to-1 functions. These results generalize facts about almost perfect nonlinear (APN) functions and almost bent (AB) functions. Two possible generalizations both of APN and AB functions are presented. The connection between plateaued and \(2^s\)-to-1 functions is investigated. The authors show that functions which are both plateaued and differentially uniform give rise to partial difference sets. Nonlinearity; Almost perfect nonlinear; Almost bent; Vectorial Boolean function; Differential uniformity Budaghyan L., Pott A.: On differential uniformity and nonlinearity of functions. Discrete Math. 309, 371--384 (2009) Cryptography, Algebraic coding theory; cryptography (number-theoretic aspects), Applications to coding theory and cryptography of arithmetic geometry On differential uniformity and nonlinearity of functions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors show that an irreducible closed semialgebraic set is the projection of an irreducible algebraic set. They discuss orders of function fields, related to this result. As applications, they give interesting examples of real algebraic sets. irreducible closed semialgebraic set; orders of function fields; real algebraic sets Andradas, C.; Gamboa, J. M., On projections of real algebraic varieties, Pacific J. Math., 121, 2, 281-291, (1986) Real algebraic and real-analytic geometry, Real-analytic and Nash manifolds, Ordered fields On projections of real algebraic 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 article studies a new tower \(\mathcal{F}\) of function fields \(F_n\) defined over \(k=\mathbb{F}_{p^{3e}}\). This tower does not beat the one introduced in [\textit{A. Bassa} et al., Acta Arith. 164, No. 2, 163--179 (2014; Zbl 1320.11104)] in the sense that its limit
\[
\lambda(\mathcal{F}/k) = \lim_{n \to \infty} \frac{N(F_n)}{g(F_n)}
\]
(where \(N(F_n)\) is the number of \(k\)-rational places and \(g(F_n)\) the genus) is less than the one found in there. Here the authors shows that
\[
\lambda(\mathcal{F}/k)= \frac{2 (p^{2e}-1)}{p^e+1}
\]
(Zink's bound). However it has better properties in terms of the asymptotic \(p\)-rank than the one (called BaGS) from \textit{A. Bassa} et al. [Mosc. Math. J. 8, No. 3, 401--418 (2008; Zbl 1156.11045)]
\[
\phi(\mathcal{F}/k) = \lim_{n \to \infty} \frac{\gamma(F_n)}{g(F_n)}
\]
where \(\gamma(F_n)\) is the \(p\)-rank of \(F_n\). Indeed it is shown in the paper that
\[
\phi(\mathcal{F}/k) \leq \frac{p^{2e} + p^e+4}{4(p^{2e}+p^e+1}
\]
with equality when \(e=1\), which is better in this case than BaGS for which
\[
\phi = \frac{2 {p+1 \choose 2}^e -2}{(p^e-1)(p^e+2)}.
\]
This question is motivated by \textit{I. Cascudo} et al. [IEEE Trans. Inf. Theory 60, No. 7, 3871--3888 (2014; Zbl 1360.14080)]. The proof for the value of \(\lambda\) relies on classical arguments. The part about \(\phi\) is based on explicit computations to determine the \(p\)-rank of \(F_2\) and then Deuring-Shafarevich formula to propagate this information in the Artin-Schreier extensions defining \(\mathcal{F}\). tower of function fields; number of rational places; ihara's constant; cartier operator; \(p\)-rank N. Anbar, P. Beelen, N. Nguyen, A new tower meeting Zink's bound with good \(p\)-rank, appeared online 18 January 2017 in Acta Arithmetica. Algebraic functions and function fields in algebraic geometry, Curves over finite and local fields, Applications to coding theory and cryptography of arithmetic geometry A new tower with good \(p\)-rank meeting Zink's bound | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 unital quadratic form is defined as a triple \((X,Q,1_X)\), where \(X\) is a module over a commutative ring of scalars \(k\), equipped with a quadratic form \(Q\) and a base point \(1_X\) with \(Q(1_X)=1\) and such that \(\alpha (1_X)=1\) for some linear form \(\alpha\) on \(X\). These unital quadratic forms may be considered as higher dimensional analogues of quadratic algebras.
The first part of this very well-written paper deals with the construction of a tensor product of two unital quadratic forms, \((X_1,Q_1,1_{X_1})\square(X_2,Q_2,1_{X_2})\), extending the tensor product of separable quadratic algebras. As a \(k\)-module, this is just \(k\oplus \left( (X_1/k1_{X_1})\otimes_k (X_2/k1_{X_2}) \right)\), but the definition of the attached quadratic form is not trivial. This product is shown to satisfy nice functorial properties. The hyperbolic plane plays the role of neutral element, and the invertible elements are given precisely by the separable quadratic algebras. Also, to any unital quadratic form \((X,Q,1_X)\), a symmetric bilinear form on \(X/k1_X\), called the vector discriminant, is assigned and shown to be multiplicative with respect to \(\square\).
In the second part of the paper, a large variety of concepts related to the unital quadratic forms are studied. First, a notion of separability for these forms is defined, which extends the notion of nondegeneracy for quadratic forms over fields. The discriminant of the tensor product is computed in terms of the discriminant of the factors. The Clifford algebra and the orthogonal group schemes associated to unital quadratic forms, as well as some cohomological characterizations, are also investigated. unital quadratic form; base point; vector discriminant; tensor product; orthogonal group scheme; cohomology Loos O. (1996). Tensor products and discriminants of unital quadratic forms over commutative rings. Mh. Math. 122: 45--98 Quadratic spaces; Clifford algebras, Jordan algebras (algebras, triples and pairs), Classical groups (algebro-geometric aspects) Tensor products and discriminants of unital quadratic forms over commutative rings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(k\) be a field of characteristic zero and \(A\) a commutative \(k\)- algebra of finite type. The author shows that there are functorial isomorphisms \(\text{HP}^{(i)}_ * (A)@>\sim>> H^{2i-*}_{\inf}(A)\). Here \(\text{HP}_ *(A)\) denotes the periodic homology of \(A\), \(\text{HP}_ *(A)= \prod_{i\in \mathbb{Z}} \text{HP}_ *^{(i)}(A)\), and, given a smooth presentation \(A= R/I\) of \(A\), \(H_{\inf}(A)\) is the infinitesimal cohomology of \(\text{Spec }A\) over \(\text{Spec }k\), i.e., the cohomology of the \(I\)-adic completion \(\widehat\Omega_ R\) of the de Rham complex of \(R\).
The author proves that the inverse system \((\text{HC}_{*+ 2m}(A), S)_ m\) satisfies the Mittag-Leffler condition. The paper contains also criteria for the degeneration of the spectral sequence associated with Connes' exact couple. cyclic homology; degeneration of spectral sequence; periodic homology; infinitesimal cohomology Emmanouil, I.: Cyclic homology and de Rham homology of commutative algebras. C. R. Math. acad. Sci. Paris, sér. I 318, No. 5, 413-417 (1994) Other (co)homology theories (cyclic, dihedral, etc.) [See also 19D55, 46L80, 58B30, 58G12], (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.), \(K\)-theory and homology; cyclic homology and cohomology, Spectral sequences, hypercohomology, Étale and other Grothendieck topologies and (co)homologies Cyclic homology and de Rham homology of commutative algebras | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 complex projective variety and let \(f\colon X \to A\) be a morphism to an abelian varierty. By the work of many people (see the introduction of the paper for a detailed account) the sheaf \(f_*\omega_X\) has many important properties:
\begin{itemize}
\item its cohomological support loci are finite unions of translations of abelian subvarieties of \(\mathrm{Pic}^0(A)\);
\item it is a GV sheaf;
\item it has a canonical decomposition into pullbacks of M-regular coherent sheaves from quotients of \(A\) twisted by torsion line bundles
\item its Fourier-Mukai transform is locally analytically quasi-isomorphic to a linear complex.
\end{itemize}
The purpose of this very interesting paper is to show that the same properties hold for the sheaves \(f_*\omega_X^{\otimes m}\) for all integers \(m\ge 2\). (This is not unexpected since, as a general rule, the pluricanonical sheaves are better behaved than the canonical sheaf).
The results are obtained as a consequence of the above properties of \(f_*\omega_X\) and of the following
Theorem A. Fix \(m\ge 2\); there exists a generically finite morphism \(g\colon Y\to X\) such that \(f_*\omega_X^{\otimes m}\) is a direct summand of \((f\circ g)_*\omega_{Y}\).
Theorem A is in turn a consequence of
Theorem B. There is an isogeny \(\phi\colon A'\to A\) such that \(f'_*\omega_{X'}^{\otimes m}\) is globally generated, where \(f'\colon X'\to A'\) is obtained from \(f\colon X\to A\) by taking base change with \(\phi\).
The authors also prove a more refined version of the decomposition into into pullbacks of M-regular coherent sheaves from quotients of \(A\) twisted by torsion line bundles in the case when \(m\ge 2\) and \(f\) is the Albanese morphism.
Some results on the pluricanonical linear series of normal varieties of maximal Albanese dimension with at most canonical singularities are obtained as an application of the previous results. direct images; pluricanonical bundles; abelian varieties; non-vanishing loci; singular Hermitian metrics; pluricanonical systems Sheaves in algebraic geometry, Algebraic theory of abelian varieties Pushforwards of pluricanonical bundles under morphisms to abelian varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(S\) be a hyperbolic surface with one cusp \(\mathcal{C}\) and whose fundamental group \(\Gamma\) is a Schottky group with \(r\ge 2\) generators. We modify the metric inside the cusp \(\mathcal{C} \), at a height \(a \ge 0\), in such a way that the parabolic group associated with \(\mathcal{C}\) is convergent. We study the influence of the parameter \(a\) on the asymptotic behavior of the orbital function of \(\Gamma \). hyperbolic surface; Schottky group; fundamental group; orbital function Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.), Coverings of curves, fundamental group Phase transition of orbital functions for negative curvature Schottky groups | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We introduce the notion of \(\pi \)-cosupport as a new tool for the stable module category of a finite group scheme. In the case of a finite group, we use this to give a new proof of the classification of tensor ideal localising subcategories. In a sequel to this paper, we carry out the corresponding classification for finite group schemes. cosupport; stable module category; finite group scheme; localising subcategory; support; thick subcategory Cohomology of groups, Representations of associative Artinian rings, Modular representations and characters, Cohomology theory for linear algebraic groups, Group schemes Stratification and \(\pi\)-cosupport: finite groups | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For the entire collection see Zbl 0549.00017.]
The authors study the Igusa compactification \(\tilde S^*_ 2(2)\) of the Siegel modular variety of degree 2 and level 2. They begin by showing that \(\tilde S^*_ 2(2)\times k\) is smooth over any algebraically closed field k of characteristic \(p>3\). This extends work of \textit{G. van der Geer} [Math. Ann. 260, 317-350 (1982; Zbl 0473.14017)] who studied the Igusa compactification over \({\mathbb{C}}\). Using the modular interpretation of the Siegel modular variety \(S^*_ 2(2)\), the authors give a geometric interpretation of the Igusa compactification, and show that \(\tilde S^*_ 2(2)\) admits a natural decomposition into the disjoint union of the moduli space of genus 2 curves with order Weierstrass points, 10 copies of \(({\mathbb{P}}^ 1-\{0,1,\infty \})^ 2\), 15 copies of a boundary component minus its singular fibres, and 15 copies of \(\cup^{3}_{i=1}{\mathbb{P}}^ 1_ i\) where the points \(0\in {\mathbb{P}}^ 1_ i\) are identified.
Fixing a prime \(p>3\), and counting \({\mathbb{F}}_{p^ n}\)-rational points on each of the above subsets, the authors compute the zeta function of \(\tilde S^*_ 2(2)\) over an algebraically closed field of characteristic p. Finally, using Deligne's proof of the Weil conjectures for varieties over finite fields, and the comparison theorems for étale cohomology, the complex cohomology of the variety \(\tilde S^*_ 2(2)\) is recovered.
In addition to the results described above, we mention that the authors also provide an excellent summary of many of the tools used in modern algebraic geometry and arithmetic. These include a description of the Weil conjectures for varieties over finite fields (Deligne's theorem), and some of the basic results of étale cohomology, and birational geometry. Igusa compactification; Siegel modular variety; moduli space of genus 2 curves; Weierstrass points; zeta function; characteristic p; Weil conjectures Ronnie Lee and Steven H. Weintraub, Cohomology of a Siegel modular variety of degree 2, Group actions on manifolds (Boulder, Colo., 1983) Contemp. Math., vol. 36, Amer. Math. Soc., Providence, RI, 1985, pp. 433 -- 488. Families, moduli, classification: algebraic theory, Theta series; Weil representation; theta correspondences, Classical real and complex (co)homology in algebraic geometry, Arithmetic ground fields for surfaces or higher-dimensional varieties, Automorphic functions, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Cohomology of a Siegel modular variety of degree 2 | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \({\mathcal D}\) be a triangulated category which is essentially small, i.e., equivalent to a small one, such as the derived category \(D({\mathcal A})\) of a small abelian category \({\mathcal A}\). The author considers full triangulated subcategories \({\mathcal A}\) of \({\mathcal D}\) such that (i) \({\mathcal A}\) is closed under isomorphisms, and (ii) each object of \({\mathcal D}\) is a direct summand of an object in \({\mathcal A}\). The author first shows that there is a bijective correspondence between such subcategories \({\mathcal A}\) in \({\mathcal D}\) and subgroups \(H\) of the Grothendieck group \(K_0({\mathcal D})\). A subcategory \({\mathcal A}\) corresponds to the image of \(K_0 ({\mathcal A})\) in \(K_0({\mathcal D})\); a subgroup \(H\) corresponds to the full subcategory \({\mathcal A}_H\) whose objects are those \(a\) in \({\mathcal D}\) such that \(\overline a\in H\).
Now let \(X\) be a quasi-compact and quasi-separated scheme, such as an affine scheme or an algebraic variety. The derived category \(D(X)_{par f}\) of perfect complexes is essentially small. A thick triangulated subcategory \({\mathcal T}\) of \(D(X)_{par f}\) is called a tensor subcategory if for each object \(e\) in \(D(X)_{par f}\) and each object \(t\) in \({\mathcal T}\) the (derived) tensor product of \(e\) and \(t\) is also in \({\mathcal T}\). The second main result gives a bijective correspondence between the tensor subcategories \({\mathcal T}\) and the subspaces \(Y\) in \(X\) which are unions of closed subspaces with quasi-compact complement. Such a subspace \(Y\) corresponds to the tensor subcategory of perfect complexes which are acyclic at each point of \(X-Y\).
Combining these results, the author obtains a classification of tensor subcategories \({\mathcal A}\) of \(D(X)_{par f}\) satisfying (i) and (ii) above: they correspond bijectively to data \((Y,H)\) where \(Y\) is a subspace of \(X\) as above and \(H\) is a \(K_0(X)\)-submodule of the Grothendieck group of perfect complexes acyclic off \(Y\). The author's motivation for developing these results was an intuition about `higher-dimensional Cartier cycles' which, in his view, would have to correspond to perfect complexes in some triangulated subcategory of \(D(X)_{par f}\). triangulated category; derived cateogry; Grothendieck group; affine scheme; algebraic variety; perfect complexes; tensor subcategories; higher-dimensional Cartier cycles R. W. Thomason, The classification of triangulated subcategories. \textit{Compositio Math.} 105 (1997), no. 1, 1--27.MR 1436741 Zbl 0873.18003 Derived categories, triangulated categories, \(K\)-theory of schemes, Grothendieck groups (category-theoretic aspects), Applications of methods of algebraic \(K\)-theory in algebraic geometry, Grothendieck groups and \(K_0\) The classification of triangulated subcategories | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 deals with the classification problem of stable holomorphic rank 2 vector bundles on \(\mathbb P^2\). Its starting point is that every such bundle \(F\) with \(c_1(F)=-1\) and \(c_2(F)=n\) is the cohomology of a self-dual monad:
\[
\mathbb C^{n-1}\otimes\mathcal O(-1)\overset{a} \longrightarrow\mathbb C^n\otimes\Omega(1)\overset{a^T(-1)} \longrightarrow\mathbb C^{n-1}\otimes\mathcal O.
\]
Using this, the rationality of the moduli scheme \(M(-1,n)\) is proved. A line \(L\subset \mathbb P^2\) is called jumping line of the second kind if \(h^0(F|L^2)\neq 0\), where \(L^2\) denotes the first infinitesimal neighbourhood of \(L\). It is proved that the set \(C(F):=\{L\in {\mathbb P^3}^*;h^0(F|L^2)\neq 0\}\) forms a curve of degree \(2(n-1)\). \(C(F)\) together with a naturally defined \(\mathcal O_{C(F)}\)-sheaf \(\theta_f\) determines the bundle completely. Moreover it is proved that the set \(S(F)\subset\mathbb P^2\) of jumping lines is contained in the singularity set of \(C(F)\). For the general bundle \(F\) the equality \(S(F)=\mathrm{Sing} C(F)\) holds. Rationality of Moduli Scheme; Holomorphic Vector Bundle; Singularity Hulek, K., Stable rank-2 vector bundles on \(\mathbf{P}_2\) with \(c_1\) odd, Math. Ann., 242, 3, 241-266, (1979) Sheaves and cohomology of sections of holomorphic vector bundles, general results, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Algebraic moduli problems, moduli of vector bundles, Rationality questions in algebraic geometry Stable rank-2 vector bundles on \(\mathbb{P}_2\) with \(c_1\) odd | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 Picard group of a scheme and related functors are studied. The representability problem for functors is of particular interest. Picard group of a scheme Picard groups, Homogeneous spaces and generalizations, Representation theory for linear algebraic groups, Riemann-Roch theorems Abelsche Schemata und Darstellbarkeit von algebraischen Gruppenräumen | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Kolchin introduced the notion of strongly normal extensions of differential fields and developed an elegant theory of finite-dimensional differential Galois extensions which extended the Picard-Vessiot theory. However, strongly normal extensions do not generalize Galois extensions of abstract fields (when considered as fields with trivial derivation).
In this paper, the author introduces several functors from the category of algebras over a commutative ring \(R\) to the category of groups. For example the Lie-Ritt functor \(\Gamma_{nR}\), which associates to each commutative \(R\)-algebra \(A\) a group \(\Gamma_n(A)\) of all infinitesimal coordinate transformations of \(n\)-variables defined over \(A\), the functor \({\mathcal F}\), which associates with each algebra \(A\) over an abstract field \(L\) the set of all differential homomorphisms from \({\mathcal L}\), the subfield of the formal Laurant series over \(L\) (namely contained in \(L[[t]] [t^{-1}]\) generated by \(L^*\) and this is shown to be independent of the choice of the transcendental basis of \(L\) over \(K\) (as abstract fields)). There is another functor \(\text{Inf-diff bir}_K(L)\), which is a group functor from the category of \(L\)-algebras to the category of groups, which is in fact a Lie-Ritt functor. The main result (Theorem 5.15) shows that when \(L\) is a strongly normal extension of \(K\) with Galois group \(G\), then \(\text{Inf-diff bir}_K(L)\) is the formal group associated with the algebraic group scheme \(G\) and that this group ignores algebraic extensions and extensions generated by constants. differential Galois theory of infinite dimension; differential fields; Lie-Ritt functor; algebraic group scheme Umemura H., Differential Galois theory of infinite dimension, Nagoya Math. J.144 (1996) 59-135. Zbl0878.12002 MR1425592 Differential algebra, Group actions on varieties or schemes (quotients) Differential Galois theory of infinite dimension | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We present an approximate implicitization method for planar curves. The computed implicit representation is a piecewise rational approximation of the distance function to the given parametric curve. The proposed method consists of four main steps: quadratic B-spline approximation of the given parametric curve, data reduction, segments-wise implicitization, multiplying with suitable polynomial factors. These segments are joined such that the collection generate a global \(C r\) spline function which approximates the distance function, for \(r = 0, 1\). approximate implicitization; distance function Computational aspects of algebraic curves, Approximation by rational functions, Spline approximation, Computer-aided design (modeling of curves and surfaces), Plane and space curves Approximate implicitization of planar curves by piecewise rational approximation of the distance 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 In this interesting paper, the author studies on the base of a generalization of Grothendieck's descent theorem in the framework of locally Krull schemes on which group schemes operate, relations among enriched principal bundles, class groups and Picard groups. In the previous work [J. Algebra 459, 76--108 (2016; Zbl 1348.13010)], the author introduced for a flat \(S\)-group scheme \(G\) and for a locally Krull \(G\)-scheme \(X\) a principal \(G\)-bundle \(\varphi : X \to Y\) and an equivariant class group \(Cl(G,X)\). It was shown, by the utilizing of Grothendieck's descent theorem, that \(Y\) is locally Krull and that `the inverse image functor \(\varphi^* \) induces an isomorphism \(Cl(Y) \to Cl(G,X)\)`. The present paper deals with an enriched version. Let \(f: G \to H\) be an fpqc homomorphism, \(N:= \ker f\), \(\varphi : X \to Y\) be a \(G\)-morphism which is also a principal \(N\)-bundle, \(Cl(G,X)\), \(Cl(H,Y)\) be equivariant class groups, \(\varphi^* : Cl(H,Y) \to Cl(G,X)\) be a morphism (really isomorphism). The main result presents and characterizes the enriched Grothendieck`s descent theorem which yields the isomorphism \(\varphi^*\) and a similar isomorphism \(\varphi^* : \mathrm{Pic}(H,Y) \to\mathrm{Pic}(G,X)\). The Grothendieck-Mumford approach consists in passing from commutative rings to corresponding affine schemes and analyzing their singularities by algebraic geometry methods. The theory of the Grothendieck descent is used for gluing schemes from affine pieces. Along with the definition of Krull rings, which the author uses, the Krull rings can be defined as corresponding rings for which there exists a theory of divisors by \textit{Z. I. Borevich} and \textit{I. R. Shafarevich} [Number theory. Translated by Newcomb Greenleaf. New York and London: Academic Press (1966; Zbl 0145.04902)]. After the introductory section the author of the paper under review gives in section 2 results on quasi-fpqc morphisms and enriched principal bundles. Section 3 deals with \(\kappa\)-schemes. Section 4 is on module sheaves over a ringed site. Big and interesting sections 5 and 6 on Grothendieck's descent and enriched Grothendieck's descent. The last short section contains observations on equivariant Picard groups and class groups. In this section, the results of the paper are applied to equivariant principal bundles, and, in particular, to the case of finite Galois extension of a field \(k\). In the last case let \(N_0\) be a finite étale \(k-\)group scheme, \(\varphi : X \to Y\) be a principal \(N_0\)-bundle and \(k^{`}/k\) be a finite Galois extension with Galois group \(H\), such that \(k^{`}\otimes N_0\) is a constant group scheme \(N\). Let \(G = N \rtimes H\). Then \(G\) and \(H\) are constant group schemes and \(\varphi\) is a principal \(N_0\)-bundle, and \(N_0\) need not be constant. group scheme; locally Krull scheme; class group; descent theory; Picard group; principal fiber bundle Group actions on varieties or schemes (quotients), Group schemes, Actions of groups on commutative rings; invariant theory Equivariant class group. II: Enriched descent theorem | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be an affine analytic complex variety. In the note the author considers the MacPherson-Chern class of a constructible function on \(X\) as an object in the Borel-Moore homology of \(X\) and discusses the problem of computing the corresponding analytic cycle on \(X\) in terms of polar multiplicities. His approach is based essentially on basic properties of the characteristic polar multiplicities established by the author in a quite general context for non-isolated singularities [see \textit{D. B. Massey}, `Lê cycles and hypersurface singularities' (1995; Zbl 0835.32002)] as well as on well-known results due to MacPherson, Brylinski, and others concerning explicit calculations of MacPherson-Chern and Chern-Mather classes, the local Euler obstruction of \(X,\) and related constructions [see \textit{R. MacPherson}, Ann. Math., II. Ser. 100, 423-432 (1974; Zbl 0311.14001); \textit{J. Brylinski} et al., C. R. Acad. Sci., Paris, Sér. I. 293, 573-576 (1981; Zbl 0492.58021)]. constructible function; perverse sheaf; Whitney stratification; polar multiplicity; Milnor number; Lê cycles; Lê numbers; MacPherson-Chern class; MacPherson-Chern cycle; Chern-Mather class; Borel-Moore homology; local Euler obstruction Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects), Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Differential topological aspects of diffeomorphisms, Multiplicity theory and related topics, Riemann-Roch theorems, Chern characters Macpherson-Chern classes and characteristic polar cycles | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\subset P^ g\) be a smooth linearly normal K-3 surface of degree \(2g-2\) \((g\geq 3)\), and let C be a smooth hyperplane section of X. By considering a general element \(\sum^{g-1}_{i=1}x_ i\) of the symmetric product \(S^{g-1}(X)\), one gets a well defined element i(\(\sum^{g-1}_{i=1}x_ i)\) by putting \(i(\sum^{g-1}_{i=1}x_ i)=X\cdot <x_ 1,...,x_{g-1}>-\sum^{g-1}_{i=0}x_ i,\) where \(X\cdot <x_ 1,...,x_{g-1}>\) is the intersection zero cycle of X with the linear subspace \(<x_ 1,...,x_{g-1}>\) of \(P^ g\) generated by \(x_ 1,...,x_{g-1}\). Thus one gets a birational involution \(i: \tilde S^{g-1}(X)\to S^{g-1}(X).\) As is well known, \(S^{g-1}(X)\) admits a natural desingularization \(\tilde S^{g-1}(X)\) (a component of the Hilbert scheme of all zero-dimensional subschemes of length \(g-1\) in X), and hence one gets an involution \(i: \tilde S^{g-1}(X)\to \tilde S^{g-1}(X).\) One can stratify \(\tilde S^{g-1}(X)\) by putting \(\tilde S_ i^{g-1}(X)=\{u\in S^{g-1}(X)| h^ 1(I_ u(C))\geq i\}\), where \(I_ u\subset {\mathcal O}_ X\) is the ideal sheaf of the effective zero cycle u. The aim of the paper under review is to study the maximal stratum \(\tilde S_ 1^{g-1}(X)-\tilde S_ 2^{g-1}(X)\) in connection with the above involution (in particular, computing its dimension). K-3 surface; intersection zero cycle; Hilbert scheme A. N. Tyurin, Special \(0\)-cycles on a polarized surface of type \(K3\) , Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987), no. 1, 131-151, 208. \(K3\) surfaces and Enriques surfaces, Special surfaces, Algebraic cycles, Projective techniques in algebraic geometry Special 0-cycles on a polarized K3 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 study graded algebras associated to 0-dimensional Gorenstein ideals \(I\subset\mathbb{C}[X_0,\dots,X_r]\) with fixed socle degree and such that \(I\) contains a complete intersection of fixed multidegree. We conjecture that among such algebras those minimizing the Hilbert function are some complete intersections. We prove an asymptotic version of this conjecture.
Our result is an important case of a conjecture of \textit{D. Eisenbud}, \textit{M. Green} and \textit{J. Harris} [in: Journées Géom. Algébr., Orsay 1992, Astérisque 218, 187-202 (1993; Zbl 0819.14001)]. In Hodge theory, it enables us to compute the biggest component of the Noether-Lefschetz locus of projective hypersurfaces of very high degree, in any dimension. graded algebras; 0-dimensional Gorenstein ideals; Hilbert function; complete intersection; Noether-Lefschetz locus Otwinowska, A., Sur la fonction de Hilbert des algèbres graduées de dimension 0, J. Reine Angew. Math., 545, 97-119, (2002) Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Complete intersections, Linkage, complete intersections and determinantal ideals On the Hilbert function of graded algebras of dimension 0 | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 part I cf. ibid. 7, 374-403 (2003; Zbl 1072.20052).]
Summary: We establish the generalized Springer correspondence for possibly disconnected groups. character sheaves; reductive algebraic groups; strata; local systems; generalized Springer correspondence Deligne, P., Kazhdan, D., Vigneras, M.-F.: Représentations des algèbres centrales simples \(p\)-adiques. In: Représentations des groupes réductifs sur un corps local Travaux en cours, Hermann, pp. 33-117 (1984) Representation theory for linear algebraic groups, Linear algebraic groups over arbitrary fields, Group actions on varieties or schemes (quotients) Character sheaves on disconnected groups. 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 \(p\) be and odd prime. Wieferich related the question of whether $2^{p-1}-1$ is divisible for $p^2$ to (the ``first case'' of) Fermat's Last Theorem for the exponent $p$. Here, we formulate an equidistribution conjecture about the sequence, indexed by odd primes $p$ of fractions $\frac{2^{p-1}-1}{p^2}$ mod $\mathbb Z$ in $\mathbb R / \mathbb Z$. We then formulate versions of this conjecture for algebraic tori, for elliptic curves, for abelian varieties and for semi-abelian varieties. equidistribution; group scheme; elliptic curve; abelian variety Elliptic curves over global fields, Arithmetic ground fields for abelian varieties, Curves of arbitrary genus or genus \(\ne 1\) over global fields, Well-distributed sequences and other variations Wieferich past and future | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Orlov's folklore conjecture states that a smooth projective surface admitting a full exceptional collection must be a rational surface. In this paper, the author, using the techniques of toric systems introduced by \textit{L. Hille} and \textit{M. Perling} [Compos. Math. 147, No. 4, 1230--1280 (2011; Zbl 1237.14043)], proves some partial results on this conjecture: he proves that a smooth projective surface with a strong exceptional collection of line bundles of maximal length should be rational (Theorem 1.2) (improving a result by \textit{M. Brown} and \textit{I. Shipman} [Mich. Math. J. 66, No. 4, 785--811 (2017; Zbl 1429.14013)]).
The second main result deals with Orlov's conjecture for surfaces with small Picard number: he proves that a smooth projective surface \(X\) with Picard number \(\rho(X)\leq 3\) admitting a full exceptional collection is rational (Theorem 1.4).
Finally, he gives a partial solution to a conjecture by \textit{S. Okawa} and \textit{H. Uehara} [Int. Math. Res. Not. 2015, No. 23, 12781--12803 (2015; Zbl 1376.13007)] about exceptional sheaves on weak del Pezzo surfaces (Theorem 1.13). toric systems; full exceptional collection; rational projective surface Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Rational and ruled surfaces, Toric varieties, Newton polyhedra, Okounkov bodies Applications of toric systems on 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 article makes a generalization of an application of the Abramovich-Vistoli theory of twisted stable maps to higher dimensions. It is a general technique to use the Abromovich-Vistoli theory to construct compactifications of moduli spaces of curves with level structure. For the following, let \(g\) be an integer \(\geq 2\).
Given \(C/S\) a smooth proper curve of genus \(g\) over a scheme \(S\). Let \(P\) be a finite set of primes which includes all residue characteristics of \(S\). For any section \(s:S\rightarrow C\) there exists a pro-object \(\pi_1(C/S,s)\) in the category of locally constant sheaves of finite groups on \(S\) whose fiber over a geometric \(\overline{t}\rightarrow S\) is equal to the maximal quotient of \(\pi_1(C_{\overline t},s_{\overline t})\) prime to \(P\).
For a finite group \(G\) of order not divisible by any of the primes in \(P\), \(\mathcal H om^{\text{ext}}(\pi_1(C/S,s),G)\) denotes the sheaf of homomorphisms \(\pi_1(C/S,s)\rightarrow G\) modulo the action of \(\pi_1(C/S,s)\) given by conjugation. Then \(\mathcal H om^{\text{ext}}(\pi_1(C/S,s),G)\) is locally constant on \(S\) and is canonically independent of the section \(s\). It turns out that for any smooth curve \(C/S\) this sheaf can be canonically defined even when there does not exist a section.
A Teichmüller structure of level \(G\) on \(C/S\) is defined as a section of \(\mathcal H om^{\text{ext}}(\pi_1(C/S),G)\) which étale locally on \(S\) can be represented by a surjective homomorphism \(\pi_1(C/S),s)\rightarrow G\) for a section \(s\). The stack \({}_G\mathcal M_g\) over \(\mathbb Z[1/|G|]\) is defined by the condition that to any \(\mathbb Z[1/|G|]\)-scheme \(S\) is associated the groupoid of pairs \((C/S,\sigma)\), where \(C\) is a smooth proper curve of genus \(g\) over \(S\) and \(\sigma\) is a Teichmüller structure of level \(G\) on \(C/S\).
The connection with the theory of Abramovich-Vistoli is described in short form as follows: Let \({}_G\mathcal K_g^\circ\) be the \(\mathbb Z[1/|G|]\)-stack associating to a \(\mathbb Z[1/|G|]\)-scheme \(S\) the groupoid of pairs \((C/S,P\rightarrow C)\) with \(C/S\) a smooth proper genus \(g\)-curve, and \(P\rightarrow S\) a principal \(G\)-bundle, such that for every geometric point \(\overline t\rightarrow S\) the fibre \(P_{\overline t}\rightarrow C_{\overline t}\) is connected. Then there is a morphism of stacks \({}_G\mathcal K_g^\circ\rightarrow\mathcal M_g\) sending the pair \((C/S,P\rightarrow C)\) to the curve \(C/S\). Choose a trivialization \(\tilde s:S^\prime\rightarrow s^\ast P\) of the \(G\)-torsor \(s^\ast P\). It defines a homomorphism \(\pi_1(C/S,s)\rightarrow G\), and the conjugacy class of this homomorphism is independent of the choice of \(\tilde s\) and also of the section \(s\). Thus a section of \(\mathcal H om^{\text{ext}}(\pi_1(C/S),G)\) is obtained, even when \(C/S\) does not admit a section. This defines a morphism of stacks of \(\mathcal M_g\), \({}_G\mathcal K_g^\circ\rightarrow{}_G\mathcal M_g\) and this map identifies \({}_G\mathcal M_g\) with the relative coarse moduli space of \({}_G\mathcal K_g^\circ\rightarrow\mathcal M_g\).
From another viewpoint, the category of \(G\)-torsors over a smooth genus \(g\)-curve \(C/S\) is equivalent to the category of morphisms \(C\rightarrow BG\). So the theory of twisted stable maps gives a natural compactification \({}_G\mathcal K_g\) of \({}_G\mathcal K_g^\circ\) over \(\mathbb Z[1/|G|]\). Forgetting the \(G\)-torsor gives a morphism \({}_G\mathcal K_g\rightarrow\overline{\mathcal M}_g\), so that passing to the associated relative coarse moduli space over \(\overline{\mathcal M}_g\) gives a compactification of \({}_G\mathcal M_g\).
The author proves that if \(G\) is a tame group scheme it is possible to give a theory of twisted stable maps. This gives an extension of the stable map spaces to schemes where \(|G|\) is not invertible, and the resulting moduli spaces \({}_G\mathcal K_g\) are proper. Considering e.g. \(G=\mu_n\), the space \({}_{\mu_n}\mathcal K_g\) is a proper moduli space parameterizing \(\mu_n\)-torsors over twisted curves. The relative coarse moduli space of \({}_{\mu_n}\mathcal K_g^\circ\rightarrow\mathcal M_g\) is equal to the \(n\)-torsion subgroup scheme of the Jacobian of the universal curve over \(\mathcal M_g\). Thus proper models over \(\mathbb Z\) of the \(n\)-torsion subgroup of the universal Jacobian can be given.
There are two natural parts in the construction of the compactification \({}_G\mathcal K_g\). There is the Deligne -Mumford compactification \(\mathcal M_g\hookrightarrow\overline{\mathcal M}_g\) of \(\mathcal M_g\), and there is the universal stable curve \(\mathcal C\rightarrow\overline{\mathcal M}_g\) restricting to the universal curve \(\mathcal C\rightarrow {\mathcal M}_g\). In the last case, viewing \(\mathcal M_g\) as the base, \({}_G\mathcal K_g\) is a compactification over \(\overline{\mathcal M}_g\) of the stack classifying \(G\)-torsors on a fixed family of curves \(\mathcal C\rightarrow\mathcal M_g\). The present article generalizes the second view to higher dimensional varieties.
Consider a flat, proper, and semi-stable morphism of log schemes \(f:(X,M_X)\rightarrow (S,M_S)\). when \(G\) is a tame flat group scheme, the author constructs a proper \(S\)-stack \({}_G\mathcal K_{X/S}\) whose restriction to the open subset \(S^\circ\subset S\) where \(X\rightarrow S\) is smooth, is the moduli stack of \(G\)-torsors on \(X^\circ\) defined as \(X^\circ=X\times_S S^\circ\).
If \(S=\overline{\mathcal M}_g\) and \(X=\overline{\mathcal C}\) is the universal curve over \(\overline{\mathcal M}_g\), the log structures \(M_{\overline{\mathcal M}_g}\) and \(M_{\overline{\mathcal C}}\) defined by the divisors at infinity define a morphism of log stacks \((\overline{\mathcal C},M_{\overline{\mathcal C}})\rightarrow (\overline{\mathcal M}_g,M_{\overline{\mathcal M}_g})\) and the present theory can be applied. In this way, the spaces of stable maps \({}_G\mathcal K_g\) is obtained. Correspondingly, for any simple extension of log structures \(j:M_S\hookrightarrow N_S\) on \(S\) there is an associated tame stack \(\mathcal X_j\rightarrow X\). Then \({}_G\mathcal K_{X/S}\) is the stack which to any \(S\) scheme \(g:T\rightarrow S\) associates the groupoid of pairs \((j:g^\ast M_S\hookrightarrow N_T,P\rightarrow\mathcal X_j),\) with \(j\) a simple extension with associated tame stack \(\mathcal X_j\rightarrow X\times_S T,\) and \(P\rightarrow\mathcal X_j\) a \(G\) torsor. Together with a suitable stability condition, this is called a \textit{twisted \(G\)-torsor}. The main property of the twisted \(G\)-torsor is a main result of the article: (i) The stack \({}_G\mathcal K_{X/S}\) is a proper algebraic stack over \(S\) with finite diagonal. (ii) The stack \({}_G\mathcal K_{X/S}\) is tame.
To be able to prove the main results, the author studies various extension results for \(G\)-torsors with \(G\) a tame group scheme. Letting \(V\) be a discrete valuation ring, \(G/V\) a finite flat tame group scheme, and \(P_\eta\) a \(G\)-torsor over the field of fractions of \(V\). After a ramified base change of \(V\), the \(G\)-torsor \(P_\eta\) extends to \(V\). The problem of when a torsor on a stack descends to the coarse moduli space is studied, including the relative setting. Finally, it is proved that \({}_G\mathcal K_{X/S}\) is an Artin stack of finite type over the base \(S\) with finite diagonal.
The article is sufficiently self contained and detailed. It is lot of parts to be proved to obtain the generalizations, and the author follows the procedure by proving point by point the different parts in a good way. Also, the article is a nice text for getting into the language of log schemes, and in particular stacks, at least if one consult the long list of relevant references. Abramovich-Vistoli theory; log scheme; Teichmüller structure; level structure; principal bundle; tame group scheme; G-torsor; twisted group torsor Martin Olsson, Integral models for moduli spaces of \?-torsors, Ann. Inst. Fourier (Grenoble) 62 (2012), no. 4, 1483 -- 1549 (English, with English and French summaries). Moduli, classification: analytic theory; relations with modular forms, Fibrations, degenerations in algebraic geometry, Algebraic moduli problems, moduli of vector bundles Integral models for moduli spaces of \(G\)-torsors | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 theorem of this paper is that the Hilbert scheme compactification of the space of twisted cubic curves is smooth. This is shown by an explicit computation of the universal deformation of the worst possible flat degeneration of a twisted cubic. Along the way the authors prove the following result, now commonly known as the Piene- Schlessinger comparison theorem: Suppose X in \({\mathbb{P}}^ n\) is defined by homogeneous polynomials \(f_ 1,...,f_ r\) of degrees \(d_ 1,...,d_ r\), respectively, and that the linear systems cut out by hypersurfaces of degrees \(d_ 1,...,d_ r\) on X are complete. Then any infinitesimal deformation of X is induced by a unique deformation of the affine cone over X. Hilbert scheme compactification of the space of twisted cubic; curves; Piene-Schlessinger comparison theorem; infinitesimal deformation; Hilbert scheme compactification of the space of twisted cubic curves R. Piene and M. Schlessinger, On the Hilbert scheme compactification of the space of twisted cubics, Amer. J. Math. 107 (1985), no. 4, 761-774. Parametrization (Chow and Hilbert schemes), Fine and coarse moduli spaces, Formal methods and deformations in algebraic geometry, Special algebraic curves and curves of low genus On the Hilbert scheme compactification of the space of twisted cubics | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(X\) a smooth quasi-projective algebraic surface and \(E\) a line bundle on \(X\). Consider \(X^{[n]}\), the Hilbert scheme of \(n\) points on \(X\), and the tautological bundle \(E^{[n]}\) on it naturally associated to \(E\). In the present paper the author relates the cohomology of \(X^{[n]}\) with values in \((E^{[n]})^{\otimes 2}\) with the cohomologies of \(X\) with values in \(E^{\otimes 2}\), \(E\) and \({\mathcal O}_X\).
This calculation is done using recent results on the McKay correspondence [\textit{T. Bridgeland, A. King} and \textit{M. Reid}, J. Am. Math. Soc. 14, No. 3, 535--554 (2001; Zbl 0966.14028)] adapted to the case of an isospectral Hilbert scheme in [\textit{M. Haiman}, J. Am. Math. Soc. 14, No. 4, 941--1006 (2001; Zbl 1009.14001)], which give a Fourier-Mukai equivalence \(\Phi\) between the derived category of the Hilbert scheme \(X^{[n]}\) and the \(S_n\)-equivariant derived category of \(X^n\). These results allow indeed to calculate the cohomologies of \(X^{[n]}\) with values in the tensor powers \((E^{[n]})^{\otimes k}\) of the tautological bundle as the hypercohomologies of \(S^n X\) with values in the invariants \(\Phi((E^{[n]})^{\otimes k})^{S_n}\). The latter groups can be calculated using polygraphs and the calculation is explicitely performed for \(k=2\) by means of a spectral sequence in order to obtain the main result. Hilbert scheme; tautological bundles; McKay correspondence; Fourier-Mukai functor Parametrization (Chow and Hilbert schemes), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Cohomology of the Hilbert scheme of points on a surface with values in the double tensor power of a tautological bundle | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(T/K\) be an algebraic torus defined over a pseudoglobal field, i.e., over an algebraic function field in one variable with pseudofinite constant field. We show that groups of \(R\)-equivalence and of \(\text{Br}\)-equivalence for \(T(k)\) are finite. Moreover, connections of these groups with the Tate-Shafarevich groups and with defects of weak approximation are analogous to the case of a global ground field. \(R\)-equivalences; algebraic tori; pseudoglobal fields; Tate-Shafarevich groups; algebraic function fields Linear algebraic groups over global fields and their integers, Other nonalgebraically closed ground fields in algebraic geometry, Galois cohomology of linear algebraic groups, Arithmetic theory of algebraic function fields On the \(R\)-equivalence on algebraic tori over pseudoglobal 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 See the preview in Zbl 0539.14006. Apéry numbers; elliptic K3-surfaces; elliptic pencils; Picard-Fuchs equation; crystalline cohomology; formal Brauer group; zeta-function; finite characteristic Stienstra J, Beukers F. On the Picard-Fuchs equation and the formal Brauer group of certain elliptic K3-surfaces. Math Ann, 1985, 271: 269--304 Structure of families (Picard-Lefschetz, monodromy, etc.), Formal groups, \(p\)-divisible groups, Special surfaces, \(p\)-adic cohomology, crystalline cohomology, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Congruences; primitive roots; residue systems On the Picard-Fuchs equation and the formal Brauer group of certain elliptic K3-surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors obtain an explicit description of the irreducible component of the Hilbert scheme parametrizing complete intersections of two quadrics in \(\mathbb{P}^n\) for \(n\geq 3\), as a suitable double blow up of the Grassmann variety of pencils of quadrics in \(\mathbb{P}^n\). As an application they compute the number 52.832.040 of elliptic quartics intersecting 16 lines and the number 47.867.287.590.090 of Del Pezzo surfaces intersecting 26 lines. Hilbert scheme; complete intersections; Grassmann variety of pencils of quadrics Avritzer, D., Vainsencher, I.: The Hilbert Scheme component of the intersection of 2 quadrics, Comm. Alg., 27 (1999), 2995--3008 Parametrization (Chow and Hilbert schemes), Complete intersections, Families, moduli of curves (algebraic), Pencils, nets, webs in algebraic geometry, Enumerative problems (combinatorial problems) in algebraic geometry The Hilbert scheme component of the intersection of two quadrics | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 provides an application of commutative algebra to error-correcting codes. It defines the notion of minimum distance function of a graded ideal \(I\) in a polynomial ring \(S=K[t_1, \dots, t_s]\), \(K\) a field, function which allows to give an algebraic formulation of the minimum distance of a projective Reed-Muller-type code (a projective Reed-Muller-type code of degree \(d\) is the image of a certain evaluation map \(ev_d: S_d\longrightarrow K^m\), where \(K\) is now a finite field) and to find lower bounds for the minimum distance of these codes.
Section 1 defines the minimum distance function \(\delta_I\) of the ideal \(I\) and summarizes the content of the paper. Sections 2 and 3 gather some concepts and results needed in the following. Section 4 studies the properties of \(\delta_I\) and Theorem 4.7 proves that \(\delta_I\) generalizes the minimum distance of projective Reed-Muller-type codes.
Then the paper considers the case of projective nested cartesian codes and a conjecture about the minimum distance of these codes, conjecture due to \textit{C. Carvalho} et al., [``Projective nested Cartesian codes'', Preprint, \url{arXiv:1411.6819}]. The present paper provides some support to that conjecture (Section 6, Theorem 6.6]. Finally Section 7 shows several examples (with procedures for Macaulay2) illustrating the results obtained. graded ideal; minimum distance function; Reed-Muller-type code; Hilbert function; Gröbner bases; Carvalho, Lopez-Neumann and López conjecture Martínez-Bernal, J., Pitones, Y., Villarreal, R.H.: Minimum distance functions of graded ideals and Reed-Muller-type codes. J. Pure Appl. Algebra 221, 251-275 (2017) Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.), Applications to coding theory and cryptography of arithmetic geometry, Geometric methods (including applications of algebraic geometry) applied to coding theory, Algebraic coding theory; cryptography (number-theoretic aspects) Minimum distance functions of graded ideals and Reed-Muller-type 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 [For the entire collection see Zbl 0584.00014.]
Suppose (A,\({\mathfrak M})\) is the local ring of a reduced curve C at an ordinary singular \(point\quad p.\) Let G(A) denote the associated graded ring of A, and let \(\{a_ i\}\) be the Hilbert function of G(A). If G(A) is a reduced ring, then p is said to be ''embedded in its tangent cone''.
In section one, we show that if \(a_ i=\min \{e(A),\left( \begin{matrix} \nu ({\mathfrak M}_ i)+i\\ \nu ({\mathfrak M}_ i)\end{matrix} \right)\}\), where e(A) is the multiplicity of A, and \(\nu\) (\({\mathfrak M})\) is the number of generators of \({\mathfrak M}\), then p is embedded in its tangent cone. This always occurs if C is a plane curve. However, p is not, in general, embedded in its tangent cone.
We show (section three) that for n sufficiently large, \((*)\quad a_ n=e(A)+b_ n-b_{n+1}\) where \(\{b_ i\}\) is the Hilbert function of G(A)/nil G(A). Finally we use (*) to give examples of 1-dimensional rings with (temporarily) decreasing Hilbert function. ordinary singular point of reduced curve; Hilbert function Singularities of curves, local rings, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Multiplicity theory and related topics, (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) Ordinary singularities of curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be an algebraic surface. Fix a divisor \(c_ 1\), an integer \(c_ 2\), and a polarization \(L\) on \(X\). Let \(\text{Spl} (c_ 1, c_ 2)\) be the moduli space of simple torsion-free rank-2 sheaves with chern classes \(c_ 1\) and \(c_ 2\), and let \({\mathcal M}_ L (c_ 1,c_ 2)\) be the Zariski open subset of \(\text{Spl} (c_ 1, c_ 2)\) consisting of \(L\)- stable locally free sheaves. For sufficiently large \((4c_ 2-c^ 2_ 1)\), Donaldson and Friedman showed that \({\mathcal M}_ L( c_ 1, c_ 2)\) is nonempty and generically smooth with dimension \((4c_ 2 - c^ 2_ 1 - 3_ \chi ({\mathcal O}_ X))\). For a \(K3\)-surface \(X\), by a well-known result of Mukai, if \(\text{Spl} (c_ 1, c_ 2)\) is nonempty, then it is smooth with dimension \((4c_ 2 - c^ 2_ 1 - 6)\) and has a symplectic structure, i.e., a nowhere-degenerate holomorphic form. Put \(d = (4c_ 2 - c_ 1^ 2 - 6)/2\), and let \(\text{Hilb}^ d(X)\) be the Hilbert scheme parametrizing all 0-cycles of \(X\) with length \(d\). In this paper, we solve the following problem (rank-2 case) raised by \textit{A. N. Tyurin} [cf. Duke Math. J. 54, 1-26 (1987; Zbl 0631.14009)].
Problem A. Assume that the moduli space \(\text{Spl} (c_ 1, c_ 2)\) is nonempty.
(i) What is the birational structure of \(\text{Spl} (c_ 1, c_ 2)\)?
(ii) Is \(\text{Spl} (c_ 1, c_ 2)\) birational to the Hilbert scheme \(\text{Hilb}^ d(X)\)?
Indeed, by Fujiki, Beauville and Mukai, \(\text{Hilb}^ d(X)\) admits a symplectic structure. If \((4c_ 2 - c^ 2_ 1) \geq 12\) and if an irreducible component \({\mathcal M}\) of \(\text{Spl} (c_ 1, c_ 2)\) contains a stable sheaf, then it contains a locally free sheaf; thus, an open subset of \({\mathcal M}\) is contained in \({\mathcal M} (c_ 1, c_ 2)\). Our first result gives the birational structures of those irreducible components which contain no stable sheaf.
Theorem B. Let \(X\) be a \(K3\)-surface. Assume that \((4c_ 2 - c^ 2_ 1) > 16\), and that \({\mathcal M}\) is an irreducible component in \(\text{Spl} (c_ 1, c_ 2)\) such that no sheaf in \({\mathcal M}\) is stable. Then,
(i) \({\mathcal M}\) is birational to either \(\text{Hilb}^ d (X)\) or \(X \times \text{Hilb}^{d - 1}(X)\);
(ii) there exists a divisor \(F\) with \((c_ 1 - 2F)^ 2 = (4c_ 2 - c^ 2_ 1) - 12\).
This is proved in section two. -- In section three, we give examples of those divisors \(F\) in
theorem B (ii). \(\text{Spl} (c_ 1, c_ 2)\) may contain infinitely many irreducible components:
Theorem C. Assume that \(c\) is an odd integer larger than three, and that \(X\) is an elliptic \(K3\)-surface. Then, \(\text{Spl} (0,c)\) contains infinitely many irreducible components if and only if the Picard number of \(X\) is larger than two.
On the other hand, if \(c\) is even and larger than four, then \(\text{Spl} (0,c)\) has finitely many irreducible components. Finally, in section four, we study \({\mathcal M}_ L (c_ 1, c_ 2)\) for an elliptic \(K3\)- surface and for arbitrary \(c_ 1\).
Theorem D. Let \(X\) be an elliptic \(K3\)-surface such that any fiber of \(j\) is irreducible and has at worst ordinary double points as singularities. If the moduli space \({\mathcal M}_ L (c_ 1,c_ 2)\) is nonempty, then it is birational to \(\text{Hilb}^ d (X)\).
This result, together with theorem B and theorem C, gives a complete answer to problem A when \(X\) is an elliptic \(K3\)-surface. elliptic \(K3\)-surface; moduli space; Hilbert scheme Qin, Zhenbo, Moduli of simple rank-\(2\) sheaves on \(K3\)-surfaces, Manuscripta Math., 79, 3-4, 253-265, (1993) Families, moduli, classification: algebraic theory, \(K3\) surfaces and Enriques surfaces, Elliptic surfaces, elliptic or Calabi-Yau fibrations, Algebraic moduli problems, moduli of vector bundles, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Parametrization (Chow and Hilbert schemes) Moduli of simple rank-2 sheaves on \(K3\)-surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let A denote the ring of integers in an algebraic number field or its p- adic completion. In order to confirm a conjecture of Ravenel, the author introduced [Trans. Am. Math. Soc. 302, 319-332 (1987; Zbl 0623.55001)] a Hopf algebroid \(E_ AT\) which generalizes the Hopf algebroid \(K_*K\). In the present paper he considers the cohomology of \(E_ AT\) in the special case of \(A={\mathbb{Z}}_ p[\zeta]\) where \(\zeta\) is a pth root of unity and \({\mathbb{Z}}_ p\) denotes the p-adic integers. As an application the author gives new proofs of results about the \(E_ 2\)-term of the K- theoretic ASS and about the odd primary Kervaire invariant elements in the ASS based on BP. Brown-Peterson cohomology; Adams spectral sequence; algebraic number field; completion; Hopf algebroid; odd primary Kervaire invariant DOI: 10.2307/2001553 Generalized cohomology and spectral sequences in algebraic topology, Bordism and cobordism theories and formal group laws in algebraic topology, Formal groups, \(p\)-divisible groups Roots of unity and the Adams-Novikov spectral sequence for formal A- 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 \(\mathbb{F}_q\) be the finite field of \(q\) elements and \(X\) a smooth subscheme of \(\mathbb{P}_{\mathbb{F}_q}^n\). In the paper under review, two results are shown concerning the density of hypersurface sections of \(X\) having a prescribed number of singular points.
The first result deals with nonsingular hypersurface sections of \(X\) containing a given subscheme \(V\). More precisely, the author shows that, if \(Z\) is a closed subscheme of \(\mathbb{P}_{\mathbb{F}_q}^n\) and \(V=X\cap Z\), then a hypersurface section of \(X\) containing \(V\) is smooth of dimension \(\dim X-1\) with positive probability, provided that the dimension and singularities of \(V\) are properly controlled.
Using this result the author shows that a reduced quasi-projective curve \(C\) over \(\mathbb{F}_q\) can be embedded into a smooth \(r\)-dimensional scheme over \(\mathbb{F}_q\) if and only if the maximal ideal at each closed point of \(C\) can be generated by \(r\) elements, thus extending the validity of a result of \textit{S. L. Kleiman} and \textit{A. B. Altman} [Commun. Algebra 7, 775--790 (1979; Zbl 0401.14002)] from infinite perfect fields to finite fields.
The second result answers a conjecture of \textit{R. Vakil} and \textit{M. M. Wood} on the density of hypersurface sections of \(X\) having a prescribed number of singularities [Duke Math. J. 164, No. 6, 1139--1185 (2015; Zbl 1461.14020)]. In [\textit{B. Poonen}, Ann. Math. (2) 160, No. 3, 1099--1127 (2005; Zbl 1084.14026)], the density of smooth hypersurface sections of \(X\) is determined, and is proved that a section has infinitely many singularities with probability 0. In this sense, the author determines the asymptotic probability of any prescribed number \(\ell\geq 0\) of singularities for a hypersurface section \(X\cap H\).
A fundamental technique for all these results is Poonen's geometric closed point sieve introduced in [Poonen, loc. cit.]. smooth scheme; hypersurface section; finite fields; density; Poonen's point sieve Hypersurfaces and algebraic geometry, Finite ground fields in algebraic geometry, Varieties over finite and local fields Random hypersurfaces and embedding curves in surfaces over finite fields | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A weight system is a system of 4 positive integers \((a,b,c,h)\) with some arithmetic conditions.
Such weight systems appear naturally in the study of Coxeter groups and hypersurface singularities, e.g. Arnold duality.
The paper gives a comprehensive study of such systems. characteristic polynomial; weight systems; Coxeter groups; hypersurface singularities; Arnold duality Kyoji Saito, ''Duality for regular systems of weights,'' Asian J. Math. 2, 983--1048 (1998). Deformations of complex singularities; vanishing cycles, Complex surface and hypersurface singularities, Singularities of surfaces or higher-dimensional varieties, Elliptic surfaces, elliptic or Calabi-Yau fibrations, \(K3\) surfaces and Enriques surfaces, Braid groups; Artin groups Duality for regular systems of weights | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Affine Deligne-Lusztig varieties are analogs of Deligne-Lusztig varieties in the context of an affine root system. We prove a conjecture stated in the paper [Compos. Math. 146, No. 5, 1339-1382 (2010; Zbl 1229.14036)] by \textit{T. J. Haines, R. E. Kottwitz, D. C. Reuman}, and the first named author, about the question which affine Deligne-Lusztig varieties (for a split group and a basic \(\sigma\)-conjugacy class) in the Iwahori case are non-empty. If the underlying algebraic group is a classical group and the chosen basic \(\sigma\)-conjugacy class is the class of \(b=1\), we also prove the dimension formula predicted in [op. cit.] in almost all cases. affine Deligne-Lusztig varieties; Weyl groups; conjugacy classes; minimal length elements; affine root systems Görtz, U.; He, X., Dimensions of Deligne-Lusztig varieties in affine flag varieties, Doc. Math., 15, 1009-1028, (2010) Representation theory for linear algebraic groups, Grassmannians, Schubert varieties, flag manifolds, Linear algebraic groups over finite fields, Reflection and Coxeter groups (group-theoretic aspects), Group actions on varieties or schemes (quotients), Linear algebraic groups over local fields and their integers Dimensions of affine Deligne-Lusztig varieties in affine flag 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 show that the Catalan-Schroeder convolution recurrences and their higher order generalizations can be solved using Riordan arrays and the Catalan numbers. We investigate the Hankel transforms of many of the recurrence solutions, and indicate that Somos-4 sequences often arise. We exhibit relations between recurrences, Riordan arrays, elliptic curves and Somos-4 sequences. We furthermore indicate how one can associate a family of orthogonal polynomials to a point on an elliptic curve, whose moments are related to recurrence solutions. convolution recurrence; generating function; Catalan number; Schröder numbers; Riordan array; Hankel transform; Somos sequence; elliptic curve; orthogonal polynomial Recurrences, Exact enumeration problems, generating functions, Elliptic curves, Matrices of integers, Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal 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 Virtual polytopes have a long history, going back at least as far as Aleksandrov's proof of the quadratic inequalities for mixed volumes, but were first formally named by \textit{A. V. Pukhlikov} and \textit{A. G. Khovanskii} [St. Petersbg. Math. J. 4, No. 2, 337--356 (1993); translation from Algebra Anal. 4, No. 2, 161--185 (1992; Zbl 0791.52010)]. The Minkowski sum
\[
K \otimes L = \{x + y \mid x \in K,\;y \in L\}
\]
of \(K,L \in \mathcal{P}\), the family of convex polytopes in euclidean space \(\mathbb{R}^d\), turns \(\mathcal{P}\) into a semigroup with cancellation. (The multiplicative notation is used because, as the authors point out, Minkowski addition behaves in some contexts much more like multiplication.) With translates (usually) identified, this semigroup embeds in the corresponding Grothendieck group \(\mathcal{P}^*\), whose elements are the virtual polytopes. A general element of \(\mathcal{P}^*\) is then written \(K \otimes L^{\otimes-1}\), with \(L^{\otimes-1}\) the inverse of \(L\) in \(\mathcal{P}^*\) with respect to \(\otimes\). Since a polytope is determined by its support function (which is piecewise linear and convex), a virtual polytope can be identified with a difference of support functions, factored out by the globally linear functions.
In this survey, after an initial discussion of how concepts such as faces and normal fans can be extended to \(\mathcal{P}^*\), the authors then look in detail at the \(2\)-dimensional case. A virtual polygon can be thought of as a (closed) \(2\)-coloured circuit of edges, with the colours indicating whether they are counted positively or negatively. In a similar way, the normal fan gives rise to a coloured star.
A polytope \(K\) can be identified with its characteristic function \(I_K\), defined by
\[
I_K(x) = \begin{cases} 1, &\text{if } x \in K, \\ 0, & \text{if } x \notin K. \end{cases}
\]
These characteristic functions generate a ring, with the usual addition -- which corresponds to the valuation property -- and multiplication defined by convolution giving the Minkowski sum; initially at least, translations are not factored out. (The basic ideas here go back to \textit{H. Groemer} [Geom. Dedicata 6, 141--163 (1977; Zbl 0394.52003)]). An important feature is that
\[
(I_K)^{\otimes-1} = \sum_F \, (-1)^{\dim K - \dim F}I_F,
\]
with the sum over all faces \(F\) of \(K \in \mathcal{P}\) including \(K\) itself (this expression is not given in the paper -- apart from the sign \((-1)^{\dim K}\), the right side is the characteristic function of the relative interior of \(K\)).
The connexions with the reviewer's polytope algebra \(\Pi\) [Adv. Math. 78, No. 1, 76--130 (1989; Zbl 0686.52005)] are considered next. Except in a minor respect, \(\Pi\) is a graded algebra over \(\mathbb{R}\), since once again translates are identified; now, the first weight space of \(\Pi\) is isomorphic to \(\mathcal{P}^*\). The relationships with fans are then discussed.
The \(3\)-dimensional case is also treated in depth. Some of the notions in this case go back to Maxwell, with stressed graphs on the sphere; we shall not go into the details.
Finally, the authors consider applications. These are to Aleksandrov's problem on the uniqueness of convex surfaces, as well as topics already mentioned such as valuations including volume and counting integer points in lattice polytopes, mixed volumes, decomposition of polytopes under Minkowski addition, and finally a brief look at projective toric varieties and associated subjects. Minkowski difference; coloured polygon; polytopal function; support functions; stressed graph; McMullen's polytope algebra; Maxwell polytope \(n\)-dimensional polytopes, Polyhedral manifolds, Toric varieties, Newton polyhedra, Okounkov bodies Virtual polytopes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 the entire collection see Zbl 0527.00041.]
In the beginning of this paper it is proposed the conjecture that reachability of a pair (F,G) over an arbitrary ring implies pole assignability if the rank of G remains constant. After the definition of weak pole assignability, the authors prove partially the above conjecture. Namely, proofs are given for the special cases of pairs (F,G) over rings of continuous functions defined on contractible domains and over rings of holomorphic functions on the closed unit polydisc of dimension q,\({\bar \Delta}^ q\). Weak pole assignability over the ring of holomorphic functions on \({\bar \Delta}^ q\) has important implications for the stabilization (independent of delay) of a large class of neutral time delay systems with q-commensurate delays. weak pole assignability; rings of continuous functions; ring of holomorphic functions; neutral time delay systems Tannenbaum, A. R.; Khargonekar, P. P.: On weak pole placement of linear systems depending on parameters. Proceedings of the MTNS-83 international symposium, 829 (1983) Pole and zero placement problems, Algebraic methods, Linear systems in control theory, Special varieties, Controllability, General commutative ring theory On weak pole placement of linear systems depending on parameters | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(X_1(4)\) be the curve of genus 0 which belongs to the group \(\Gamma_1 (4)\) of all matrices in \(SL_2 (\mathbb{Z})\) which are congruent to \(\left (\begin{smallmatrix} 1 & * \\ 0 & 1 \end{smallmatrix} \right)\) mod 4. By means of Jacobi theta series, the authors construct a generator \(j_{1,4}\) and a normalized generator \(N(j_{1,4})\) for the function field of \(X_1(4)\). The arithmetical relevance is shown in two topics. The first one concerns moonshine [\textit{J. H. Conway} and \textit{S. P. Norton}, Bull. Lond. Math. Soc. 11, 308-339 (1979; Zbl 0424.20010); \textit{R. E. Borcherds}, Invent. Math. 109, 405-444 (1992; Zbl 0799.17014)], and it states that \(N(j_{1,4})\) corresponds to the Thompson series of type \(4C\). The other topic is complex multiplication and class field theory. It is shown that certain class fields over imaginary quadratic fields can be constructed by adjoining values of \(j_{1,4}\). modular function; normalized generator of a function field; moonshine; complex multiplication; class fields over imaginary quadratic fields Chang Heon Kim and Ja Kyung Koo, Arithmetic of the modular function \?_{1,4}, Acta Arith. 84 (1998), no. 2, 129 -- 143. Modular and automorphic functions, Relationship to Lie algebras and finite simple groups, Holomorphic modular forms of integral weight, Algebraic numbers; rings of algebraic integers, Class field theory, Special algebraic curves and curves of low genus Arithmetic of the modular function \(j_{1,4}\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 difficult to sum up all the results appearing in this nice, wide-ranging, and rich paper. We will use in this review the introduction given by the author, adding some commentaries and complements when necessary. Submersive morphisms of schemes are morphisms of schemes inducing the quotient topology on the target. They appear naturally in many situations, such as when studying quotients, homology, descent and the fundamental group of schemes. Questions related to submersive morphisms of schemes can often be resolved by using topological methods using the description of schemes as locally ringed spaces. Corresponding questions for algebraic spaces are much harder, because an algebraic space is not fully described as a locally ringed space. The first proper treatment of submersive morphisms seems to be due to \textit{A. Grothendieck} [Séminaire de géométrie algébrique du Bois Marie 1960/61 (SGA 1). Revêtements étales et groupe fondamental. Lecture Notes in Mathematics. 224. Berlin-Heidelberg-New York: Springer-Verlag (1971; Zbl 0234.14002)], with applications to the fundamental group of a scheme. He showed that submersive morphisms are morphisms of descent for the fibered category of étale morphisms, with effectiveness for the fibered category of quasi-compact and separated étale morphisms, in some special cases of universally submersive morphisms. The main result of the paper consists in several very general effectiveness results, extending significantly Grothendieck's work. For example, any universal submersion of Noetherian schemes is a morphism of effective descent for quasi-compact étale morphisms. As an application, these effectiveness results imply that strongly geometric quotients are categorical in the category of algebraic spaces.
Later on, \textit{G. Picavet} singled out a subclass of submersive morphisms [``Submersion et descente'', J. Algebra 103, 527--591 (1986; Zbl 0626.14014)]. This paper is written in the affine scheme context and defines subtrusive morphisms as morphisms such that specializations of points have a lifting from the target space to the domain space. For arbitrary morphisms of schemes, Rydh adds the condition that these morphisms are submersive in the constructible topology. The class of subtrusive morphisms is natural in many respects. For example, over a locally Noetherian scheme, every submersive morphism is subtrusive. Picavet gave an example showing that a finitely presented universally submersive morphism needs not to be subtrusive.
A key result of Rydh's paper, missing in Picavet's paper, is that every finitely presented universally subtrusive morphism is a limit of finitely presented submersive morphisms of Noetherian schemes, allowing the author to eliminate Noetherian hypotheses. These facts show that the class of subtrusive morphisms is an important and very natural extension of submersive morphisms between Noetherian schemes.
A crucial tool in this article is the structure theorem for universally subtrusive morphisms: Let \(f: X \to Y\) be a universally subtrusive morphism of finite presentation. Then there are a morphism \(g: X' \to X\) and a factorization \(f_2 \circ f_1\) of \(f\circ g\), where \(f_1\) is faithfully flat of finite presentation and \(f_2\) is proper, surjective, and of finite presentation. The author derives from this result many other useful results.
As a first application, it is shown in Section 4 that universally subtrusive morphisms of finite presentation are morphisms of effective descent for locally closed subsets.
Section 5 establishes that quasi-compact universally subtrusive morphisms are morphisms of effective descent for the fibered category of quasi-compact and separated étale schemes. This result hold also for algebraic spaces, not necessarily separated.
In Section 6, the author shows that the class of subtrusive morphisms is stable under inverse limits and that subtrusive (universally open) morphisms descend under inverse limits.
The rest of the paper is mainly devoted to descent results, in particular to weakly normal descent for universally submersive morphisms. The results of Appendix A about étale morphisms and Henselian pairs are the core of the proof that proper morphisms are morphisms of effective descent for étale morphisms. Appendix B describes properties of absolute weak normalizations.
To end we observe that some results are generalizations to the non-Noetherian case of \textit{V. Voevodsky}'s results in [``Homology of schemes'', Sel. Math., New Ser. 2, No. 1, 111--153 (1996; Zbl 0871.14016)]. algebraic space; blow-up; constructible topology; patch topology; effective descent; étale morphism; finitely presented morphism; flatification; geometric quotient; \(h\)-topology; integral morphism; schematically dominant morphism; scheme; proper morphism; \(S\)-topology; submersion; subtrusion; universally closed; totally integrally closed scheme; (absolute) weak normalization Rydh, D., Submersions and effective descent of étale morphisms, Bull. Soc. Math. France, 138, 2, 181-230, (2010) Schemes and morphisms, Integral dependence in commutative rings; going up, going down, Integral closure of commutative rings and ideals, Étale and flat extensions; Henselization; Artin approximation, Étale and other Grothendieck topologies and (co)homologies, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Submersions and effective descent of étale morphisms | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(C\) denote a smooth, geometrically irreducible projective curve of genus 4 defined over the finite field \({\mathbb F}_8\). It was already known that such a curve cannot have more than 27 rational points, and that such curves with 25 rational points do exist. In an appendix to this article, K. Lauter shows that \(C\) cannot have exactly 26 rational points by ruling out all the possible zeta functions such a curve would have. The author shows that \(C\) cannot have 27 rational points, thus establishing that the maximum number of rational points for such a curve is 25.
The proof uses the fact that the canonical embedding of \(C\) would be the intersection of an irreducible cubic surface and an irreducible quadric surface in projective 3-space over \({\mathbb F}_8\). Up to isomorphism, there are only three irreducible quadric surfaces in \({\mathbb P}^3\) over a finite field. Let \(Q\) denote one of these quadrics and let \(\text{Fix}(Q)\) denote the subgroup of \(\text{PGL}_4({\mathbb F}_8)\) that leaves \(Q\) invariant. The author shows that if \(S\) is a subset of \(Q\) with 27 rational points, no four of which are collinear, then there exists \(\sigma\in\text{Fix}(Q)\) such that \(\sigma(S)\) contains one of several lists of points of \(Q\). This fact is used to reduce the number of cubics that can also contain these 27 points, resulting in a list of possible cubics that can be searched by a computer. While the computer search turns up many cubics and quadrics that intersect in 27 rational points, none of these intersections gives a geometrically irreducible curve. The author gives a precise accounting of which ``bad'' curves with 27 points can occur. number of rational points; zeta function; quadric surface; cubic surface Rational points, Finite ground fields in algebraic geometry, Curves over finite and local fields The maximum number of points on a curve of genus 4 over \({\mathbb F}_ 8\) is 25. With an appendix by Kristin Lauter. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Drinfeld Zastava space \(Z^{\underline{d}}\) is a natural closure of the moduli space of degree \({\underline{d}}\) based maps from the projective line to the Kashiwara flag scheme of the affine Lie algebra \(\widehat{\mathfrak{sl}}_n\). Drinfeld Zastava space is affine and singular. Another natural closure of the same object is the quasiprojective smooth affine Laumon space. The latter carries a natural Poisson structure which, in fact, descends to the Drinfeld Zastava space. The main aim of the paper under review is to quantize this Poisson structure.
The authors introduce an affine, reduced, irreducible, normal quiver variety \(Z\) which maps to the Drinfeld Zastava space bijectively at the level of complex points. Using this reformulation the authors describe the Poisson structure on \(Z\) in terms of Hamiltonian reduction of certian Poisson subvariety of a dual space of a (nonsemisimple) Lie algebra. The quantum Hamiltonian reduction of the corresponding quotient of the universal enveloping algebra of this Lie algebra gives the desired quantization \(Y\) of the coordinate ring of \(Z\). The authors show that \(Y\) is a quotient of the affine Borel Yangian for generic values of the parameter. Drinfeld Zastava; moduli space; projective line; Kashiwara flag scheme; Lie algebra; quiver variety; Poisson structure; Hamiltonian reduction; quantization; Yangian M. Finkelberg and L. Rybnikov, \textit{Quantization of Drinfeld Zastava}, arXiv:1009.0676. Geometric invariant theory, Root systems Quantization of Drinfeld Zastava in type \(A\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(R=\mathbb{K}[\mathbb{P}^N]\), where \(\mathbb{K}\) is an algebraically closed field of any characteristic and \(N\geq 2\). Let \(I=\bigcap_i I(P_i)^{m_i}\), where \(P_i\) are distinct points in \(\mathbb{P}^N\), \(I(P_i)\) is the ideal of all forms that vanish at \(P_i\) and the multiplicity \(m_i\) is a nonnegative integer.
For this type of ideals we define the \(m\)-th symbolic power as \[I^{(m)}=\bigcap_i I(P_i)^{mm_i}.\] The containment problem is to decide for which \(m\) and \(r\) the symbolic power \(I^{(m)}\) is contained in the ordinary power \(I^r\).
\textit{C. Bocci} and \textit{B. Harbourne} [J. Algebr. Geom. 19, No. 3, 399--417 (2010; Zbl 1198.14001); Proc. Am. Math. Soc. 138, No. 4, 1175--1190 (2010; Zbl 1200.14018)] introduced an asymptotic quantity, knows as the resurgence, whose computation is clearly linked to the containment problem.\\
Definiton.\\
Given a nonzero proper homogeneous ideal \(I\) in \(R\), the resurgence of \(I\), denoted by \(\varrho (I)\), is defined as the quantity \[\varrho (I)=\sup \{\frac{m}{r} : I^{(m)}\nsubseteq I^r\}.\] In general, directly computing \(\varrho (I)\) is quite difficult and it has been determined only in very special cases.
In the paper authors compute the resurgence of \(I(Z)\) for two classes of fat point subschemes. First they study the subscheme \[Z=\sum_{i=1}^n m_iP_i\] in \(\mathbb{P}^N\), where the points \(P_i\) are distinct and collinear and they prove that in this case \[I(Z)^{(m)}=I(Z)^m\] for all \(m\in \mathbb{N}\), thus \(\varrho(I(Z))=1\).
Then they consider the subscheme \[Z=m_0P_0+m_1P_1+m_2P_2,\] where \(P_i\) are noncollinear points in \(\mathbb{P}^N\) and \(m_0\leq m_1\leq m_2\) are nonnegative integer. In this situation they classify all fat point ideals with \(m\)-symbolic defect zero for all \(m\) (and hence that \(\varrho(I(Z))=1\)).
Let's recall that the \(m\)-symbolic defect of a homogeneous ideal \(I\) of \(R\) is the minimal number of generators of the \(R\)-module \(I^{(m)}/I^m\). The authors formulate the following statement.
Theorem.
For \(I\) as before, \(\mathrm{sdefect}(I(Z),m)=0\) forall \(m\in\mathbb{N}\) if and only if one of the following conditions holds:
\begin{itemize}
\item[1.] \(m_0+m_1\leq m_2\),
\item[2.] \(m_0+m_1>m_2\) and \(m_0+m_1+m_2\) is even.
\end{itemize}
In other cases they compute resurgence of the ideal. More precisely, if \(m_0+m_1>m_2\) and \(m_0+m_1+m_2\) is odd, then \[\varrho (I(Z))=\frac{m_0+m_1+m_2+1}{m_0+m_1+m_2}.\] containment; fat points scheme; resurgence; symbolic powers Configurations and arrangements of linear subspaces, Ideals and multiplicative ideal theory in commutative rings, Polynomial rings and ideals; rings of integer-valued polynomials On the containment problem for fat points | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper, the authors compute the sum of Betti numbers of smooth Hilbert schemes on the complex projective spaces. For fixed positive integer \(n\) and polynomial \(p(t)\), let \(\mathbb P^{n[p]}\) denote the Hilbert scheme of closed subschemes of the complex projective space \(\mathbb P^{n}\) with Hilbert polynomial \(p(t)\). It is previously known that the Hilbert scheme \(\mathbb P^{n[p]}\) is nonempty if and only if \(p(t)= \sum_{i=1}^r \binom{t+\lambda_i - i}{\lambda_i - 1}\) for some integer partition \(\lambda = (\lambda_1, \ldots, \lambda_r)\) satisfying \(\lambda = (n+1)\), \(r = 0\), or \(n \ge \lambda_1 \ge \ldots \ge \lambda_r \ge 1\). Moreover, \(\mathbb P^{n[p]}\) is smooth if and only if one of the following seven cases is true:
\begin{itemize}
\item[1.] \(n \le 2\);
\item[2.] \(\lambda_r \ge 2\);
\item[3.] \(\lambda = (1)\) or \(\lambda = (n^{r-2}, \lambda_{r-1}, 1)\) where \(r \ge 2\);
\item[4.] \(\lambda = (n^{r-s-3}, \lambda_{r-s-2}^{s+2}, 1)\) where \(r \ge s+3\);
\item[5.] \(\lambda = (n^{r-s-5}, 2^{s+4}, 1)\) where \(r \ge s+5\);
\item[6.] \(\lambda = (n^{r-3}, 1^3)\) where \(r \ge 3\);
\item[7.] \(\lambda = (n+1)\) or \(r = 0\).
\end{itemize}
The main theorem of the paper presents explicit formulas for the sum \(H_{n, \lambda}\) of the Betti numbers of \(\mathbb P^{n[p]}\) in the above seven cases except Case~2. For instance,
\[
H_{n, \lambda} = \binom{n+r-2}{r-2} \binom{n+1}{\lambda_{r-1}} (n + 1 - \lambda_{r-1})(\lambda_{r-1} + 1)
\]
when \(\lambda = (n^{r-2}, \lambda_{r-1}, 1)\) is in Case 3 with \(n > \lambda_{r-1} > 1\).
The main ideas in the proofs are to use the \(\mathrm{PGL}(n + 1)\)-action on \(\mathbb P^{n[p]}\) induced from the \(\mathrm{PGL}(n + 1)\)-action on \(\mathbb P^{n}\) and to apply the classical theorem of A.~Bialynicki-Birula. These ideas enable the authors to translate the computation of the ranks of the homology groups into counting saturated monomial ideals and then to translate that into counting choices of orthants in an \((n + 1)\)-dimensional lattice. Hilbert scheme; Betti number; cohomology; homology; saturated monomial ideal Parametrization (Chow and Hilbert schemes), Syzygies, resolutions, complexes and commutative rings The sum of the Betti numbers of smooth Hilbert schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author studies two \(dg\)-operads, dual in the sense of Ginzburg-Kapranov, which are related to the moduli spaces of the title. The algebras described by these operads are, respectively, the author's ``gravity'' algebras [Commun. Math. Phys. 163, No. 3, 473-489 (1994; Zbl 0806.53073)] and the algebras discovered by \textit{R. Dijkgraaf}, \textit{H. Verlinde} and \textit{E. Verlinde} [Nuclear Phys. B 352, No. 1, 59-86 (1991)], which the author rechristens ``polycommutative''. The latter have a sequence of operations \(A^{\otimes n} \to A\) satisfying an appropriate generalization of associativity. An important class of examples is provided by the quantum cohomology of compact Kähler manifolds.
As in much other recent work, a key role is played by the moduli spaces \({\mathcal M}_{0,n}\) of \(n\)-punctured Riemann spheres and the Knudsen-Deligne-Mumford compactification \(\overline {\mathcal M}_{0,n}\). The relevant technical tools include the spectral sequence of the natural stratification of the compactified moduli space and mixed Hodge theory which for genus 0 is pure. The author is building on much of his earlier work, especially that with Kapranov, cf. cyclic and modular operads. Graphs and trees play a significant part, although the given combinatorial definition of graph is far from perspicuous. He also obtains new formulas for the characters of the homology of \({\mathcal M}_{0, n}\) and of \(\overline {\mathcal M}_{0,n}\) as \({\mathcal S}_n\)-modules. \(dg\)-operads; gravity algebras; polycommutative; punctured Riemann spheres; stratification of compactified moduli space; homology characters; moduli spaces; quantum cohomology; compact Kähler manifolds; Knudsen-Deligne-Mumford compactification; spectral sequence; mixed Hodge theory Getzler, E.: Operads and moduli spaces of genus \(0\) Riemann surfaces. In Dijkgraaf, R., Faber, C., van der Gerr, G. (eds.) The Moduli Space of Curves, volume 129 of \textit{Progress in Mathematics}, pp. 199-230. Birkhäuser, Basel (1995) Monoidal categories (= multiplicative categories) [See also 19D23], Families, moduli of curves (algebraic), Riemann surfaces; Weierstrass points; gap sequences, Homological algebra in category theory, derived categories and functors, Applications of differential geometry to physics, Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Other \(n\)-ary compositions \((n \ge 3)\), Quantization in field theory; cohomological methods Operads and moduli spaces of genus 0 Riemann surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In 1992, \textit{J. Denef} and \textit{F. Loeser} [J. Am. Math. Soc. 5, No. 4, 705--720 (1992; Zbl 0777.32017)] introduced the topological zeta function \(Z_{\mathrm{top},f}\) of a holomorphic germ \(f:(\mathbb C^n,0)\to (\mathbb C,0)\). This function can be computed from an embedded resolution of singularities of the germ \(f\). If \(f\) is a non-degenerate function, then it is possible to compute \(Z_{\mathrm{top},f}\) from the Newton polyhedron of \(f\) (see the article of Denef and Loeser cited before and the article of \textit{W. Veys} [Manuscr. Math. 87, No. 4, 435--448 (1995; Zbl 0851.14012)]). Both ways give rise to a set of candidate poles of \(Z_{\mathrm{top},f}\), containing all poles. In this note, the authors show how to determine from the Newton polyhedron of a non-degenerate plane curve which candidate poles are actual poles. topological zeta function; Newton polyhedron Ann Lemahieu and Lise Van Proeyen, Poles of the topological zeta function for plane curves and Newton polyhedra, C. R. Math. Acad. Sci. Paris 347 (2009), no. 11-12, 637 -- 642 (English, with English and French summaries). Singularities of curves, local rings Poles of the topological zeta function for plane curves and Newton polyhedra | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Ever since the famous paper of \textit{A. Andreotti} and \textit{A. L. Mayer} [Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 21, 189--238 (1967; Zbl 0222.14024)] it has been known that valuable information is concealed in the stratification of \({\mathcal A}_g\) (the moduli space of principally polarised abelian varieties of dimension~\(g\)) according to the dimension of the singular locus of the theta divisor. It is well concealed, though: even after another forty years we still do not know even the dimensions of the Andreotti-Mayer loci \(N_k=\{(X_\tau,\Theta_\tau)\mid \dim\text{Sing}\Theta_\tau\geq k\}\subset{\mathcal A}_g\), which in any case are reducible. This paper works with a different, but related, stratification: by the multiplicity of points of \(\Theta\) rather than the dimension.
As usual we let \(\Theta\) denote the universal theta divisor and \(\Theta_{\text{sing}}\) its singular locus. We put
\[
G_k=\{(X_\tau,\Theta_\tau)\mid \dim(X_\tau \cap \Theta_{\text{sing}})\geq k\}
\]
which, by the heat equation, is the same as
\[
\{(X_\tau,\Theta_\tau)\mid \dim\{z\in X_\tau \mid \text{mult}_z\Theta_\tau\geq 2\}\geq k\}.
\]
It is quite hard even to get much information about \(G_0\). Also define
\[
(\partial\theta)_{\text{null}}=\{(X_\tau,\Theta_\tau)\mid X_\tau[2]^{\text{odd}} \cap \Theta_{\text{sing}}\neq\emptyset\}
\]
and
\[
(\partial^k\theta)_{\text{null}}=\{(X_\tau,\Theta_\tau)\mid \exists x\in X_\tau[2]\;\text{mult}_x\Theta\in k+2{\mathbb N}\},
\]
analogously to the locus \(\theta_{\text{null}}=\{(X_\tau,\Theta_\tau)\mid X_\tau[2]^{\text{even}} \cap \text{Sing}\Theta\neq\emptyset\}\), which is a component of \(N_0\).
Working on \({\mathcal A}(4,8)\) to avoid orbifold (or stack) difficulties, the authors write down modular forms cutting out some components of \((\partial^2\theta)_{\text{null}}\). They show that for \(k\leq g-4\) the locus \((\partial^k\theta)_{\text{null}}\) is irreducible: one component is \(\theta_{\text{null}}^{(g-k)}\times{\mathcal A}_1\times\cdots\times{\mathcal A}_1\), which has codimension \(gk+1-{{1}\over{2}}(k^2+k)\) in \({\mathcal A}_g\). More generally they say when \(X\times{\mathcal A}_1\) is an irreducible component of \((\partial^k\theta)_{\text{null}}\) in \({\mathcal A}_{g+1}\) if \(X\) is an irreducible component of \((\partial^k\theta)_{\text{null}}\) in \({\mathcal A}_g\). Similarly they exhibit two decomposible irreducible components of~\(G_0\).
The last part of the paper includes several conjectures on the codimensions of these loci and some evidence for them. According to these, \((\partial\theta)_{\text{null}}\) and \(G_0\) should have codimension~\(g\) and \((\partial^2\theta)_{\text{null}}\) should have codimension~\(2g-2\), while the codimension of \(\Theta_{\text{sing}}\) in the universal family \({\mathcal X}_g\) should be \(2g\) and \(\text{codim}G_k\) should be at least \(g+k\) (all for \(g\geq 4\) at any rate).
These conjectures are related to one another, and by examining rank~\(1\) degenerations the authors are able to prove a small part of them, and show equivalences among other parts, on the assumption that every component of the relevant loci does have rank~\(1\) degenerations. For \(\Theta_{\text{sing}}\) the issue is instead whether it is pure-dimensional or not, and it is easy to show that the conjectures on \(\text{codim}\Theta_{\text{sing}}\) and on \(\text{codim}G_k\), for given~\(g\), are equivalent. abelian variety; theta divisor; theta function; singularities Grushevsky, S., Salvati Manni, R.: The loci of abelian varieties with points of high multiplicity on the theta divisor. Geom. Dedic. 139, 233--247 (2009) Theta functions and abelian varieties, Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms, Algebraic moduli of abelian varieties, classification, Theta functions and curves; Schottky problem The loci of abelian varieties with points of high multiplicity on the theta divisor | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(p\) be a prime and let \(\mathbb F_ q\) be a field with \(q=p^ a\) elements. Consider the following codes over \(\mathbb F_ p\):
\[
C_ R(q)=\left\{c(a,b)=\left(\text{Tr}\left(ax+{b\over R(x)}\right)\right)_{x\in\mathbb F_ q-Z}: a,b\in\mathbb F_ q\right\}.
\]
Here \(R(x)=\sum_ i a_ ix^{p^ i}\in\mathbb F_ q[x]\) and \(Z\) denotes the \(\mathbb F_ p\)-vector space of zeroes of \(R\) in \(\mathbb F_ q\). By Tr we denote the trace map from \(\mathbb F_ q\) to \(\mathbb F_ p\).
The authors show that the subcodes \(\{c(a,b): a,b\in\mathbb F_ q\) for which \(\text{Tr}(ax)=0\) for all \(x\in Z\}\) have covering radius \(\left(1- {1\over p}\right)\bigl(q-p^{\dim(Z)}\bigr)\). In the special case \(p=2\) and \(R(x)=x^ 2+x\) the authors determine the weight distributions of the codes \(C_ R(q)\). Briefly, the weights are related to the number of \(\mathbb F_ q\)-rational points on the curves in the family. \(Y^ 2+Y=aX+b/X(X+1)\), \(a,b\in\mathbb F_ q\). For \(a,b\in\mathbb F_ q^*\), this curve has genus 2 and its Jacobian is isogenous to a product of two elliptic curves. The final answer involves the number of elliptic curves over \(\mathbb F_ q\) with a given number of rational points and is given in terms of certain products of class numbers of integral binary quadratic forms. rational points; field; codes; subcodes; covering radius; weight distributions; curves; elliptic curves; integral binary quadratic forms Van Der Geer, G.; Van Der Vlugt, M.: Trace codes and families of algebraic curves. Math. Z. 209, 307-315 (1992) Geometric methods (including applications of algebraic geometry) applied to coding theory, Applications to coding theory and cryptography of arithmetic geometry, Finite ground fields in algebraic geometry, Rational points Trace codes and families of algebraic curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We consider mirror symmetry (A-side \textit{vs} B-side, namely singularity side) in the framework of quantum differential systems. We focus on the logarithmic non-resonant case, which describes the geometric situation and we show that such systems provide a good framework in order to generalize the construction of the rational structure given by Katzarkov, Kontsevich and Pantev for the complex projective space. As an application, we give a closed formula for the rational structure defined by the Lefschetz thimbles on the flat sections of the Gauss-Manin connection associated with the Landau-Ginzburg models of weighted projective spaces (a class of Laurent polynomials). As a by-product, using a mirror theorem, we get a rational structure on the orbifold cohomology of weighted projective spaces. The formula on the B-side is more complicated than the one on the A-side (the latter agrees with one of Iritani's results), depending on numerous combinatorial data which are rearranged after the mirror transformation. mirrow symmetry; quantum differential systems Douai, A, Quantum differential systems and rational structures, Manuscr. Math., 145, 285-317, (2014) Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Mirror symmetry (algebro-geometric aspects), Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms Quantum differential systems and construction of rational structures | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 1956, Abhyankar stated his local weak simultaneous resolution theorem for algebraic surfaces. By using this theorem, he was able to eliminate the use of Zariski's factorization theorem for proving the local uniformization theorem for algebraic surfaces (a step in Zariski's plan for proving resolution of singularities). The author thinks that generalizing the local weak simultaneous resolution theorem could be useful to show local uniformization for higher-dimensional varieties since Zariski's factorization theorem is not true for these varieties. In this sense, he gives a partial generalization which does not account for all possible rational ranks. The concrete result (section 3 of the paper) which is proved from a monomialization theorem for ordered semigroups (section 2) is the following:
Let \(K\) be an \(n\)-dimensional function field over an algebraically closed field \(k\). Assume that either the characteristic of \(k\) is zero or that it is positive and \(n\leq 3\). Let \(L/K\) be a finite algebraic extension. Let \(v\) be a valuation of \(L/k\) with \(k\)-dimension zero, rational rank \(r\), and valuation ring \(V\). Let \((S,N)\subset L\) be an \(n\)-dimensional local ring over \(k\) which is birationally dominated by \(V\). Let \(R=S\cap K\), \(M=N\cap K\), \(U=V\cap K\) and \(Q=MS\).
(1) If \(r=n\), then \(S\) can be replaced by an iterated monoidal transform along \(v\) so that \(Q\) is \(N\)-primary;
(2) If \(r<n\), then \(S\) can be replaced by an iterated monoidal transform along \(v\) so that \(\text{ht} Q\geq r+1\). In particular, if \(r=n-1\), then for such a transform \(Q\) is \(N\)-primary. simultaneous resolution; valuations; monomialization; local uniformization; function field DOI: 10.1006/jabr.1996.7014 Valuations and their generalizations for commutative rings, Arithmetic theory of algebraic function fields, Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Algebraic functions and function fields in algebraic geometry, Regular local rings Local weak simultaneous resolution for high rational ranks | 0 |
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