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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 We introduce graded \(\mathbb{E}_\infty\)-rings and graded modules over them, and study their properties. We construct projective schemes associated to connective \(\mathbb{N}\)-graded \(\mathbb{E}_\infty\)-rings in spectral algebraic geometry. Under some finiteness conditions, we show that the \(\infty\)-category of almost perfect quasi-coherent sheaves over a spectral projective scheme \(\text{Proj}\,(A)\) associated to a connective \(\mathbb{N}\)-graded \(\mathbb{E}_\infty\)-ring \(A\) can be described in terms of \(\mathbb{Z}\)-graded \(A\)-modules. graded \(\mathbb{E}_\infty\)-ring; projective scheme; quasi-coherent sheaf; spectral algebraic geometry Fundamental constructions in algebraic geometry involving higher and derived categories (homotopical algebraic geometry, derived algebraic geometry, etc.), Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.), Derived categories, triangulated categories On graded \(\mathbb{E}_\infty\)-rings and projective schemes in spectral algebraic geometry | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Given a (smooth, projective, geometrically connected) curve over a number field, one expects its Hasse-Weil \( L\)-function, a priori defined only on a right half-plane, to admit meromorphic continuation to \( \mathbb{C}\) and satisfy a simple functional equation. Aside from exceptional circumstances, these analytic properties remain largely conjectural. One may formulate these conjectures in terms of zeta functions of two-dimensional arithmetic schemes, on which one has non-locally compact ``analytic'' adelic structures admitting a form of ``lifted'' harmonic analysis first defined by Fesenko for elliptic curves. In this paper we generalize his global results to certain curves of arbitrary genus by invoking a renormalizing factor which may be interpreted as the zeta function of a relative projective line. We are lead to a new interpretation of the gamma factor (defined in terms of the Hodge structures at archimedean places) and a (two-dimensional) adelic interpretation of the ``mean-periodicity
correspondence'', which is comparable to the conjectural automorphicity of Hasse-Weil \( L\)-functions. scheme of finite type; zeta function; local field; Hasse-Weil \(L\)-function; complete discrete valuation field; adeles Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture Zeta integrals on arithmetic 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 We study the vanishing of \(h^1 (\mathcal I_Z (a_1, a_2))\) for a zero-dimensional scheme \(Z \subset\mathbb P_1 \times\mathbb P_1\), when \(\deg(Z) \leq 4a_2 + 1\) and \(a_1 \geq a_2\). Hilbert function; postulation; smooth quadric surface; zero-dimensional scheme Projective techniques in algebraic geometry, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series On the Hilbert function of zero-dimensional schemes of the smooth quadric 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 rewrite the (extended) Ooguri-Vafa partition function for colored HOMFLY-PT polynomials for torus knots in terms of the free-fermion (semi-infinite wedge) formalism, making it very similar to the generating function for double Hurwitz numbers. This allows us to conjecture the combinatorial meaning of full expansion of the correlation differentials obtained via the topological recursion on the Brini-Eynard-Mariño spectral curve for the colored HOMFLY-PT polynomials of torus knots.
This correspondence suggests a structural combinatorial result for the extended Ooguri-Vafa partition function. Namely, its coefficients should have a quasi-polynomial behavior, where nonpolynomial factors are given by the Jacobi polynomials (treated as functions of their parameters in which they are indeed nonpolynomial). We prove this quasi-polynomiality in a purely combinatorial way. In addition to that, we show that the \((0,1)\)- and \((0,2)\)-functions on the corresponding spectral curve are in agreement with the extension of the colored HOMFLY-PT polynomials data, and we prove the quantum spectral curve equation for a natural wave function obtained from the extended Ooguri-Vafa partition function. HOMFLY-PT polynomials; torus knots; free fermions; Ooguri-Vafa partition function; spectral curve; Chekhov-Eynard-Orantin topological recursion; Hurwitz numbers; Jacobi polynomials Topological field theories in quantum mechanics, Relationships between algebraic curves and physics, Mirror symmetry (algebro-geometric aspects), Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.), Higher-dimensional knots and links Combinatorial structure of colored HOMFLY-PT polynomials for torus knots | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 tensorial Bernstein basis for multivariate polynomials in \(n\) variables has a number \(3^n\) of functions for degree \(2\). Consequently, computing the representation of a multivariate polynomial in the tensorial Bernstein basis is an exponential time algorithm, which makes tensorial Bernstein-based solvers impractical for systems with more than \(n = 6\) or \(7\) variables. This article describes a polytope (Bernstein polytope) with a number \(O({n \choose 2})\) of faces, which allows to bound a sparse, multivariate polynomial expressed in the canonical basis by solving several linear programming problems. We compare the performance of a subdivision solver using domain reductions by linear programming with a solver using a change to the tensorial Bernstein basis for domain reduction. The performance is similar for \(n = 2\) variables but only the solver using linear programming on the Bernstein polytope can cope with a large number of variables. We demonstrate this difference with two formulations of the forward kinematics problem of a Gough-Stewart parallel robot: a direct Cartesian formulation and a coordinate-free formulation using Cayley-Menger determinants, followed by a computation of Cartesian coordinates. Furthermore, we present an optimization of the Bernstein polytope-based solver for systems containing only the monomials \(x_i\) and \({x_i}^2\). For these, it is possible to obtain even better domain bounds at no cost using the quadratic curve \((x_i,{x_i}^2)\) directly. Bernstein polynomials; algebraic systems; subdivision solver; linear programming; simplex algorithm General methods in interval analysis, Computational aspects of higher-dimensional varieties, Effectivity, complexity and computational aspects of algebraic geometry Optimizations for tensorial Bernstein-Based solvers by using polyhedral bounds | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems These two volumes represent the second edition of the author's well-known and beautiful introductory book on algebraic geometry (1972; Zbl 0258.14001) which has been translated into several other languages (e.g. into English 1974, second edition 1977). This new edition differs from the first one because it was reworked and completed with great care. Here we only want to emphasize the differences between the two editions.
Chapter I begins with some new sections such as: an elementary discussion about plane curves, their singularities and the projective plane. Then some considerations about the zeta function, the theorem of Abhyankar- Moh, the Jacobian conjecture, the Grassmann variety and its Plücker embedding, associative algebras, determinantal varieties and the Tsen theorem and its application to the rationality of surfaces, are added.
Chapter II contains the following new things: more examples of smooth varieties, the varieties associated to associative algebras, Puiseux expansions, singularities of maps, the generic irreducibility or smoothness of morphisms, etc.
In chapter III the author added considerations about pencils of conics, a more detailed discussion about the cubic curves with emphasis to some arithmetical questions, and in chapter IV, the inequality of Riemann-Roch for surfaces, the geometry of the smooth cubic surface in \({\mathbb{P}}^ 3\), the singularities of a curve on a surface and their resolutions, and Du Val singularities of surfaces. The first volume ends with a (new) appendix of algebraic prerequisites.
The second volume contains the following new paragraphs: (1) The classification of the geometric objects, universal schemes, and the Hilbert scheme (in chapter VI; for this reason the paragraph in chapter I about Chow coordinates has been removed); (2) Connectivity of the fibers of a morphism of algebraic varieties; (3) The topology of the singularities of curves (both in chapter VII), and (4) Kähler manifolds and the Hodge theorem (in chapter VIII). A few exercises from the old edition disappeared, but many others have been included.
All in all these changes and completions made this remarkable textbook more updated and even more interesting. This second edition is highly recommended to every mathematician. plane curves; zeta function; Jacobian conjecture; Puiseux expansions; singularities; pencils of conics; universal schemes; Hilbert scheme I. R. Shafarevich, \textit{Basic Algebraic Geometry} (Nauka, Moscow, 1988; Springer, Berlin, 2013), Vol. 1. Research exposition (monographs, survey articles) pertaining to algebraic geometry, Foundations of algebraic geometry, Local theory in algebraic geometry, Cycles and subschemes Basic algebraic geometry. (Osnovy algebraicheskoj geometrii.) Vol. 1: Algebraic manifolds in projective space. (Tom 1: Algebraicheskie mnogoobraziya v proektivnom prostranstve). Vol. 2: Schemes. Complex manifolds. (Tom 2: Skhemy. Kompleksnye mnogoobraziya). 2nd ed., rev. and compl | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 use geometric invariant theory and the language of quivers to study compactifications of moduli spaces of linear dynamical systems. A general approach to this problem is presented and applied to two well known cases: We show how both Lomadze's and Helmke's compactification arise naturally as a geometric invariant theory quotient. Both moduli spaces are proven to be smooth projective manifolds. Furthermore, a description of Lomadze's compactification as a Quot scheme is given, whereas Helmke's compactification is shown to be an algebraic Grassmann bundle over a Quot scheme. This gives an algebro-geometric description of both compactifications. As an application, we determine the cohomology ring of Helmke's compactification and prove that the two compactifications are not isomorphic when the number of outputs is positive. quivers; geometric invariant theory; quot scheme; linear dynamical systems Bader, M.: Quivers, geometric invariant theory, and moduli of linear dynamical systems, Linear algebra appl. 428, 2424-2454 (2008) Linear systems in control theory, Algebraic methods, Algebraic systems of matrices, Geometric invariant theory, Representations of quivers and partially ordered sets Quivers, geometric invariant theory, and moduli of linear 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 prove that the general isolevel set of the ultra-discrete periodic Toda lattice is isomorphic to the tropical Jacobian associated with the tropical spectral curve. This result implies that the theta function solution obtained in the authors' previous paper is the complete solution. We also propose a method to solve the initial value problem. tropical geometry; Riemann's theta function; Toda lattice; general isolevel set; tropical spectral curve Inoue, R., Takenawa, T.: Tropical Jacobian and the generic fiber of the ultra-discrete periodic Toda lattice are isomorphic. http://arXiv.org/abs/0902.0448v1[nlin.SI] , 2009 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Relationships between algebraic curves and integrable systems, Jacobians, Prym varieties Tropical Jacobian and the generic fiber of the ultra-discrete periodic Toda lattice are isomorphic | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 direct and inverse isoenergy spectral problems are solved for a class of multidimensional periodic difference operators. It is proved that the inverse spectral problem is solvable in terms of theta functions of curves added to the spectral variety under compactification, and multidimensional analogs of the Veselov-Novikov relations are found.'' multidimensional scattering problem; Bloch function; spectral data; inverse spectral problem Linear difference operators, Relationships between algebraic curves and integrable systems, Difference operators, Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions Isoenergy spectral problem for multidimensional difference operators | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The main aim of the authors is to show that an algebro-geometrical approach for the open Toda lattice based on the concept of the Baker-Akhiezer function can be used in the case of reducible singular algebraic curves. As a consequence, the authors provide a solution to the inverse spectral problem for a finite Jacoby matrix which is different from the classical Stieltjes solution. Baker-Akhiezer function; Jacoby matrix; inverse spectral problem I. Krichever and K. Vaninsky, \textit{The periodic and open Toda lattice} , Mirror symmetry IV (2000), 139---158, AMS/IP Stud. Adv. Math. 33, Amer. Math. Soc., Providence, RI, 2002; arXiv:hep-th/0010184. Lattice dynamics; integrable lattice equations, Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions, Curves in algebraic geometry, Jacobi (tridiagonal) operators (matrices) and generalizations The periodic and open Toda lattice | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\mathcal{S}(M)\) be the ring of (continuous) semialgebraic functions on a semialgebraic set \(M\) and \(\mathcal{S}^\ast(M)\) its subring of bounded semialgebraic functions. In this work we compute the size of the fibers of the spectral maps \(\mathrm{Spec}(\mathrm{j})_1:\mathrm{Spec}(\mathcal{S}(N))\to\mathrm{Spec}(\mathcal{S}(M))\) and \(\mathrm{Spec}(\mathrm{j})_2:\mathrm{Spec}(\mathcal{S}^\ast(N))\to\mathrm{Spec}(\mathcal{S}^\ast(M))\) induced by the inclusion \(\mathrm{j}:N\hookrightarrow M\) of a semialgebraic subset \(N\) of \(M\). The ring \(\mathcal{S}(M)\) can be understood as the localization of \(\mathcal{S}^\ast(M)\) at the multiplicative subset \(\mathcal{W}_M\) of those bounded semialgebraic functions on \(M\) with empty zero set. This provides a natural inclusion \(\mathfrak{i}_M:\mathrm{Spec}(\mathcal{S}(M))\hookrightarrow\mathrm{Spec}(\mathcal{S}^\ast(M))\) that reduces both problems above to an analysis of the fibers of the spectral map \(\mathrm{Spec}(\mathrm{j})_2:\mathrm{Spec}(\mathcal{S}^\ast(N))\to\mathrm{Spec}(\mathcal{S}^\ast(M))\). If we denote \(Z:=\mathrm{Cl}_{\mathrm{Spec}(\mathcal{S}^\ast(M))}(M\backslash N)\), it holds that the restriction map \(\mathrm{Spec}(\mathrm{j})_2|:\mathrm{Spec}(\mathcal{S}^\ast(N))\backslash\mathrm{Spec}(\mathrm{j})_2^{-1}(Z)\to\mathrm{Spec}(\mathcal{S}^\ast(M))\backslash Z\) is a homeomorphism. Our problem concentrates on the computation of the size of the fibers of \(\mathrm{Spec}(\mathrm{j})_2\) at the points of \(Z\). The size of the fibers of prime ideals ``close'' to the complement \(Y:=M\backslash N\) provides valuable information concerning how \(N\) is immersed inside \(M\). If \(N\) is dense in \(M\), the map \(\mathrm{Spec}(\mathrm{j})_2\) is surjective and the generic fiber of a prime ideal \(\mathfrak{p}\in Z\) contains infinitely many elements. However, finite fibers may also appear and we provide a criterium to decide when the fiber \(\mathrm{Spec}(\mathrm{j})_2^{-1}(\mathfrak{p})\) is a finite set for \(\mathfrak{p}\in Z\). If such is the case, our procedure allows us to compute the size \(s\) of \(\mathrm{Spec}(\mathrm{j})_2^{-1}(\mathfrak{p})\). If in addition \(N\) is locally compact and \(M\) is pure dimensional, \(s\) coincides with the number of minimal prime ideals contained in \(\mathfrak{p}\). semialgebraic set; semialgebraic function; Zariski spectrum; spectral map; \texttt{sa}-tuple; suitably arranged \texttt{sa}-tuple; singleton fiber; finite fiber; infinite fiber Fernando, J.F., On the size of the fibers of spectral maps induced by semialgebraic embeddings, Math. nachr., 289, 14-15, 1760-1791, (2016) Semialgebraic sets and related spaces, Fields related with sums of squares (formally real fields, Pythagorean fields, etc.), Triangulation and topological properties of semi-analytic and subanalytic sets, and related questions, Real-valued functions in general topology On the size of the fibers of spectral maps induced by semialgebraic embeddings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper studies the displacement map (associated to a continuous family of ovals)
\[
D(t,\varepsilon)=\varepsilon M_1(t)+\varepsilon^2 M_2(t) +\varepsilon^3 M_3(t)+ \ldots
\]
related to small polynomial perturbations \(dF+\varepsilon \omega=0\) in the case when the Hamiltonian function \(F(x,y)\) is a product of \(d+1\) linear functions \(f_i(x,y)=a_ix+b_iy+c_i\), \(i=0,1,\ldots,d\) and assuming that the corresponding straight lines \(\ell_i: f_i(x,y)=0\) are in a generic position. Then the Hamiltonian vector field has \(a_1=\frac12d(d-1)\) centers and \(a_2=\frac12d(d+1)\) saddles. The main result states that if \(k\geq 2\) and \(M_1=\ldots=M_{k-1}\equiv 0\), \(M_k(t)\not\equiv 0\), then \(M_k(t)\) belongs to a \({\mathbb C}[t,1/t]\)-module generated by \(2a_1\) Abelian integrals \(I_i(t)\) and \(a_1\) non-Abelian integrals \(I_{i,j}^*(t)=\int_{\delta(t)}\varphi_id\varphi_j\), \(1\leq i<j\leq d\) containing logarithmic terms. Abelian integrals; Poincare-Pontryagin functions; Melnikov functions; polynomial systems; cohomology decompositions; displacement function Uribe, M, Principal Poincaré-Pontryagin function associated to polynomial perturbations of a product of (d+1) straight lines, J Differ Equ, 246, 1313-1341, (2009) Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.), 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.), Topological structure of integral curves, singular points, limit cycles of ordinary differential equations Principal Poincaré-Pontryagin function associated to polynomial perturbations of a product of \((d+1)\) straight lines | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this article we develop a broad generalization of the classical Bost-Connes system, where roots of unity are replaced by an algebraic datum consisting of an abelian group and a semi-group of endomorphisms. Examples include roots of unity, Weil restriction, algebraic numbers,Weil numbers, CM fields, germs, completion ofWeil numbers, etc. Making use of the Tannakian formalism, we categorify these algebraic data. For example, the categorification of roots of unity is given by a limit of orbit categories of Tate motives while the categorification of Weil numbers is given by Grothendieck's category of numerical motives over a finite field. To some of these algebraic data (e.g. roots of unity, algebraic numbers, Weil numbers, etc), we associate also a quantum statistical mechanical system with several remarkable properties, which generalize those of the classical Bost-Connes system. The associated partition function, low temperature Gibbs states, and Galois action on zero-temperature states are then studied in detail. For example, we show that in the particular case of the Weil numbers the partition function and the low temperature Gibbs states can be described as series of polylogarithms. quantum statistical mechanical systems; Gibbs states; zeta function; polylogarithms; Tannakian categories; Weil numbers; motives; Weil restriction Noncommutative algebraic geometry, (Equivariant) Chow groups and rings; motives, Quantum equilibrium statistical mechanics (general) Bost-Connes systems, categorification, quantum statistical mechanics, and Weil numbers | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 see \textit{B. A. Dubrovin}, \textit{I. M. Krichever}, \textit{S. P. Novikov}, ibid. 4, 179-248 (1985; Zbl 0591.58013).]
A general group-theoretic scheme for the construction of Hamiltonian systems and their solutions is exposed. integrable Hamiltonian systems; Hamilton-Jacobi theory; spectral theory; Korteweg-de Vries equation; hyperelliptic curves Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems, Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Partial differential equations of mathematical physics and other areas of application, Curves in algebraic geometry, Applications of PDEs on manifolds Integrable systems. 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 Here we extend the definition and the main properties of separators of a connected component \(mP\) of a zero-dimensional scheme \(Z:= mP\cup Z''\subset\mathbb P^n\) introduced by \textit{E. Guardo, L. Marino} and \textit{A. Van Tuyl} [J. Algebra 324, No. 7, 1492--1512 (2010; Zbl 1216.13010)] if \(Z''\) is a disjoint union of fat point. zero-dimensional scheme; fat point; Hilbert function Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Syzygies, resolutions, complexes and commutative rings, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Projective techniques in algebraic geometry Separations of zero-dimensional schemes 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 Let \(X_1\) be a smooth geometrically connected projective curve over the finite field of \(q\)-elements \(\mathbb{F}_q\). Let \(S_1\subseteq X_1\) be a reduced divisor consisting of \(N_1\) closed points. Set \(\mathbb{F}\) to be the algebraic closure of \(\mathbb{F}_q\). Let \((X,S):=(X_1, S_1)\otimes_{\mathbb{F}_q}\mathbb{F}\), \(l\) a prime number not dividing \(q\). There is an equivalence of categories between the category of \(\bar{\mathbb{Q}}_l\)-lisse sheaves and the category of continous finite dimensional \(\bar{\mathbb{Q}}_l\)-representations of \(\pi_1^{\text{ét}}(X-S,x)\), where \(x\in (X-S)(\mathbb{F})\) is a geometric point on which the equivalence functor depends. Since \(X-S\) is obtained from \(X_1-S_1\) by base change, the Galois group \({\text{Gal}}(\mathbb{F}/\mathbb{F}_q)\) acts on the scheme \(X-S\) and hence on the set of isomorphism classes of \(\bar{\mathbb{Q}}_l\)-lisse sheaves, or equivalently on the set of isomorphism classes of continous finite dimensional \(\bar{\mathbb{Q}}_l\)-representations of \(\pi_1^{\text{ét}}(X-S,x)\), via transport of structures. There are several equivalent ways to describe this action. One way is to look at the fundamental exact sequence of the étale fundamental groups
\[
1\to \pi_1^{\text{ét}}(X-S,x)\to \pi_1^{\text{ét}}(X_1-S_1,x) -^{{\phi}}\rightarrow {\text{Gal}}(\mathbb{F}/\mathbb{F}_q)\to 1.
\]
For any \(\sigma\in{\text{Gal}}(\mathbb{F}/\mathbb{F}_q)\) we choose an element \(\tau\in \phi^{-1}(\sigma)\), then for any continous finite dimensional representation \(\rho: \pi_1^{\text{ét}}(X-S,x)\to \text{GL}(V)\) we have \(\sigma(\rho)=( g\mapsto \rho(\tau^{-1}g\tau))\) for all \(g\in\pi_1^{\text{ét}}(X-S,x)\). As an action of \({\text{Gal}}(\mathbb{F}/\mathbb{F}_q)\) on the \textit{isomorphism classes} of \(\bar{\mathbb{Q}}_l\)-lisse sheaves, the so defined action does not depend on the choice of the element \({\tau\in \phi^{-1}(\sigma)}\). Let \(F_1\) be the fraction field of \(X_1\), \(F=F_1\otimes_{\mathbb{F}_q}\mathbb{F}\) be the fraction field of \(X\), \(s\in S\). Then the choice of a place \(\bar{s}\) of \(\bar{F}\) above \(s\) defines an inertia group \(I_s\subset {\text{Gal}}(\bar{F}/F)\). A \(\bar{\mathbb{Q}}_l\)-lisse sheaf is said to have ``\textit{Principal unipotent local monodromy at \(s\)}'' if the composition \(I_s\subset {\text{Gal}}(\bar{F}/F)\twoheadrightarrow \pi_1^{\text{ét}}(X-S,x)-^{\rho}\rightarrow {\text{GL}}(V)\) factors through the largest pro-\(l\) quotient (\(\cong \mathbb{Z}_l\)) of \(I_s\) with an element of \(I_s\) with image \(a\) in \( \mathbb{Z}_l\) acting on \(V\) as \text{exp}\((aN)\), where \(N\) is nilpotent with one Jordan block. Let \(\mathcal{T}^{(n)}(X,S)\) be the set of isomorphism classes of rank \(n\) irreducible \(\bar{\mathbb{Q}}_l\)-smooth sheaves on \(X-S\), with principal unipotent local monodromy at each \(s\in S\). \(\mathcal{T}^{(n)}(X,S)\) as a subset of the isomorphism classes of \(\bar{\mathbb{Q}}_l\)-lisse sheaves is stable under the \({\text{Gal}}(\mathbb{F}/\mathbb{F}_q)\)-action. Let \(T(X_1,S_1,n)\) denote the number of fixed points of \(\mathcal{T}^{(n)}(X,S)\) by the geometric Frobenius \(\text{Frob}\in{\text{Gal}}(\mathbb{F}/\mathbb{F}_q)\). For each \(m\geq 1\), let \((X_m,S_m):=(X_1,S_1)\otimes_{\mathbb{F}_q}\mathbb{F}_{q^m}\), then \(T(X_1,S_1,n,m):= T(X_m,S_m,n)\), where \((X_m,S_m)\) is viewed as a pair over \(\mathbb{F}_{q^m}\).
The aim of the article under review is to give a computation of the number \(T(X_1,S_1,n,m)\). It starts with a formula for \(T(X_1,S_1,n)\), under the assumption that \(n\) and \(N_1\) are \(\geq 2\), in terms of \(N_1\), \(n\), \(q\), the degrees \(\deg(s)\) for \(s\in S_1\) and the coefficients of the polynomial \(f(t):= \det(1-\text{Frob}\cdot t, H^1(X))\), where \({\text{Gal}}(\mathbb{F}/\mathbb{F}_q)\) acts on the \(l\)-adic cohomology group \(H^1(X)\) by transport of structures. However, the first formula is not helpful to understand how the number of fixed points varies with \(m\). One problem is that when one replaces \((X_1,S_1)/\mathbb{F}_q\) by \((X_m,S_m)/\mathbb{F}_{q^m}\), the number \((n/S_1):=\{\text{the largest divisor of }n\) that is prime to all \(\deg(s)\) for \(s\in S_1\) local systems; \(\ell\)-adic smooth sheaf; Lefschetz fixed point formula; automorphic representations; function fields; \(\mathrm{GL}(n)\); principal unipotent local monodromy; trace formula P. Deligne and Y. Flicker, Counting local systems with principal unipotent local monodromy, Ann. of Math. (2) 178 (2013), 921-982. Étale and other Grothendieck topologies and (co)homologies, Cohomology of arithmetic groups, Finite ground fields in algebraic geometry, Modular and Shimura varieties, Structure of families (Picard-Lefschetz, monodromy, etc.) Counting local systems with principal unipotent local monodromy | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper is a summary of results obtained in the author's habilitation thesis concerning the development of a spectral theory for simply periodic, \(2\)-dimensional, complex-valued solutions of the sinh-Gordon equation:
\[
\Delta u+\sinh(u)=0.
\]
One of the most salient differences between the spectral theory for doubly periodic solutions and for simply periodic solutions of the sinh-Gordon equation is that in the former case the spectral curve is of finite geometric genus and can be compactified, whereas in the latter case, it generally has infinite geometric genus.
In this paper, spectral data for the above solutions are defined for periodic Cauchy data on a line and the space of spectral data is described by an asymptotic characterization. Using methods of asymptotic estimates, the inverse problem for the spectral data of such Cauchy data is answered. Finally, a Jacobi variety for the spectral curve is constructed, which is used to study the asymptotic behavior of the spectral data corresponding to actual simply periodic solutions of the sinh-Gordon equation on strips of positive height. sinh-Gordon equation; spectral theory; integrable systems; inverse problem; Jacobi variety S. Klein, A Spectral Theory for Simply Periodic Solutions of the Sinh-Gordon Equation. Manuscript, submitted as habilitation thesis. Minimal surfaces in differential geometry, surfaces with prescribed mean curvature, Variational problems concerning minimal surfaces (problems in two independent variables), Jacobians, Prym varieties, Representations of entire functions of one complex variable by series and integrals, Second-order elliptic equations, Inverse problems for PDEs, Convergence and divergence of infinite products Spectral data for simply periodic solutions of the sinh-Gordon equation | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Nonsingular intersections of three real six-dimensional quadrics are considered. Such algebraic varieties are referred to for brevity as real four-dimensional triquadrics. Necessary and sufficient conditions for a real four-dimensional triquadric to be a \(GM\)-variety are established. six-dimensional quadric; \(GM\) variety; triquadric; spectral curve; spectral bundle; index function; cohomology group; Stiefel-Whitney class Real algebraic sets, Hypersurfaces and algebraic geometry, Classical real and complex (co)homology in algebraic geometry Real four-dimensional \(GM\)-triquadrics | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 set of notes is based on a lecture I gave at ``50 years of finite geometry -- a conference on the occasion of Jef Thas's 70th birthday,'' in November 2014. It consists essentially of three parts: in a first part, I introduce some ideas which are based in the combinatorial theory underlying \(\mathbb{F}_1\), the field with one element. In a second part, I describe, in a nutshell, the fundamental scheme theory over \(\mathbb{F}_1\) which was designed by \textit{A. Deitmar} [Beitr. Algebra Geom. 49, No. 2, 517--525 (2008; Zbl 1152.14001)]. The last part focuses on zeta functions of Deitmar schemes, and also presents more recent work done in this area. field with one element; Deitmar scheme; loose graph; zeta function; Weyl geometry Arithmetic algebraic geometry (Diophantine geometry), Varieties over finite and local fields, Schemes and morphisms, Finite ground fields in algebraic geometry Counting points and acquiring flesh | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 minimal immersions of tori in the 3-dimensional sphere \(S^3(1)\). The starting point of the study is the work of \textit{N. J. Hitchin} [J. Differ. Geom. 31, No.~3, 627--710 (1990; Zbl 0725.58010)], who established an explicit bijection between harmonic maps to tori in the 3-sphere and spectral curve data (which contains an hyperelliptic curve \(\Sigma\) and a line bundle over \(\Sigma\)).
The main result of the paper shows that for any strictly positive number \(g\), there are countably many spectral curves of arithmetic genus \(g\) giving rise to minimal immersions from rectangular tori to \(S^3\). As a corollary it is shown that for each positive integer \(n\), there exist countably many real \(n\)-dimensional families of minimal immersions from rectangular tori to \(S^3\). Each of these families consists of maps from a fixed torus.
The proof of the theorem naturally divides into 2 cases depending on whether \(g\) is even or odd. Whereas the even case is quite similar to previous results for minimal tori in \(\mathbb R^3\), the odd case needs more work.
The paper also contains a list of some remaining open questions and problems related to minimal tori in \(S^3\). immersions of tori; minimal surfaces; integrable systems; spectral curves Carberry E., Minimal tori in \({\mathbb{S}^{3}}\), Pacific J. Math. 233 (2007), no. 1, 41-69. Differential geometry of immersions (minimal, prescribed curvature, tight, etc.), Global submanifolds, Relationships between algebraic curves and integrable systems Minimal tori in \(S^{3}\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For the entire collection see Zbl 0626.00011.]
Let C be a nodal curve of the form \(C=C_ 1\cup C_ 2\), whose \(C_ 1\) and \(C_ 2\) are nonsingular (complete) curves (in characteristic zero). Consider a line bundle L on C and denote \(L_ i=L| C_ i\), \(V_ i=Im(H^ 0(L)\to H^ 0(L_ i))\). The authors give several results concerning the connection between the deformation theory of \(| L|\) and that of \((L_ 1,V_ 1)\), \((L_ 2,V_ 2)\). In particular, one gets conditions which assure that \(| L|\) (or subsystems of it) is smoothable (limit of linear systems on smooth curves).
As application one constructs components of the Hilbert scheme \(H_{d,g,r}\) which are smooth of the expected dimension and expected number of moduli for which the ratio d/g\(\to (3r-3)/(5r-4)\) as \(d\to \infty\) when \(r\geq 4\), respectively d/g\(\to 5/8\) when \(d\to \infty\) and \(r=3.\)
The construction of examples of good families of curves on projective spaces was the motivation of the developed theory and the obtained results refine results of \textit{E. Sernesi} [Invent. Math. 75, 25-57 (1984; Zbl 0541.14024)]. smoothing of linear systems on curves; nodal curve; line bundle; deformation theory; Hilbert scheme; good families of curves on projective spaces Chang , M.-C. Rav , Z. , '' Deformations of complete linear systems on reducible curves '', Proc. of Symp. in Pure Math. A.M.S. 46 ( 1987 ), 63 - 75 . Zbl 0659.14003 Divisors, linear systems, invertible sheaves, Families, moduli of curves (algebraic), Special algebraic curves and curves of low genus Deformations and smoothing of complete linear systems on reducible curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The subject of this work are linear systems in \({\mathbb P}^n\) of type \({\mathcal L} = {\mathcal L}_{n,d}(-\sum _{i=1}^h m_iP_i)\), \(P_i\in {\mathbb P}^n\), \(m_i\in {\mathbb N}\); which are given by hypersurfaces of degree \(d\) passing through \(h\) generic points \(P_i\) with multiplicities \(\geq m_i\).
Even for \(n=2\), the dimension of \({\mathcal L}\) is not known in general, but several equivalent conjectures (the first due to B. Segre) state that the dimension is the expected one except in the cases when the linear system contains a multiple fixed component which is a \((-1)\)-curve. In a previous paper, the author proposed yet other two equivalent forms for such conjectures, stating that \({\mathcal L}_{2,d}(-\sum _{i=1}^h m_iP_i)\) is special (i.e. it does not have the expected dimension), if and only if it is ``numerically special'' or ``cohomologically special'', where the main interest for those two concepts is that they could be generalized to \(n\geq 3\), where no general conjecture for the dimension of \({\mathcal L}\) is known.
We say that \({\mathcal L}\) is numerically special if it exists an \(\alpha\)-special effect variety \(Y\) for \({\mathcal L}\), i.e. an irreducible variety \(Y\) such that \({\mathcal L}-\alpha Y\) has positive virtual dimension greater than \({\mathcal L}\) (a few more conditions are required if \(\dim Y = n-1\)). We say that \({\mathcal L}\) is cohomologically special, instead, if it exists a \(h^1\)-special effect variety \(Y\), i.e. an irreducible \(Y\) such that \({\mathcal L}- Y\) has positive dimension, \(h^0({\mathcal L}|_Y)=0\) and \(h^1({\mathcal L}|_Y)>h^2({\mathcal L}-Y)\) (an example showing that the two definitions do not coincide is given).
The author conjectures that for \({\mathcal L}\) to be special is equivalent to being numerically special and also to being cohomologically special. In the case \(n=2\), \(\alpha\)-special effect varieties and \(h^1\)-special effect varieties are actually \((-1)\)-curves, while for \(n\geq 3\) examples are given using rational normal curves, hypersurfaces or linear spaces, and those covers most known examples of special \({\mathcal L}\)'s.
Moreover, the case of linear systems in a product \({\mathbb P}^{n_1}\times {\mathbb P}^{n_2}\times\dots\times {\mathbb P}^{n_t}\) is studied and examples (known and new) are given of systems whose speciality is due to special effect varieties. linear systems; multiple points; fat poins; Hilbert function Bocci, C.: Special effect varieties in higher dimension. Collect. Math. \textbf{56}(3), 299-326 (2005). ISSN: 0010-0757 Divisors, linear systems, invertible sheaves, Projective techniques in algebraic geometry, Singularities of curves, local rings Special effect varieties in higher 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 This is the first part of a survey on integrable dynamical systems. In the first chapter the authors deal with Hamiltonian systems and classical methods to integrate them. The second chapter deals with modern ideas for integrating evolution systems (for example Korteweg-de Vries equation), in particular methods from algebraic geometry are considered. An English translation of the article is announced to appear in 1987 (Encyclopedia of Mathematical Sciences, Springer Verlag). integrable Hamiltonian systems; Hamilton-Jacobi theory; algebraic- geometric spectral theory; hyperelliptic curves; Korteweg-de Vries equation Современные проблемы математики. Фундаментал\(^{\приме}\)ные направления, Том 4, Акад. Наук СССР, Всесоюз. Инст. Научн. и Техн. Информ., Мосцощ, 1985 [ МР0842907 (87ј:58032)]; Транслатион бы Г. Щассерманн; Транслатион едитед бы В. И. Арнол\(^{\приме}\)д анд С. П. Новиков. Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems, Partial differential equations of mathematical physics and other areas of application, Curves in algebraic geometry, Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Applications of PDEs on manifolds Dynamical systems. IV: Integrable systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We compute the coefficients of the polynomials \(C_n(q)\) defined by the equation
\[
1 + \sum_{n\geq 1} \, \frac{C_n(q)}{q^n} \, t^n = \prod_{i\geq 1}\, \frac{(1-t^i)^2}{1-(q+q^{-1})t^i + t^{2i}}.
\]
As an application we obtain an explicit formula for the zeta function of the Hilbert scheme of \(n\) points on a two-dimensional torus and show that this zeta function satisfies a remarkable functional equation. The polynomials \(C_n(q)\) are divisible by \((q-1)^2\). We also compute the coefficients of the polynomials \(P_n(q) = C_n(q)/(q-1)^2\): each coefficient counts the divisors of \(n\) in a certain interval; it is thus a non-negative integer. Finally we give arithmetical interpretations for the values of \(C_n(q)\) and of \(P_n(q)\) at \(q = -1\) and at roots of unity of order 3, 4, 6. infinite product; modular forms; zeta function; Hilbert scheme Combinatorial aspects of partitions of integers, Parametrization (Chow and Hilbert schemes), Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Enumerative problems (combinatorial problems) in algebraic geometry, \(q\)-calculus and related topics, Partition identities; identities of Rogers-Ramanujan type, Finite ground fields in algebraic geometry Complete determination of the zeta function of the Hilbert scheme of \(n\) points on a two-dimensional torus | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This is a survey of recent works of several people on the relation between the Waring's problem (concerning the representation of the \(j\)-forms of \(R=K[X_1,\ldots, X_n]\) as a sum of \(j\)-powers of linear forms), families PGOR\((T)\) of graded Artin algebra quotients of \(R\) having Hilbert function \(T\) and the Hilbert scheme parametrizing locally Gorenstein punctual subschemes of projective space [see also \textit{A. Iarrobino} and \textit{V. Kanev}, ``Power sums, Gorenstein algebras, and determinantal loci'', Lect. Notes Math. 1721 (1999; Zbl 0942.14026)]. The author studies the dimension, the closure and the irreducibility problems on the families PS\((s,j,n)\subset {\mathbb P}^N\), \(N={n+j\choose n}\) parametrizing the degree \(j\) homogeneous polynomials \(f\) of \(R\) which can be written as \(f=L_1^j+\cdots +L_s^j\) for some linear forms \(L_i\), the determinantal varieties of the catalecticant matrices Cat\(_F(u,v,n)\), \(u+v=j\) (the catalecticant matrices extend the well known Hankel matrices of case \(n=2\)), PGOR\((T)\), the quoted Hilbert scheme, etc.. Hilbert scheme; Waring's problem; catalecticant matrices; Hankel matrices; Hilbert function Iarrobino, A. 1996. Gorenstein Artin Algebras, Additive Decompositions of Forms and the Punctual Hilbert scheme. Proceedings of Hanoi conference in Commutative Algebra. 1996. Edited by: Eisenbud, D. and Tuan Hoa, Le. Springer-Verlag. to appear Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Parametrization (Chow and Hilbert schemes), Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Determinantal varieties Gorenstein Artin algebras, additive decompositions of forms and the punctual Hilbert scheme | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(Z \subset \mathbb{P}^2\) be a zero-dimensional scheme. Fix \(t \in \mathbb N\). In this paper we study the following question: find assumptions on \(Z\) and \(t\) such that \(h^1 (\mathcal{I}_A (t)) < h^1 (\mathcal{I}_Z (t))\) for all \(A\subsetneq Z\) and check if \(t\) does not exist for a certain class of schemes \(Z\). zero-dimensional scheme; plane curve; Hilbert function Projective techniques in algebraic geometry Zero-dimensional schemes in the plane | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 welcome monograph on algebraic K-theory in the way it was founded by D. Quillen in the early 1970's. It is based on lectures given by the author at the Tata Institute during 1986-1987 and consists of an Introduction, seven chapters on what may be called classical K-theory, an extensive account of the Merkurev-Suslin Theorem, a final chapter on localization for singular varieties and three appendices on topology, category theory and exact couples, respectively. After an introductory first chapter on Milnor's K-theory of \(K_ 0\), \(K_ 1\) and \(K_ 2\) of rings, the notion of symbols (norm residue symbol, Galois symbol, differential symbol) and Matsumoto's theorem on \(K_ 2\) of a (commutative) field, the reader is progressively led, via the plus- construction, Quillen's first definition of \(K_ i(R)\), R an associative ring, as the homotopy group \(\pi_ i(BGL(R)^+)\), Loday's theory of the H-space structure of \(BGL(R)^+\) and his natural products \(K_ i(R)\otimes K_ j(R)\to K_{i+j}(R)\), to the higher algebraic K-theory of (spectra of) rings and schemes. To this end the notion of the classifying space of a small category is extensively discussed. If \({\mathcal C}\) is a small exact category (exact meaning additive and embedded as a full additive subcategory of an abelian category \({\mathcal A}\) and ``closed under extensions'' in \({\mathcal A})\) one can construct another small category Q\({\mathcal C}\) with the same objects as \({\mathcal C}\) and suitably defined morphisms, with classifying space BQ\({\mathcal C}\), and then one defines, after Quillen, the higher algebraic K-groups \(K_ i({\mathcal C})\) as the homotopy groups \(\pi_{i+1}(BQ{\mathcal C},\{0\})\), \(i\geq 0\), where 0 is a zero-object of \({\mathcal C}\) so that \(\{\) \(0\}\) is a point of BQ\({\mathcal C}\). Then, if \({\mathcal P}\) is a full additive subcategory of the (small) exact category \({\mathcal M}\), closed under extensions and taking subobjects in \({\mathcal M}\) and such that any object of \({\mathcal M}\) has a finite resolution by objects of \({\mathcal P}\), one has the Resolution Theorem which says that \(K_ i({\mathcal P})\simeq K_ i({\mathcal M})\) for all \(i\geq 0\). Also, for a full abelian subcategory \({\mathcal B}\) of an abelian category \({\mathcal A}\), closed under taking subobjects, quotients and finite products and such that each object of \({\mathcal A}\) admits a finite filtration with consecutive quotients in \({\mathcal B}\), one has the Devissage Theorem: \(K_ i({\mathcal B})\simeq K_ i({\mathcal A})\). As a third result one has the Localization Exact Sequence
\[
...\to K_{i+1}({\mathcal C})\to K_ i({\mathcal B})\to K_ i({\mathcal A})\to K_ i({\mathcal C})\to...\to K_ 0({\mathcal A})\to K_ 0({\mathcal C})\to 0,
\]
where \({\mathcal B}\) is a Serre subcategory of the abelian category \({\mathcal A}\), and \({\mathcal C}\) is the quotient category \({\mathcal A}/{\mathcal B}\). Now, for an arbitrary scheme X, let \({\mathcal P}={\mathcal P}(X)\) denote the category of locally free sheaves of finite rank on X. Then \({\mathcal P}\) is an exact category and one defines \(K_ i(X)=K_ i({\mathcal P}(X))\). For a noetherian scheme X, let \({\mathcal M}={\mathcal M}(X)\) be the category of coherent sheaves on X. Then one defines \(G_ i(X)\) (or \(K_ i'(X))\) by \(G_ i(X)=K_ i({\mathcal M}(X))\). For a regular noetherian scheme X one obtains, by the Resolution Theorem, \(K_ i(X)=G_ i(X)\). For a (noetherian) ring R one defines \(K_ i(R)=K_ i(Spec(R))\) \((G_ i(R)=K_ i'(R)=G_ i(Spec(R)))\). This definition agrees with the one given by the plus-construction. \(K_ i\) is a contravariant functor from the category of schemes to abelian groups and \(G_ i\) is a contravariant functor from the category of noetherian schemes and that morphisms to abelian groups. Tensoring with vector bundles induces an action of \(K_ 0(X)\) on \(K_ i(X)\) and \(G_ i(X)\) when X is a noetherian scheme. For a proper morphism \(f:X\to Y\) between noetherian schemes one has a direct image map \(f_*: G_ i(X)\to G_ i(Y)\) under suitable finiteness conditions. Similarly for \(f_*: K_ i(X)\to K_ i(Y)\). Also, there is a projection formula \(f_*(x)\cdot y=f_*(x\cdot f^*(y))\), \(x\in K_ 0(X)\), \(y\in K_ i(Y)\) or \(G_ i(Y)\), where \(\cdot\) denotes the action of \(K_ 0(Y)\) on \(K_ i(Y)\) or \(G_ i(Y)\), respectively.
Further basic topics discussed are the existence of a Mayer-Vietoris sequence for \(G_ i\), the Homotopy Property which says that a flat map with affine fibers between noetherian schemes (e.g. a vector bundle) induces isomorphisms on the \(G_ i's\) of the source and target schemes, the Projective Bundle Theorem, the Brown- Gersten-Quillen Spectral Sequence, Gersten's Conjecture giving a long exact sequence relating \(G_ i(X)\), X the spectrum of a regular local ring, to the \(K_{i-j}(k(x^{(j)}))\), \(j=0,1,...,i\), where the \(k(x^{(j)}))\) are the residue fields of points of codimension j in X, its verification in some special cases, Bloch's formula \(H^ p_{Zar}(X,{\mathcal K}_{pX})=CH^ p(X)\), \(p\geq 0\), where \({\mathcal K}_{p,X}\) is the sheaf associated to the presheaf \(U\to K_ p(U)\), and the \(K_ i\) of Severi-Brauer schemes. Chapter 8 deals with the Merkurev- Suslin Theorem. It says that, for a field F of characteristic p and a positive integer n such that \((p,n)=1\), the Galois symbol \(R_{n,F}: K_ 2(F)\otimes_ ZZ/nZ\to H^ 2(F,\mu_ n^{\otimes 2})\) is an isomorphism. This theorem is one of the milestones in K-theory and leads to interesting applications for the Chow groups of algebraic varieties, e.g., it is proved that the n-torsion subgroup of the Chow group \(CH^ 2\) of a smooth, quasi-projective variety over an algebraically closed field of characteristic p, \((p,n)=1\), is finite. For a smooth affine surface it is zero, and for a smooth projective surface it coincides with the n-torsion subgroup of the Albanese variety (Roitman's Theorem). The final chapter deals with a localization theorem of Quillen and a generalization due to Levine. As a corollary one obtains: Let X be a normal quasi-projective surface with only quotient singularities and let \(\pi\) : \(Y\to X\) be a resolution of singularities. Then \(\pi^*: CH^ 2(X)\to CH^ 2(Y)\) is an isomorphism.
Summarizing, one can say that it is a pleasure to read this mathematically beautiful book but a critical remark may be made with respect to its typographic presentation. The table of contents lacks precision as a table of reference, there is no index, the text is poorly subdivided into sections, and the editor should have taken the trouble to have it rewritten in a more fashionable way (only the Introduction meets the usual standards) before publishing it in this expensive series. algebraic \(K\)-theory; Merkurev-Suslin Theorem; Milnor's \(K\)-theory of \(K_ 0\), \(K_ 1\) and \(K_ 2\) of rings; symbols; plus-construction; homotopy group; higher algebraic \(K\)-theory; classifying space of a small category; small exact category; higher algebraic \(K\)-groups; Resolution Theorem; Devissage Theorem; Localization Exact Sequence; Serre subcategory; noetherian scheme; sheaves; Mayer-Vietoris sequence; Homotopy Property; Projective Bundle Theorem; Brown-Gersten-Quillen Spectral Sequence; Gersten's Conjecture; long exact sequence; Chow groups of algebraic varieties; Albanese variety; localization theorem Srinivas, V.: Algebraic K-theory. Progress in Math. vol. 90, Birkhäuser (1991) Introductory exposition (textbooks, tutorial papers, etc.) pertaining to \(K\)-theory, Research exposition (monographs, survey articles) pertaining to \(K\)-theory, Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects), Higher algebraic \(K\)-theory, Grothendieck groups and \(K_0\), Whitehead groups and \(K_1\), Steinberg groups and \(K_2\), \(Q\)- and plus-constructions, \(K\)-theory of schemes, Applications of methods of algebraic \(K\)-theory in algebraic geometry Algebraic \(K\)-theory | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A classical way to study a finite set of points in a projective space \(\mathbb{P}^r\) over an algebraically closed field is to look at relations between its Castelnuovo function (i.e. the first difference function of the Hilbert function of its homogeneous coordinate ring) of geometric properties of the point set. In this extended abstract (containing no proofs), the authors refine that method as follows.
Given a sequence \(X= (P_1,\dots, P_s)\) of points in \(\mathbb{P}^r\), let \(S= (d_1,\dots, d_s)\in \mathbb{N}^s\) be defined by \(d_1=0\) and \(d_k=\) least degree of a hypersurface separating \(P_k\) from \(P_1,\dots, P_{k-1}\) for \(k>1\). Then the multiplicity sequence \(\gamma_S(n)= \#\{i\mid d_i=n\}\) equals the Castelnuovo function of \(X\) and does not depend on the order of the points. Hence it makes sense to study which sequences \(S\) are realizable and to try to classify all point sets \(X\) with given Castelnuovo function \(\Delta HF_X\) according to their sequences \(S\).
Here the authors announce some steps in this direction by examining the effect of neighbour transposition in the point sequence on the degree sequence. They discover that a set \(X\) gives rise to only one non-decreasing sequence \(S\) if and only if \(X\) is in uniform position. Moreover, maximal growth of the Castelnuovo function is shown to correspond to sequences of the form \(S= (0,1,\dots, n+h, S')\) with non-decreasing sequence \(S'\). All results are amply illustrated by examples. 0-dimensional scheme; Hilbert function; geometric properties of the point set; Castelnuovo function Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Cycles and subschemes A new approach to the Hilbert function of a 0-dimensional projective scheme | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Given a system of polynomial equations with parameters, we present a new algorithm for computing its Dixon resultant \(R\). Our algorithm interpolates the monic square-free factors of \(R\) one at a time from monic univariate polynomial images of \(R\) using sparse rational function interpolation. In this work, we use a modified version of the sparse multivariate rational function interpolation algorithm of Cuyt and Lee.
We have implemented our new Dixon resultant algorithm in Maple with some subroutines coded in C for efficiency. We present timing results comparing our new Dixon resultant algorithm with Zippel's algorithm for interpolating \(R\) and a Maple implementation of the Gentleman \& Johnson minor expansion algorithm for computing \(R\). Dixon resultant; parametric polynomial systems; resultant; sparse rational function interpolation; Kronecker substitution Symbolic computation and algebraic computation An interpolation algorithm for computing Dixon resultants | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 discusses an autonomous 4-dimensional integral system of Painlevé type. The study of such system is noticeably simplified if it is possible to build the so called Lax pair for it. In this paper, the authors present a systematic way to construct such Lax pairs. The key statement of the paper is the following one.
\textbf{Theorem 1.} For the 4-dimensional autonomous Painlevé-type equations, the Jacobian of the generic spectral curve has no nontrivial endomorphism. integrable systems; Lax pair; Painlevé divisors; spectral curve Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies, Relationships between algebraic curves and integrable systems, Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms, Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests Uniqueness of polarization for the autonomous 4-dimensional Painlevé-type systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(K\) be a complete field extension of \({\mathbb Q}_p\). A very important invariant attached to a \(p\)-adic (ordinary) differential equation on a rigid analytic curve over \(K\) is the radii of convergence of its solutions. It has been shown to be a powerful tool in the study of \(p\)-adic differential equations. However, explicit computation of the radius of convergence is often very difficult.
The paper concerns first order differential equations \(L(f) = \frac d{dx} f - Pf =0\) with \(P\) in \(K[x]\). In this case, the question at hand is equivalent to computing the radius of convergence of the exponential series \(\mathrm{exp}(Q)\), where \(Q\) is the formal integral of \(P\).
The author first points out two criteria for the product \(fg\) of two exponential series \(f\) and \(g\) to have its radius of convergence equal to the smaller of the radii of convergence of convergence of the two factors \(f\) and \(g\). The author then computes the radius of convergence in a special case, namely that of the \textit{Robba exponentials}, \(e_{p^m, \pi}(x) = \mathrm{exp}(\pi_m x + \pi_{m-1} p^{-1} x^p + \cdots + \pi_0p^{-m} x^{p^m})\), where \(\pi_0, \pi_1, \dots\) is a sequence of primitive \(p^\infty\)-torsion elements of a Lubin-Tate formal group. After that, the author explains an algorithm to compute the radius of convergence for a general exponential series: one can write it as a product of finitely many variants of Robba exponentials. Using the two general criteria explained earlier, the author shows that the radius of convergence of a general exponential series is equal to the minimum among the radii of convergence of all (variants of) Robba exponentials appearing in the above factorization, which are handled by the aforementioned special case.
From this explicit algorithm, the author deduces that, viewing the radius of convergence of the solution to a given first order differential equation as a function on the Berkovich affine line, its value is entirely determined by the values on a finite subtree of the Berkovich affine line. One may view this as an explicit example of a general continuity theorem of the radius of convergence on Berkovich curves, proved by \textit{F. Baldassarri} [Invent. Math. 182, No. 3, 513--584 (2010; Zbl 1221.14027)]. In the final subsection, the author carries out the algorithm in the special case \(L(f) = xf' - \pi(px^p +ax) f= 0\), where \(\pi^{p-1} = p\) and \(a\) lies in some valued extension of \(\mathbb Q_p\).
This paper is very reader-friendly because the author has detailed his computations quite explicitly, and has also made it mostly self-contained. Robba exponentials; Berkovich points; radius of convergence function [4] G. Christol, `` The radius of convergence function for first order differential equations {'', in \(Advances in non-Archimedean analysis\), Contemp. Math., vol. 551, Amer. Math. Soc., Providence, RI, 2011, p. 71-89. &MR 28 | &Zbl 1238.} \(p\)-adic differential equations, Computational aspects of field theory and polynomials, Rigid analytic geometry The radius of convergence function for first order differential 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 The aim of this paper is to show that the deformation calculus on spaces of Hitchin's spectral covers can be naturally induced from a much more transparent deformation theory on moduli spaces of holomorphic or meromorphic abelian differentials on Riemann surfaces. Using the embedding of the moduli space of generalized \(GL(n)\) Hitchin's spectral covers to the moduli space of meromorphic abelian differentials the authors show how variational formula for the period matrix, the canonical bi-differential and the prime form on the moduli spaces of generalized Hitchin's systems (when the coefficients of the equation defining the spectral cover are allowed to be meromorphic differentials) can be deduced from variational formula on moduli spaces of meromorphic abelian differentials derived in [\textit{C. Kalla} and \textit{D. Korotkin}, Commun. Math. Phys. 331, No. 3, 1191--1235 (2014; Zbl 1295.30096); \textit{A. Kokotov} and \textit{D. Korotkin}, J. Differ. Geom. 82, No. 1, 35--100 (2009; Zbl 1175.30041)]. In the special case of regular Hitchin's systems they reproduce residue formula for the canonical bi-differential obtained in [\textit{D. Baraglia} and \textit{Z. Huang}, ``Special Kähler geometry of the Hitchin system and topological recursion'', Preprint, \url{arXiv:1707.04975}] and for the period matrix (given by the Donagi-Markman cubic [\textit{R. Donagi} and \textit{E. Markman}, Lect. Notes Math. 1620, 1--119 (1996; Zbl 0853.35100)]). They also derive residue formula for variations of Bergman tau function of spaces of spectral covers for the holomorphic case. This paper is organized as follows: Section 1 is an introduction to the subject and a description of the results. Section 2 deals with spaces of generalized spectral covers. Section 3 is devoted to variational formula and Bergman tau function on moduli spaces of meromorphic abelian differentials. Section 4 deals with variational formula on spaces of generalized Hitchin's covers. Section 5 is devoted to the higher-order derivatives with respect to moduli on the space \(\mathcal{H}_{\widehat{g}}\) which can be obtained by a simple iteration of first derivatives, and the spaces of spectral covers \(\mathcal{M}^n_H\). While higher derivatives of the period matrix, tau-function and canonical bi-differential on the space \(\mathcal{H}_{\widehat{g}}\) are given by a simple formula, their restriction to the space \(\mathcal{M}^n_H\) is much less trivial. As an example of such computation the authors find the second derivatives of the period matrix. deformation calculus; Hitchin's spectral covers; moduli spaces; abelian differentials; Riemann surfaces; period matrix; Bergman tau function Families, moduli of curves (algebraic), Families, moduli of curves (analytic), Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Period matrices, variation of Hodge structure; degenerations, Differentials on Riemann surfaces Spaces of abelian differentials and Hitchin's spectral covers | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For part II see Duke Math. J. 48, 35-47 (1981; Zbl 0474.14030).]
The author characterizes in section 1 the Hilbert function of a Cohen- Macaulay (CM) scheme \(V^ d_ e\subset {\mathbb{P}}^{\nu}\) \((\nu >d\geq 1)\) of given degree e lying on a unique hypersurface of given degree k and such that its general curvilinear section \(V^ 1=V^ d\cap {\mathbb{P}}^{r+1}\), \(r=\nu -d\geq 2\), has minimal (arithmetic) genus (see proposition 1.3). He also describes a more geometric construction of the reduced irreducible codimension two CM varieties \(V^ d_ e\subset {\mathbb{P}}^{\nu}\) whose general curvilinear section has minimal genus among the CM curves of degree e in \({\mathbb{P}}^ 3\) (see the proof of theorem 1.7). In section 2 d-dimensional CM schemes of degree \(e=\left( \begin{matrix} m-1+\nu -d\\ \nu -d\end{matrix} \right)\) in \({\mathbb{P}}^{\nu}\), whose general curvilinear sections have minimal genus, are considered. The corresponding part of the Hilbert scheme is a connected smooth variety which is for \(\nu =d+2\) birationally equivalent to \(\prod^{d+1}_{i=1}Sym^ e({\mathbb{P}}_ i^{\nu -d}) \) in a canonical way (theorem 2.6). Cohen-Macaulay scheme; Hilbert function; codimension two; CM varieties; minimal genus; CM curves; CM schemes Treger, R, On equations defining arithmetically Cohen-Macaulay schemes III, J. Algebra, 125, 58-65, (1989) Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Low codimension problems in algebraic geometry, Families, moduli of curves (algebraic) On equations defining arithmetically Cohen-Macaulay schemes. III | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors study the Hilbert function of schemes \(\mathbb{Z}\) of ``fat'' points in \(\mathbb{P}^3\), whose support lies on a rational normal curve \(C\) of degree \(r\). More precisely, let \(I_Z\) be a homogeneous ideal of type \({\mathfrak p}_1^{m_1}\cap \cdots\cap {\mathfrak p}^{m_s}_s\), where each \({\mathfrak p}_i\) is the homogeneous ideal in \(R=k[x_0, \dots,x_r]\) of a point \(P_i\) and \(m_1,\dots,m_s\) are integers. In this paper the authors consider the following conjectures:
Conjecture A. The value of the Hilbert function \(H(R/I_Z,t)\) does not depend on the choice of the points on \(C\);
Conjecture B. If \(W\) is a scheme of fat points supported on a set of points in linear general position, then \(H(R/I_Z,t)\leq H(R/I_W,t)\).
Conjecture A was proved in the case \(r=2,3\) by \textit{M. V. Catalisano} and \textit{A. Gimigliano}, who gave an implicit algorithm (depending only on the numbers \(m_1,\dots,m_s)\) for computing \(H(R/I_Z,t)\) [see in particular J. Algebra 183, No. 1, 245-265 (1996; Zbl 0863.14028)].
Here, for the case \(g\geq 4\), the authors conjecture an algorithm (depending only on the \(m_i\)'s) to determine inductively \(H(R/I_Z,t)\). They also study the Hilbert function of the infinitesimal neighborhoods of a rational normal curve and they find the value where it coincides with the Hilbert polynomial. -- The main geometrical idea is to consider the base locus of the linear system \((I_Z)_t\). The authors prove that their algorithm works in several cases (and hence conjecture A in those cases): Either for \(s<r+2\), or for \(s\geq r+3\) and \(t\geq m_1+m_{r+3}-1\) (assuming \(m_1\geq \cdots\geq m_s)\); moreover, in case \(m_1=\cdots =m_s=m\) and either for \(s\leq r+2\) or for \(s>(m-1)(r-2)\); this implies conjecture A for \(m=2\) and any number of points [see also \textit{K. A. Chandler} in: Zero-dimensional schemes, Conf. Ravello 1992, 65-79 (1994; Zbl 0830.14019)]. Conjecture B is proved in the case \(r=2\). value of the Hilbert function; scheme of fat points; infinitesimal neighborhoods of a rational normal curve; Hilbert polynomial; number of points Catalisano, M. Virginia; Ellia, P.; Gimigliano, A., Fat points on rational normal curves, J. Algebra, 216, 2, 600-619, (1999) Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Jacobians, Prym varieties, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series Fat points on rational normal 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 Our object is the theory of ``\(\pi\)-exponentials'' \textit{Andrea Pulita} developed in his thesis [Équations différentielles \(p\)-adiques d'ordre un et applications. PhD thesis, 2006], generalising Dwork's and Robba's exponentials and extending \textit{S. Matsuda}'s work [Duke Math. J. 77, No. 3, 607--625 (1995; Zbl 0849.12013)]: We start with an abstract algebra statement about the structure of the kernel of iterations of the Frobenius endomorphism on the ring of Witt vectors with coordinates in the ring of integers of an ultrametric extension of \(\mathbb Q_p\). Provided sufficiently (ramified) roots of unity are available, it is, unexpectedly simply, a principal ideal with respect to an explicit generator essentially given by Pulita's \(\pi\)-exponential. This result is a consequence and a reformulation of core facts of Pulita's theory. It happened to be simpler to prove directly than reformulating Pulita's results.
Its translation in terms of series is very elementary, and gives a criterion for solvabilty and integrality for \(p\)-adic exponential series of polynomials. We explain how to deduce an explicit formula of their radius of convergence, and even the function radius of convergence. We recover this way, in elementary terms, with a new proof, and important simplifications, an algorithm of \textit{G. Christol} [Contemp. Math. 551, 71--89 (2011; Zbl 1238.12006)] based similarly on Pulita's work. One concrete advantage is: one can easily prove rigorous complexity bounds about the implied algorithm from our explicit formula. We also add there and there refinements and observation, notably hinting some of the finer informations that can also given by the algorithm.
One of the appendix produce a computation which gives finer estimates on the coefficients of these series. It should provide useful in proving complexity bounds for various computational use involving these series. It is not apparent yet in the present work, but should be in latter projected developments, the series under consideration are the base object for some exponential sums on finite fields via \(p\)-adic approach, namely via rigid cohomology with rank one coefficients.
Convergence radius and coefficients estimates are involved studying the efficiency of computational implementations of these objects. The understanding of convergence radius of \(p\)-adic differential equations is a subject undergoing active developments, and we here provide a fine theoretical and computational study of the simplest of cases.
This initiates a projected series of articles. We start here, with the case of the affine line as a base space, a Witt vectors paradigm. This provides an alternative purely algebraic approach of Pulita's theory; the richness of Witt vectors theory allow suppleness and efficiency in working with \(\pi\)-exponentials, which will prove efficient later in the series. \(\pi\)-exponentials; \(p\)-adic differential equations; kernel of Frobenius endomorphism of Witt vectors over a \(p\)-adic ring; radius of convergence function; algorithm [17] R. Richard, `` Des \(\pi\)-exponentielles I: vecteurs de Witt annulés par Frobénius et algorithme de (leur) rayon de convergence {'', \(Rend. Semin. Mat. Univ. Padova\)133 (2015), p. 125-158. &MR 33} \(p\)-adic differential equations, Witt vectors and related rings, Local ground fields in algebraic geometry On \(\pi\)-exponentials. I: Witt vectors that vanish under Frobenius and algorithm of (their) radius of convergence | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Consider a system in Lax form, \(dA/dt=[A,B]\) where \(A\) and \(B\) are \((r\times r)\) matrix functions, polynomial in a variable \(x(t)\). The coefficient \(A_d\) of the leading term \(A_dx^d\) in the \(x\) polynomial defining \(A\) is a constant of motion. Define the function \(Q(x,y)=\det [A(x)-yI]\), and the isospectral manifold
\[
V_Q^M= \bigl\{A(x): Q(x,y)=0,\;A_d=M\bigr\}
\]
Consider the affine curve \(X=\{(x,y)\in \mathbb{C}^2: Q(x,y)=0\}\) and its compactification \(\overline X\), called the spectral curve for \(A\). Also, denote as \(\mathbb{P}_M\) the centralizer of \(M\) in the projective group \(\mathbb{P} GL_r (\mathbb{C})\).
If \(X\) is smooth, then so is \(V_Q^M\), and a number of results are available in the literature. In particular, \textit{P. van Moerbeke} and \textit{D. Mumford} [Acta Math. 143, 93-154 (1979; Zbl 0502.58032] showed that, in this case, the quotient \(V_Q^M/ \mathbb{P}_M\) corresponds to a Zariski open subset in the Jacobian \(J(X)\); and that if \(X\) is non-smooth, \(V_Q^M/\mathbb{P} M\) corresponds to a Zariski open subset in a generalized Jacobian. Work by \textit{M. R. Adams}, \textit{J. Harnad} and \textit{J. Hurtubise} [Commun. Math. Phys. 134, 555-585 (1990; Zbl 0717.58051)] and by \textit{A. Beauville} [Acta Math. 164, No. 3/4, 211-235 (1990; Zbl 0712.58031)] showed that \(V_Q^M/ \mathbb{P}_M\) can also be described in terms of an affine part of the standard Jacobian \(J(X)\).
Assume \(X\) is smooth and at \(q_i\in \overline X\), with \(x(q_i)=\infty\), \(\overline X\) is locally a normal crossing of several branches. Denote by \({\mathcal F}=\{(\mathbb{C}^r,M)\), \((E,K)\}\) the data at infinity of \(\overline X\), with \(M\in\text{End} (\mathbb{C}^r)\), \(E\) a vector space and \(K\in\text{End}(E)\); we denote by \(\pi\) the projection on \(E\). Assume \(M\) is diagonalizable, and \(K\) is diagonalizable with distinct eigenvalues. With this, one can introduce
\[
{\mathcal M}_Q^{\mathcal F}=\bigl\{A\in V_Q^M:[\pi A_{d-1}]_E= K\bigr\}.
\]
Then the author considers the (singular) curve \(Y\) obtained from the smooth compactification of \(X\) by identifying the ``infinite'' points on \(X\), and denote by \(\Theta(Y)\) the theta divisor formed by special line bundles on \(Y\) of degree equal to the arithmetic genus of \(Y\).
The main result of the paper is that the manifold \({\mathcal M}^{\mathcal F}_Q\) is smooth and bi-holomorphic to a Zariski open subset of the generalized Jacobian \(J(Y)-\Theta(Y)\). integrable systems; spectral curve; Lax form; isospectral manifold Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Relationships between algebraic curves and integrable systems, Jacobians, Prym varieties, Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics Jacobians of singular spectral curves and completely integrable systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Finding pairs of commuting differential operators with rational coefficients is a classical problem in the theory of ODE's. As it was shown in [\textit{J. L. Burchnall} and \textit{T. W. Chaundy}, Proc. Lond. Math. Soc. (2) 21, 420--440 (1923; JFM 49.0311.03)] that if two such operators, $L_{n} = \sum_{i=0}^{n}u_{i}(x)\partial_{x}^{i}$ and $L_{m} = \sum_{i=0}^{m}v_{i}(x)\partial_{x}^{i}$, commute, then there exists a nonzero polynomial $R(z, w)$ such that $R(L_{n}, L_{m}) = 0$. The curve $\Gamma$ defined by $R(z, w) = 0$ is called the \textit{spectral curve} of the pair $L_{n}, L_{m}$; its genus is called the genus of the commuting pair of operators. If $L_{n}\psi = z\psi$ and $L_{m}\psi = w\psi$, then $(z, w)\in\Gamma$. For almost all $(z, w)\in\Gamma$, the dimension of the space of common eigenfunctions for generic $(z, w)\in\Gamma$ is called the \textit{rank} of the given pair of operators $L_{n}$ and $L_{m}$ (it is a common divisor of $m$ and $n$). \par The paper under review presents new pairs of self-adjoint commuting differential operators of rank two with rational coefficients (examples of such operators of arbitrary genus can be found in [\textit{A. E. Mironov}, Adv. Math. Sci. 67, 309--321 (2014; Zbl 1360.35227)]; examples of commuting differential operators of arbitrary genus and arbitrary rank were obtained in [\textit{O. I. Mokhov}, Adv. Math. Sci. 67, 323--336 (2014; Zbl 1360.35228)]). The author also proves that any curve of genus 2 written as a hyperbolic curve is a spectral curve of a pair of commuting differential operators with rational coefficients. integrable systems; commuting differential operators; Weyl algebra; spectral curve Commutative rings of differential operators and their modules, Special algebraic curves and curves of low genus, General theory of partial differential operators, Relationships between algebraic curves and integrable systems Commuting differential operators of rank 2 with rational coefficients | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 defines a new notion of dagger algebra that axiomatizes some of the properties of the dagger algebras used to define the Monsky-Washnitzer cohomology of affine smooth algebraic varieties over finite fields. This new notion is obtained via the theory of bornological algebras. Bornological algebras are functional analytic objects, similar to topological algebras but with better formal properties.
The paper starts with a section that recalls the basics of the theory of bornological spaces needed for understanding the rest of the text. Therefore it can be considered to be a self-contained article accessible to readers with minimal background on the subject. Moreover, in this section, the authors already discuss new results as they are adapting the classical theory to allow the base ring to be a complete discrete valuation ring instead of a complete valued field. The subsequent sections introduce key concepts needed to define dagger algebras: the notion of a separated bornological module, the notion of a complete bornological module and the completion functor, the spectral radius of a bounded subset of a bornological algebra, and the notion of bornological torsion-freeness (for a bornological algebra). The text contains proofs of the main properties of these concepts together with key examples and counter-examples that help the reader to develop his own intuition. Eventually, in Section 5 the abstract notion of dagger algebra is defined as a bornological algebra (over a fixed base complete valuation ring) that is complete, bornologically torsion-free, and whose bounded subsets have spectral radius \(1\). Then, it is shown that any bornological algebra has a unique dagger completion to which it maps. The last two sections of the paper are devoted to examples of dagger algebras coming from monoid algebras and crossed product algebras. bornological space; dagger algebra; \(p\)-adic cohomology; spectral radius; bornological torsion-free; complete bornological space \(p\)-adic cohomology, crystalline cohomology, Rigid analytic geometry, Generalizations (algebraic spaces, stacks), Non-Archimedean analysis, Bornologies and related structures; Mackey convergence, etc., (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) Dagger completions and bornological torsion-freeness | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 cohomology of the Hilbert scheme of points \(\mathrm{Hilb}_n(S)\) of non-singular quasi-projective surfaces \(S\) has been described in terms of vertex operators on \(\mathcal{F}:=\bigoplus_nH^*(\mathrm{Hilb}_n(S,\mathbb{Q}))\) introduced by Nakajima and Grojnowski. In this paper the authors introduce another natural set of vertex operators on \(\mathcal{F}\) that depends on a line bundle \(\mathcal{L}\) over \(S\). These operators are defined in terms of the Chern classes of the alternating Ext-groups between two ideal sheaves twisted by \(\mathcal{L}\), and can be characterized by their commutation relations with the Nakajima operators. Moreover, the authors give an explicit formula for them in terms of the latter.
For the \((\mathbb{C}^*)^2\)--equivariant cohomology of \(\mathrm{Hilb}_n(\mathbb{C}^2)\) the trace of these vertex operators becomes the Nekrasov instanton partition function with matter in the adjoint representation. Equivariant localization reduces the above mentioned formula to a Pieri-type formula for the Jack symmetric functions. Hilbert scheme of points; Nakajima operators; Ext-groups; Nekrasov instanton partition function; Jack symmetric functions E. Carlsson and A. Okounkov, \textit{Exts and Vertex Operators}, arXiv:0801.2565. Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) Exts and vertex operators | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For all integers \(n\), \(d\), \(g\) such that \(n\geq 4\), \(d\geq n+1\), and \((n+2)(d-n-1)\geq n(g-1)\), we define a good (i.e. generically smooth of dimension \((n+1)d+(3-n)(g-1)\) and with the expected number of moduli) irreducible component \(A(d,g;n)\) of the Hilbert scheme of smooth and nondegenerate curves in \(\mathbb{P}^n\) with degree \(d\) and genus \(g\). For most of these \((d, g)\), we prove that a general \(X\in A(d,g;n)\) has maximal rank. We cover, in this way, a range of \((d,g,n)\) outside the Brill-Noether range. curve in projective spaces; normal bundle; Hilbert scheme; Hilbert function Plane and space curves, Special divisors on curves (gonality, Brill-Noether theory), Projective techniques in algebraic geometry, Families, moduli of curves (algebraic) Good components of curves in projective spaces outside the Brill-Noether range | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 review and explore recently discovered, interesting new connections between the three topics mentioned in the title of the paper. In J. Algebra 174, No. 3, 1080-1090 (1995; Zbl 0842.14002), \textit{J. Emsalem} and \textit{A. Iarrobino} observed that there is a relationship between fat points in \(\mathbb{P}^n\) (i.e. zero-dimensional subschemes \(\mathbb{X}\) defined by ideals of the form \(I_{\mathbb{X}}= {\mathfrak p}_1^{\alpha_1} \cap\cdots\cap {\mathfrak p}_s^{\alpha_s}\)) and ideals generated by powers of linear forms. More precisely, for \(j\geq \max\{\alpha_1,\dots, \alpha_s\}\), the \(j\)-th graded piece of the Macaulay inverse system \((I_{\mathbb{X}})^{-1}\) equals the corresponding graded piece of a certain ideal generated by \(s\) powers of linear forms.
In the case of linearly independent linear forms \(\{L_1,\dots, L_s\}\) in two indeterminates, the authors compute explicit formulas for the Hilbert function of \(J= (L_1^{\alpha_1},\dots, L_s^{\alpha_s})\), for the socle degree of \(k[y_0,y_1]/J\), and thus for the minimal graded free resolution of \(J\).
In the paper: Compos. Math. 108, No. 3, 319-356 (1997; Zbl 0899.13016), \textit{A. Iarrobino} also observed that there is a relationship between splines (i.e. piecewise polynomial functions satisfying certain smoothness conditions) on a \(d\)-dimensional simplicial complex \(\Delta\) and the ideals generated by powers of the linear forms defining hyperplanes incident to the interior faces of \(\Delta\). The authors recall a certain chain complex whose top homology module is precisely the module \(C^\alpha (\widehat{\Delta})\) of mixed splines on the cone \(\widehat{\Delta}\) over \(\Delta\) which are smooth of order \(\alpha\). In the case of planar splines \(\Delta\subset \mathbb{R}^2\), they are then able to provide formulas for the number of splines in \(C^\alpha (\widehat{\Delta})_k\) of sufficiently large degrees \(k\gg 0\).
The paper ends with a discussion of the obstacles which have to be overcome in order to get higher-dimensional versions of those results. zero-dimensional scheme; Macaulay inverse system; mixed spline; ideal of linear forms; fat points; Hilbert function; number of splines Geramita, A. V.; Schenck, H., Fat points, inverse systems, and piecewise polynomial functions, J. Algebra, 204, 1, 116-128, (1998) Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Numerical computation using splines Fat points, inverse systems, and piecewise polynomial 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 This paper is dedicated to the use of Kähler differential algebras for the study of 0-dimensional subschemes of \({\mathbb P}^n_K = {\mathbb P}^n\), \(K\) being a field with char \(K = 0\).
Let \(S=K[X_0,\ldots , X_n]\) and \(X\subset {\mathbb P}^n\) be a 0-dimensional scheme whose ideal is \(I_X\) and its graded coordinate ring is \(R_X = S/I_X\); let \(R_X\otimes R_X = \bigoplus_{i\geq 0}(\bigoplus_{j+k=i}(R_X)_j\otimes (R_X)_k)\) be its enveloping algebra and \(J\subset R_X\otimes R_X\) the kernel of \(\mu: R_X\otimes R_X \rightarrow R_X\), where \(\mu(f\otimes g) = fg\) (\(J\) is generated by \(\{x_i\otimes 1 - 1\otimes x_i\}\), \(i=0,\ldots,n\), where \(x_i\) is the image of \(X_i\) in \(R_X\)).
Then \(\Omega^1_{R_X} = \frac{J}{J^2}\) is called the \textit{module of Kähler differential 1-forms}, while \(\Omega^m_{R_X} = \wedge ^m \Omega^1_{R_X}\) is called the \textit{module of Kähler differential \(m\)-forms}; eventually, the graded algebra \(\Omega_{R_X} = \bigoplus_{m\geq 0}\Omega^m_{R_X}\) is the \textit{Kähler differential algebra of} \(R_X\) (here \(\Omega^0_{R_X} = R_X\)).
The following example shows how much the modules \(\Omega^m_{R_X}\) are related to the geometry of \(X\): let \(X,Y\subset {\mathbb P}^2\) be two reduced schemes made of six points and let \(X\) be contained in a smooth conic, while \(Y\) is made of three points on a line and other three points on a different line (six points on a degenerate conic). Then the Hilbert functions agree for \(R_X\) and \(R_Y\) and also for \(\Omega^1_{R_X}\) and \(\Omega^1_{R_Y}\), but \(\Omega^2_{R_X},\Omega^3_{R_X}\) have different Hilbert functions from \(\Omega^2_{R_Y},\Omega^3_{R_Y}\); hence the study of 2-forms and 3-forms can distinguish the two cases.
In the paper, a few general bounds on Hilbert functions and regularity indeces of \(\Omega^m_{R_X}\) are found; more detailed results are given when \(X\) is a scheme of fat points (namely when \(X\) is in \({\mathbb P}^1\), or has support on a conic in \({\mathbb P}^2\), or for uniform fat points schemes). Kähler differential algebra; 0-dimensional schemes; fat point scheme; Hilbert function Modules of differentials, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Structure of families (Picard-Lefschetz, monodromy, etc.) Kähler differential algebras for 0-dimensional schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A short overview of recent developments in the theory of dynamics of rational maps on projective spaces is presented. The authors expose the important notions and results obtained for canonical height, preperiodic points, and the generalized Mahler formula established in [\textit{J. Pineiro} and the authors, Geometric methods in algebra and number theory. Basel: Birkhäuser, Prog. Math. 235, 219--250 (2005; Zbl 1101.11020)]. The Mahler formula is one of the few global formulas for the height of a point. It established a first relation between the canonical height and the bad reduction. Another result of the authors relates the bad reduction to bounds of dynamical systems.
Then phenomena of Shafarevich type in dynamics, and further questions are discussed. Another direction is the study of dynamics over a function field.
See also the results of the authors published in [A Shafarevich-Faltings theorem for rational functions. Pure Appl. Math. Q. 4, No. 3, 715--728 (2008; Zbl 1168.14020)]. dynamics of rational maps on projective spaces; canonical height; preperiodic points; generalized Mahler formula; bad reduction; bounds of dynamical systems; phenomena of Shafarevich type in dynamics; dynamics over function field Heights, Arithmetic varieties and schemes; Arakelov theory; heights, Height functions; Green functions; invariant measures in arithmetic and non-Archimedean dynamical systems Algebraic dynamics | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this article we give a survey on computational algebraic geometry today. It is not our goal to guide research in this area, but to help algebraic geometers to decide whether nowadays' computational techniques and computer algebra systems provide useful tools for their own research. To illustrate some of the main techniques we focus on rather small examples. But we also give hints on the actual size of the computations which can be carried through today, and we quote further survey articles for more information in this direction.''
Examples feature MACAULAY2, SINGULAR, the MAPLE packages SCHUBERT and CASA, and SURF, a system for visualizing curves and surfaces. Topics covered include basic applications of Gröbner bases (e.g. division, elimination, Hilbert polynomials), local rings, homological algebra and computation of coherent cohomology, \(D\)-module computations, primary decomposition, normalization, deformation theory, invariant rings, construction of special varieties. There is a large bibliography. Bibliography; computer algebra systems; Gröbner bases; ideal membership; Hilbert function; elimination; Milnor numbers; Beilinson monads; \(D\)-modules; primary decomposition; normalization; Puiseux expansion; rational parametrization; deformations; invariant rings; special varieties; intersection theory; syzygy conjectures; Zariski's conjecture; visualisation; complexity Computational aspects of algebraic curves, Computational aspects of algebraic surfaces, Software, source code, etc. for problems pertaining to algebraic geometry, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Computational aspects of higher-dimensional varieties Computational algebraic geometry today | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 original Hitchin systems [\textit{N. Hitchin}, Duke Math. J 54, 91-114 (1987; Zbl 0627.14024)] are completely integrable Hamiltonian systems on the cotangent bundle of the moduli space of stable principal \(G\)-bundles (of fixed degree, for a complex reductive group \(G\)) over a compact Riemann surface \(\Sigma\). An element of this cotangent bundle can be thought of as a principal \(G\)-bundle \(E\to\Sigma\) equipped with a ``Higgs field'' \(\varphi\), a holomorphic 1-form over \(\Sigma\) with values in the coadjoint bundle of \(E\). The pair \((E,\varphi)\) is called a Higgs bundle. The total space has the natural symplectic structure of a cotangent bundle and the Hamiltonians which define the systems arise from the \(G\)-invariant polynomials of the coadjoint representation. The most important feature of these systems is that they are ``algebraically completely integrable'', meaning that their Lagrangian submanifolds are (open subsets) of abelian varieties. Hitchin used these systems for computing certain cohomology groups, but for some time it was unclear whether these systems had recognizable realisations as systems of differential equations.
The article under review is a relatively short, readable survey of the author's contributions towards understanding generalized Hitchin systems. These systems, introduced by \textit{F. Bottacin} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 28, No. 4, 391-433 (1995; Zbl 0864.14004)] and \textit{E. Markman} [Compos. Math. 93, 255-290 (1994; Zbl 0824.14013)], allow for the Higgs field to be meromorphic. The total space \(\mathcal{M}\) of these Higgs bundles is no longer a cotangent bundle, but it can be given a Poisson structure by realising it (over open sets) as the quotient of a cotangent bundle by the Hamiltonian action of a certain Lie group. The first section of this survey describes this in clear detail. This section shows that what we win from this generalization is the ability to recognize a number of familiar systems (for example, systems which arise from \(R\)-matrix structures on loop algebras) as generalized Hitchin systems.
The second section describes attempts by \textit{J. C. Hurtubise} and \textit{E. Markman} [Commun. Math. Phys. 223, 533-552 (2001; Zbl 0999.37048)] to use generalized Hitchin systems to give geometry to the Calogero-Moser systems: There is one of these for each root system. The article discusses how this is done by using, instead of the natural semi-simple Lie group associated to each root system, a semi-direct product of the maximal torus and the sum of the root spaces. This approach produces a phase space for Calogero-Moser but does not give a clean explanation of why it is an algebraically completely integrable system. One gets the impression that the Calogero-Moser systems are yet to be fully understood.
The third section discusses the role of \(Gl(N)\)-Hitchin systems and ``abelianisation''. This is the process of associating to each Higgs bundle \((E,\varphi)\) (which we now think of as the associated rank \(n\) vector bundle equipped with the Higgs field) a line bundle over a covering curve \(\pi: S\to\Sigma\), called the spectral curve since it is the curve of eigenvalues of the Higgs field. This curve lives in the total space of the line bundle \(K_\Sigma(D)\) of holomorphic 1-forms with divisor of poles no worse than \(D\). The Jacobi variety of \(S\) (or, to be precise, a Zariski open subset of it) lives as a Lagrangian submanifold of the Poisson manifold \(\mathcal{M}\): This is how one sees that the Hitchin systems are algebraically completely integrable. The author then explains how on each symplectic leaf of \(\mathcal{M}\) the family of spectral curves is encoded by a surface \(Q\). Each spectral curve embeds in \(Q\) and there is a symplectic isomorphism from a Hilbert scheme of points on \(Q\) to the family of Jacobi varieties of the spectral curves: When restricted to any spectral curve this map is just the Abel map. This isomorphism provides a way of separating variables for the Hamiltonian system.
The final section is a brief review of what is known about abelianisation when the group is an arbitrary reductive group. Hitchin system; spectral curve; moduli space; Higgs field; Higgs bundle; algebraically completely integrable systems; Poisson structure; Calogero-Moser systems Hurtubise, J., The geometry of generalized Hitchin systems, Integrable Systems: From Classical to Quantum, 55-76, (2000), American Mathematical Society, Providence, RI Analytic theory of abelian varieties; abelian integrals and differentials, 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, Hamilton's equations The geometry of generalized Hitchin systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(k\) be any field, \(\text{Hilb}^{p(z)}_{\mathbb{P}^n_k}\) the Hilbert scheme of subschemes of \(\mathbb{P}^n\) with Hilbert polynomial \(p(z)\). \textit{F. J. Macaulay} showed [Proc. Lond. Math. Soc. (2) 26, 531-555 (1927; JFM 53.0104.01)] that there exists a unique saturated lexicographic ideal \(L\) such that \(k[x_0, \dots,x_n]/L\) has Hilbert polynomial \(p(z)\). We show that the scheme corresponding to \(L\) is parametrized by a smooth point on the Hilbert scheme. In the process we calculate the dimension of the unique component through this point explicitly, and describe explicitly the subscheme corresponding to the general point of this component. smoothness; lexicographic point; Hilbert scheme; JFM 53.0104.01; Hilbert polynomial A. Reeves - M. Stillman, Smoothness of the lexicographic point. J. Algebraic Geom., 6 (2) (1997), pp. 235-246. Zbl0924.14004 MR1489114 Parametrization (Chow and Hilbert schemes), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series Smoothness of the lexicographic point | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 0741.00055.]
This is essentially an expository paper. It gives a unitary reorganization of the material contained in several papers by \textit{R. Maggioni} and the author. In this reorganization several proofs have been improved and some unnecessary hypotheses have been removed.
If \(S\subset\mathbb{P}^ 3\) is a smooth cubic surface, a class of linearly equivalent curves of \(S\) is given by a 7-tuple of integers: the purpose of the paper is to give as much information as possible on the curves lying on \(S\) just using these 7-tuples. So, the dimensions of the cohomology groups \(H^ i(D)\) \((i=0,1,2)\) are computed in \(\S2\) for any divisor \(D\subset S\) and the Hilbert function of any curve (:=effective divisor) \(C\subset S\) is determined in \(\S3\). In sections 4 and 5 the graded Betti numbers of a minimal free resolution of the homogeneous ideal \(I(C)\subset k[x_ 0,x_ 1,x_ 2,x_ 3]\) of any curve \(C\subset S\) are determined. Sections 6 and 7 are devoted to the Rao module \(M(C)\) of any curve \(C\subset S\) linearly equivalent to a smooth one. In \(\S6\) the degrees of the elements of a minimal system of generators of \(M(C)\) are determined; in \(\S7\) a first insight into its structure is given: we see that \(M(C)\) has a nested structure.
Finally, appendix 2 contains some remarks on the smooth cubic surfaces containing Eckardt points. curves on cubic surface; Hilbert function; Rao module; Eckardt points Salvatore Giuffrida, Graded Betti numbers and Rao modules of curves lying on a smooth cubic surface in \?³, The Curves Seminar at Queen's, Vol. VIII (Kingston, ON, 1990/1991) Queen's Papers in Pure and Appl. Math., vol. 88, Queen's Univ., Kingston, ON, 1991, pp. Exp. A, 61. Plane and space curves, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Special surfaces, Projective techniques in algebraic geometry Graded Betti numbers and Rao modules of curves lying on a smooth cubic surface in \(\mathbb{P}{}^ 3\). Appendix 1: Some commutative diagrams. --- Appendix 2: Eckardt points | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(C\subset \mathbb {P}^{g-1}\) the canonical model of a smooth genus \(g\) curves. If \(g\ge 11\) the authors shows that the second syzygy scheme of \(C\) is \(\ne C\) if and only if \(C\) is a bielliptic curve. For lower \(g\) they classify the second syzygy scheme of the \(4\)-gonal curves. For the proof they first exclude the curves with Clifford index \(\ge 3\) and then analize the \(4\)-gonal curves. curves; syzygy; bielliptic curve; gonality; second syzygy scheme; tetragonal curve; Clifford index Special divisors on curves (gonality, Brill-Noether theory), Syzygies, resolutions, complexes and commutative rings A characterization of bielliptic curves via syzygy schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper, we present an explicit formula that connects the Kontsevich-Witten tau-function and the Hodge tau-function by differential operators belonging to the \(\widehat{GL(\infty)}\) group. Indeed, we show that the two tau-functions can be connected using Virasoro operators. This proves a conjecture posted by \textit{A. Alexandrov} [Lett. Math. Phys. 104, No. 1, 75--87 (2014; Zbl 1342.37066)]. Kontsevich-Witten tau-function; Hodge tau-function; Virasoro operators Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations, Virasoro and related algebras Connecting the Kontsevich-Witten and Hodge tau-functions by the \(\widehat{GL(\infty)}\) operators | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The topological zeta function and Igusa's local zeta function are respectively a geometrical invariant associated to a complex polynomial \(f\) and an arithmetical invariant associated to a polynomial \(f\) over a \(p\)-adic field. When \(f\) is a polynomial in two variables we prove a formula for both zeta functions in terms of the so-called log canonical model of \(f^{-1} \{0\}\) in \(\mathbb{A}^2\). This result yields moreover a conceptual explanation for a known cancellation property of candidate poles for these zeta functions. Also in the formula for Igusa's local zeta function appears a remarkable non-symmetric `\(q\)-deformation', of the intersection matrix of the minimal resolution of a Hirzebruch-Jung singularity. curve singularities; log canonical model; topological zeta function; Igusa's local zeta function Willem Veys.- Zeta functions for curves and log canonical models. Proceedings of the London Mathematical Society, 74 :360-378 (1997). Zbl0872.32022 MR1425327 Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants, Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.), Minimal model program (Mori theory, extremal rays), Local ground fields in algebraic geometry Zeta functions for curves and log canonical models | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We develop an idea of ``formal function along a subspace'', a notion that is related to Grothendieck's ``formal completion along a subscheme'', but expressed in concrete analytic terms using ``smooth coordinate charts''. Formal functions are families of formal power series parametrized by functions from a given class, satisfying natural commutativity relations between formal differentiation and differentiation with respect to the parameters. One of our main results is that, in generic coordinates for a chart \(U\), the standard basis of a local ideal \({\mathcal I}_{X,a}\) of a closed subspace \(X\) of \(U\) (the standard basis is a special set of generators of \({\mathcal I}_{X,a})\) extends as formal functions to generators along the Samuel stratum \(S\) of \(a\). \((S\) is the subset of \(X\) of points of the same singularity-type as \(a\), as measured by the Hilbert-Samuel function.) The simplest example of this phenomenon is ``implicit differentiation'' in elementary analysis. Our result can be used to reduce resolution of singularities over an infinite field, in general, to a ``hypersurface case''. formal function along a subspace; implicit differentiation; formal power series; standard basis of a local ideal; resolution of singularities Edward Bierstone and Pierre D. Milman, Standard basis along a Samuel stratum, and implicit differentiation, The Arnoldfest (Toronto, ON, 1997) Fields Inst. Commun., vol. 24, Amer. Math. Soc., Providence, RI, 1999, pp. 81 -- 113. Formal neighborhoods in algebraic geometry, Local structure of morphisms in algebraic geometry: étale, flat, etc., Global theory and resolution of singularities (algebro-geometric aspects), Modifications; resolution of singularities (complex-analytic aspects), Formal methods and deformations in algebraic geometry, Formal power series rings, Relevant commutative algebra Standard basis along a Samuel stratum, and implicit differentiation | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 tame Galois branched cover of curves \(\pi:X\to Y\) with finite Galois group \(G\) whose representations are rational, we compute the dimension of the (generalized) Prym variety \(\text{Prym}_\rho(X)\) corresponding to each irreducible representation \(\rho\) of \(G\). This formula can be applied to the study of algebraic integrable systems using Lax pairs, in particular systems associated with Seiberg-Witten theory. However, the formula is much more general and its computation and proof are entirely algebraic. Prym variety; algebraic integrable systems; Lax pairs; Seiberg-Witten theory Ksir, A.: Dimensions of Prym Varieties. Int. J. Math. Math. Sci. 26 (2001), no. 2, 107-116. Jacobians, Prym varieties, Relationships between algebraic curves and integrable systems, Yang-Mills and other gauge theories in quantum field theory, Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions Dimensions of Prym 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 quadratic stable range property is discussed. The ring \(A\) has quadratic stable range \(1\) (\(\text{qsr}(A) = 1\)) if every primitive quadratic form over \(A\) represents a unit. The property is motivated by the ring of holomorphic functions on a connected noncompact Riemann surface. The authors prove the following results:
(1) if \(\text{qsr} (A)=1\) then the stable range of \(A\) equals \(1\) and \(\text{Pic} (A)=1\).
(2) \(\text{qsr} (A)=1\) iff \(\text{Pic}(T)=1\) for every quadratic \(A\)-algebra \(T\).
They also classify quadratic forms over Bezout domains of characteristic not 2 satisfying a very strong approximation property (defined in the paper). This classification applies to the ring of holomorphic functions mentioned above. quadratic stable range; Picard group; quadratic form; holomorphic function; Riemann surface; Bezout domain D. R. Estes and R. M. Guralnick, ''A stable range for quadratic forms over commutative rings,'' J. Pure Appl. Algebra, 120, No. 3, 255--280 (1997). Quadratic forms over local rings and fields, Quadratic forms over global rings and fields, Quadratic and bilinear forms, inner products, Picard groups, Rings and algebras of continuous, differentiable or analytic functions A stable range for 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 The paper describes the Lamé differential operators \(L_n\) with a full set of algebraic solutions, where
\[
L_n=(\frac{d} {dx}^2+\frac 12 \sum^3_{i=1}\frac{1}{x-e_i}\frac{d}{dx}-\frac {n(n+1)x+B}{4\prod^3_{i=1}(x-e_i)}
\]
For each finite group \(G\), it describes the possible values of the degree parameter \(n\) such that the Lamé operator has the projective monodromy group \(G\). Previous investigations were concerned with the case \(n=1\) and determined the number of essentially different Lamé operators with finite monodromy. This paper continuous this investigation and describes completely these operators for arbitrary \(n\in Q\). The main result of this paper is contained in the following theorem:
(1) There is no Lamé operator with cyclic projective monodromy group.
(2) There is no Lamé operator with tetrahedral projective monodromy group.
(3) If the projective monodromy group of the Lamé operator \(L_n\) is octahedral, then \(n\in\frac 12(Z+ \frac 12)\cup\frac 13(Z+\frac 12)\).
(4) If the projective monodromy group of the Lamé operator \(L_n\) is icosahedral, then \(n\in\frac 13(Z+ \frac 12)\cup\frac 15(Z+\frac 12)\).
(5) If the projective monodromy group of the Lamé operator \(L_n\) is dihedral, then \(n\in Z\). If \(n\in Z\) and the projective monodromy group is finite, then this group is dihedral of order at least 6.
The details surrounding this result are clearly presented included with several tables presenting the results. The paper is very well written. Belyi function; dessin d'enfant; Lamé differential operators Litcanu, R, Lamé operators with finite monodromy - a combinatorial approach, J. Diff. Eqs., 207, 93-116, (2004) Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms, Coverings of curves, fundamental group Lamé operators with finite monodromy -- a combinatorial approach | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Reductive groups over finite fields are classified by root systems in a lattice with an action of the Frobenius; a way to get these data from the connected reductive algebraic group \(G\) is to take the projective variety \({\mathcal B}\) of Borel subgroups of \(G\), the set of isomorphism classes of G- equivariant line bundles over \({\mathcal B}\) forms a lattice \(X\); the subgroups containing a given \(B\in {\mathcal B}\) are not conjugate under \(G\), and the orbits of the minimal ones by conjugation under \(G\) form a finite set, each one giving \({\mathcal B}\) as the total space of a \({\mathbb{P}}_ 1\)-fibration over it, hence by taking the tangent bundle along the projections we get elements of \(X\): this is the basis of the root system. Now, each orbit of \(G\) in \({\mathcal B}\times {\mathcal B}\) defines an automorhism of X using the two projections on \({\mathcal B}\), and these orbits form the Weyl group \(W\) of \(G\). The Galois group of the algebraic closure of \(F_ q\) is naturally \({\bar {\mathbb{Z}}}\), generated topologically by the Frobenius F, and the set \(H^ 1({\bar {\mathbb{Z}}},W)\) classifies the maximal tori defined over \(F_ q\) in \(G\).
For each \(w\in W\), let \({\mathcal B}_ w\) be the set of Borel subgroups \(B\in {\mathcal B}\) for which \({}^ FB\) is in position w with respect to B; with a maximal tori \(T\subset B\) defined over \(F\), \textit{P. Deligne} and \textit{G. Lusztig} constructed a variety \({\mathcal B}^ T_ w\) projecting over \({\mathcal B}_ w\) with fibers \(T(F_ q)\) and compatible action of \(G(F_ q)\) so the alternate sum of the \(\ell\)-adic cohomology groups give a virtual representation of \(G(F_ q)\) commuting with the action of \(T(F_ q)\): this leads to the representations \(R^ T_{\theta}\) for the characters \(\theta\) of \(T(F_ q)\) in \({\bar {\mathbb{Q}}}^ x_{\ell}\) [Ann. Math., II. Ser. 103, 103-171 (1976; Zbl 0336.20029)]. Up to equivalence, all the irreducible representations of \(G(F_ q)\) occur in these \(R^ T_{\theta}\), when T and \(\theta\) vary. What was not given in this fundamental article, is an explicit formula for the multiplicities of the irreducible components in the \(R^ T_{\theta}\)'s. The book answers this question, completely in the case \(G\) has a connected center (since, Lusztig obtained the general case).
One of the main tools is the étale intersection cohomology of \textit{P. Deligne, A. A. Beilinson} and \textit{J. Bernstein} [Astérisque 100 (1982; Zbl 0536.14011)], applied to the closures of the varieties \({\mathcal B}_ w\), the Schubert cells. Another one is a deep understanding of the Weyl groups and their Hecke algebras; some properties on them are obtained through the theory of primitive ideals of enveloping algebras of complex reductive Lie algebras; the book uses systematically the results obtained by its author and by D. Kazhdan and its author in the theory of Weyl and Coxeter groups. He shows how the classification of irreducible representations reduces to the classification of unipotent representations of the ``endoscopic'' groups, where the solution comes from the Hecke algebra of the corresponding Weyl group. Also, the author, using the Springer correspondence [\textit{T. A. Springer}, Invent. Math. 36, 173-207 (1976; Zbl 0374.20054)], gives a parametrisation of the irreducible representations in terms of the special conjugacy classes of the dual group of \(G\). root systems; connected reductive algebraic group; projective variety; Borel subgroups; line bundles; orbits; tangent bundle; Weyl group; maximal tori; \(\ell\)-adic cohomology groups; virtual representation; characters; irreducible representations; multiplicities; irreducible components; intersection cohomology; Schubert cells; Weyl groups; Hecke algebras; enveloping algebras; complex reductive Lie algebras; unipotent representations G. Lusztig. Characters of reductive groups over a finite field, Ann. Math. Studies 107, Princeton University Press, 1984. ''BN13N22'' -- 2018/1/30 -- 14:57 -- page 225 -- #27 2018] QUANTIZATIONS OF REGULAR FUNCTIONS ON NILPOTENT ORBITS 225 Representation theory for linear algebraic groups, Linear algebraic groups over finite fields, Research exposition (monographs, survey articles) pertaining to group theory, Cohomology theory for linear algebraic groups, Universal enveloping (super)algebras, Étale and other Grothendieck topologies and (co)homologies, Group actions on varieties or schemes (quotients) Characters of reductive groups over a finite field | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For the entire collection see Zbl 0621.00009.]
This note is supplementary to the authors' paper in Math. Ann. 279, No.3, 435-448 (1988; Zbl 0657.14003) and consists largely of applications of a paper by \textit{E. D. Davis} [Ann. Univ. Ferrara, Nuova Ser., Sez. VII 32, 93-107 (1986; Zbl 0639.14033)].
Let S be the surface obtained by blowing-up a finite set of points Z of \({\mathbb{P}}^ 2\). The theorem of the title asserts that under certain conditions on generality and size of Z a certain divisor on S is very ample. It is shown that a generalization and sharpening of that theorem is a rather simple consequence of a characterization of ``punctured complete intersections'' in \({\mathbb{P}}^ 2\) related to the classical Cayley-Bacharach theorem. That term means ``scheme of \(colength\quad 1\) in a complete intersection'' - so to speak, ``a complete intersection minus one point''. linear systems of curves; generality on pointsets in projective 2-space; very ample divisor; blowing-up a finite set of points; Cayley-Bacharach theorem Divisors, linear systems, invertible sheaves, Projective techniques in algebraic geometry, Families, moduli of curves (algebraic) Bese's very ampleness theorem and punctured complete intersections | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The numerical range of an \(n\times n\) matrix is determined by an \(n\) degree hyperbolic ternary form. Helton-Vinnikov confirmed conversely that an \(n\) degree hyperbolic ternary form admits a symmetric determinantal representation. We determine the types of Riemann theta functions appearing in the Helton-Vinnikov formula for the real symmetric determinantal representation of hyperbolic forms for the genus \(g=1\). We reformulate the Fiedler-Helton-Vinnikov formulae for the genus \(g=0,1\), and present an elementary computation of the reformulation. Several examples are provided for computing the real symmetric matrices using the reformulation. determinantal representation; hyperbolic form; Riemann theta function; numerical range Computational aspects of algebraic curves, Norms of matrices, numerical range, applications of functional analysis to matrix theory, Theta functions and curves; Schottky problem Computing the determinantal representations of hyperbolic 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 We study the topos of sets equipped with an action of the monoid of regular \(2 \times 2\) matrices over the integers. In particular, we show that the topos-theoretic points are given by the double quotient \(\mathrm{GL}_2(\hat{\mathbb{Z}}) \backslash \mathrm{M}_2(\mathbb{A}_f) / \mathrm{GL}_2(\mathbb{Q})\), so they classify the groups \(\mathbb{Z}^2 \subseteq A \subseteq \mathbb{Q}^2\) up to isomorphism. We determine the topos automorphisms and then point out the relation with Conway's big picture and the work of \textit{A. Connes} and \textit{C. Consani} [C. R., Math., Acad. Sci. Paris 352, No. 12, 971--975 (2014; Zbl 1315.11054)] on the arithmetic site. As an application to number theory, we show that classifying extensions of \(\mathbb{Q}\) by \(\mathbb{Z}\) up to isomorphism relates to Goormaghtigh conjecture. arithmetic site; monoid; topos; topos automorphism; Adele ring; topos-theoretic point; torsion-free abelian group; zeta function; Goormaghtigh conjecture Relations with algebraic geometry and topology, Arithmetic algebraic geometry (Diophantine geometry), Relations with noncommutative geometry, Generalizations (algebraic spaces, stacks), Topoi An arithmetic topos for integer matrices | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We study the Hitchin system on singular curves. We consider curves obtainable from the projective line by matching at several points or by inserting cusp singularities. It appears that on such singular curves, all basic ingredients of Hitchin integrable systems (moduli space of vector bundles, dualizing sheaf, Higgs field, etc.) can be explicitly described, which can be interesting in itself. Our main result is explicit formulas for the Hitchin Hamiltonians. We also show how to obtain the Hitchin integrable system on such curves by Hamiltonian reduction from a much simpler system on a finite-dimensional space. We pay special attention to a degenerate curve of genus two for which we find an analogue of the Narasimhan-Ramanan parameterization of the moduli space of \(SL(2)\) bundles as well as the explicit expressions for the symplectic structure and Hitchin-system Hamiltonians in these coordinates. We demonstrate the efficiency of our approach by rederiving the rational and trigonometric Calogero-Moser systems, which are obtained from Hitchin systems on curves with a marked point and with the respective cusp and node. integrable systems; Hitchin systems; singular curves; Calogero-Moser system; Narasimhan-Ramanan parameterization Talalaev, D. V. and Chervov, A. V., ``Hitchin system on singular curves,'' e-print (2004). Relationships between algebraic curves and integrable systems, Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Vector bundles on curves and their moduli, Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics Hitchin system on singular 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 present several conjectures on multiple \(q\)-zeta values and on the role which they play in certain problems of enumerative geometry. multiple \(q\)-zeta value; \(q\)-deformation; Hilbert scheme; CW/DT correspondence Okounkov, A, Hilbert schemes and multiple \(q\)-zeta values, Funct. Anal. Appl., 48, 138-144, (2014) Parametrization (Chow and Hilbert schemes), Binomial coefficients; factorials; \(q\)-identities, Enumerative problems (combinatorial problems) in algebraic geometry, Multiple Dirichlet series and zeta functions and multizeta values Hilbert schemes and multiple \(q\)-zeta values | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 the theory of algebraic groups to the super context from a functorial point of view.This means that the authors study the quotient sheaf \(\widetilde{G/H}\) of an algebraic supergroup \(G\) by a closed supersubgroup \(H\). Most of the text is used to explain the scheme theory and the geometric invariant theory in the ``super'' setting.
The ground field \(K\) is assumed to have characteristic different from \(2\), and the vector spaces graded by \(\mathbb Z_2\) form a symmetric tensor category \(\text{SMod}_K\). Objects from this category get the prefix ``super'', e.g. superalgebra, Hopf superalgebra and so on. All superalgebras including the Hopf superalgebras are assumed to be supercommutative, in the meaning that \(ab=(-1)^{|a||b|}ba\), were \(a\) and \(b\) are homogeneous elements of degree \(|a|\) and \(|b|\) respectively. A \(K\)-functor (resp. a supergroup) is a set-valued (resp. group-valued) functor defined on the category \(\text{SAlg}_K\) of superalgebras. The sheafification of \(K\)-functors includes the subclasses of affine superschemes, contained in superschemes, contained in \(K\)-sheaves.
By an algebraic supergroup one understands an algebraic affine supergroup, which is a supergroup \(G\) represented by a finitely generated Hopf superalgebra \(K[G]\). A closed supersubgroup of \(G\) is a supergroup \(H\) represented by a quotient Hopf superalgebra of \(K[G]\). For such \(G\), \(H\), the functor that associated the set \(G(R)/H(R)\) of right cosets to a superalgebra \(R\) is called the naive quotient, and is denoted \((G/H)_{(n)}\). The sheafification of this functor is denoted by \(\widetilde{G/H}\). It is proved that \(\widetilde{G/H}\) is an affine supergroup if \(H\) is normal in \(G\), and this article considers the situation when \(H\) is not necessarily normal in \(G\). If the functor is restricted to the category \(\text{Alg}_K\) of purely even algebras (the ordinary commutative algebras) we get the ordinary algebraic group denoted \(G_{\text{res}}\) with its closed subgroup \(H_{\text{res}}\). The article considers the relation between \(\widetilde{G/H}\) and the quotient \(\widetilde{G_{\text{res}}/H_{\text{res}}}\) in the classical situation.
The main questions the article is about to answer are the following: 1) is \(\widetilde{G/H}\) necessarily a superscheme?, and 2) is \(\widetilde{G/H}\) affine when the algebraic group \(H_{\text{res}}\) is geometrically reductive? These questions are asked by Brundan, who give another definition of the quotient \(G/H\), and the authors finally ask under which conditions their quotient coincides with the Brundan quotient.
The main result of the article is the affirmative answer of question 1): Let \(G\) be an algebraic supergroup, and let \(H\) be a closed supergroup. Then the \(K\)-sheaf \(\widetilde{G/H}\) is a Noetherian superscheme. This result also holds in the situation with the sheafification \(\widetilde{H\backslash G}\) of the left coset functor.
The Brundan quotient of \(G\) by \(H\) is a pair \((X,\pi)\) of a Noetherian superscheme \(X\) and a morphism \(\pi:G\rightarrow X\) with given properties. It is proved that the quotient in the main theorem above fulfills this, so that also question 3) is answered affirmatively. The second question is proven to be true in the introductory parts of the article.
It is already proved by Demazure and Gabriel that \(\widetilde{G_{\text{res}}/H_{\text{res}}}\) is a Noetherian scheme, and the authors proof reduces to the this result by the relations to \(\widetilde{G/H}\). The method uses both Hopf algebraic and geometric methods, the geometric methods is represented by a comparison theorem generalizing the analogous classical comparison theorem of Demazure and Gabriel to the super context. In addition to the functorial approach, it is used a supergeometry through geometric superspaces which are topological spaces with structure sheaves of superalgebras. Then the comparison theorem shows that the two approaches are equivalent in some sense.
The article uses most of its space to give meaning to the above. Direct limits are defined to construct geometric superspaces from \(K\)-functors, where basic conditions on super(co)algebras and super(co)modules are considered. Then geometric superspaces can be defined by direct limits applied on the \(K\)-functors. This is straight forward, but there is a need to prove all results in the noncommutative situation. The above mentioned comparison theorem is proved, and to be able to define the quotient superschemes, that is its open affines, the supergrassmannian is defined.
The quotient dur sheaf \(\widetilde{\widetilde{X/G}}\) associated to an affine superscheme \(X\) with the action of an affine supergroup \(G\) is studied, and it is applied to the case \(U/H\) where \(U\) is an affine open supersubscheme of \(G\) which is stable under right multiplication with \(H\). An important result states necessary and sufficient conditions for the action of \(G\) on \(X\) to be free so that \(X/G\) is affine. The proof of this result uses the bosonization technique of braided Yetter-Drinfeld modules to a more general situation than actually needed.
The article defines the ``super'' setting. This is, roughly speaking, the category of \(\mathbb Z_2\)-graded objects. Then the most needed algebraic geometric theory is generalized to this noncommutative setting. This includes Direct limits, supermodules, and supercomodules, all treated in the functorial setting by the so called \(K\)-functors (groups) to the category of sets (groups). Most of the concepts are relatively easily translated to the super setting. Of particular interest is the comparison theorem, geometric superspaces, the supergrassmannian and the superinvariant theory. The bozonization technique involves braided groups and Hopf algebraic methods.
All in all the article is a nice generalization to the super setting. The printing is a bit disturbing, for instance, the ``less-that-or-equal''-sign is substituted by ``6'', and the square at the end of proofs are replaced by ``2''. The authors are probably not to blame for this, and the article is a very nice base for the algebraic geometric study of supergeometry. Also, it gives proper hints in the noncommutative setting. algebraic supergroup; super Hopf algebra; super; superscheme; super quotient scheme; K-functor; geometric superspace; Yetter-Drinfeld moduled; supermodule; supercomodule; bozonization A. N. Grishkov and A. N. Zubkov, ''Solvable, reductive and quasireductive supergroups,'' submitted to \textit{J. Alg.}; see arXiv: math.RT/1302.5648. Supervarieties, Noncommutative algebraic geometry, Group schemes Quotient sheaves of algebraic supergroups are superschemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 very well-written article, the authors establish several results relating finite schemes and secant varieties over arbitrary fields. Some of these results were previously known only over the complex numbers. The paper is in part expository and contains background material on scheme theory, apolarity theory, Castelnuovo-Mumford regularity, Hilbert schemes, and secant varieties. Let \(\mathbb K\) be a field and \(R\) be a finite scheme over \(\mathbb K\). One of the main objectives is to study the \textit{smoothability} of \(R\) both as an abstract scheme and as an embedded scheme in some algebraic variety \(X\). The condition of smoothability can be easily seen over an algebraically closed field: a finite scheme \(R\) is smoothable if and only if it is a flat limit of distinct points. Theorem 1.1 gives the equivalence between the abstract smoothability and the embedded smoothability in some algebraic variety \(X\), whenever \(X\) is smooth. Moreover, smoothability over \(\mathbb K\) is equivalent to smoothability in the algebraic closure of \(\mathbb K\) (Proposition 1.2). Let \(\mathbb K\) be an algebraically closed field. Let \(X\) be an algebraic variety \(\mathbb K\) and let \(r\) be an integer. Condition \((\star)\) holds if every finite \textit{Gorenstein} subscheme over \(\mathbb K\) of \(X\) of degree at most \(r\) is smoothable in \(X\). One of the main results is Theorem 1.7. This relates the scheme theoretic condition above with the possibility of giving \textit{set-theoretic equations} for secants of sufficiently high Veronese embeddings of \(X\), by determinantal equations from vector bundles on \(X\). If condition \((\star)\) does not hold, then those equations are not enough to cut them. Interestingly, the locus of determinantal equations from vector bundles contain more general loci than secants: the \textit{cactus varieties}. This containment is the ultimate reason for the failure of present methods to give good enough lower bounds on tensor ranks. smoothable; secant varieties; finite Gorenstein scheme; cactus variety; Veronese reembedding; Hilbert scheme Buczyński, J.; Jelisiejew, J., Finite schemes and secant varieties over arbitrary characteristic, Differential Geom. Appl., 55, 13-67, (2017) Determinantal varieties, Local deformation theory, Artin approximation, etc., Parametrization (Chow and Hilbert schemes), Schemes and morphisms, Homogeneous spaces and generalizations Finite schemes and secant varieties over arbitrary characteristic | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We study periodic points and orbit length distribution for endomorphisms of abelian varieties in characteristic \(p>0\). We study rationality, algebraicity and the natural boundary property for the dynamical zeta function (the latter using a general result on power series proven by Royals and Ward in the appendix), as well as analogues of the prime number theorem, also for tame dynamics, ignoring orbits whose order is divisible by \(p\). The behavior is governed by whether or not the action on the local \(p\)-torsion group scheme is nilpotent. abelian variety; inseparability; fixed points; Artin-Mazur zeta function; recurrence sequence; natural boundary Arithmetic dynamics on general algebraic varieties, Asymptotic results on counting functions for algebraic and topological structures, Positive characteristic ground fields in algebraic geometry, Isogeny, Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics, Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc. Dynamics on abelian varieties in positive characteristic | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In dieser Abhandlung untersucht Cayley diejenigen abwickelbaren Flächen, deren Rückkehrkante auf einem Kreiscylinder gelegen ist und durch die Abwickelung in einen Kreisbogen übergeht. Die Coordinaten eines Punktes der Rückkehrkante hängen von elliptischen Functionen eines Parameters ab. surface; torse; elliptic function Euclidean analytic geometry, Foundations of algebraic geometry On a torse depending on elliptic 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 Eine transcendente Curve kann nur einem einzigen der in vorstehendem Referate (JFM 06.0393.02) angegebenen Systeme angehören. systems of curves; algebraic differential equations Curves in Euclidean and related spaces, Enumerative problems (combinatorial problems) in algebraic geometry On transcendental planar curves, that may belong to a \((\mu, \nu )\) system. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\mathbb X\subset \mathbb P^2_k\) be a general set of \(s\) distinct points, with \(k\) an algebraically closed field of characteristic \(0\). Let \(R=k[x_0,x_1,x_2]\) be the graded ring associated to \(\mathbb P^2\) and \(I_X=p_1\cap \dots \cap p_s\subset R\) be the defining ideal of \(s\). Let \(2\mathbb X\) be the scheme of \(\mathbb P^2\) with defining ideal \({p_1^2}\cap \dots\cap{p_s^2}\).
\textit{A. V. Geramita}, \textit{J. Migliore} and \textit{L. Sabourin} [J. Algebra 298, No. 2, 563--611 (2006; Zbl 1107.14048)] asked whether it is possible to find the Hilbert functions of all the non-reduced subschemes defined by ideals of the type \({p_1^2}\cap \dots\cap{p_s^2}\), whose underlying reduced subscheme was any set of \(s\) distinct points.
This difficult problem has been taken up in several papers see [\textit{A. V. Geramita}, \textit{B. Harbourne} and \textit{J. Migliore}, Collect. Math. 60, No. 2, 159--192 (2009); erratum ibid. 62, No. 1, 119--120 (2011; Zbl 1186.14008)] and [\textit{A. V. Geramita}, \textit{B. Harbourne} and \textit{J. Migliore}, ``Hilbert functions of fat point subschemes of the plane: the two-fold way'', \url{arxiv:1101.5140}] obtaining satisfying results in the case \(s\leq 9\) (in this case, the exponents of the ideals \(p_i\) can be any positive integer).
In the general case, the Hilbert function \(H_{2\mathbb X}\) has the upper bound \(H_{2\mathbb X}(t)\leq \min\{\binom{t+2}{2},3s\}\) and this bound is achieved for any \(s\neq 2,5\) and for \(X\) set of general \(s\) points having the generic Hilbert function \(H_{\mathbb X}(t)=\min\{\binom{t+2}{2},s\}.\)
Geramita, Migliore and Sabourin [Zbl 1107.14048] ask whether, in the case that the Hilbert function of \(X\) is the generic Hilbert function, there is a lower bound and wether such a bound is achieved for some \(2\mathbb X\). In their paper they give an affirmative answer in the case \(s=\binom{d}{2}\). This implies that the first open case is \(s=11\).
In this paper the authors give an affirmative answer in the case \(s=11\) and give a supportive evidence that the problem might have a positive answer for any \(s\), by proving that for any \(s\) the assertion holds for the ``first half'' of the Hilbert function. Hilbert function; fat points; infinitesimal neighborhood Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Syzygies, resolutions, complexes and commutative rings, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Configurations and arrangements of linear subspaces Hilbert functions of double point schemes in \(\mathbb P^{2}\) | 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 \(X\) be an algebraic variety defined over a field \(K\) and denote by \(X(K)/R\) the set of \(K\)-rational points of \(X\) modulo rational equivalence as originally defined by \textit{Yu. I. Manin} [Cubic forms. Algebra, geometry, arithmetic. Transl. from the Russian by M. Hazewinkel. 2nd ed. North-Holland Mathematical Library, Vol. 4. Amsterdam-New York-Oxford: North-Holland (1986; Zbl 0582.14010)]. In this paper, the authors consider absolutely simple adjoint linear algebraic groups \(G\). By Weil's classification, these groups are of type \(A_n\), \(B_n\), \(C_n\) or \(D_n\) (non-trialitarian in case \(D_4\)). Now \(G(K)/R\) is naturally equipped with a group structure and one would like to know if this group is trivial. \textit{A. S. Merkurjev} [Publ. Math., Inst. Hautes Étud. Sci. 84, 189-213 (1996; Zbl 0884.20029)] showed that this is so if \(G\) is of inner type \(A_n\) or of type \(B_n\), but that triviality generally fails for type \(D_n\) (\(n\geq 3\)).
In the present paper, the authors consider the situation where \(K\) is a function field of a smooth geometrically integral curve over a nondyadic local field. They show that triviality holds over such fields if the absolutely simple adjoint group is of type \(^2A_n^*\) (i.e. where the underlying central simple algebra has square-free index), \(C_n\) and \(D_n\) (non-trialitarian in case \(D_4\)). The main ingredients in the proofs are Merkurjev's description of \(G(K)/R\) as the quotient group of similitudes of a certain Hermitian form [loc. cit.], the fact that the \(u\)-invariant of such a field \(K\) is at most \(8\) due to \textit{R. Parimala} and \textit{V. Suresh} [Ann. Math. (2) 172, No. 2, 1391-1405 (2010; Zbl 1208.11053)], the first author's own classification results for Hermitian forms over algebras with involution [J. Algebra 385, 294-313 (2013; Zbl 1292.11056)], and a theorem proved also in the present paper and of interest in its own right, namely, that the Rost invariant of \(SL_1(A)\) for a central simple algebra \(A\) of index at most \(4\) over such a field \(K\) has trivial kernel. algebraic groups; adjoint groups; R-equivalence; nondyadic local fields; function fields of curves; algebras with involution; Hermitian forms; Rost invariant R. Preeti and A. Soman, Adjoint groups over \Bbb Q_{\?}(\?) and R-equivalence, J. Pure Appl. Algebra 219 (2015), no. 9, 4254 -- 4264. Linear algebraic groups over local fields and their integers, Quadratic forms over general fields, Bilinear and Hermitian forms, Classical groups, Galois cohomology of linear algebraic groups, Rational points, Other nonalgebraically closed ground fields in algebraic geometry, Finite-dimensional division rings, Rings with involution; Lie, Jordan and other nonassociative structures Adjoint groups over \(\mathbb Q_p(X)\) and R-equivalence. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(l\) be a prime number, and let \(\mathbb F\) be a finite field of characteristic \(l\). Let \(X\) be a smooth, geometrically irreducible curve defined over a finite field \(k\) of characteristic \(p\neq l\), and let \(\bar{\rho}:\pi_1(X)\rightarrow \text{GL}_n({\mathbb F})\) be a continuous representation. Under some technical assumptions the authors prove that \(\bar{\rho}\) lifts to a representation \(\rho:\pi_1(X\setminus T)\rightarrow \text{GL}_n(W({\mathbb F}))\), where \(T\) is a finite set of places of \(X\) and \(W({\mathbb F})\) are the Witt vectors over \(\mathbb F\). As an application of results proved by \textit{L. Lafforgue} [Invent. Math. 147, No.~1, 1--241 (2002; Zbl 1038.11075)] they find that any such \(\bar{\rho}\) is automorphic, which should be seen as an analogue of \textit{J.-P. Serre's} conjecture, that any two-dimensional, continuous, irreducible, odd representation of the absolute Galois group \(G_{\mathbb Q}\) over \(\mathbb F\) arises from a new form [Duke Math. J. 54, 179--230 (1987; Zbl 0641.10026)]. function field of positive characteristic; arithmetic fundamental group; Galois representation; automorphic representation G. Böckle and C. Khare, Finiteness results for mod \(l\) Galois representations over function fields, Galois representations, Representation-theoretic methods; automorphic representations over local and global fields, Coverings of curves, fundamental group, Galois cohomology Mod \(\ell\) representations of arithmetic fundamental groups. I: An analog of Serre's conjecture for 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 \(X\) be a smooth complex projective curve and \(G\) be a complex reductive group with Langlands dual group \({}^LG\). In its most naive form, the geometric Langlands correspondence seeks to parametrize Hecke eigensheaves on the moduli stack \(\mathrm{Bun}_{G}\) by flat \({}^LG\)-connections on \(X\).
In this paper, following a question of R. Langlands, the authors formulate a conjectural function-theoretic version of the Langlands correspondence for complex curves, more akin to the classical formulation of the Langlands correspondence as a spectral problem for Hecke operators.
Assume for simplicity that \(G\) is simple and simply-connected. In that case the canonical bundle \(K\) on \(\mathrm{Bun}_{G}\) has a square root \(K^{1/2}\). Denote by \(\overline{K}^{1/2}\) the anti-holomorphic complex conjugate of \(K^{1/2}\). The authors propose to study sections of the \(C^{\infty}\)-line bundle \(\Omega^{1/2}:=K^{1/2}\otimes \overline{K}^{1/2}\) of half-densities instead of functions on \(\mathrm{Bun}_{G}\). The line bundle \(\Omega^{1/2}\) admits an action of the algebra \(\mathcal{A} = D_{G} \otimes_{\mathbb{C}} \overline{D}_{G}\) where \(D_{G}\) is the algebra of global regular differential operators acting on \(K^{1/2}\). This algebra comes with a natural anti-linear involution and we denote by \(\mathcal{A}_{\mathbb{R}}\) the \(\mathbb{R}\)-algebra of invariants of that involution.
Denote by \(\mathrm{Bun}^{\circ}_{G}\) the coarse moduli space classifying stable \(G\)-bundles whose automorphism group is \(Z(G)\). Consider the space
\[
\mathcal{H} = L^2(\mathrm{Bun}_{G})
\]
defined as the completion of the space of smooth compactly supported sections of \(\Omega^{1/2}\) on \(\mathrm{Bun}_{G}^{\circ}\).
The authors make the following conjectures. See Conjectures 1.9.--1.11. for more details.
\begin{enumerate}
\item There is an \(\mathcal{A}\)-invariant extension \(S(\mathcal{A}) \subset \mathcal{H}\) of \(V\) such that \((\mathcal{A}_{\mathbb{R}}, S(\mathcal{A}))\) is a strongly commuting family of unbounded essentially self-adjoint operators on \(\mathcal{H}\). This allows one to define the joint spectrum \(\mathrm{Spec}_{\mathcal{H}}(\mathcal{A})\) of \(\mathcal{A}\) on \(\mathcal{H}\), see \S 11.
\item The joint spectrum \(\mathrm{Spec}_{\mathcal{H}}(\mathcal{A})\) of \(\mathcal{A}\) on \(\mathcal{H}\) is discrete. By a result of Beilinson and Drinfeld it is therefore parametrized by a countable subset \(\Sigma\) of the space of \({}^L G\)-opers on \(X\) and the joint \(\mathcal{A}\)-eigen sections form a basis of \(L^2(\mathrm{Bun}_{G})\).
\item The set \(\Sigma\) is contained in the set of \({}^LG\)-opers on \(X(\mathbb{C})\) which are defined over \(\mathbb{R}\).
\end{enumerate}
Generalizing the set-up to bundles with parabolic structures, the authors prove their conjectures in the abelian case \(G=\mathrm{GL}_1\) and in the case \(G=\mathrm{SL}_2\) and \(X=\mathbb{P}^1\) with at least four marked points. In these cases they prove that the \({}^LG\)-opers \(\Sigma\) coming from the spectrum \(\mathrm{Spec}_{\mathcal{H}}(\mathcal{A})\) are not only contained in the set of \({}^LG\)-opers defined over \(\mathbb{R}\), but that they actually coincide. Langlands program; spectral problem; oper; differential operator Geometric Langlands program (algebro-geometric aspects), General topics in linear spectral theory for PDEs, Langlands-Weil conjectures, nonabelian class field theory, Geometric Langlands program: representation-theoretic aspects An analytic version of the Langlands correspondence for complex 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 use algebraic methods to study systems of linear partial differential equations with constant coefficients. equations and systems with constant coefficients; Hilbert schemes General theory of PDEs and systems of PDEs with constant coefficients, Overdetermined systems of PDEs with constant coefficients, Parametrization (Chow and Hilbert schemes) Linear differential operators with constant coefficients and Hilbert schemes of points | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The moduli space of principally polarised abelian surfaces is rational, and a uniformisation is given by three Igusa functions, \(j_1\), \(j_2\) and \(j_3\) on the Siegel upper half-plane. These are (in one sense) analogues of the \(j\)-function in one variable, and there are now quite efficient methods to compute \(j(z)\) to arbitrary precision fairly fast. One of the better methods is to use the Eisenstein series for \(g_2\) and \(g_3\) and the formula \(j(z)=1728{{g_2(z)^2}\over{g_2(z)^3-27g_3(Z)^2}}\), but it is still better to use the Dedekind \(\eta\) function and the formula \(j(z)=\left({{(\eta(z/2)/\eta(z))^{24}+16} \over{(\eta(z/2)/\eta(z))^8}}\right)^3\), because the \(q\)-expansion of \(\eta\) is very sparse.
The Igusa functions have well-known expressions in terms of theta functions but the authors want to find alternative expressions that will have similar advantages for computation. No analogue of the \(\eta\) function is available, but analogues of the Eisenstein series are, and it was computed by Igusa that
\[
j_1= 486{{\chi^5_{12}}\over{\chi^6_{10}}};\quad j_2={{27E_4\chi^3_{12}}\over{8\chi^4_{10}}};\quad j_3={{3E_6\chi^2_{12}}\over{32\chi^3_{10}}} +{{9E_4\chi^3_{12}}\over{8\chi^4_{10}}},
\]
where \(\chi_{10}\) and \(\chi_{12}\) are cusp forms defined in terms of the Siegel Eisenstein series \(E_w\) by
\[
\chi_{10}={{43867}\over{2^{12}\cdot 3^5\cdot 5^2\cdot 7\cdot 53}}(E_4E_6-E_{10})
\]
and
\[
\chi_{12}={{131\cdot 593}\over{2^{13}\cdot 3^7\cdot 5^3\cdot 7^2\cdot 337}}(3^2\cdot 7^2E_4^3+2\cdot 5^3E_6^2-691 E_{12}).
\]
So the idea is to use these formulas for computation of the Igusa functions. It is only partly successful, in that the algorithm thus obtained turns out to be asymptotically slower than using theta functions, but there are some compensating numerical advantages in precision. What is perhaps more interesting is that there are many intermediate results about the computation of the Siegel forms themselves.
The main such result says that the coefficients of \(E_w\) corresponding to matrices with coefficients bounded by \(A,\,B,\,C\in{\mathbb N}\) can be computed in time \(O((ABC)^{1+\varepsilon})\), with an explicitly given coefficient depending on~\(w\). The method uses the description of \(E_w\) as elements of the Maass Spezialschar, i.e., as lifts of Jacobi forms of weight~\(w\) and index~\(1\). Although this is also sufficient to compute \(\chi_{10}\) and \(\chi_{12}\) because of the formulae above, it is also possible to take the same direct approach for them, as they are also in the Spezialschar.
Sections~1 and~2 of the paper explain this background. Section~3 sets up the well-known relation between Eisenstein series and Jacobi forms, and gives a formula for the coefficients: most of this can be found in the book of Eichler and Zagier, but a minor correction is needed. The formula involves the Cohen numbers \(H(w-1,D)\) and they depend on special values of \(L\)~functions: Section~4 explains how to compute them and this is the part that dominates the time of computation overall. Section~5 does the same thing for \(\chi_{10}\) and \(\chi_{12}\), but there is more work to be done to bound the Fourier coefficients, and the values of the \(L\)~functions that arise, in an efficient way. Section~6 analyses the speed of convergence of the Eisenstein series precisely, and in Section~7 the authors illustrate their method with two examples. Igusa function; moduli space; Eisenstein series; abelian surfaces Bröker, R.; Lauter, K.: Evaluating igusa functions. Math. comp. 83, 2977-2999 (2014) Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms, Moduli, classification: analytic theory; relations with modular forms Evaluating Igusa functions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this talk I explain the proof given by \textit{A. Marian} and \textit{D. Oprea} [Invent. Math. 168, No. 2, 225--247 (2007; Zbl 1117.14035)], as well as \textit{P. Belkale} [J. Am. Math. Soc. 21, No. 1, 235--258 (2008; Zbl 1132.14028)], of the strange duality or rank-level duality, which establishes a duality between two spaces of generalized theta functions, i.e., spaces of global sections of line bundles over moduli spaces of semi-stable vector bundles over a complex smooth projective curve. vector bundle; moduli space; generalized theta function; Gromov-Witten invariant Christian Pauly, La dualité étrange [d'après P. Belkale, A. Marian et D. Oprea], Astérisque 326 (2009), Exp. No. 994, ix, 363 -- 377 (2010) (French, with French summary). Séminaire Bourbaki. Vol. 2007/2008. Vector bundles on curves and their moduli, Relationships between algebraic curves and physics, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) The strange duality (after P. Belkale, A. Marian and D. Opera) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 under review concerns quotients of affine varieties by torus actions. For an affine variety \(X\) endowed with an action of a torus \(T\) the authors study its toric Chow quotient, toric Hilbert scheme and the Chow morphism between them restricted to the main components. They introduce the main component \(H_0\) of the toric Hilbert scheme, which parametrizes general \(T\)-orbit closures in \(X\) and their flat limits. They also give a geometric description of the Altmann-Hausen family of \(T\)-varieties over the normalization \((X/_CT)_0^{\mathrm{norm}}\) of the main component of the toric Chow quotient of \(X\) by \(T\), introduced by \textit{K. Altmann} and \textit{J. Hausen} [Math. Ann. 334, No. 3, 557--607 (2006; Zbl 1193.14060)]. Define
\[
W_x = \overline{\{(x,q(x)) : x \in X^{\mathrm{ss}}\}} \subseteq X\times (X/_CT)_0,
\]
where \(q\) is the quotient map and \(X^{\mathrm{ss}}\) is the set of points which are semistable with respect to all characters of \(T\). Then the Altmann-Hausen family is shown to be the normalization of \(W_X\) together with the projection to \((X/_CT)_0^{\mathrm{norm}}\).
The main result of the article describes the relation between the universal family \(U_0 \rightarrow H_0\) and the family \(W_X \rightarrow (X/_CT)_0\). The authors prove that the restriction of the toric Chow morphism \(H_0 \rightarrow (X/_CT)_0\), coming from a generalization of the construction given by \textit{M. Haiman} and \textit{B. Sturmfels} [J. Algebr. Geom. 13, No. 4, 725--769 (2004; Zbl 1072.14007)], lifts to a birational projective morphism \(U_0 \rightarrow W_0\). The last section concerns the case when \(X\) is a toric variety and \(T\) is a subtorus of its big torus. The results from previous sections are rephrased in terms of fans. In particular, an explicit description of the fan of the Altmann-Hausen family is given. torus action; toric variety; toric Chow quotient; toric Hilbert scheme O.V. Chuvashova, N.A. Pechenkin, Quotients of an affine variety by an action of a torus. February 2012. ArXiv e-prints arXiv:1202.5760. Geometric invariant theory, Parametrization (Chow and Hilbert schemes), Toric varieties, Newton polyhedra, Okounkov bodies Quotients of an affine variety by an action of a torus | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Enriques surfaces can be characterised as those compact complex surfaces that are not simply connected with a \(K3\) surface as universal cover. In this article, the authors introduce Enriques manifolds as those compact complex manifolds that are not simply connected with a hyperkähler manifold as universal cover. A hyperkähler (or holomorphic symplectic) manifold is a simply connected compact Kähler manifold \(X\) such that \(H^{0}(X, \Omega_{X}^{2})\) is generated by a closed non-degenerate \(2\)-form. Their theory largely runs parallel to the theory of \(K3\) surfaces [\textit{D. Huybrechts}, Invent. Math. 135, No. 1, 63--113 (1999; Zbl 0953.53031)]. A two-dimensional Enriques manifold is an Enriques surface in the usual sense.
Among others, the following basic properties of Enriques manifolds are shown: if \(Y\) is an Enriques manifold with universal cover \(X\rightarrow Y\), then \(\dim(Y)=2n\), \(\pi_1(Y)\) is a finite cyclic group of order \(d\mid n+1\) and \(\varphi(d)<b_2(X)\) (\(\varphi\) is Euler's phi-function). The order of \(\pi_1(Y)\) is called the index of \(Y\).
The main part of this paper consists of the construction of examples of Enriques manifolds. They are obtained as quotients of moduli spaces of sheaves on certain \(K3\) surfaces and as quotients of Hilbert schemes of points on bielliptic surfaces. All the examples constructed here have index \(2\), \(3\), or \(4\). It remains open which indices can actually occur. This seems mainly due to the small number of known examples of hyperkähler manifolds.
The authors continue to study Enriques manifolds in [Pure Appl. Math. Q. 7, No. 4, 1631--1656 (2011; Zbl 1316.32011)].
Enriques manifolds were introduced and studied independently by \textit{S. Boissière} et al. [J. Math. Pures Appl. (9) 95, No. 5, 553--563 (2011; Zbl 1215.14046)]. Enriques surfaces; hyperkähler manifold; Hilbert scheme; bielliptic surface Oguiso, K.; Schröer, S., Enriques manifolds, J. Reine Angew. Math., 661, 215-235, (2011) Moduli, classification: analytic theory; relations with modular forms, Special Riemannian manifolds (Einstein, Sasakian, etc.), Hyper-Kähler and quaternionic Kähler geometry, ``special'' geometry, Calabi-Yau theory (complex-analytic aspects), Parametrization (Chow and Hilbert schemes), \(K3\) surfaces and Enriques surfaces Enriques 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 \(C\) be an algebraic space curve over the algebraically closed field \(k\) (i.e. a closed locally Cohen-Macaulay subscheme of dimension 1 in \(\mathbb{P}^3 (k))\) and let \({\mathcal I}_C\) be its ideal sheaf. The Rao function of \(C\) is defined by \(\rho_C(n) =h^1({\mathcal I} (n))\) and the Rao module is \(M_C= \bigoplus H^1 (\mathbb{P}^3, {\mathcal I}_C(n))\). In their previous book: ``Sur la classification des courbes gauches'', Astérisque 184-185 (1990; Zbl 0717.14017), \textit{M. Martin-Deschamps} and \textit{D. Perrin} defined the postulation character of \(C\) by \(\gamma_C (n)= \partial^3 (h^0(I_C(n)) -h^0({\mathcal O}_p (n))\) where \(\partial f\) is the difference function of \(f\). The following invariants are also used: \(s_0= \inf \{n\in\mathbb{Z} \mid h^0 ({\mathcal I}_C(n)) \neq 0\}\) and \(e=\sup \{n\in \mathbb{Z} \mid h^1({\mathcal O}_C(n)) \neq 0\}\).
In this note, there is defined an order relation on the set of couples \((\gamma, \rho)\); this order relation uses the numerical invariants and the functions \(\gamma, \rho\). If \(C\) and \(C_0\) are two minimal curves with the same Rao function \(\rho\) \((\rho \neq 0)\) and if the Rao-module of \(C_0\) is trivial then the invariants of \(C\) are less then the ones of \(C_0\). admissible cohomology; space curve; Rao function; Rao module [H3]\textsc{R. Hartshorne},\textit{Questions of Connectedness of the Hilbert Scheme of Curves in}\textbf{P}\^{}\{3\} preprint. Plane and space curves, Étale and other Grothendieck topologies and (co)homologies, Special algebraic curves and curves of low genus Bounds for the invariants of minimal curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author shows that Tate's conjecture holds for elliptic modular surfaces under the partial semi-simplicity conjecture. The principal result obtained here is as follows. Let \(E(N)\) be the modular elliptic surface over the modular curve \(X(N)\) associated with the principal congruence group of level \(N\) with \(N\ge3\). Let \(p\) be a prime number congruent \(1\) modulo \(N\) and assume the partial semi-simplicity conjecture.
Then Tate's conjecture holds for the reduction \(E(N)_{\mathbb F_p}\) of \(E(N)\) modulo \(p\) and the Mordell-Weil group of a generic fibre of \(E(N)_{\mathbb F_p}\rightarrow X(N)_{\mathbb F_p}\) is isomorphic to \((\mathbb Z/N\mathbb Z)^2\). This extends a result of \textit{T. Shioda} [in: Manifolds, Proc. int. Conf. Manifolds relat. Top. Topol., Tokyo 1973, 357--364 (1975; Zbl 0311.14007)] to \(N>4\). Let \(Br(E(N)_\mathbb Q)=H^2_{\text{ét}}(E(N)_\mathbb Q,\mathbb G_m)\) and \(V=V_pBr(E(N)_\mathbb Q)=\text{Hom}(\mathbb Q_p/\mathbb Z_p,Br(E(N)_\mathbb Q)\otimes \mathbb Q\). Let \(D=D_{\text{cris}}(V)\) be the filtered \(\varphi\)-module associated with \(V\), where \(\varphi\) is the Frobenius. It is sufficient to show \(D^{\varphi=1}=0\) to prove that Tate's conjecture holds for \(E(N)_{\mathbb F_p}\). For details, see section 2.2 in this article. The proof of that \(D^{\varphi=1}=0\) uses the theory of Hecke operator, the Eichler-Shimura congruence relation between the \(p\)th Hecke operator and Frobenius endomorphism. The condition that \(p\equiv 1 \mod N\) implies that the automorphism \(I_p\) of \(X(N)\) given by multiplying the level structure by \(p\) is trivial. The desired result is deduced from the triviality of \(I_p\) and the fact that \(V\) is a Hodge-Tate representation with weight \(\pm 1\). Further, by using a result of Serre on the \(\ell\)-adic representation of newforms, the author shows that the partial semi-simplicity conjecture holds for the reduction of \(E(N)_L\) at a set of places of density \(1\) for every number field \(L\). elliptic curves; modular forms; \(p\)-adic cohomology; zeta function; elliptic surfaces Elliptic curves over global fields, Holomorphic modular forms of integral weight, \(p\)-adic cohomology, crystalline cohomology, Elliptic surfaces, elliptic or Calabi-Yau fibrations On Tate's conjecture for the elliptic modular surface of level \(N\) over a prime field of characteristic \(1\mod N\) | 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 This paper contains applications to abelian schemes of previous differential algebraic work by the author. The first application applies to an abelian scheme \(A\) of relative dimension \(g\) over the ring of integers \(R\) of a finite unramified extension \(K\) of the \(p\)-adic field (odd \(p\)). Let \(k\) denote the residue field of \(R\) and assume \(|k|=p^\nu\); let \(A_0\) be the closed fibre of \(A\), which is assumed ordinary, and let \(p\) be the characteristic polynomial of the \(\nu^{\text{th}}\) power of the Frobenius on \(A_0\). Let \(R_{ur}\) be the maximal unramified extension of \(R\) and let \(p^\infty A(R_{ur})\) be all the infinitely \(p\) divisible points of \(R\) in \(R_{ur}\). Then under an appropriate condition on the matrix of Serre-Tate parameters of \(A\) (which is automatically satisfied if \(g=1\) and also in the most degenerate and most generic cases for the parameter matrix) and the assumption that \(P\) has only simple roots, the author proves that all points of \(p^\infty A(R_{ur})\) are torsion.
The second application concerns abelian schemes \(A\) of relative dimension \(1\) (not necessarily with ordinary reduction) over an absolutely unramified DVR \(R\) with quotient field \(K\) and algebraically closed residue field \(k\) of odd prime characteristic \(p\). Let the automorphism \(\phi: R \to R\) lift the Frobenius. In previous work, the author defined the operator \(\delta: R \to R\) by \(\delta x=(\phi(x)-x^p)/p\) and \(p\) derivatives of \(x\) of order \(i\) to be \(\delta^i x\). A \(\delta\) character of order \(n\) is a group homomorphism \(\psi: A(R) \to R\) which is Zariski locally representable as a \(p\)-adic limit of \(R\) polynomials in the affine coordinates and their \(p\) derivatives of order \(\leq n\). Let \(\psi\) be the uniquely determined (up to \(K\) scalar multiple) \(\delta\) character of minimal order. Then the author proves that if the closed fibre \(A_0\) is supersingular then \(p^\infty A(R)= \text{Ker } \psi\). abelian scheme; \(p\)-adic field; Frobenius; derivatives; differential algebra Buium A.: Differential characters and characteristic polynomial of Frobenius. J. Reine Angew. Math. 485, 209--219 (1997) Arithmetic ground fields for abelian varieties, Modules of differentials, Local ground fields in algebraic geometry, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure Differential characters and characteristic polynomial of Frobenius | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 algebraic-geometric structure of the elliptic Gaudin two-puncture model previously obtained. We identify this system with the system of pole dynamics of finite-gap solutions of the matrix Davey-Stewartson equation. We also obtain the action-angle variables and construct explicit solutions of this system in terms of theta functions. We discuss the geometry of degenerations of this system. elliptic Gaudin two-puncture model; finite-gap solutions; matrix Davey-Stewartson equation; theta functions; Hitchin system; algebraic-geometric symplectic form; inverse spectral sproblem Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions, Relationships between algebraic curves and integrable systems, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Other completely integrable equations [See also 58F07], Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Groups and algebras in quantum theory and relations with integrable systems The elliptic Gaudin system with spin | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper is concerned with the zeros of the numerical function
\[
\lambda(g,q,t):= q+1+\lfloor 2\sqrt q\rfloor g-N_q(g)-t ,
\]
where \(g\) and \(t\) are non-negative integers, \(q\) a prime power, and \(N_q(g)\) is the maximum of the number of \(\mathbb F_q\)-rational points that projective, geometrically irreducible, non-singular algebraic curves of genus \(g\) defined over \(\mathbb F_q\) can have. We have \(\lambda(g,q,0)\geq 0\) for any \(g\) and \(q\) by \textit{J.-P. Serre}'s improvement of the Hasse-Weil bound [C. R. Acad. Sci., Paris, Sér. I 296, 397-402 (1983; Zbl 0538.14015)]. Finding \(t\) such that \(\lambda(g,q,t)=0\) once \(g\) and \(q\) are fixed, or equivalently computing \(N_q(g)\), is in general a difficult task. Tables for this function, with \(g\) and \(q\) small, were constructed by van der Geer and van der Vlugt and it is regularly updated at [\url{http://www.wins.uva.nl/~geer}].
We have \(\lambda(g,q,1)\geq 0\) whenever \(g>(q^2-q)/(\lfloor 2\sqrt q\rfloor^2+\lfloor 2\sqrt q\rfloor -2q)\) by a refined version of a result due to \textit{Y. Ihara} [J. Fac. Sci., Univ. Tokyo, Sect. I A 28, 721-724 (1981; Zbl 0509.14019)]. \textit{J.-P. Serre} [Rational points on curves over finite fields; Notes by F. Gouvea of lectures at Harvard University (1985)] showed that \(\lambda(g,q,1)\neq 0\) if \(g>2\). In general a method to find \(t\) such that \(\lambda(g,q,t)\geq 0\) is via Oesterlé's optimization method of Serre-Weil's explicit formulae (loc. cit.)
Theorem 1 in this paper shows that \(\lambda(g,q,1)\geq 0\) for either \(g=4, q=8\), or \(g\geq 3, q=32\), or \(g\geq 4, q=2^{13}\), or \(g\geq 3, q=27\), or \(g\geq 4, q=243\), or \(g\geq 4, q=125\). Then for these \(g\)'s and \(q\)'s, \(\lambda(g,q,2)\geq 0\). The proof of this theorem is done via Galois descent method by taking advantage of the special value of \(q\). A similar result holds if \(q\in\{x^2+1,x^2+x+1\}\) for some integer \(x\) (loc. cit.).
Theorem 2 shows that \(\lambda(g,q,2)\neq 0\) whenever \(g\geq 3\) and \(q=2^{2s}\geq 16\). This result relies on Honda-Tate theory of abelian varieties [Sémin. Bourbaki 1968/69, Exp. No. 352, 95-110 (1971; Zbl 0212.25702)]. Now \(\lambda(g,q,1)\geq 0\) whenever \(q\) is an even power and either \((\sqrt q^2-\sqrt q+4)/6<g<(\sqrt q-1)^2/4\) or \((\sqrt q-1)^2/4<g<\sqrt q(\sqrt q-1)/2\): This is a result on maximal curves [see \textit{R. Fuhrmann} and \textit{F. Torres} [Manuscr. Math. 89, 103-106 (1996; Zbl 0857.11032)] and \textit{G. Korchmáros} and \textit{F. Torres} [math.AG/0008202]. Therefore if \(q=2^{2s}\geq 16\) and \(g\) belongs to the intervals above, \(\lambda(g,q,3)\geq 0\).
Finally the numbers \(\lambda(5,3,5), \lambda(7,3,8), \lambda(5,9,4), \lambda(6,8,3)\) are non-negative by the aforementioned Oesterlé method; equivalently \(N_3(5)\leq 14\), \(N_3(7)\leq 17\), \(N_9(5)\leq 36\), \(N_8(6)\leq 36\). Theorem 3 in the paper shows that all these inequalities are in fact strict. This result is proved by studying the possible zeta functions that the underlying curve might have. Details of the method are given in \textit{K. Lauter} [Proc. Am. Math. Soc. 128, No. 2, 369-374 (2000)] where it is proved that \(N_3(5)<14\). Hasse-Weil-Serre bound; zeta function of curves over finite fields; rational points K. Lauter, Geometric methods for improving the upper bounds on the number of rational points on algebraic curves over finite fields, Institut de Mathématiques de Luminy, preprint, 1999, pp. 99--29. Curves over finite and local fields, Finite ground fields in algebraic geometry, Arithmetic ground fields for curves Improved upper bounds for the number of rational points on algebraic curves over finite fields | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Addressing a general audience the paper covers work done since around 1970 on integrable systems in classical and quantum mechanics, related to the teaching and research of J.-L. Verdier. The central themes are the roles of the Korteweg-de Vries (soliton) and the Kadomtsev-Petviashvili equations, and the interplay of 19th century concepts (algebraic curves, elliptic and theta functions) with cohomology. ``The whole theory of elliptic functions can be deduced from the stationary KdV equation; also one may say that the whole theory of abelian functions is contained in KP\dots'' -- Few historical remarks and occasional references to sources are scattered over the paper. No bibliography. integrable systems; Korteweg-de Vries; elliptic functions; abelian functions Bennequin, D., Hommage à Jean-Louis Verdier: au jardin des systèmes intégrables, inIntegrable Systems: The Verdier Memorial Conference (Luminy, 1991), pp. 1--36. Birkhäuser, Boston, MA, 1993. Development of contemporary mathematics, History of algebraic geometry, Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), KdV equations (Korteweg-de Vries equations), History of global analysis In memory of Jean-Louis Verdier: In the garden of integrable systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems ''Le but de ce travail est de calculer le groupe de Brauer des corps de fonctions rationnelles à coefficients complexes \({\mathbb{C}}(t_ 1,...,t_ n)\). Dès que n est supérieur ou égal à 2, ce groupe est non nul et il se décompose en une somme directe non dénombrable de groupes \(\chi\) (K) associés à des corps de fonctions complexe K. Plus précisément, on a une formule de récurrence
\[
Br({\mathbb{C}}(t_ 1,...,t_ n)) \simeq Br({\mathbb{C}}(t_ 1,...,t_{n-1})) \oplus \left\{ \oplus_{f\in P_{n- 1}} \chi (K_ f)\right\},
\]
où \(P_{n-1}\) est l'ensemble des polynômes unitaires irréductibles à coefficients dans \({\mathbb{C}}(t_ 1,...,t_{n-1})\) et où \(K_ f\) est l'extension de \({\mathbb{C}}(t_ 1,...,t_{n-1})\) obtenue en adjoignant une racine de f.
Chaque groupe \(\chi(K)\) est calculé à l'aide d'un modèle géométrique de K, c'est-à-dire à l'aide d'une variété algébrique lisse X ayant K pour corps des fonctions. On obtient la suite exacte
\[
0 \to H^1(X,{\mathbb{Q}}/{\mathbb{Z}}) \to \chi(K) \to Div(X)\otimes_{\mathbb{Z}}{\mathbb{Q}}/{\mathbb{Z}} @>c>> {^{c}H^2} (X,{\mathbb{Q}}/{\mathbb{Z}}),
\]
qui exprime \(\chi (K)\) en terme du groupe \(Div(X)\) des diviseurs de \(X\) et de la cohomologie \(H^ i(X,{\mathbb{Q}}/{\mathbb{Z}})\) de \(X\) muni de la topologie transcendante, l'homomorphisme \(c\) étant induit par la première classe de Chern. -- On montre ensuite que pour tout corps de fonctions, il existe un modèle \(X\) avec \(H^ 1(X,{\mathbb{Q}}/{\mathbb{Z}})\) divisible. Pour \(f\in P_ i\), choisissons un tel modèle \(X_ f\) de l'extension de \({\mathbb{C}}(t_ 1,...,t_ i)\) obtenue en adjoignant une racine de \(f\) et notons \(Div_ 0(X_ f)\) le groupe de ses diviseurs dont la première classe de Chern est nulle. On établit la formule
\[
Br({\mathbb{C}}(t_ 1,...,t_ n)) = \oplus^{n-1}_{i=1} \oplus_{f\in P_ i} \left\{H^ 1(X_ f,{\mathbb{Q}}/\quad {\mathbb{Z}}) \oplus Div_0(X_ f)\otimes_{{\mathbb{Z}}}{\mathbb{Q}}/{\mathbb{Z}}\right\}.
\]
Elle met en évidence que \(Br({\mathbb{C}}(t_ 1,...,t_ n))\) est divisible et que sa classe d'isomorphie est indépendante de n, dès que \(n\geq 2\). On montrera ensuite comment (une fois un modèle X d'un corps de fonctions fixé) on peut construire des algèbres simples centrales sur \({\mathbb{C}}(t_ 1,...,t_ n)\) à partir d'éléments de Div(X) et de \(H^ 1(X,{\mathbb{Q}}/{\mathbb{Z}})\). On étudiera pour terminer ce dernier groupe, qui est un invariant birationnel pour les variétés complètes lisses. Il est isomorphe au groupe des classes de diviseurs de torsion de X; c'est ce qui nous permettra d'en décrire géométriquement une partie dans quelques exemples.'' Brauer group of rational function field over complex field Brauer groups of schemes, Galois cohomology Groupe de Brauer et topologie des variétés algébriques complexes lisses. (Brauer group and topology of smooth complex 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 Let \((T,l)\) be a principally polarized abelian surface, and \(\hat{T}\) its dual. The Fourier--Mukai transform is the equivalence \(\Phi_{P}: D^b(T) \to D^b(\hat{T})\) whose kernel is the Poincaré line bundle \(P\), considered as an object of \(D^b(T \times \hat{T})\). Such transform is a very powerful tool for the study of stable vector bundles on \(T\) and their moduli spaces.
In this paper, the author investigates the family of curves in the linear systems \(|l|\) and \(|2l|\) employing Fourier--Mukai techniques. Given an ideal sheaf \(I_X\), for \(X\) a closed zero-dimensional subscheme of \(T\), and the polarization line bundle \(L\), the main technical tool is the notion of \textit cohomology jumping schemes \textrm associated to \(I_X\). These are the determinantal loci of the zero-degree component of the Fourier--Mukai transform of the map \(L^i \to O_X\), for \(i>0\). The results are obtained computing the spaces of linear sections of \(L^i P_{\hat{x}} I_X\), where \(P_{\hat{x}}\) is the line bundle on \(T\) corresponding to a given point \(\hat{x}\) of \(\hat{T}\). These spaces describe indeed the divisors in the linear system \(L^i\) containing \(X\). The proof proceeds by induction on the length of \(X\), considering first, separately, the cases where \(X\) has length \(\leq 5\), and then the more general case.
As a consequence, the author studies moduli spaces of some rank 2 stable sheaves, showing, for example, that the compactification of the moduli space of stable sheaves with Chern character \((2,0,-3)\) is isomorphic to \({\mathrm{Hilb}}^6(T \times \hat{T})\). Also, he shows that his arguments provide a new proof of the following: if \(C\) is a smooth curve of genus 5 on \(T\), then there is no \(g^1_3\) on \(D\). ideal sheaf; Fourier-Mukai; divisor; abelian surface; Hilbert scheme; stable sheaf Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Enumerative problems (combinatorial problems) in algebraic geometry, Configurations and arrangements of linear subspaces, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Algebraic moduli problems, moduli of vector bundles, Picard schemes, higher Jacobians, Divisors, linear systems, invertible sheaves A Fourier-Mukai approach to the enumerative geometry of principally polarized abelian 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 We compute the multiplicity of the discriminant of a line bundle \({\mathcal L}\) over a nonsingular variety \(S\) at a given section \(X\), in terms of the Chern classes of \({\mathcal L}\) and of the cotangent bundle of \(S\), and the Segre classes of the jacobian scheme of \(X\) in \(S\). For \(S\) a surface, we obtain a precise formula that expresses the multiplicity as a sum of a term due to the non-reduced components of the section, and a term that depends on the Milnor numbers of the singularities of \(X_{\text{red}}\). Also, under certain hypotheses, we provide formulas for the ``higher discriminants'' that parametrize sections with a singular point of prescribed multiplicity. As an application, we obtain criteria for the various discriminants to be ``small''. higher discriminants; multiplicity of the discriminant of a line bundle; Chern classes; cotangent bundle; Segre classes of the jacobian scheme P. Aluffi and F. Cukierman, Manuscripta Math., 78, 245--258 (1993); M. Chardin, J. Pure Appl. Algebra, 101, 129--138 (1995); L. Ducos, J. Pure Appl. Algebra, 145, 149--163 (2000); L. Busé and C. D'Andrea, C. R. Math. Acad. Sci. Paris, 338, 287--290 (2004); C. D' Andrea, T. Krick, and A. Szanto, J. Algebra, 302, 16--36 (2006); arXiv:math/0501281v3 (2005). Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Divisors, linear systems, invertible sheaves, Jacobians, Prym varieties, Picard schemes, higher Jacobians Multiplicities of discriminants | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper we present results obtained from the study of an invariant formulation of completely integrable \(\mathbb{C}P^{N-1}\) Euclidean sigma models in two dimensions defined on the Riemann sphere, having finite actions. Surfaces connected with the \(\mathbb{C}P^{N-1}\) models, invariant recurrence relations linking the successive projection operators, and immersion functions of the surfaces are discussed in detail. We show that immersion functions of 2D-surfaces associated with the \(\mathbb{C}P^{N-1}\) model are contained in 2D-spheres in the \(\mathfrak{su}(N)\) algebra. Making use of the fact that the immersion functions of the surfaces satisfy the same Euler-Lagrange equations as the original projector variables, we derive surfaces induced by surfaces and prove that the stacked surfaces coincide with each other, which demonstrates the idempotency of the recurrent procedure. We also demonstrate that the \(\mathbb{C}P^{N-1}\) model equations admit larger classes of solutions than the
ones corresponding to rank-1 Hermitian projectors. This fact allows us to generalize the Weierstrass formula for the immersion of 2D-surfaces in the \(\mathfrak{su}(N)\) algebra and to show that, in general, these surfaces cannot be conformally parametrized. Finally, we consider the connection between the structure of the projective formalism and the possibility of spin representations of the \(\mathfrak{su}(2)\) algebra in quantum mechanics. sigma models; soliton surfaces; integrable systems; Weierstrass formula for immersion Model quantum field theories, Compact Riemann surfaces and uniformization, Spaces of embeddings and immersions, Applications of Lie (super)algebras to physics, etc., Spinor and twistor methods applied to problems in quantum theory, Clifford algebras, spinors, Riemann surfaces; Weierstrass points; gap sequences Analysis of \(\mathbb{C}P^{N-1}\) sigma models via soliton 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 Some years ago, Florian Pop showed that a field which is finitely generated over its prime field is determined up to isomorphism by its absolute Galois group (allowing a finite purely inseparable extension in positive characteristic). This theorem, whose pedigree can be traced back to investigations by Neukirch concerning Galois groups of number fields in the early 1970's, gives a positive answer to the so-called ``birational anabelian conjecture'' of A. Grothendieck formulated in 1983. In work in progress [Prog. Math. 181, 519--532 (2000; Zbl 1022.14006); ``Pro-1 birational anabelian geometry over algebraically closed fields. I'', preprint, \\url{arxiv:math.AG/0307076}] \textit{F. Pop} extends the above result to fields of finite type and of dimension at least 2 over the algebraic closure of the prime field; the case of dimension 2 was also considered recently by Bogomolov et Tschinkel.
The lecture will survey the known results in the area and then present the main ideas entering Pop's proofs. absolute Galois group; function field; anabelian geometry Szamuely, T., Groupes de Galois de corps de type fini (d'après pop), Astérisque, 294, 403-431, (2004) Separable extensions, Galois theory, Global theory and resolution of singularities (algebro-geometric aspects), Arithmetic ground fields for curves, Arithmetic ground fields for surfaces or higher-dimensional varieties Galois groups of fields of finite type (following Pop). | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We formulate a general question regarding the size of the iterated Galois groups associated with an algebraic dynamical system and then we discuss some special cases of our question. Our main result answers this question for certain split polynomial maps whose coordinates are unicritical polynomials. arithmetic dynamics; arboreal Galois representations; iterated Galois groups Dynamical systems over global ground fields, Heights, Galois theory, Global ground fields in algebraic geometry, Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps, Height functions; Green functions; invariant measures in arithmetic and non-Archimedean dynamical systems A question for iterated Galois groups in arithmetic dynamics | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let C be any reduced and irreducible curve, lying on a smooth cubic surface \(S\subset {\mathbb{P}}^ 3\). In this paper we determine the Hilbert function of C. Moreover we characterize some kinds of curves on S: the arithmetically Cohen-Macaulay curves, the maximal rank curves and the extremal ones. space curve on a smooth cubic surface; maximal rank curves; extremal curves; Hilbert function DOI: 10.1007/BF01762395 Special algebraic curves and curves of low genus, Projective techniques in algebraic geometry, Cycles and subschemes The Hilbert function of a curve lying on a smooth cubic 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 \textit{S. Lang} [Bull. Am. Math. Soc., New Ser. 80, 779-787 (1974; Zbl 0298.14014)] discussed the higher dimensional analogue of Mordell's conjecture for curves of genus \(\geq 2\) in terms of hyperbolic manifolds and posed a relative formulation of the problem for algebraic families of hyperbolic varieties: If there are an infinite number of cross sections, then the family contains split subfamilies and almost all cross sections are due to constant ones.
In this paper the author gives some affirmative answer to the above problem (main theorem). Then, assuming the conditions of the main theorem, he gives an analogue of Mordell's conjecture over function fields which was proved by \textit{J. I. Manin} [Izv. Akad. Nauk SSSR, Ser. Mat. 27, 1395-1440 (1963; Zbl 0166.169)] and \textit{H. Grauert} [Publ. Math., Inst. Hautes Étud. Sci. 25, 131-149 (1965; Zbl 0137.405)] who didn't assume these conditions. hyperbolic fibre space; higher dimensional analogue of Mordell's conjecture for curves; hyperbolic manifolds; algebraic families of hyperbolic varieties; Mordell's conjecture over function fields Noguchi, J.Hyperbolic fiber spaces and Mordell's conjecture over function fields, Publ. Research Institute Math. Sciences Kyoto University21, No. 1 (1985), 27--46. Hyperbolic and Kobayashi hyperbolic manifolds, Holomorphic bundles and generalizations, Families, moduli, classification: algebraic theory Hyperbolic fibre spaces and Mordell's conjecture 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 Key distribution schemes play a significant role in key assignment schemes which allow participants in a network to communicate by means of symmetric cryptography in a secure way without the need of a unique key for every pair of participants. It is assumed that an adversary can eavesdrop on all communication and can corrupt up to \(t\) vertices in the network. It follows that, in general, the sender needs to transmit at least \(t+1\) shares of the message over different paths to the intended receiver and that each participant needs to possess at least \(t+1\) encryption keys. We do assume that vertices in the network will forward messages correctly (but only the corrupted vertices will collude with the adversary to retrieve the message).
We focus on two approaches. In the first approach, the goal is to minimize the number of keys per participant. An almost complete answer is presented. The second approach is to minimize the total number of keys that are needed in the network. The number of communication paths that are needed to guarantee secure communication becomes a relevant parameter. Our security relies on the random oracle model. key distribution; secure communication; ad hoc networks; privacy; graph theory; block designs; Steiner systems; combinatorics Y. Desmedt, N. Duif, H. van Tilborg, and H. Wang, ''Bounds and constructions for key distribution schemes,'' Adv. Math. Commun., vol. 3, no. 3, pp. 273--293, Aug. 2009. Cryptography, Algebraic coding theory; cryptography (number-theoretic aspects), Applications to coding theory and cryptography of arithmetic geometry Bounds and constructions for key distribution 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 categorification program was initiated by I. Frenkel with the aim of extending 3-dimensional topological field theories to dimensions 4 and higher [\textit{L. Crane} and \textit{I. Frenkel}, J. Math. Phys. 35, No. 10, 5136--5154 (1994; Zbl 0892.57014)]. This program was extended by the first author in his work on categorified tangle invariants [\textit{M. Khovanov}, Algebr. Geom. Topol. 2, 665--741 (2002; Zbl 1002.57006)].
The present paper is a part of ongoing research by the authors on categorification of quantum groups and their representations. A categorification of quantum \(sl(2)\) obtained previously by the second author is generalised to \(sl(n)\). More precisely the authors construct a linear 2-category whose Grothendieck category coincides with the idempotent form of quantum \(sl(n)\). Note that the interpretation of elements of Lustig's canonical basis of the idempotent form as classes of indecomposible ob jects established for \(sl(2)\) is still an open problem for \(sl(n)\). This category has potential applications in representation theory. The authors expect this category to manifest itself as a symmetry of various categories of interest in representation theory, ranging from derived categories of coherent sheaves on quiver varieties to categories of modules over cyclotomic and degenerate affine Hecke algebras. categorification; quantum group; quantum \(sl(n)\); iterated flag variety; 2-representation; 2-category Khovanov, M.; Lauda, A., A categorification of quantum \(\mathfrak{sl}_n\), Quantum Topol., 1, 1, 1-92, (2010) Quantum groups (quantized enveloping algebras) and related deformations, Quantum groups and related algebraic methods applied to problems in quantum theory, Grassmannians, Schubert varieties, flag manifolds, Ring-theoretic aspects of quantum groups A categorification of quantum \(\text{sl}(n)\) | 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 Let \(K/F\) be an abelian Galois extension of fields. The group \(\text{Dec}(K/F)\) is the subgroup of the relative Brauer group \(\text{Br}(K/F)\) generated by the relative Brauer groups \(\text{Br}(L/F)\) of all the cyclic extensions \(L/F\) contained in \(K\). This group was introduced by the reviewer [J. Algebra 70, 420-436 (1980; Zbl 0473.16004)] in relation with the construction of indecomposable division algebras of prime exponent. If \(K/F\) is elementary abelian of degree 4, then it is known that \(\text{Dec}(K/F)\) is the 2-torsion subgroup in \(\text{Br}(K/F)\).
In the present paper, the author explicitly constructs for each prime \(p\) and each integer \(n\geq 1\) (\(n \geq 2\) if \(p = 2\)) a field \(F\), an abelian extension \(K/F\) with Galois group \(({\mathbb{Z}}/p^ n \mathbb{Z})\times (\mathbb{Z}/p\mathbb{Z})\) and a central simple algebra \(A\) of exponent \(p\) split by \(K\) whose Brauer class is not in \(\text{Dec}(K/F)\). The base field \(F\) is a rational function field in three variables over a field of characteristic zero containing sufficiently many roots of unity. The methods of proof are essentially valuation-theoretic. abelian Galois extensions; relative Brauer groups; cyclic extensions; indecomposable division algebras of prime exponent; central simple algebras; Brauer class; rational function fields Finite-dimensional division rings, Equations in general fields, Valued fields, Brauer groups of schemes, Separable extensions, Galois theory Dec groups for arbitrarily high exponents | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Was ist ``undergraduate algebraic geometry''? Im Falle des vorliegenden Buches der gelungene Versuch, auf gut 100 Seiten eine elementare Einführung in das fundamentale, technisch komplizierte, hochabstrakte und daher anspruchsvolle Gebiet der Algebraischen Geometrie zu geben, das der Autor selbst ``a monolithic block'' nennt, which is ``colonising adjacent areas of mathematics''. In der Tat, ``algebraic geometry was able to absorb practically all the advances made in topology, homological algebra, number theory, etc.'', ganz zu schweigen von Kategorientheorie, Komplexer Analysis, Differentialgeometrie usw. Der Versuch, einen elementaren Zugang zu diesem Gebiet zu finden, scheint daher von vornherein zum Scheitern verurteilt zu sein. Daß er vom Autor dennoch erfolgreich unternommen worden ist, verdient Beachtung. Schon die existierenden, zumeist recht umfangreichen Lehrbücher über ``Graduate Algebraic Geometrie'' weisen Kompromisse auf, indem z.B. die Stoffauswahl eingeschränkt und/oder die Darstellung teilweise skizzenhaft ist, und ``étudier les EGAs'' \((=\acute Elements\) de géométrie algébrique, Publ. Math., Inst. Hautes Étud. Sci.) nach Grothendieck erfordert das Durcharbeiten eines großen Stapels von paperbacks, ``many of which still remain to be written up in an approachable way''. Wie hat nun der Autor dieses Problem gelöst? Durch konsequente, exemplarische, dabei klassiche Stoffauswahl und ausführliche Erörterung geschickt ausgewählter Beispiele. Behandelt werden u.a. ebene Kegelschnitte, elliptische Kurven, das Geschlecht von Kurven (recht kurz), affine Varietäten nebst Nullstellensatz, Funktionen auf Varietäten, projektive Varietäten, birationale Äquivalenz, Nicht-Singularität und Dimension. Dabei sind etwa die Ausführungen über Kegelschnitte exemplarisch für beliebige rationale Kurven, werden die kubischen Flächen als Musterbeispiele von rationalen del Pezzo Flächen untersucht und dienen elliptische Kurven als einfachste Beispiele abelscher Varietäten.
Im Hinblick auf den Gegensatz ``Computation versus Theory'' hat sich der Autor für die Betonung von ``Computation'' entschieden: ``When general theory proves the existence of some construction, then doing it in terms of explicit coordinate expressions is a useful exercise that helps one to keep a grip on reality''. In einem lesenswerten Anhang wird die neueste Entwicklung der Algebraischen Geometrie aufgezeigt und kommentiert. Hier geht der Autor auf die modernen Begriffe ein, indem er z.B. erläutert, inwieweit affine Schemata allgemeiner sind als affine Varietäten. Wünschenswert wäre eventuell noch die Aufnahme der Garbentheorie im Zusammenhang mit rationalen Funktionen auf Varietäten sowie die Behandlung quasiprojektiver Varietäten im Anschluß an die projektiven Varietäten gewesen, aber das hätte wohl den Umfang des Bändchens zu sehr vergrößert. Der Stil des nicht immer leicht lesbaren Buches ist locker und witzig (der Leser findet u.a. psychologische Betreuung, und selbst Expräsident Reagan kommt im Text vor). Selbst wer einen weniger flapsigen Tonfall bevorzugt, wird Gefallen an dieser kleinen, doch inhaltsreichen Monographie finden. curve; variety; genus; singularity; dimension; rational function; tangent space; surface; birational equivalence Reid, M.: Undergraduate algebraic geometry. London mathematical society student texts (1988) Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Curves in algebraic geometry, Varieties and morphisms, Singularities in algebraic geometry, Surfaces and higher-dimensional varieties Undergraduate algebraic geometry | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For the entire collection see Zbl 0607.00004.]
This is an overview of two papers of the author [``Fields of definition of algebraic varieties in characteristic zero'', Compos. Math. 61, 339- 352 (1987) and ``Differential function fields and moduli of algebraic varieties'', Lect. Notes Math. 1226 (1986; Zbl 0613.12018)]. For an algebraic function field F/K the author defines the concept of being strongly normal, weakly normal or having no movable singularity - concepts which partly were defined by \textit{E. R. Kolchin} in his book ``Differential algebra and algebraic groups'' (1973; Zbl 0264.12102). The main results are theorem 1 (strongly normal \(\Leftrightarrow\) weakly normal and having no movabled singularity), theorem 2 (having no movable singularity \(\Rightarrow\) there exists a strongly normal extension \(E/K',\) where \(K'/K\) is finite and E is an extension of K'F) and theorem 6 (giving examples were the concepts of being strongly normal and weakly normal are the same). algebraic function field; strongly normal; weakly normal; movable singularity Abstract differential equations, Arithmetic theory of algebraic function fields, Algebraic functions and function fields in algebraic geometry, Singularities of curves, local rings Movable singularities and differential Galois theory | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Der Grundkörper ist \(k=\mathbb{C}\), und \({\mathcal H}:=H_{d,g}= \text{Hilb}^P (\mathbb{P}^3_k)\), \(P(T)= dT-g+1\), ist das (volle) Hilbertschema der Raumkurven vom Grad \(d\) und Geschlecht \(g\). Für \(3\leq d\leq 11\) definiert man \(g(d)\) durch die Tabelle
\[
\begin{matrix} d & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11\\ g(d) & -2 & 0 & 1 & 2 & 4 & 6 & 9 & 11 & 15\end{matrix}
\]
und für \(d\geq 12\) durch die Formel \(g(d)={1\over 6}d(d-3)\). Die vorliegende Arbeit bringt zunächst eine Ergänzung und verschiedene Verbesserungen zu früheren Arbeiten des Autors [\textit{G. Gotzmann}, ``Der kombinatorische Teil der ersten Chowgruppe eines Hilbertschemas von Raumkurven'' und ``Der algebraische Teil der ersten Chowgruppe eines Hilbertschemas von Raumkurven'' (Münster 1994; Zbl 0834.14004 und Münster 1997; Zbl 0954.14002)], und man erhält als Zusammenfassung:
Satz I. Wenn \(d\geq 3\) und \(g\leq g(d)\) ist, dann ist
(i) \(\dim_\mathbb{Q} A_1({\mathcal H})\otimes_\mathbb{Z} \mathbb{Q}=3\);
(ii) \(\text{Pic} ({\mathcal H}) \simeq \mathbb{Z}^3\oplus \mathbb{C}^r\), mit \(r=\dim_kH^1 ({\mathcal H},{\mathcal O}_{\mathcal H})\);
(iii) \(\text{NS}({\mathcal H})\simeq\mathbb{Z}^3\).
Der Satz ist eine Folgerung aus etwas genaueren Ergebnissen in Abschnitt 6, wo explizite Basen von \(A_1({\mathcal H})\otimes\mathbb{Q}\) und \(\text{NS}({\mathcal H})\) bestimmt werden. -- Man kann die in Satz I genannten Ergebnisse für \({\mathcal H}\) auf die zugehörige universelle \({\mathcal C}\subset{\mathcal H}\times\mathbb{P}^3\) übertragen, und man erhält:
Satz II. Wenn \(d\geq 3\) und \(g\leq g(d)\) ist, dann ist
(i) \(\dim_\mathbb{Q} A_1({\mathcal C})\otimes_\mathbb{Z} \mathbb{Q}=4\);
(ii) \(\text{Pic}({\mathcal C})\simeq \mathbb{Z}^4\oplus \mathbb{C}^r\), mit \(r=\dim_kH^1 ({\mathcal C},{\mathcal O}_{\mathcal C})\);
(iii) \(\text{NS}({\mathcal C})\simeq\mathbb{Z}^4\). Néron-Severi group; Hilbert scheme; universal curve; Picard group Parametrization (Chow and Hilbert schemes), Plane and space curves The Néron-Severi group of a Hilbert scheme of space curves and the universal curve | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The problem of studying linear systems of plane curves is one of the oldest and most important in algebraic geometry. In its more general formulation the problem (which still so far remains unsolved) is the following: given a set of (distinct or infinitely near) points \(p_ 1,...,p_ n\) in the projective plane (over any algebraically closed field) to analyse the properties of the linear system of curves of a given degree d passing through \(p_ 1,...,p_ n\) with assigned multiplicities \(m_ 1,...,m_ n\), determining in particular the dimension of the system and its base locus. As well known this amounts to study all complete linear systems on the rational surface V obtained from the plane with a (minimal) sequence of blowing-ups which substitute to the points \(p_ 1,...,p_ n\) a suitable configuration of exceptional divisors. - The present paper gives a relevant contribution to this question in case the points \(p_ 1,...,p_ n\) are non singular points of an irreducible plane cubic (which is the same as saying that the surface V possesses an irreducible anticanonical divisor). This is an important case because for instance, among other things, it includes the study of linear systems of curves on some interesting classes of surfaces: Del Pezzo surfaces, some rational elliptic surfaces, rational surfaces with infinitely many exceptional divisors, etc. For this reason it has been analysed in the past by various authors, too many to recall here (some of them are quoted in the paper), but nobody provided (like in this paper is done) complete answers to the problem asked at the beginning.
We describe now the results. First the author defines certain types of divisor classes on the blown-up surface V: the standard, the almost excellent and the excellent classes. It would be too long to repeat the definitions here, but to make things short, let us say that, in the case the points \(p_ 1,...,p_ n\) are all distinct, the standard classes essentially correspond to proper transforms on V of plane curves of degree d passing through \(p_ 1,...,p_ n\) with multiplicities \(m_ 1,...,m_ n\), such that \(d\geq m_ 1\geq...\geq m_ n\) and \(d\geq m_ 1+m_ 2+m_ 3\), while the almost excellent [excellent] classes have to verify moreover the inequality \(3d+(m_ 1+...+m_ n)\geq 0\) \([3d+(m_ 1+...+m_ n)>0]\). The results are now the following:
(1) any standard divisor class on V is effective and for any excellent divisor class \({\mathcal F}\) one has \(h^ 1(V,{\mathcal F})=h^ 2(V,{\mathcal F})=0\), thus \(h^ 0(V,{\mathcal F})\) can be computed by the Riemann-Roch theorem;
(2) the linear systems corresponding to almost excellent, non excellent classes are completely described and their dimension, of course, is computed;
(3) for any divisor class \({\mathcal F}\) either \(h^ 0(V,{\mathcal F})=0\) or \({\mathcal F}={\mathcal G}+{\mathcal C}\), where \({\mathcal G}, {\mathcal C}\), are classes of effective divisors, \({\mathcal G}\) is almost excellent with \(h^ 0(V,{\mathcal F})=h^ 0(V,{\mathcal G})\), and \({\mathcal C}\) is a non negative sum of exceptional divisors classes, of so called nodal classes (i.e. classes with self intersection -2 and 0 intersection with the canonical system) and of the canonical class;
(4) the linear systems with fixed components or base points are completely determined.
It should be stressed that the proof of (3) is constructive, in the sense that it gives an algorithm for effectively computing \({\mathcal G}\) and \({\mathcal C}\), thus \(h^ 0(V,{\mathcal F})\), once \({\mathcal F}\) is given.
In conclusion the reviewer wants to mention that related results have been obtained by the same author in Math. Ann. 272, 139-153 (1985; Zbl 0545.14003)] and more recently by \textit{A. Gimigliano}, ''On linear systems of plane curves'' (Thesis, Queen's Univ. 1987). Moreover the questions studied in this article have proved to be relevant in another classical problem, namely the determination of all possible degrees and genera of smooth, non degenerate curves in a projective space [see \textit{J. Rathmann}, ''The genus of algebraic space curves'' (Thesis, Univ. California, Berkeley 1986); the reviewer and \textit{E. Sernesi}, ''Curves on surfaces of degree \(2r-\delta\) in \({\mathbb{P}}^ r\)'' (preprint); the reviewer, ''On the degree and genus of smooth curves in a projective space'' (preprint)]. effective divisor class; almost excellent effective divisors; linear systems of plane curves B. Harbourne, Complete linear systems on rational surfaces, Trans. Amer. Math. Soc., 289 (1985), no. 1, 213--226.Zbl 0609.14004 MR 0779061 Divisors, linear systems, invertible sheaves, Riemann-Roch theorems, Rational and unirational varieties Complete linear systems on rational 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 fundamental theorems of classical Brill-Noether theory of smooth projective curves have not been extended yet to stable curves, due to the technical difficulties of combinatorial nature. The goal of this paper is to extend some of them to binary curves. A binary curve is a stable nodal curve having two irreducible rational components, intersecting at \(g+1\) points. Their moduli space \(B_g\subset \overline{M_g}\) is irreducible of dimension \(2g-4\). The analogue of theorems of Riemann, Clifford and Martens are proved to hold for any binary curve and for line bundles parametrized by the compactified Jacobian scheme. An analogue of Brill-Noether theorem is proved for general binary curves and for \(r\leq 2\). More precisely, let
\[
B_d(g)=\{(d_1,d_2) \mid d_1+d_2=d, \frac{d-g-1}{2}\leq d_i\leq \frac{d-g+1}{2}, i=1,2\}
\]
be the set of balanced multidegrees.
If \(\underline{d}\in B_d(g)\), put \(W^r_{\underline{d}}(X)= \{L\in Pic^{\underline{d}}(X)\mid h^o(L)\geq r+1\}\), where \(X\) is a binary curve. Then it is proved that, for a general binary curve \(X\) and for \(r\leq 2\), \(\dim W^r_{\underline{d}}(X)\leq \rho_d^r(g)\), the usual Brill-Noether number, and equality holds for some \(\underline{d}\). Moreover \(\dim \overline{W^r_{\underline{d}}(X)}= \rho_d^r(g)\). stable curve; moduli space; Clifford theorem; Picard scheme; Brill-Noether Caporaso L.: Brill-Noether theory of binary curves. Math. Res. Lett. 17(2), 243--262 (2010) Families, moduli of curves (algebraic), Divisors, linear systems, invertible sheaves, Special divisors on curves (gonality, Brill-Noether theory) Brill-Noether theory of binary curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The articles of this volume will be reviewed individually under the abbreviation ''Systèmes différentiels et singularités, Colloq. Luminy/France 1983, Astérisque 130''. Systèmes différentiels; Singularités; C.I.R.M.; Colloque; Luminy/France; differential systems; singularities Conference proceedings and collections of articles, Proceedings, conferences, collections, etc. pertaining to algebraic geometry, Proceedings, conferences, collections, etc. pertaining to several complex variables and analytic spaces, Proceedings, conferences, collections, etc. pertaining to global analysis Systèmes différentiels et singularités, C.I.R.M. (Luminy, France), 27 juin - 9 juillet 1983 (Colloque) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In the Congress ICDEA2019 in London, we give two examples of QRT-families of biquadratic curves \(Q_1(x,y)-\lambda Q_2(x,y)=0\), with \(Q_1\) of degree 4 and \(Q_2\) of degree 2, each of them has genus zero; these examples contrast with many examples published of QRT-families, where almost all curves have genus one. After a brief summary of these examples (the details will be published in Sarajevo Journal of Mathematics), we give an example with \(Q_1\) of degree 4 and \(Q_2\) of degree 3. We prove that, for the QRT-map \(T\) associated to this family, the orbit of every point not in the union of three lines and an hyperbola converges to a fixed point. Finally we present an example with \(Q_1\) and \(Q_2\) of degrees 4, where there are some bifurcations in the behaviour of the QRT-map. QRT maps; genus of curves; dynamical systems Integrable difference and lattice equations; integrability tests, Completely integrable discrete dynamical systems, Rational and birational maps, Relationships between algebraic curves and integrable systems QRT-families of degree four biquadratic curves each of them has genus zero, associated 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 Siehe das ausführliche Referat über die diesen Arbeiten zugrunde liegende vorläufige Mitteilung [Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. I, N. F. 1, 119--129 (1935; Zbl 0013.19701)]. Teil I vgl. [J. Reine Angew. Math. 175, 55--62 (1936; Zbl 0014.14903)]. abstract elliptic function fields; automorphisms; meromorphisms; addition theorem Hasse, H.: Zur theorie der abstrakte elliptischen funktionenkörper. II. automorphismen und meromorphismen. Das additionstheorem. J. reine angrew. Math. 175, 69-88 (1936) Arithmetic theory of algebraic function fields, Algebraic functions and function fields in algebraic geometry Zur Theorie der abstrakten elliptischen Funktionenkörper. II: Automorphismen und Meromorphismen. Das Additionstheorem | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 0745.00062.]
The author constructs an extension \(F_ R= \mathbb{F}_ q (x, y_ 1,y_ 2)\) over the rational function field \(\mathbb{F}_ q(x)\) by \(y^ q_ 1- y_ 1= x^{q_ 0} (x^ q -x)\), \(y^ q_ 2- y_ 2= x^{q_ 0} (y^ q_ 1- y_ 1)\), where \(q=3^{2s+1}\), \(q_ 0= 3^ s\), \(\mathbb{F}_ q\) is the field with \(q\) elements; and proves that the automorphism group \(\text{Aut}(F_ R/ \mathbb{F}_ q)= \text{Aut} (F_ R/ \overline{\mathbb{F}}_ q)\) is the Ree group \({}^ 2G_ 2(q)\), that \(F_ R\) has the maximal number \(q^ 3+1\) of \(\mathbb{F}_ q\)-rational places (comparing to its genus \(g\)), and that the number \(N_ m\) of places of degree one in \(F_ R/ \mathbb{F}_{q^ m}\) reaches the Hasse-Weil bound \(q^ m+1 +2g \sqrt{q^ m}\) when \(m\equiv 6\pmod {12}\). rational function field; automorphism group; Ree group; Hasse-Weil bound Pedersen, J.P.: A function field related to the Ree group. In: Coding Theory and Algebraic Geometry, Lecture Notes in Mathematics, vol. 1518, pp. 122--132. Springer, Berlin (1992) Arithmetic theory of algebraic function fields, Simple groups, Algebraic functions and function fields in algebraic geometry A function field related to the Ree group | 0 |
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