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The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In dieser Arbeit studieren wir Kontraktionen von Schemata und wenden sie auf Probleme über algebraische Flächen und relative Kurven an. Unter einer Kontraktion wollen wir im Folgenden einen eigentlichen Morphismus \(f:X\to Y\) mit \({\mathcal O}_Y\to f_*({\mathcal O}_X)\) bijektiv verstehen. Topologisch betrachtet sind dies überaus einfache Abbildungen, da der Raum \(Y\) der Quotient von \(X\) nach der Relation \(f(x_1)= f(x_2)\) ist. Die Existenz oder Nicht-Existenz einer Kontraktion ist meistens nicht einfach nachzuweisen. Ein Hilfsmittel zur Konstruktion von projektiven Kontraktionen sind invertierbare \({\mathcal O}_X\)-Moduln \({\mathcal L}\), für die eine genügend hohe Potenz \({\mathcal L}^{\otimes n}\) mit \(n>0\) global erzeugt ist. Wir bezeichnen solche Moduln als kontraktiv und fassen sie als Verallgemeinerungen von amplen Moduln auf. Das geometrische Problem, ob eine Kontraktion existiert, lässt sich häufig auf das algebraische Problem zurückführen, ob ein invertierbarer Modul kontraktiv ist. Obschon dies noch kein leichtes Problem ist, lässt es sich mit den kohomologischen Methoden der Garbentheorie behandeln und ist in vieler Hinsicht zugänglicher.
Diesem Ansatz folgend werden wir kohomologische Kriterien für kontraktive Garben entwickeln, die in Analogie zu den bekannten kohomologischen Kriterien für ample Garben stehen, und diese Kriterien in den einfachsten Spezialfällen, nämlich algebraische Flächen und relative Kurven, anwenden. contraction; contractive module; contractive sheaf Rational and birational maps, Local structure of morphisms in algebraic geometry: étale, flat, etc. Contractions of schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper broadly compares the notions of on the one hand Gröbner basis schemes, and on the other hand border basis schemes.
The main results of the paper include a proof that Gröbner basis schemes can be viewed as weighted projective schemes as well as a proof that Gröbner basis schemes can be viewed as sections of border basis schemes.
Several consequences are also explored. For example the author concludes that despite the non-uniquness of Buchberger's Algorithm, using it to construct a Gröbner basis scheme leads to a canonical ideal. Another consequence is that if the point in the Gröbner basis scheme corresponding to the unique monomial ideal is smooth then the scheme is an affine space.
In the paper it is also determined when a border basis scheme will equal the Gröbner basis scheme. The author finishes by posing two open problems in the field. Groebner basis scheme; border basis scheme L. Robbiano, \textit{On border basis and Gröbner basis schemes}, Collect. Math., 60 (2009), pp. 11--25. Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Algebraic moduli problems, moduli of vector bundles, Deformations and infinitesimal methods in commutative ring theory, Formal methods and deformations in algebraic geometry On border basis and Gröbner basis schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We show that the dual character of the flagged Weyl module of any diagram is a positively weighted integer point transform of a generalized permutahedron. In particular, Schubert and key polynomials are positively weighted integer point transforms of generalized permutahedra. This implies several recent conjectures of \textit{C. Monical} et al. [Sel. Math., New Ser. 25, No. 5, Paper No. 66, 37 p. (2019; Zbl 1426.05175)]. Schubert polynomial; key polynomial; dual character of the flagged Weyl module; Newton polytope; integer point transform; generalized permutahedron Symmetric functions and generalizations, Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Combinatorial aspects of matroids and geometric lattices, Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.), Representations of finite symmetric groups, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) Schubert polynomials as integer point transforms of generalized permutahedra | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 continue their study of Hitchin systems, started in [\textit{A. Chervov} and \textit{D. Talalaev} [Theoret. Math. Phys. 140, 1043--1072 (2004)]. They consider curves that can be obtained from a projective line by gluing two subschemes and they describe the related moduli space of vector bundles, the dualizing sheaf, and module endomorphisms. Finally they show that these constructions lead to Hitchin systems and prove their integrability. The exposition is followed by numerous examples. Hitchin systems; gluing subschemes; integrable systems; singular algebraic curves; \(r\)-matrix Vector bundles on curves and their moduli, Relationships between algebraic curves and integrable systems, Singularities of curves, local rings, Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics Hitchin systems on singular curves. II: Gluing subschemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Fix integers \((n,d,r,s)\) and let \(M\) be a general \(r\)-dimensional subspace of degree \(d\) homogeneous polynomials in \(n+1\) variables. Let \(A\) be the variety of \(s\)-sided polar polyhedra of \(M\). Here the author proves that \(A\) is irreducible, rational and of the expected dimension if \((n,d,r,s) = (2,4,2,8)\), \((2,3,4,7)\), \((3,2,6,7)\), and \((2,3,7,0)\). He proves that \(A\) is finite and computes its cardinality for \(5\) quadruples. In all cases for the proofs he uses a specific feature of the minimal free resolution of \(s\) general points in \(\mathbb {P}^n\). Waring's problem; apolar scheme; homogeneous polynomials Projective techniques in algebraic geometry, Classical problems, Schubert calculus Apolar schemes of algebraic 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 The paper under review deals with the scheme structure of the jet schemes of determinantal varieties. Given a variety \(X\) over an algebraically closed field \(k\) of characteristic zero, the study of the arc space and jet schemes \({\mathcal J}_m(X), m\geq 0\) on \(X\) was suggested in \textit{J. F. Nash, jun.} [Duke Math. J. 81, No. 1, 31--38 (1996; Zbl 0880.14010)] as an approach to the study of its singularities. A nice result in \textit{M. Mustaţă} [Invent. Math. 145, No. 3, 397--424 (2001; Zbl 1091.14004)] states that if \(X\) is a local complete intersection variety, then all the jet schemes \({\mathcal J}_m(X)\) are irreducible if and only if \(X\) has only rational singularities. Since determinantal varieties only have rational singularities, it follows that all the jet schemes of the variety of singular square matrices are irreducible. A natural question arises: are the jet schemes of any determinantal variety irreducible?
In this paper, the author proves that the second jet scheme and all the odd jet schemes of \textit{essentially} all determinantal varieties are reducible, thus giving a negative answer to the above question. The idea of the proof is based on the computation of the dimensions of the schemes of jets centered on the singular locus of \(X\). Similar results appear in \textit{T. Košir} and \textit{B. A. Sethuraman} [J. Pure Appl. Algebra 195, No. 1, 75--95 (2005; Zbl 1085.14043)]. In addition, a formula for the number of irreducible components and their dimensions in the case of the variety of matrices with rank at most one is given (cf. Example 4.7 of \textit{M. Mustaţă} [J. Am. Math. Soc. 15, No. 3, 599--615 (2002; Zbl 0998.14009)]). As an application, the log canonical threshold of the pair \((\mathbb{A}^{rs},X)\) is computed. determinantal variety; jet schemes; complete intersections; rational singularities Cornelia Yuen, Jet schemes of determinantal varieties, Algebra, geometry and their interactions, Contemp. Math., vol. 448, Amer. Math. Soc., Providence, RI, 2007, pp. 261 -- 270. Determinantal varieties, Complete intersections, Singularities in algebraic geometry Jet schemes of determinantal varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The present article is an overview about group schemes of multiplicative type in the style of SGA 3 [\textit{M. Demazure} and \textit{A. Grothendieck}, Schémas en groupes II: Groupes de type multiplicatif, et structure des schemas en groupes généraux. Lecture Notes in Mathematics. 152. Springer-Verlag (1970; Zbl 0209.24201)]. It is based on lectures given by the author during a summer school at the CIRM in Luminy. The article starts with a brief introduction on group schemes over some base \(S\), e.g. \(S = \mathrm{Spec}(\mathbb{Z})\). A group scheme over \(S\) is locally diagonalisable if every point of \(S\) has an open neighbourhood \(U\) over which \(G\) is diagonalisable. A group scheme is of multiplicative type if it is locally diagonalizable in the \(fpqc\)-topology. Important examples are the tori, group schemes which are locally isomorphic to \(\mathbb{G}_m^r\) for some \(r \in \mathbb{N}\) in the \(fpqc\)-topology. The author discusses some properties of group schemes of multiplicative type and their classification. Additional topics are: Representations of group schemes of multiplicative type, operations on affine schemes, Hochschild cohomology and infinitesimal properties of group schemes of multiplicative type. group schemes; tori; torsors; Hochschild cohomology Oesterlé, J.; Brochard, S. (ed.); Conrad, B. (ed.); Oesterlé, J. (ed.), Schémas en groupes de type multiplicatif, No. 42/43, 63-91, (2014), Paris Group schemes, Étale and other Grothendieck topologies and (co)homologies Group schemes of multiplicative type | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We use coverings by smooth projective varieties then apply nonabelian Hodge techniques to study the topology of proper Deligne-Mumford stacks as well as more general simplicial varieties. Simpson C., Local systems on proper algebraic \textit{V}-manifolds, Pure Appl. Math. Q. 7 (2011), no. 4, 17675-1759. Generalizations (algebraic spaces, stacks), Stacks and moduli problems, Transcendental methods, Hodge theory (algebro-geometric aspects), Topology of analytic spaces Local systems on proper algebraic \(V\)-manifolds | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper under review concerns the problem of enumerating nodal curves on smooth complex surfaces: it is the fourth on the topic by the Authors. Here they complete the proof of a theorem about a family of curves on a family of surfaces, announced and partly proved in [\textit{S. L. Kleiman} and \textit{R. Piene}, Math. Nachr. 271, 69--90 (2004; Zbl 1066.14063)]. The set up is a smooth projective family of surfaces \(\pi: F\rightarrow Y\) and a relative effective divisor \(D\) on \(F/Y\), where \(Y\) is an equidimensional Cohen-Macaulay scheme over an algebraically closed field of characteristic \(0\): the first part of the theorem, which was proved in the aforementioned paper, describes a natural cycle \(U(D, r)\) on \(Y\) that enumerates the fibers \(D_y\) with precisely \(r\) ordinary nodes, for each \(r\geq 1\). The second part of the theorem, which is proved in the present paper, asserts that, for \(r\leq 8\), the rational equivalence class \(u(D, r)\) of \(U (D, r)\) is given by a computable universal polynomial in the pushdowns to the parameter space of products of the (relative) Chern classes of the family: \(u(D,r)= \frac{1}{r!}P_r(a_1(D),\dots,a_r(D))\cap[Y]\), where \(P_r\) is the \(r\)th Bell polynomial. In the proof, a central role is played by the induced pairs \((F_i/X_i,D_i)\) of a given pair \((F/Y,D)\), (see [\textit{S. Kleiman} and \textit{R. Piene}, Contemp. Math. 241, 209--238 (1999; Zbl 0953.14031)] for construction and elementary properties): the key to prove the theorem is in fact a recursive relation for the class \(u(D, r)\) in terms of the classes \(u(D_i, r_i)\) of the induced pairs \((F_i/X_i,D_i)\). enumerative geometry; nodal curves; nodal polynomials; Bell polynomials; Enriques diagrams; Hilbert schemes Enumerative problems (combinatorial problems) in algebraic geometry, Divisors, linear systems, invertible sheaves, Jacobians, Prym varieties, Algebraic theory of abelian varieties Node polynomials for curves on surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems If X is a 0-dimensional subscheme of a smooth quadric \(Q\cong {\mathbb{P}}^ 1\times {\mathbb{P}}^ 1\) we investigate the behaviour of X with respect to the linear systems of divisors of any degree (a,b). This leads to the construction of a matrix of integers which plays the role of a Hilbert function of X; we study numerical properties of this matrix and their connection with the geometry of X. Further we put into relation the graded Betti numbers of a minimal free resolution of X on Q with that matrix, and give a complete description of the arithmetically Cohen- Macaulay 0-dimensional subschemes of Q. postulation; arithmetically Cohen-Macaulay 0-dimensional subschemes of a smooth quadric; linear systems of divisors; Hilbert function Giuffrida, S; Maggioni, R; Ragusa, A, On the postulation of \(0\)-dimensional subschemes on a smooth quadric, Pac. J. Math., 155, 251-282, (1992) Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Projective techniques in algebraic geometry, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series On the postulation of 0-dimensional subschemes on a smooth quadric | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Given an abelian algebraic group \(A\) over a global field \(F, {\alpha}\in A(F)\), and a prime \({\ell}\), the set of all preimages of \({\alpha}\) under some iterate of [\({\ell}\)] generates an extension of \(F\) that contains all \({\ell}\)-power torsion points as well as a Kummer-type extension. We analyze the Galois group of this extension, and for several classes of \(A\) we give a simple characterization of when the Galois group is as large as possible up to constraints imposed by the endomorphism ring or the Weil pairing. This Galois group encodes information about the density of primes \(\mathfrak{p}\) in the ring of integers of \(F\) such that the order of (\({\alpha}\) mod \(\mathfrak {p})\) is prime to \({\ell}\). We compute this density in the general case for several classes of \(A\), including elliptic curves and one-dimensional tori. For example, if \(F\) is a number field, \(A/F\) is an elliptic curve with surjective 2-adic representation and \({\alpha}\in A(F)\) with \({\alpha}{\notin}2A(F(A\)[4])), then the density of \(\mathfrak{p}\) with (\({\alpha}\) mod \(\mathfrak{p}\)) having odd order is 11/21. [6]R. Jones and J. Rouse, Galois theory of iterated endomorphisms, Proc. London Math. Soc. (3) 100 (2010), 763--794. Galois representations, Group varieties, Arithmetic ground fields for abelian varieties Galois theory of iterated endomorphisms | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In 1986 Eisenbud and Harris developed a theory to study the degeneration of a linear series on a family smooth curves as the curves degenerate to a certain type of reducible curve, the so-called compact type curve; all curves being defined over a base field of characteristic zero. Since then, some generalizations of this theory have appeared [see e.g. \textit{E.\ Esteves}, Mat. Contemp. 14, 21-35 (1998; Zbl 0928.14007) and \textit{M. Teixidor i Bigas}, Duke Math. J. 62, No. 2, 385--400 (1991; Zbl 0739.14006)].
In the present paper, the author proposes another generalization, removing the assumption on the characteristic of the base field and working with a more general family of curves, which he calls a smoothing family. This approach is more functorial in nature, and seems to be better suited to generalizations to higher-dimensional varieties and higher-rank vector bundles. In short, the author associates a functor to a smoothing family and the main result of paper states that this functor is representable by a projective scheme. The author also presents results on smoothing linear series from the special fiber when the dimension is expected, including the cases of positive and mixed characteristic. limit linear series; deformations Osserman, B.: A limit linear series moduli scheme. Ann. Inst. Fourier \textbf{56}(4), 1165-1205 (2006a) Special divisors on curves (gonality, Brill-Noether theory), Families, moduli of curves (algebraic) A limit linear series moduli scheme | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors investigate the problem how to extend their effective versions of Hilbert's Nullstellensatz in characteristic zero [Electron. Res. Announc. Am. Math. Soc. 2, No.2, 82--91 (1996; Zbl 0872.14042); Am. J. Math. 121, No. 4, 723--796 (1999; Zbl 0944.14002)] to the case of arbitrary characteristic. By using analytic division-interpolation methods, they are successful to prove characteristic-free effective versions of the Nullstellensatz. Effectivity, complexity and computational aspects of algebraic geometry, Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA)), Diophantine approximation, transcendental number theory Division-interpolation methods and Nullstellensätze. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(\mathcal{O}_K\) be a discrete valuation ring of mixed characteristics \((0, p),\) with residue field \(k.\) Using work by Sekiguchi and Suwa [On the unified Kummer-Artin-Schreir-Witt Theory, Preprint no. 111, Preprint series of the Laboratoire de Mathematiques Pures de Bordeaux (1999)], the authors of the paper under review construct some finite flat \(\mathcal{O}_K\)-models of the scheme \(\mu_{p^{n}, K}\) of \(p^n-\)th roots of unity, which they call Kummer group schemes. They carefully set out the general framework and algebraic properties of this construction. When \(k\) is perfect and \(\mathcal{O}_K\) is a complete totally ramified extension of the ring of Witt vectors \(W(k),\) they provide a parallel study of the Breuil-Kisin modules of finite flat models of \(\mu_{p^{n}, K},\) in such a way that the construction of Kummer groups and Breuil-Kisin modules can be compared. They construct these objects for \(n \leq 3.\) This leads the authors of the paper under review to conjecture that all finite flat models of \(\mu_{p^{n}, K}\) are Kummer group schemes.
Models of schemes have investigated in paper by \textit{P. Deligne} [in: Sem. Bourbaki 1970/71, Lect. Notes Math. 244, 123--165 (1971; Zbl 0225.14007)]. Models of unipotent group schemes have introduced and studied in papers by \textit{B. Yu. Veĭsfeĭler} and \textit{I. V. Dolgachev} [Izv. Akad. Nauk SSSR, Ser. Mat. 38, 757--799 (1974; Zbl 0314.14015)], by \textit{F. A. Bogomolov} [Math. USSR, Izv. 13, 499--555 (1979); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 42, 1227--1287 (1978; Zbl 0439.14002)], by \textit{W. C. Waterhouse} and \textit{B. Weisfeiler} [J. Algebra 66, 550--568 (1980; Zbl 0452.14013)].
A group scheme is finite of order \(m\) over an affine scheme \(S\) if it is locally free of rank \(m\) over \(S.\) Let \(H_K\) be a group scheme over \(K.\) Any flat \({\mathcal O}_K-\)group scheme \(G\) such that \( G_K \simeq H_K \) is called a model of \(H_K.\)
The authors of the paper under review consider models of \((\mathbb{G}_{m,K})^n\) constructed by successive extensions of affine, smooth, one-dimensional models of \(\mathbb{G}_{m,K}\) with connected fibers, called filtered group schemes. Kummer group schemes are defined as the kernels \(G\) of some well-chosen isogenies \(\mathcal{E} \to \mathcal{F} \) between filtered group schemes.
The range of topics of the paper is indicated by the titles of the sections: 1. Introduction; 2. Breuil-Kisin modules of finite flat group schemes; 3. The loop of \(\mu-\)matrices; 4. Relating lattices and matrices; 5. Computation of \(\mu-\)matrices for \(n = 3;\) 6. Sekiguchi-Suwa Theory; 7. Kummer group schemes; 8. Computation of Kummer group schemes for \(n = 3\) .
The authors most interesting results are the statement of Theorem 8.3.3 on computation of Kummer group schemes for \(n = 3,\) comments after it and proof of the theorem. group schemes; roots of unity; Breuil-Kisin module Mézard, A.; Romagny, M.; Tossici, D., Models of group schemes of roots of unity, Ann. inst. Fourier (Grenoble), 63, 3, 1055-1135, (2013) Group schemes, Valuation rings, Global ground fields in algebraic geometry Models of group schemes of roots of unity | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 describes several attempts to define schemes ``defined over \(\mathbb{F}_1\)'' in eight sections. Section one addresses the quetion whether there exists a large site containing Grothendieck schemes, in which one can define ``absolute Descartes powers'' \(\mathrm{Spec}\mathbb{Z}\times_{\Upsilon}\times\cdots\times_{\Upsilon}\mathrm{Spec}\mathbb{Z}\) over some deeper base ``\(\Upsilon\)'' than \(\mathbb{Z}\) so that \(\mathrm{Spec}\mathbb{Z}\times_{\Upsilon}\mathrm{Spec}\mathbb{Z}\) becomes a surface.
In Section two the author explains Deitmar's ``monoidal scheme theory'' in some detail, and in Section three he describes some fundamental examples, like affine and projective spaces. In Section four he introduces the notion of a loose graph and shows how one can construct projective completion from the loose graphs. Section five presents his own theory of \(\Upsilon\)-schemes, which is another approach to \(\mathbb{F}_1\) schemes. In Section six and seven he reviews several attempts to define absolute motives and their absolute zeta functions. In the final section eight he describes the approach of \textit{A. Connes} and \textit{C. Consani} [J. Number Theory 131, No. 2, 159--194 (2011; Zbl 1221.14002)] to understand the adèle class space through hyperring extension theory, in which a connection is revealed with certain group actions on projective spaces. motives; monodies; zeta functions; loose graphs Thas, K., The combinatorial-motivic nature of \(\mathbb{F}_1\)-schemes, Preprint Schemes and morphisms, Finite ground fields in algebraic geometry The combinatorial-motivic nature of \(\mathbb F_1\)-schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper summarizes recent results concerning singularities with respect to the Mather-Jacobian log discrepancies over an algebraically closed field of arbitrary characteristic. The basic point is that the inversion of adjunction with respect to Mather-Jacobian discrepancies holds under arbitrary characteristic. Using this fact, we will reduce several geometric properties of the singularities to jet scheme problems and try to avoid discussions that are distinctive to characteristic \(0\). Singularities of surfaces or higher-dimensional varieties, Arcs and motivic integration, Jets in global analysis Singularities in arbitrary characteristic via jet schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper is devoted to the development of Galois theory for extensions of fields of nonzero characteristic. For such fields, one can construct (see [\textit{A. Maurischat}, Trans. Am. Math. Soc. 362, No. 10, 5411--5453 (2010; Zbl 1250.13009)]) a Galois theory similar to Kolchin's theory for extensions of differential fields, if one replaces ordinary derivations by the so-called higher derivations. Since the fields of constants arising in this case are usually not algebraically closed, finite group schemes are used instead of algebraic groups. The paper contains the main concepts and basic results of this theory as an introduction. Its ``main part is to find computable criteria when higher derivations are iterative derivations, and furthermore when an iterative derivation on the function field of an abelian variety is compatible with the addition map''. For the inverse problem of Galois theory, it is shown ``that torsion group schemes of abelian varieties in positive characteristic occur as iterative differential Galois groups of extensions of iterative differential fields''. differential Galois theory; group schemes; elliptic curves Abstract differential equations, Inverse Galois theory, Algebraic theory of abelian varieties, Group schemes Torsion group schemes as iterative differential Galois groups | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A monomial curve \(C=C(a,b,c) \subset \mathbb{P}^3\) (over an algebraically closed field of characteristic 0) is the image of \(\mathbb{P}^1\) by the map: \([s:t] \mapsto [s^c:s^{c-a} t^a: s^{c-b} t^b: t^c]\), where \(c>b>a\) are relatively prime positive integers. Monomial curves have been extensively studied in recent years (many references on the subject are listed in this paper). The author assumes the point of view of liaison theory and shows that monomial curves are directly linked to schemes supported on one or two lines (theorem 2.1) and that, in particular, the case of one line follows from the existence of non-negative integer solutions of a certain linear equation with coefficients rational functions of \(a,b,c\) and of their greatest common divisors (theorem 1.5, corollary 1.6, theorem 1.7). This analysis is sufficient to conclude that an arithmetically Cohen-Macaulay monomial curve in \(\mathbb{P}^3\) is directly linked to a scheme supported on a line or is complete intersection (corollary 1.11). The discussion about two conjectures regarding the deficiency module \(M(C)= \bigoplus_{n\in \mathbb{Z}} H^1(I_C(n))\) concludes the paper: \(C\) is \((b-a-1)\)-Buchsbaum (conjecture 3.2) and \(M(C)\) is not direct sum of two nonzero modules (conjecture 3.4). -- Since the homogeneous ideal \(I\) of a monomial curve is generated by reduced binomials (i.e. a binomial without any monomial factor), the author introduces some partial orderings on reduced binomials to show that the multiplication by a power of the generators of the ideal of one or two lines maps the generators of \(I\) into the ideal of a suitable complete intersection. The matter of this paper was studied by the author since his doctoral dissertation [``Applications of liaison theory to schemes supported on lines, growth of the deficiency module and low rank vector bundle'' (Ph.D. thesis, Duke Univ. 1994)]. linkage; liaison theory; monomial curves; Cohen-Macaulay monomial curve; orderings on reduced binomials Linkage, Plane and space curves, Special algebraic curves and curves of low genus Monomial curves and schemes supported on 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 The present work studies deformations of projective Stanley-Reisner schemes associated to combinatorial manifolds. It continues work by the authors, who introduced in [\textit{K. Altmann} and \textit{J. A. Christophersen}, Manuscr. Math. 115, No. 3, 361--378 (2004; Zbl 1071.13008)] the cotangent cohomology of Stanley-Reisner rings for arbitrary simplicial complexes. The two main achievements of the present paper are the following. First, the authors give detailed descriptions of first order deformations and obstruction spaces and even dimension formulas in some cases (sections 4 and 5, theorems 4.6, 5.5, 5.6 and 5.7). Second, versal spaces of algebraic deformations are given for certain Stanley-Reisner surfaces, namely 2-dimensional combinatorial manifolds with vertex valencies not greater than six (section 6). Stanley-Reisner schemes; combinatorial manifolds; deformation theory Altmann, K; Christophersen, JA, Deforming Stanley-Reisner schemes, Math. Ann., 348, 513-537, (2010) Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes, Formal methods and deformations in algebraic geometry Deforming Stanley-Reisner 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 Here we introduce and discuss a definition of abstract Birkhoff interpolation problem for linear series on a smooth and projective curve \(X\). However, our definition depends on the choice of a finite covering \(f:X\to \mathbb P^1\). linear series on curves; Weierstrass points; Weierstrass multiple loci; ramification points Riemann surfaces; Weierstrass points; gap sequences, Projective techniques in algebraic geometry, Applications to coding theory and cryptography of arithmetic geometry An abstract Birkhoff interpolation problem for smooth projective 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 provide a real algebraic symbolic-numeric algorithm for computing the real variety \(V_{\mathbb R}(I)\) of an ideal \(I \subseteq \mathbb R [\mathbf x]\), assuming \(V_{\mathbb R}(I)\) is finite (while \(V_{\mathbb C}(I)\) could be infinite). Our approach uses sets of linear functionals on \(\mathbb R [\mathbf x]\), vanishing on a given set of polynomials generating \(I\) and their prolongations up to a given degree, as well as on polynomials of the real radical ideal \(^{\mathbb R} \sqrt I\) obtained from the kernel of a suitably defined moment matrix assumed to be positive semidefinite and of maximum rank. We formulate a condition on the dimensions of projections of these sets of linear functionals, which serves as a stopping criterion for our algorithm; this new criterion is satisfied earlier than the previously used stopping criterion based on a rank condition for moment matrices. This algorithm is based on standard numerical linear algebra routines and semidefinite optimization and combines techniques from previous work of the authors together with an existing algorithm for the complex variety. real solving; finite real variety; numerical algebraic geometry; semidefinite optimization Lasserre J.B., Laurent M., Rostalski P.: A prolongation-projection algorithm for computing the finite real variety of an ideal. Theoret. Comput. Sci. 410, 2685--2700 (2009) Real algebraic sets, Real algebra, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Real and complex fields, Symbolic computation and algebraic computation, Semidefinite programming A prolongation-projection algorithm for computing the finite real variety of an ideal | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems There has been enormous progress on the subgroup structure of finite and algebraic groups. In particular, much more is known about maximal subgroups. In this paper, the author focuses on two applications of this information. One is about generation of simple finite and algebraic groups. The basic idea is to describe the maximal subgroups of simple groups in a useful way, i.e., first to classify the maximal subgroups preserving some natural structure and then to show that essentially all others are almost simple subgroups acting irreducibly. For instance, the author proves that if \(G\) is a finite simple group which is not isomorphic to \(B_2(q)\) (\(q=2^n\) or \(3^n\)) or \(^2B_2(2^{2k+1})\), then with finitely many exceptions, \(G\) can be generated by an element of order 2 and an element of order 3. The second application is to coverings of curves. Many problems in this area can be translated to those about finite permutation groups (in particular, primitive permutation groups). One can use the knowledge on maximal subgroups to determine the possible group-theoretic solutions to the arithmetic and geometric data. Some conjectures are also presented. simple algebraic groups; subgroup structure; generation of finite simple groups; covers of curves; maximal subgroups; primitive permutation groups Robert M. Guralnick, Some applications of subgroup structure to probabilistic generation and covers of curves, Algebraic groups and their representations (Cambridge, 1997) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 517, Kluwer Acad. Publ., Dordrecht, 1998, pp. 301 -- 320. Finite simple groups and their classification, Probabilistic methods in group theory, Generators, relations, and presentations of groups, Linear algebraic groups over arbitrary fields, Maximal subgroups, Simple groups, Coverings of curves, fundamental group, Primitive groups Some applications of subgroup structure to probabilistic generation and covers of curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Here we propose a partial multivariate polynomial interpolation problem in which we only prescribe some of the first order partial derivates
along the coordinate axes. Projective techniques in algebraic geometry, Vector and tensor algebra, theory of invariants, Numerical interpolation A partial multivariate polynomial interpolation problem | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Spectrahedra are the feasible sets of semidefinite programming and provide a central link between real algebraic geometry and convex optimization. In this expository paper, we review some recent developments on effective methods for handling spectrahedra. In particular, we consider the algorithmic problems of deciding emptiness of spectrahedra, boundedness of spectrahedra as well as the question of containment of a spectrahedron in another one. These problems can profitably be approached by combinations of methods from real algebra and optimization. spectrahedron; spectrahedral computation; real algebraic geometry; convex algebraic geometry; containment Convex sets in \(n\) dimensions (including convex hypersurfaces), Effectivity, complexity and computational aspects of algebraic geometry, Real algebraic sets, Symbolic computation and algebraic computation, Semidefinite programming Some recent developments in spectrahedral computation | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(k\) be a finite field of characteristic \(p\) and let \(X \rightarrow \operatorname{Spec} k[[t]]\) be a semistable family of varieties over \(k\). We prove that there exists a Clemens-Schmid type exact sequence for this family. We do this by constructing a larger family defined over a smooth curve and using a Clemens-Schmid exact sequence in characteristic \(p\) for this new family. \(p\)-adic cohomology, crystalline cohomology, Varieties over finite and local fields A Clemens-Schmid type exact sequence over a local basis | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems \(A\) désigne le spectre de \(\mathbb{Z}[t]\). L'A. définit, alors, la topologie primitive pour une catégorie donnée \({\mathcal C}\) de schémas de la façon suivante: On suppose qu'un recouvrement pour un objet de \({\mathcal C}\) est un singleton \((U,X)\) tel que l'application \(U\mapsto X\) soit surjective et puisse être considérée comme la composée d'un plongement ouvert dans \(X \times A^n\) et de la projection naturelle. L'étude d'une telle topologie, nourrie de très nombreux exemples, est menée de manière exhaustive; ainsi, une suite de faisceaux de groupes abéliens y est exacte si et seulement si elle est exacte dans son évaluation par rapport à un anneau primitif quelconque. Cette théorie permet de lever une conjecture de Gersten: Cependant, l'intérêt principal de ce travail est de montrer que la topologie primitive diffère peu par ses propriétés de la topologie de Zariski, qu'elle permet de trouver des résultats tout à fait analogues, mais d'autres encore, ce qui atteste la richesse du sujet abordé. topology of a scheme Walker, M.: The primitive topology of a scheme. J. algebra 201, 656-685 (1998) Schemes and morphisms, Global topological rings The primitive topology of a scheme | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this book the authors indicate a general procedure for obtaining multiple point formulas for morphisms \(f:V\to W\) of non-singular complex varieties, under quite general conditions. Intuitively a \(k\)-tuple point of \(f\) is a point \(x\) such that \(f^{-1}f(x)\) consists of at least \(k\) points, and the \(k\)-tuple locus is the subvariety of \(V\) consisting of all such points. A \(k\)-tuple point formula is a formula that expresses the class of the \(k\)-tuple locus in a suitable intersection theory for \(V\), in terms of polynomials in the Chern classes of the virtual normal bundle \(\nu(f) =f^\ast TW-TV\). For the formulas to be meaningful certain relations between the numbers \(\dim(V)\), \(\dim(W)\) and \(k\) must be satisfied. To obtain \(k\)-tuple formulas it is necessary to have a manageable parameter space for \(k\)-tuple points in \(V\). In this space one should be able to define a class of \(k\)-tuple points of \(f\), that, at least formally, represents the subvariety consisting of \(k\)-tuple points of \(V\) that are \(k\)-tuple points of the morphism \(f\). Unfortunately the variety consisting of \(k\)-tuple points of \(V\) that are \(k\)-tuple points of \(f\) often contain excess components that correspond to improper \(k\)-tuple points of \(f\). In order that the \(k\)-tuple formulas should have a geometric meaning it is necessary to give procedures for removing the improper ones. The authors give parameter spaces for the \(k\)-tuple points of \(V\) for \(k=2,3,4\), and define the class of \(k\)-tuple points of \(f\) in the Chow ring of these spaces. They also perform the necessary calculations in the Chow ring to obtain double- and triple-point formulas under quite general conditions on \(f\). We give some details of the construction of the parameter spaces used for double and triple points. To this end we denote by \(H^i(V)\) the Hilbert scheme of colength \(k\) points of a scheme \(V\). For the double points the authors use the two-sheeted cover \(\widehat{H^2(V)}\) of \(H^2(V)\). They use this space to obtain the well known double point formula [see, e.g., \textit{W. Fulton}, ``Intersection theory'' (1998; Zbl 0885.14002)]. The construction of the parameter space and the calculations to obtain the formula are performed in chapters 1 and 2. For the triple points the situation is much more complicated. The parameter space \(\widehat {H^3(V)}\) of triple points of \(V\) is the locus in \(V^3\times H^2(V)^3\times H^3(V)\) consisting of points of the form \((p_1, p_2, p_3, d_{12}, d_{23}, d_{13}, t)\) satisfying the relations:
\[
\begin{cases} p_i\subset d_{ij} \subset t\\ p_j=\text{Res}(p_i,d_{ij})\\ p_k=\text{Res}(d_{ij},t) \quad \text{with} \quad \{i,j,k\} =\{1,2,3\}, \end{cases}
\]
where \(\text{Res}(\eta,\xi)\) denote the residual closed point of the \((k-1)\)-tuple \(\eta\) contained in the \(k\)-tuple \(\xi\). The authors follow a method of \textit{Z. Ran} [Acta Math. 155, 81-101 (1985; Zbl 0578.14046)] and \textit{T. Gaffney} [Math. Ann. 295, 269-289 (1993; Zbl 0841.14002)], and define a class that formally represents the triple points of \(f\). In chapter 3 they obtain the well known triple points formula of \textit{S. L. Kleiman} [Acta Math. 147, 13-49 (1993; Zbl 0479.14004)] and \textit{F. Ronga} [Compos. Math. 53, 211-223 (1984; Zbl 0563.57014)] under quite general conditions (see also \textit{W. Fulton}, loc. cit.). Many of the computations are tedious and are performed in chapter 4. The intersections of the locus of triple points corresponding to the class of triple points have excess intersections in the presence of \(S_2\)-singularities, and are difficult to handle. The authors indicate how this can be done, by treating the case \(\dim(V)=2\), \(\dim(W)=3\), when \(f\) has \(S_2\) singularities. A similar treatment of the triple points have been given by \textit{P. le Barz} [Duke Math. 5, 57, 925-946 (1988; Zbl 0687.14042) and Bull. Soc. Math. Fr. 112, 303-324 (1984; Zbl 0561.14021)]. The novel part of the book is the case of quadruple points. Here there is a real leap in difficulty. The second part of the book is devoted to this case. We shall not even indicate the construction of the parameter space. The reader who enjoys concrete, detailed geometry, and who likes to draw pictures representing ideals of the same colength, having various properties, will enjoy this part of the work. multiple point formulas; Hilbert scheme; Chow ring; Chern class of virtual normal bundle Danielle, D.; Le Barz, P.: Configuration spaces over Hilbert schemes and applications. Lecture notes in math. 1647 (1996) Parametrization (Chow and Hilbert schemes), Research exposition (monographs, survey articles) pertaining to algebraic geometry, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Special surfaces Configuration spaces over Hilbert schemes and applications | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A method to compute the degrees of the Segre classes of a subscheme of complex projective space is presented. The method is based on generic residuation and intersection theory. Finally, the authors provide a symbolic implementation using the software system Macaulay2. The numerical implementation, using numerical homotopy methods and the regenerative cascade algorithm, is also implemented in the software package Bertini. Segre classes; computational algebraic geometry; numerical homotopy methods Eklund, D., Jost, C., Peterson, C.: A method to compute Segre classes of subschemes of projective space. J. Algebra Appl. (2011, to appear). arXiv:1109.5895 [math.AG] Computational aspects and applications of commutative rings, Computational aspects in algebraic geometry, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Numerical computation of solutions to systems of equations, General theory of numerical methods in complex analysis (potential theory, etc.) A method to compute Segre classes of subschemes of projective space | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\subset {\mathbb P}^n\) be a generically reduced projective scheme. A fundamental goal in computational algebraic geometry is to compute information about \(X\) even when defining equations for \(X\) are not known. We use numerical algebraic geometry to develop a test for deciding if \(X\) is arithmetically Gorenstein and apply it to three secant varieties. Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Computational aspects of higher-dimensional varieties, Symbolic computation and algebraic computation Numerically testing generically reduced projective schemes for the arithmetic Gorenstein property | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 projective minimal Gorenstein 3-fold of general type with \(\mathbb{Q}\)-factorial terminal singularities. The author classifies minimal Gorenstein 3-folds of general type according to the birationality of 3-canonical system on \(X\). Let \(\phi_m\) be the \(m\)-canonical map corresponding to the complete linear system \(|mK_X|\). Let \(d\) be the dimension of \(\phi_1(X)\). If \(\phi_3\) on \(X\) is not birational, then \(X\) must be one of the following types:
(1) \(p_g(X)\leq 7\);
(2) \(p_g\geq 8\), while: {\parindent=6mm \begin{itemize}\item[(a)] \(d=3\), \(X\) contains a surface which has a family of curves of genus 2; \item[(b)] \(d=2\), \(X\) contains a surface which has a family of genus \(\leq 3\); \item[(c)] \(d=1\), \(X\) contains a surface \(S\) such that either \(S\) has a family of curves of genus \(\leq 3\) or \(S\) satisfies \(p_g(S)=1\), \(K_{S_0}^2\leq 9\), where \(S_0\) is the minimal model of \(S\).
\end{itemize}} projective minimal Gorenstein 3-fold; birationality Zhou, Y, The 3-canonical system on 3-folds of general type, Osaka J Math, 48, 91-98, (2011) Rational and birational maps, Families, moduli, classification: algebraic theory, \(3\)-folds The 3-canonical system on 3-folds of general type | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(D\) be a smooth Cartier divisor on a smooth quasi-projective surface \(S\). The Hilbert scheme \((S \setminus D)^{[n]}\) of \(n\) points on \(S \setminus D\) is not proper, but \textit{J. Li} and \textit{B. Wu} have constructed a compactification relative to \(D\) [``Good degeneration of Quot-schemes and coherent systems'', Preprint, \url{arXiv:1110.0390}], called the relative Hilbert scheme. The author uses the moduli stack of stable ideal sheaves and the stack of expanded degenerations of \textit{J. Li} [J. Differ. Geom. 57, 509--578 (2001; Zbl 1076.14540)] to produce the generating function for the normalized Poincaré polynomial of the relative Hilbert scheme of points analogous to the generating function for the Hilbert scheme of \(n\) points given by \textit{L. Göttsche} and \textit{W. Soergel} [Math. Ann. 296, No. 2, 235--245 (1993; Zbl 0789.14002)]. When \(S = \mathbb P^2\) and \(D \subset \mathbb P^2\) is a line, the cohomology groups of the relative Hilbert scheme are computed and it is shown that the natural map from the Chow group to the Borel-Moore homology is an isomorphism. Hilbert scheme of points; relative Hilbert scheme; Poincaré polynomial Iman Setayesh, Relative Hilbert scheme of points, ProQuest LLC, Ann Arbor, MI, 2011. Thesis (Ph.D.) -- Princeton University. Parametrization (Chow and Hilbert schemes), Families, moduli, classification: algebraic theory Relative Hilbert scheme 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 author studies the structure of polycyclic-by-finite affine group schemes over fields of characteristic \(p>0.\) He goes to great length to define (finite and infinite) cyclic affine (unipotent) group schemes, but does not define his notion of polycyclic affine group schemes. He also defines closed subgroup schemes and normal closed subgroup schemes, but not precisely his notion of extensions of affine group schemes. So the reader needs a background considerably above the level of the definitions of the first chapter. On the other hand the motivation and quotation of all theorems is contained in the first chapter, the proofs and technical details being postponed to the following chapters. This makes the paper quite easy to read for a reader with the required background.
The motivation of the paper is the observation that the finiteness condition of being algebraic for affine group schemes over a field corresponds essentially to the finiteness condition of being finitely generated for groups. This is reasonable in characteristic zero but too restrictive in characteristic \(p>0.\) The class of polycyclic-by-finite affine group schemes is one class which shows the need for a less restrictive condition. One of the key conditions for a reasonable finiteness condition is that the space of primitive elements be finite dimensional. Chain conditions, decompositions, and representations of polycyclic-by-finite affine group schemes especially over algebraically closed fields are investigated. Hopf algebra; polycyclic-by-finite affine group schemes; characteristic p Crawley-Boevey, W. W.: Polycyclic-by-finite affine group schemes. Proc. London math. Soc. 52, No. 3, 495-516 (1986) Group schemes, Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act, Group varieties, Solvable groups, supersolvable groups Polycyclic-by-finite affine group schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The notion of adeles on algebraic surfaces was first introduced and studied by Parshin and then generalised to arbitrary noetherian schemes by Beilinson. In the paper under review the author introduces the notion of a category \(C_n\) which is closely related to the iterated functor \(\lim_{\leftrightarrow}\) invented by Beilinson. In particular, he proves that an adelic space on a \(n\)-dimensional noetherian scheme is an object of \(C_n\). He also computes the endomorphism algebra of a \(n\)-dimensional local field. local fields; adeles Osipov, D. V., Adeles on \textit{n}-dimensional schemes and categories \(C_n\), Internat. J. Math., 18, 3, 269-279, (2007) Formal neighborhoods in algebraic geometry, Categories of topological spaces and continuous mappings [See also 54-XX] Adeles on \(n\)-dimensional schemes and categories \(C_{n}\) | 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 \(\mathbb{H}_{ab}(H)\) be the equivariant Hilbert scheme parametrizing the zero-dimensional subschemes of the affine plane \(k^2\), fixed under the one-dimensional torus \(T_{ab}= \{(t^{-b}, t^a),\, t\in k^*\}\) and whose Hilbert function is \(H\). This Hilbert scheme admits a natural stratification in Schubert cells which extends the notion of Schubert cells on Grassmannians. However, the incidence relations between the cells become more complicated than in the case of Grassmannians. In this paper, we give a necessary condition for the closure of a cell to meet another cell. In the particular case of Grassmannians, it coincides with the well known necessary and sufficient incidence condition. There is no known example showing that the condition wouldn't be sufficient. Hilbert scheme; zero-dimensional scheme; Schubert cell; Grassmannian Evain, L., Incidence relations among the Schubert cells of equivariant Hilbert schemes, \textit{Math. Z.}, 242, 4, 743-759, (2002) Parametrization (Chow and Hilbert schemes), Grassmannians, Schubert varieties, flag manifolds Incidence relations among the Schubert cells of equivariant punctual Hilbert schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a connected smooth projective curve of genus \(g\) over a field \(K\). For any ample line bundle \(L\) over \(X\) of degree \(\text{deg}(L) \geq 2g+1\), there exist nonzero global sections \(s_0, \dots, s_N \in H^0(X, L)\) such that the associated morphism
\begin{align*}
X \longrightarrow \mathbb{P}^N, \ \ \ \ \ p \mapsto (s_0(p): \cdots : s_N(p))
\end{align*}
gives an embedding of the curve into projective space.
The purpose of this paper is to find an analogous statement for a Berkovich skeleton in nonarchimedean geometry: Assume that \(K\) is an algebraically closed field which is complete with respect to a non-trivial nonarchimedean absolute value. Let \(X^{\text{an}}\) denote the Berkovich analytification of \(X\) and let \(\Gamma\) be a skeleton of \(X^{\text{an}}\). Skeletons are certain polyhedral complexes with integral structure such that one can deformation retract \(X^{\text{an}}\) onto these skeletons.
If \(g \geq 2\) and \(\text{deg}(L) \geq 3g-1\), then
\begin{align*}
X^{\text{an}} \longrightarrow \mathbb{TP}^N, \ \ \ \ \ p=(p, |\,\cdot \,|) \mapsto (-\text{log} |s_0(p)|: \cdots : -\text{log} |s_N(p)|)
\end{align*}
is a faithful tropicalization of \(\Gamma\) into tropical projective space; in the sense that, its restriction to \(\Gamma\) is a homeomorphism onto its image preserving integral structures.
As an application, they show for suitable affine curve \(X\), the Berkovich analytification \(X^{\text{an}}\) is homeomorphic to the inverse limit of tropicalizations over affine embeddings \(X \hookrightarrow \mathbb{A}^N\) given by polynomials of degree at most some effective bound. algebraic curves; linear system; faithful tropicalization; skeleton; Berkovich space; nonarchimedean geometry Foundations of tropical geometry and relations with algebra, Rigid analytic geometry, Divisors, linear systems, invertible sheaves Effective faithful tropicalizations associated to linear systems on curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For the entire collection see Zbl 0588.00014.]
It is shown that, if X is a 1-dimensional scheme, then \(SK_ 1(X)\cong H^ 1_{Za}(X;\underline K_ 2)\) if X is affine, reduced or Cohen- Macaulay. In fact, if X is a 1-dimensional Cohen-Macaulay scheme there is an exact sequence
\[
0\quad \to \quad H^ 1_{Za}(X;\underline K_{n+1})\quad \to \quad K_ n(X)\quad \to \quad H^ 0_{Za}(X;\underline K_ n)\quad \to \quad 0.
\]
The author gives several examples related to these results and related to the question of the existence of a Gersten type of spectral sequence \(E_ 2=H^ i_{Za}(X;\underline K_{n+i})\Rightarrow K_ n(X),\) of the type established by Quillen in the smooth case.
The results in this paper, on Cohen-Macaulay schemes, are superseded in a recent preprint of \textit{M. Levine}. K-theory; 1-dimensional Cohen-Macaulay scheme Weibel, C.: K-theory of 1-dimensional schemes. AMS contemp. Math. 55, 811-818 (1986) Applications of methods of algebraic \(K\)-theory in algebraic geometry, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects), Curves in algebraic geometry K-theory of 1-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 It is a classical fact (see for instance [\textit{D. N. Bernstein}, Funct. Anal. Appl. 9, 183--185 (1976; Zbl 0328.32001); translation from Funkts. Anal. Prilozh. 9, No. 3, 1--4 (1975)]) that the number of solutions in \(({\mathbb C}\setminus\{0\})^k\) of a given system of \(k\) Laurent polynomials in \(k\) variables which is \textit{generic} with respect to having the set of nonzero coefficients in finite sets \(A_1,\ldots, A_k\subset{\mathbb Z}^k\) is a finite number called the \textit{mixed volume} of the convex hulls in \({\mathbb R}^k\) of these sets. For \(k+1\) polynomials in \(k\) variables, the system will be solvable generically only if some additional combinatorial conditions are imposed in the finite sets \(A_1,\dots, A_{k+1}\subset{\mathbb Z}^k\), and in general the variety of solvable systems has codimension \(1\) in the space of the coefficients of the polynomials (cf. [\textit{B. Sturmfels}, J. Algebr. Comb. 3, No. 2, 207--236 (1994; Zbl 0798.05074)], [\textit{C. D'Andrea} and \textit{M. Sombra}, Proc. Lond. Math. Soc. (3) 110, No. 4, 932--964 (2015; Zbl 1349.14160)]).
In the paper under review the autor extends the combinatorial definitions of \textit{defect} and \textit{essential} given in [\textit{B. Sturmfels}, J. Algebr. Comb. 3, No. 2, 207--236 (1994; Zbl 0798.05074)] and some of their properties, to systems of \(n\) polynomials in \(k\), variables with \(n\geq k,\) and considers in all cases generic \textit{consistent} (i.e. with solutions) systems of polynomial equations. The main result of this paper is a description of this set of solutions which generalizes the cases \(n=k\) and \(n=k+1\). This description not only involves not geometric, but also arithmetic and combinatorial invariants as those considered in [\textit{B. Sturmfels}, J. Algebr. Comb. 3, No. 2, 207--236 (1994; Zbl 0798.05074)], [\textit{C. D'Andrea} and \textit{M. Sombra}, Proc. Lond. Math. Soc. (3) 110, No. 4, 932--964 (2015; Zbl 1349.14160)]. Applications to root number counting and computation of Euler characteristic and geometric genus of generic varieties are given as a consequence of this result. Newton polyhedra; Laurent polynomials; generically inconsistent systems; resultants Toric varieties, Newton polyhedra, Okounkov bodies Discrete invariants of generically inconsistent systems of Laurent polynomials | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors study the geometry and topology of the rank stratification of the solution variety for polynomial system solving. This variety is the set of pairs (system, solution) such that the derivative of the system at the solution has a given rank. It is a smooth manifold and it is endowed with a natural Riemannian structure. The authors' approach is to study the gradient flow of the Frobenius condition number defined on each stratum. solution variety; polynomial system solving; Frobenius number Beltrán, C; Shub, M, On the geometry and topology of the solution variety for polynomial system solving, Found Comput Math, 12, 719-763, (2017) Dynamics induced by flows and semiflows, Geodesics in global differential geometry, Singularities of surfaces or higher-dimensional varieties, Numerical computation of solutions to systems of equations, Global methods, including homotopy approaches to the numerical solution of nonlinear equations On the geometry and topology of the solution variety for polynomial system solving | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(F\) be a number field. The author defines a topology \(\overline{\mathrm{Spec}(\mathcal O_F)}_W\) so that the groups \(H^i_c(\overline{\mathrm{Spec}(\mathcal O_F)}_W,\mathbb Z)\) recover the groups introduced by Lichtenbaum in his study of Weil-étale cohomology groups. Let \(\mathcal X\) be a separated scheme of finite type over \(\mathbb Z\). Artin-Verdier defines a topo \(\overline{\mathcal X}_{\mathrm{et}}\) so that there are complementary open and closed immersions
\[
\mathcal X_{\mathrm{et}}\to \overline{\mathcal X}_{\mathrm{et}} \leftarrow Sh(\mathcal X_\infty),
\]
where \(\mathcal X_\infty\) is the topological quotient space \(\mathcal X(\mathbb C)/ {\mathrm{Gal}(\mathbb C/\mathbb R)}\). For \(\mathcal Y=\mathcal X\) or \(\overline{\mathcal X}\), let
\[
{\mathcal Y}_W={\mathcal Y}_{\mathrm{et}}\times _{\overline{\mathrm{Spec}(\mathbb Z)}_{\mathrm{et}}} \overline{\mathrm{Spec}(\mathbb Z)}_W.
\]
In the case where \(\mathcal X\) is regular and proper over \(\mathbb Z\), the author proves that the cohomology groups of \(\mathcal X_W\) satisfy some (but not all) properties that one hopes for the Weil-étale cohomology groups. Weil-étale cohomology Flach, M.; Morin, B., On the Weil-étale topos of regular arithmetic schemes, Documenta Mathematica, 17, 313-399, (2012) Étale and other Grothendieck topologies and (co)homologies, Zeta functions and \(L\)-functions, \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture, Grothendieck topologies and Grothendieck topoi On the Weil-étale topos of regular arithmetic 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 For any prime power \(q\) and any dimension \(s\geq 1\), we present a construction of \((t,s)\)-sequences in base \(q\) with finite-row generating matrices such that, for fixed \(q\), the quality parameter \(t\) is asymptotically optimal as a function of \(s\) as \(s\to \infty \). This is the first construction of \((t,s)\)-sequences that yields finite-row generating matrices and asymptotically optimal quality parameters at the same time. The construction is based on global function fields. We put the construction into the framework of \((u,\mathbf {e},s)\)-sequences that was recently introduced by Tezuka. In this way we obtain in many cases better discrepancy bounds for the constructed sequences than by previous methods for bounding the discrepancy. low-discrepancy sequence; \((t, s)\)-sequence; finite-row generating matrices; global function field Hofer, R.; Niederreiter, H., A construction of \((t, s)\)-sequences with finite-row generating matrices using global function fields, Finite fields appl., 21, 97-110, (2013) Pseudo-random numbers; Monte Carlo methods, Special sequences, Irregularities of distribution, discrepancy, Finite ground fields in algebraic geometry A construction of \((t,s)\)-sequences with finite-row generating matrices using global function fields | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This article studies primitive multiple schemes on a smooth irreducible variety. A primitive multiple scheme is a Cohen-Macaulay scheme \(Y\) such that the associated reduced scheme \(X = Y_{\textrm{red}}\) is smooth and irreducible and \(Y\) can be locally embedded in a smooth variety of dimension \(\textrm{dim}(X)+1\). Let \(Y = X_n\) be a primitive multiple scheme of multiplicity \(n\) with \(X = (X_n)_{\textrm{red}}\). Each primitive multiple scheme structure has a filtration \(X = X_1 \subset X_2 \subset ... \subset X_{n-1} \subset X_n\) where each \(X_i\) is a primitive multiple scheme of multiplicity \(i\). If \(\mathscr{I}_{X_j}\) denotes the ideal sheaf of \(X_j\) inside \(X_n\), then there exist a line bundle \(L\) on \(X\) such that \(\mathscr{I}_{X_j}/\mathscr{I}_{X_{j+1}} = L^j\).
\begin{itemize}
\item[(1)] It has been shown in [\textit{J.-M. Drézet}, Adv. Geom. 7, No. 4, 559--612 (2007; Zbl 1135.14017)] that \(X_n\) is locally trivial and is hence parametrized by \(H^1(X,\mathscr{G}_n)\) where \(\mathscr{G}_n(U)\) is the group of automorphisms of \(\mathscr{O}_X(U)[t]/(t^n)\).
\item[(2)] In \textbf{Proposition} \(4.3.1\), it is shown that \(\mathbb{P}(H^1(T_X \otimes L))\) parametrizes primitive double schemes on \(X\) recovering results of [\textit{Bayer, Dave; Eisenbud, David} Trans. Am. Math. Soc. 347, No. 3, 719-756 (1995). Zbl 0853.14016]. It is then shown in \textbf{Proposition} \(4.3.2\) that obstructions to extending \(X_n\) to a primitive multiple scheme \(X_{n+1}\) lie in \(H^2((\Omega_{X_2|_X})^* \otimes L^n)\) and if there exists such an extension, all such extensions are parametrized by the orbits of the action of \(\textrm{Aut}(X_n)\) on \(H^1((\Omega_{X_2|_X})^* \otimes L^n)\).
\item[(3)] The ideal sheaf \(\mathscr{I}_{X,X_n}\) of \(X\) inside \(X_n\) is a line bundle on \(X_{n-1}\). It has been shown in \textbf{Proposition} \(4.5.2\) that the obstructions to extending \(X_n\) to \(X_{n+1}\) such that \(\mathscr{I}_{X,X_{n+1}}|_{X_{n-1}} = \mathscr{I}_{X,X_n}\) lie in \(H^2(T_X \otimes L^n)\).
\item[(4)] Suppose that \(\mathbb{E}\) is a vector bundle on \(X_n\). Let \(\mathbb{E}|_X = E\). Suppose \(X_n\) can be extended to \(X_{n+1}\). It is shown in \textbf{Proposition} \(7.1.1\) that the obstructions to extending \(\mathbb{E}\) to \(X_{n+1}\) lie in \(H^2(E \otimes E^* \otimes L^n)\).
\item[(5)] Suppose \(\tau_n = \Omega_{X_n}^*\). Let \(\textrm{Aut}_0(X_n)\) is the group of automorphisms of \(X_n\) inducing identity on \(L\). It is shown in \textbf{Theorem} \(6.3.2\) that there is a natural bijection \(\textrm{Aut}_0(X_n) = H^0(X_n, \mathscr{I}_X\tau_n)\) which is not a group homomorphism unless \(n = 2\).
\item[(6)] It has been shown in \(7.2.3\) that if \(H^0(X_n, \mathscr{I}_X\tau_n) = 0\) and hence \(\textrm{Aut}(X_n) = \mathbb{C}^*\), the extensions of \(X_n\) to \(X_{n+1}\) are parametrized by a weighted projective space.
\item[(7)] As a result of the analysis, it has been shown that for \(X = \mathbb{P}^n\), \(n \geq 3\) there exists only trivial primitive multiple schemes while for \(n = 2\) there exists exactly two non-trivial primitive multiple schemes: \(X_2\) of multiplicity \(2\) corresponding to the line bundle \(\mathscr{O}_{\mathbb{P}^2}(-3)\) and \(X_4\) of multiplicity \(4\) corresponding to the line bundle \(\mathscr{O}_{\mathbb{P}^2}(-1)\), both of which are non-projective (and have trivial dualizing bundle). On the other hand, for rank two projective bundles on a curve there exists infinite nested sequences of primitive multiple schemes (of increasing multiplicity) all of which are projective.
\end{itemize} multiple structures; obstructions; sheaves of nonabelian groups Algebraic moduli problems, moduli of vector bundles, Formal neighborhoods in algebraic geometry Primitive multiple schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a quasi-compact scheme and \(x\in X \). By \(\operatorname{cp}(X)\) and \(\operatorname{cl}(x)\), we denote the set of closed points of \(X\) and the closure of the subset \(\{x\}\). Let \(S\) be a nonempty subset of \(\operatorname{cp}(X)\). We define the \(S\)-Zariski topology graph on the scheme \(X\), denoted by \(\Gamma_Z(X,S)\), as an undirected graph whose vertex set is the set \(X\), for two distinct vertices \(x\) and \(y\), there is an arc from \(x\) to \(y\), denoted by \(x\sim y \), whenever \(\operatorname{cl}(x)\cap \operatorname{cl}(y)\cap S\neq\emptyset \). In this paper, we study the connectivity properties of the graph \(\Gamma_Z(X,S)\), we establish the relationship between the connectivity of the graph \(\Gamma_Z(X,S)\) and the structure of irreducible components of the scheme \(X\). Also, we characterize when the complement graph of the Zariski topology graph \(\Gamma_Z(X,S)\) is a complete multipartite graph. scheme; irreducible component; closed point; generic point Planar graphs; geometric and topological aspects of graph theory, Connectivity, Schemes and morphisms The Zariski topology graph on 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 New techniques for dealing with problems of numerical stability in computations involving multivariate polynomials allow a new approach to real world problems. Using a modelling problem for the optimization of oil production as a motivation, we present several recent developments involving border bases of polynomial ideals. After recalling the foundations of border basis theory in the exact case, we present a number of approximate techniques such as the eigenvalue method for polynomial system solving, the AVI algorithm for computing approximate border bases, and the SOI algorithm for computing stable order ideals. To get a deeper understanding for the algebra underlying this approximate world, we present recent advances concerning border basis and Gröbner basis schemes. They are open subschemes of Hilbert schemes and parametrize flat families of border bases and Gröbner bases. For the reader it will be a long, tortuous, sometimes dangerous, and hopefully fascinating journey from oil fields to Hilbert schemes. oil field; polynomial system solving; eigenvalue method; Buchberger-Möller algorithm; border basis; approximate algorithm; border basis scheme; Gröbner basis scheme; Hilbert scheme; \texttt{CoCoA} Kreuzer, M.; Poulisse, H.; Robbiano, L., From oil fields to Hilbert schemes, (Robbiano, L.; Abbott, J., Approximate Commutative Algebra, Texts and Monographs in Symbolic Computation, (2009), Springer-Verlag Wien), 1-54 Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.), Solving polynomial systems; resultants, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Parametrization (Chow and Hilbert schemes) From oil fields to Hilbert schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(Q\to Y\) be a family of 2-dimensional quadrics over a 3-dimensional base with \(Q\) and \(Y\) smooth and consider its relative Fano scheme of lines \(\rho: M\to Y\). In this paper, a criterion for the smoothness of \(M\) is given and the bounded derived category \(\mathcal D^b(M)\) of coherent sheaves on \(M\) is studied.
Let \(D_r\subset Y\) be the locus of quadrics of corank at least \(r\). Under the assumptions that the generic fibre of \(Q\to Y\) is smooth, \(D_3=\emptyset\), and \(D_2\) consists of finitely many points \(y_1,\dots,y_N\) all of which are ordinary double points of \(D_1\), the author proves that \(M\) is smooth and there is a semiorthogonal decomposition
\[
\mathcal D^b(M)=\bigl\langle \mathcal D^b(X^+),\mathcal D^b(Y,\mathcal B_0),\{\mathcal O_{\Sigma_i^+}\}_{i=1}^N\bigr\rangle
\]
obtained as follows. For \(i=1,\dots,N\), the fibre \(\rho^{-1}(y_i)\) is the union of two planes and \(\mathbb P^2\cong \Sigma_i^+\subset M\) is chosen as one of them. Then it is shown that the structure sheaves \(\mathcal O_{\Sigma_i^+}\) form a completely orthogonal exceptional collection. The Stein factorisation \(M\to X\to Y\) of \(\rho\) consists of a generically conic bundle \(M\to X\) and the double cover \(X\to Y\) ramified over \(D_1\). Let \(M^+\) be the flip of \(M\) in the planes \(\Sigma_i^+\). The author proves that the induced map \(M^+\to X\) factors as \(M^+\to X^+\to X\) where \(M^+\to X^+\) is a \(\mathbb P^1\)-fibration and \(X^+\to X\) is a small resolution of singularities. It follows that \(\mathcal D^b(X^+)\) can be embedded into \(\mathcal D^b(M)\) via the pull-back along \(M^+\to X^+\) followed by the canonical embedding \(\mathcal D^b(M^+)\subset \mathcal D^b(M)\). Finally, \(\mathcal B_0\) is the sheaf of the even part of the Clifford algebra associated to the quadric fibration \(Q\to Y\) and it is shown that \(\mathcal D^b(Y,\mathcal B_0)\) is equivalent to the left orthogonal complement of \(\mathcal D^b(X^+)\) in \(\mathcal D^b(M^+)\). The results are used by \textit{C. Ingalls} and the author in [``On nodal Enriques surfaces and quartic double solids'', \url{arXiv:1012.3530}] to provide a description of the derived category of a nodal Enriques surface. quadric fibration; Fano scheme of lines; derived category; semiorthogonal decomposition; Clifford algebra A. Kuznetsov, Scheme of lines on a family of 2-dimensional quadrics: geometry and derived category, math. AG/arXiv:1011. 4146. Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Brauer groups of schemes, Quadratic spaces; Clifford algebras, Fibrations, degenerations in algebraic geometry, Families, moduli, classification: algebraic theory, Grassmannians, Schubert varieties, flag manifolds Scheme of lines on a family of 2-dimensional quadrics: geometry and derived category | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For a reduced projective scheme over the ring of integers of a number field, the set of places over which the fibres of the scheme are not reduced is a finite set. We give an explicit upper bound for the product of the norms of places in this set. For this purpose, we introduce a generalization of the notion of height over the adelic ring. We reduce the general case of a scheme of pure dimension to the case of a hypersurface by using the theory of Chow varieties. The case of a hypersurface is then treated with the help of the resultant of the equation of the hypersurface with some partial derivatives of the equation. Global ground fields in algebraic geometry, Parametrization (Chow and Hilbert schemes) Non-reduced fibers of an arithmetic 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 By \textit{T. Ekedahl}'s result [in: The Grothendieck Festschrift, Vol. II, Prog. Math. 87, 197-218 (1990; Zbl 0821.14010)] one is able to extend the methods and results of \textit{A. A. Beilinson, J. Bernstein} and \textit{P. Deligne} [``Faisceaux pervers'', Astérisque 100 (1992; Zbl 0536.14011)] to any base field. Thus one may construct a (bounded) derived category \(D^b_c(X,{\mathbb{Z}}_{\ell})\) of constructible \({\mathbb{Z}}_{\ell}\)-sheaves on a variety over any base field. \(D^b_c(X,{\mathbb{Z}}_{\ell})\) has a canonical \(t\)-structure whose heart is the category of constructible \({\mathbb{Z}}_{\ell}\)-sheaves. However, Ekedahl's formalism does not allow to define a well-behaved notion of weights, and one knows the difference [\textit{U. Jannsen}, ``Mixed motives and algebraic \(K\)-theory'', Lect. Notes Math. 1400 (1988; Zbl 0691.14001)] in case the characteristic of the base field is \(p>0\), which was treated by Beilinson, Bernstein and Deligne (loc. cit.), and the characteristic zero case, in particular, number fields. This situation is remedied in the underlying paper.
One fixes a 1-dimensional irreducible regular noetherian connected base scheme \(U_0\). Let \(B=\text{ Spec}(K)\) be its generic point, and for base change, introduce the category \(\mathcal U\) dual to the category of all open subschemes of \(U_0\). A \(U_0\)-scheme \(X\) is called horizontal if it is flat and of finite type over \(U_0\). For such \(X\) let \({\mathcal U}X\) denote the direct system of schemes \(X\times_{U_0}U\) for \(U\) in \({\mathcal U}\). One has an equivalence of categories:
\[
\{\text{schemes of finite type over }B\}\leftrightarrow\{\text{direct limits of \({\mathcal U}X\) for horizontal \(X\) over }U_0\},
\]
in particular, the limit of \({\mathcal U}X\) is representable by \(X_B\). For a horizontal \(U_0\)-scheme \(X\) one defines the derived category \(D^b_c({\mathcal U}X,{\mathbb{Z}}_{\ell})\) as the direct 2-limit of the categories \(D^b_c(X',{\mathbb{Z}}_{\ell})\) in the sense of Ekedahl for \(X'\) in \({\mathcal U}X\). \(D^b_c({\mathcal U}X,{\mathbb{Q}}_{\ell})\) will be the category obtained by tensoring all morphisms in \(D^b_c({\mathcal U}X,{\mathbb{Z}}_{\ell})\) by \({\mathbb{Q}}\). Much of the theory and results from the Beilinson-Bernstein-Deligne book (loc. cit.) carry over to \(D^b_c({\mathcal U}X,{\mathbb{Z}}_{\ell})\), in particular, it has a \(t\)-structure and the objects in its heart are called \textit{perverse \({\mathbb{Z}}_{\ell}\)-sheaves} on \({\mathcal U}X\). Also, the perverse sheaves in \(D^b_c({\mathcal U}X,{\mathbb{Q}}_{\ell})\) form an artinian and noetherian category whose objects are of finite length. For an immersion \(j:V\rightarrow X\) of horizontal \(U_0\)-schemes with \(V\) smooth over \(U_0\) in \(\mathcal U\), and for an irreducible smooth \({\mathbb{Q}}_{\ell}\)-sheaf \(L\) on \(V\), the direct image \(j_{!*}(L[\dim(V)])\) is a simple perverse \({\mathbb{Q}}_{\ell}\)-sheaf on \({\mathcal U}X\). Conversely, any simple perverse \({\mathbb{Q}}_{\ell}\)-sheaf on \({\mathcal U}X\) is of this kind.
An interesting question is whether the natural functor \(\eta^*:D_c^b({\mathcal U}X,\mathbb{Z}_\ell)\to D_c^b(X_B,\mathbb{Z}_\ell)\) induced by \(\eta:B\to U_0\) is an equivalence of categories. It follows from \textit{U. Jannsen}'s book (loc. cit.) that this is not the case. One has the result that the heart of the canonical \(t\)-structure on \(D^b_c({\mathcal U}X,{\mathbb{Z}}_{\ell})\) is equivalent to the full subcategory of constructible \({\mathbb{Z}}_{\ell}\)-sheaves on \(X_B\) that extend to some model of \(X_B\). These sheaves are called horizontal. \(\eta^*:D^b_c({\mathcal U}X,{\mathbb{Z}}_{\ell})\to D^b_c(X_B,{\mathbb{Z}}_{\ell})\) and also \(\overline{\eta}^*:D^b_c({\mathcal U}X,{\mathbb{Z}}_{\ell})\to D^b_c(X_{\overline{B}},{\mathbb{Z}}_{\ell})\) are faithful on \({\mathbb{Q}}_{\ell}\)-perverse sheaves.
To define weights in the case of a number field \(K\) take \(U_0=\text{ Spec}({\mathcal O}_K[{1/\ell]})\). Thus the residue fields of the closed points of \(U_0\) are finite. This makes the obvious extension of the notion of mixedness for \({\mathbb{Q}}_{\ell}\)-sheaves on a horizontal \(U_0\)-scheme \(X\) as in the book of Beilinson, Bernstein and Deligne (loc. cit.) possible. Let \(D^b_m({\mathcal U}X,{\mathbb{Q}}_{\ell})\) denote the subcategory of \(D^b_c({\mathcal U}X,{\mathbb{Q}}_{\ell})\) consisting of objects \(C\) such that all \(H^i(C)\) become mixed after base change by suitable \(U\in{\mathcal U}\). Then \(D^b_m({\mathcal U}X,{\mathbb{Q}}_{\ell})\) is stable under the six Grothendieck functors. The perverse \(t\)-structure induces a \(t\)-structure on \(D^b_m({\mathcal U}X,{\mathbb{Q}}_{\ell})\), and any subquotient of a mixed perverse \({\mathbb{Q}}_{\ell}\)-sheaf on \({\mathcal U}\) is mixed.
The notion of weight for an object \(C\) of \(D^b_m({\mathcal U}X,{\mathbb{Q}}_{\ell})\) can be carried over from the cited book ``Faisceaux pervers''. Many facts thereof apply in this situation, but contrary to the case treated in this book, there are non-trivial morphisms [see the book by \textit{U. Jannsen} (loc. cit.)] in \(D^b_m({\mathcal U}X,{\mathbb{Q}}_{\ell})\) from \(C\) of weight \(\leq w\) to \(L\) with weights \(>w\).
The morphisms in \(D^b_c({\mathcal U}X,{\mathbb{Q}}_{\ell})\) are given by direct limits over Jannsen's continuous étale cohomology groups: For a constructible \({\mathbb{Z}}_{\ell}\)-sheaf \(F\) on the horizontal \(U_0\)-scheme \(X\), one has
\[
\text{Hom}_{D^b_c({\mathcal U}X,{\mathbb{Z}}_{\ell})}({\mathbb{Z}}_{\ell},F[i])=\lim_{\mathcal U}H^i_{cont}(X_U,F|{}_U).
\]
For a \(B\)-scheme \(X_B\), any horizontal model \(X\) of \(X_B\) over an open part of \(U_0\) and a horizontal \({\mathbb{Z}}_{\ell}\)-sheaf \(F\) on \(X_B\), one defines the horizontal étale cohomology
\[
H^i_{hor}(X_B,F)=\text{Hom}_{D^b_c({\mathcal U}X,{\mathbb{Z}}_{\ell})}({\mathbb{Z}}_{\ell},F[i]).
\]
This leads to a good cohomology theory with long exact sequences, Poincaré duality and Chern class morphisms. Embedding \(K\) into \({\mathbb{C}}\) gives rise to a comparison between horizontal perverse \({\mathbb{Z}}_{\ell}\)-sheaves on \(X_B\) and Saito's Hodge modules. perverse \(\ell\)-adic sheaves; derived category; weights; mixedness; canonical \(t\)-structure; perverse \(t\)-structure; horizontal étale cohomology; horizontal perverse sheaves; Hodge modules doi:10.1023/A:1000273606373 Étale and other Grothendieck topologies and (co)homologies, Derived categories, triangulated categories, \(p\)-adic cohomology, crystalline cohomology, Galois cohomology, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials Mixed perverse sheaves for schemes over number 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 In this article, the authors study the dimension of the vector space of splines, or piecewise polynimial functions, of a fixed maximun polynomial degree and which are defined on planar polygonal partitions. The term ``mixed smoothness'' refers to the choice of different orders of smoothness across various edges of the partition. The approach follows the homological methods introduced to the study of splines in [\textit{L. J. Billera}, Trans. Am. Math. Soc. 310, No. 1, 325--340 (1988; Zbl 0718.41017)].
Starting from a spline space whose lower homology modules vanish, the authors present sufficient conditions that ensure that the same will be true for the spline space obtained after relaxing the smoothness conditions on certain edges of the partition.
The results can be used to compute the dimensions of T-meshes with holes, therefore extending the results on mixed non-uniform bi-degree T-meshes presented in [\textit{D. Toshniwal} et al., Adv. Comput. Math. 47, No. 1, Paper No. 16, 42 p. (2021; Zbl 1473.13026)] and [\textit{D. Toshniwal} and \textit{N. Villamizar}, Comput. Aided Geom. Des. 80, Article ID 101880, 9 p. (2020; Zbl 07207288)]. splines; polygonal meshes; spline dimension formulas; mixed smoothness Syzygies, resolutions, complexes and commutative rings, Spline approximation, Numerical computation using splines, Geometric aspects of numerical algebraic geometry Counting the dimension of splines of mixed smoothness. A general recipe, and its application to planar meshes of arbitrary topologies | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author discusses some results on kernels of locally nilpotent derivations of a polynomial ring over an algebraically closed field of characteristic zero \(k\). Locally nilpotent derivations of \(k[x_1,\ldots,x_n]\) correspond to algebraic actions of the additive group \((k,+)\) on affine \(n\)-space. The kernel of such a locally nilpotent derivation corresponds to the invariant set of polynomials with respect to the corresponding \((k,+)\) action. An interpretation of the Cancellation problem in terms of the kernel of a locally nilpotent derivations is given. The Cancellation problem asks : if \(X\) is an affine variety and \(X\times{\mathbb{A}}^1\) is isomorphic to affine \(n\)-space, then is \(X\) isomorphic to affine \(n-1\)-space. A slice of a locally nilpotent derivation \(D\) is an element \(f\) such that \(D(f)=1\).
The Cancellation problem can be interpreted as the following question: Let \(D\) be a locally nilpotent \(k\)-derivation on \(k[x_1,\ldots,x_n]\) and suppose that \(D\) has a slice. Then is the kernel of \(D\) isomorphic to a polynomial ring in \(n-1\) variables?
The theorems 3.2 and 3.4 of the present article claim the following.
(1) Let D be a locally nilpotent derivation with a slice on \(k[x_1,...,x_n]\) with \(n\geq 4\) and such that \(x_1,...,x_{n-3}\) are in the kernel of \(D\). Then the kernel of \(D\) is a polynomial ring of dimension \(n-1\).
(2) Let D be a triangulable locally nilpotent derivation with a slice on \(k[x_1,...,x_n] \). Then the kernel of \(D\) is a polynomial ring of dimension \(n-1\). However, the proofs of these two results are not complete. cancellation problem; \(k^*\)-action; fix-pointed action; triangular derivation Masuda, K.: Torus actions and kernels of locally nilpotent derivations with slices, Affine algebraic geometry in honor of Professor masayoshi miyanishi (2007) Group actions on affine varieties, Group actions on varieties or schemes (quotients), Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) Torus actions and kernels of locally nilpotent derivations with slices | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \({\mathfrak O}\) be the ring of integers of a local field \(K\) of characteristic 0 with a residue field \(k\) of characteristic \(p>0\) and with ramification index \(e \leq p-1\); let \(\Gamma = \text{Gal} (\overline {K}/K)\). In this paper we obtain an explicit construction of finite commutative group schemes \(G\) over the ring \({\mathfrak O}\) that are annihilated by multiplication by \(p\); we obtain a description of them with the help of generalized finite Honda systems. We investigate the functor \(G \mapsto G (\overline K)\) (including the case \(e = p - 1)\), and we get sufficient (also necessary for \(e = 1)\) conditions under which a given \(\mathbb{F}_ p [\Gamma]\)-module can be realized as a \(\Gamma\)-module of \(\overline K\)-points \(G (\overline K)\) of the group scheme of \(G\) of the type considered. small ramification; characteristic \(p\); group schemes; generalized finite Honda systems Abrashkin, V.: Group schemes over a discrete valuation ring with small ramification, Leningrad Math. J. \textbf{1}(1), 57-97 Group schemes, Local ground fields in algebraic geometry, Galois theory, Valuation rings Group schemes over a discrete valuation ring with small ramification | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a smooth projective variety defined over an algebraically closed field \(k\). Let \(X^{[n]} := \text{Hilb}(X)\) be the Hilbert scheme of zero-dimensional subschemes of \(X\) of length \(n\). The main theme of the book is to study the Hilbert schemes \(X^{[n]}\), with particular reference to understand the cohomology and the Chow ring of \(X^{[n]}\).
For \(n = 1,2\), the Hilbert scheme \(X^{[n]}\) is very well understood. In fact, \(X^{[1]}\) is just \(X\) itself and \(X^{[2]}\) can be obtained by blowing up \(X \times X\) along the diagonal and taking the quotient by the natural involution that exchanges factors in \(X \times X\). The Hilbert scheme \(X^{[n]}\) is smooth if \(n \leq 3\) or \(\dim X \leq 2\). If \(X\) is a curve, then \(X^{[n]}\) is isomorphic to the \(n\)-th symmetric product, \(X^{(n)}\), of \(X\). In general, in any dimension, the natural set- theoretic map \(X^{[n]} \to X^{(n)}\) sending each 0-scheme to its support (with multiplicities) gives a desingularization of \(X^{(n)}\) if \(X^{[n]}\) is smooth. When \(\dim X = 2\), the canonical bundle of \(X^{[n]}\) is the pullback of the dualizing sheaf of \(X^{(n)}\).
In chapter 1 some fundamental facts and background material are recalled. In \S1.1 the definition and the most important properties of \(X^{[n]}\) are given. In \S1.2 the author discusses the Weil conjectures in the form he needs to compute Betti numbers of Hilbert schemes. In \S1.3 the punctual Hilbert scheme, which parametrizes subschemes concentrated in a point of a smooth variety is studied.
Chapter 2 is devoted to the computation of the Betti numbers of Hilbert schemes. In \S\S2.1, 2.2 the structure of the closed subscheme of \(X^{[n]}\) which parametrizes subschemes of length \(n\) on \(X\) concentrated on a variable point of \(X\), and the punctual Hilbert schemes \(\text{Hilb}(k[[x,y]])\) are studied. \S2.3 contains the explicit computation of the Betti numbers of \(S^{[n]}\) for a smooth surface \(S\), using the Weil conjectures. The Betti numbers of all the \(S^{[n]}\) are computed as simple power series expressions in terms of the Betti numbers of \(S\). Similar results are obtained in \S2.4, where the Betti numbers of higher order Kummer varieties \(KA_ n\) are also computed. These varieties were previously defined by A. Beauville, as new examples of Calabi-Yau manifolds. \S2.5 is devoted to the computation of the Betti numbers for triangle varieties, which parametrize length 3 0-dimensional subschemes. The main powerful tool used throughout this chapter are the Weil conjectures.
The second part of the book, chapters 3 and 4, is devoted to the computation of the cohomology and the Chow ring of Hilbert schemes. In chapter 3, \S\S3.1, 3.2, varieties of second and higher order data are constructed and studied. Such varieties are needed to give precise solutions to classical problems in enumerative algebraic geometry concerning contacts of families of subvarieties of projective space. As an application (see \S\S3.2, 3.3) a formula for the numbers of higher order contacts of a smooth projective variety with linear subvarieties in the ambient space is computed.
The last chapter, chapter 4, is the most elementary and classical of the book. In this chapter the Chow ring of relative Hilbert schemes of projective bundles is studied. In \S4.1 the author constructs the embeddings of relative Hilbert schemes into Grassmannian bundles and studies them. The case of the relative Hilbert scheme of a \(\mathbb{P}^ 1\)- bundle over a smooth variety is studied in more detail. In \S4.2 the Chow ring of the variety \(\widetilde{\text{Hilb}}^ 3(\mathbb{P}^ 2)\), parametrizing triangles in \(\mathbb{P}^ 2\) with a marked side is computed. \S4.3 is devoted to a generalization of this result to a relative situation. In \S4.3 the author studies the relative Hilbert scheme \(\text{Hilb}^ 3(\mathbb{P}(E)/X)\) of subschemes of length 3 in the fibers of the projectivization \(\mathbb{P}(E)\) of a vector bundle \(E\).
The various chapters are rather independent from each other. To read this book the reader only needs to know the basics of algebraic geometry. The book is therefore of interest not only to experts but also to graduate students and researchers in algebraic geometry not familiar with Hilbert schemes of points. -- This book is also very well organized and very nicely written. Finally, it contains a wide up-to-date bibliography on the topic. Bibliography; punctual Hilbert scheme; Betti numbers; higher order Kummer varieties; triangle varieties; Chow ring Göttsche, L. 1994.Hilbert Schemes of Zero-dimensional Subschemes of Smooth Varieties, Lecture Notes in Math. #1572 1--196. Berlin: Springer-Verlag. Parametrization (Chow and Hilbert schemes), Topological properties in algebraic geometry, Algebraic cycles, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry Hilbert schemes of zero-dimensional subschemes of smooth 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 In this dissertation, the author studies invariants of Hilbert schemes of zero-dimensional subschemes of smooth varieties. He simplifies his arguments from Math. Ann. 286, No. 1-3, 193-207 (1990; Zbl 0679.14007) to compute the Betti numbers of the Hilbert scheme \(\text{Hilb}^n (X)\) of zero-dimensional subschemes of length \(n\) of a smooth projective surface. The proof is based on an extension of a cell decomposition of the local Hilbert scheme presented by \textit{G. Ellingsrud} and \textit{S. A. Strømme} [in Invent. Math. 87, 343-352 (1987; Zbl 0625.14002)], and on finding a good reduction mod \(p\) and using the Weil conjectures. The author proceeds to use similar methods to find the Betti numbers of \(\text{Hilb}^n (X)\) for Kummer varieties \(X\) of higher order, and for several kinds of Hilbert schemes of triangles.
The second part of the thesis deals with cases in which the Chow ring of Hilbert schemes can be computed. The author succeeds in the case of varieties of second and higher order data and applies his formulas to give enumerative results about contact varieties of projective varieties with linear spaces in \(\mathbb{P}^N\). He also describes the Chow ring of \(\text{Hilb}^3 (\mathbb{P} ({\mathcal E}),X)\), the relative Hilbert scheme of a projective bundle of a vector bundle \({\mathcal E}\) of rank 3 over a smooth variety \(X\), in analogy with the results of \textit{G. Elencwajg} and \textit{P. Le Barz} [Compos. Math. 71, No. 1, 85-119 (1989; Zbl 0705.14004)]. Also several varieties of triangles are treated. The results get rather messy and were determined using the aid of a computer.
The exposition is clear and on a very high level. The reader is assumed to have a thorough knowledge of a good portion of the work contained in the 108 references. invariants of Hilbert schemes of zero-dimensional subschemes; Betti numbers; Kummer varieties; Chow ring Göttsche, L.: Hilbert schemes of zero-dimensional subschemes of smooth varieties. Lect. Notes Math. vol. 1572, Berlin Heidelberg New York: Springer 1993 Parametrization (Chow and Hilbert schemes), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Algebraic cycles, Algebraic moduli problems, moduli of vector bundles Hilbert schemes of zero-dimensional subschemes of smooth 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 \(R\) be a commutative ring with identity and \(M\) be a unital \(R\)--module. The prime spectrum of \(M\), the set of all prime submodules of \(M\) which is denoted by \(\text{Spec}(M)\), forms a topological space under the Zariski topology [\textit{C.-P. Lu}, Houston J. Math. 25, No. 3, 417--432 (1999; Zbl 0979.13005)]. Let \(X=\text{Spec}(M)\), then for every open subset \(U\subseteq X\), one may assign a subring \(\mathcal{O}_X(U)\) of \(\prod_{\mathfrak{p}\in\mathcal{O}(U)}R_{\mathfrak{p}}\), where \(\mathcal{O}(U)=\{(P:_RM) \mid P\in U\}\subseteq\text{Spec}(R)\).
The main result of the paper, Theorem 2.10, states that \((X,\mathcal{O}_X)\) is a scheme when \(X\) is a \(T_0\)-space and \(M\) is faithful and primeful \(R\)-module. Moreover, if \(R\) is Noetherian ring, then \((X,\mathcal{O}_X)\) is a Noetherian scheme. It is investigated by the authors that for each \(P\in X\), the stalk \(\mathcal{O}_P\) of the sheaf \(\mathcal{O}_X\) is isomorphic to \(R_{\mathfrak{p}}\), where \(\mathfrak{p}=(P:M)\). If \(M\) is faithful and primeful, then for \(f\in R\), the ring \(\mathcal{O}_X(X_f)\) is isomorphic to \(R_f\), where \(X_f\) is the open subset \(X\setminus V(rM)\).
Let \(N\) be another \(R\)-module and \(\varphi:M\longrightarrow N\) be a homomorphism. Looking for an induced morphism between locally ringed spaces of \(\mathcal{O}_{\text{Spec}(N)}\) and \(\mathcal{O}_X\) is the subject of the other results of the authors. They show that when \(\varphi\) is epimorphism, such an induced map exists. They also discuss the problem for a ring homomorphism \(\Phi:R\longrightarrow S\) and an \(S\)-module \(N\), and prove that \(\Phi\) induces a morphism of locally ringed spaces
\[
(\text{Spec}(N),\mathcal{O}_{\text{Spec}(N)})\longrightarrow(X,\mathcal{O}_X),
\]
when \(M\) is primeful, \(X\) is a \(T_0\)-space and \((0:_RM)\subseteq (0:_RN)\). prime submodule; Zariski topology; primeful module; sheaf of rings; scheme Theory of modules and ideals in commutative rings, General commutative ring theory, Schemes and morphisms A scheme over prime spectrum of modules | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Here we use the so-called Horace Method to study interpolation problems for multihomogeneous polynomials over finite fields. multihomogeneous polynomial; Hasse derivatives Projective techniques in algebraic geometry, Finite fields and commutative rings (number-theoretic aspects), Finite ground fields in algebraic geometry Multigraded polynomials and interpolation over a finite field | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Authors' abstract: We prove a closed formula for leading Gopakumar-Vafa BPS invariants of local Calabi-Yau geometries given by the canonical line bundles of toric Fano surfaces. It shares some similar features with Göttsche-Yau-Zaslow formula: Connection with Hilbert schemes, connection with quasimodular forms, and quadratic property after suitable transformation. In Part I of this paper we will present the case of projective plane, more general cases will be presented in Part II. Gopakumar-Vafa BPS invariant; Gromov-Witten invariants; local Calabi-Yau; Hilbert scheme; quasimodular form S. Guo, J. Zhou, Gopakumar-Vafa BPS invariants, Hilbert schemes and quasimodular forms. II, preprint. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Calabi-Yau manifolds (algebro-geometric aspects) Gopakumar-Vafa BPS invariants, Hilbert schemes and quasimodular forms. I | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In an earlier work [Math. Ann. 255, 107-122 (1981; Zbl 0437.12010)] the author showed how Klein's various parametrizations of the modular equation could be used to iteratively construct ring class fields of discriminant \(d=d_ 0b^{2t}\) \((<0)\) for \(b=2,3,4,5\). These cases correspond to Euclid's regular polyhedra (of genus 0). Here the author completes the work for \(b=7\), possibly the last case for which an explicit formulation is possible. It corresponds to the 168-tesselation and Klein's curve of genus 3. representation of prime by principal form; parametrizations of the modular equation; ring class fields; discriminant; Euclid's regular polyhedra; 168-tesselation; Klein's curve of genus 3 Class field theory, Modular and automorphic functions, Representation problems, Arithmetic ground fields for curves Iterated ring class fields and the 168-tesselation | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 holomorphic Euler characteristics of tautological sheaves on Hilbert schemes of points on surfaces. In particular, we establish the rationality of K-theoretic descendent series. Our approach is to control equivariant holomorphic Euler characteristics over the Hilbert scheme of points on the affine plane. To do so, we slightly modify a Macdonald polynomial identity of Mellit. Hilbert schemes; tautological bundles; Macdonald polynomials Parametrization (Chow and Hilbert schemes), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Symmetric functions and generalizations \(K\)-theoretic descendent series for Hilbert schemes of points on surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper investigates compact algebraic surfaces in \(\mathbb{R}^3\) and \(\mathbb{R} P^3\). It shows that any compact orientable surface is diffeomorphic to an irreducible subvariety of \(\mathbb{R}^3\) with degree equal to the sum of the Betti numbers. For the nonorientable case, a connected nonorientable surface must be the connected sum of an odd number \(2g+1\) of \(\mathbb{R} P^2\)'s to be imbedded in \(\mathbb{R} P^3\) and it is shown that it is diffeomorphic to a real variety of degree \(\leq 4g+4\). Specialized structures on manifolds (spin manifolds, framed manifolds, etc.), Topology of real algebraic varieties, Topology of Euclidean 2-space, 2-manifolds An explicit algebraic structure for compact surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(<\) be a monomial well-ordering on \(R=K[x_0, \dots,x_n]\), \(K\) an algebraically closed field For a monomial ideal \(I_0\) with Hilbert polynomial \(p(t)\) consider \(M_{I_0}= \{J\subset R\mid J\) homogeneous and saturated ideal and \(\text{in}(J)=I_0\}\). Here \(\text{in}(J)\) denotes the initial ideal of \(J\). It is proved that \({\mathcal M}_{I_0}\) carries the structure of a locally closed subscheme in \(\text{Hilb}^{p(t)}_{\mathbb{P}^n}\). \({\mathcal M}_{I_0}\) is affine if \(I_0\) is saturated. In this case, an explicit construction of the coordinate ring is possible. The set of all \({\mathcal M}_{I_0}\), \(I_0\) monomial with Hilbert polynomial \(p(t)\), leads to a natural stratification of \(\text{Hilb}_{\mathbb{P}^n}^{p(t)}\).
As an application, the singular locus of the component of \(\text{Hilb}^{4t+1}_{\mathbb{P}^4}\) containing the arithmetically Cohen-Macaulay curves of degree 4 is described. Hilbert scheme; stratification; Gröbner basis; Hilbert polynomial R. Notari and M.L. Spreafico, A stratification of Hilbert schemes by initial ideals and applications , Manuscr. Math. 101 (2000), 429-448. MR1759253 %(2001b:14008)
Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Computational aspects of higher-dimensional varieties, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) A stratification of Hilbert schemes by initial ideals and applications | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For an isogeny \(X \to Y\) of degree \(n\) between two abelian varieties of degree \(g\), the kernel is a finite flat group scheme of order \(n^{2g} .\) This is used to motivate the study of finite flat group schemes. The paper is designed to be an introduction to the subject, described in the language of schemes. At first, there are no restrictions on the group schemes or their base scheme \(S\). However, the attention is quickly turned towards group schemes that are commutative, and later finite over an affine base scheme \(S\)=Spec(\(R\)). The author constructs group schemes as group objects in the category of schemes over \(S\). Examples are given and basic properties are discussed, including those of the representing Hopf algebra of an affine group scheme. Results of Grothendieck are used to establish quotient objects in the category [\textit{A. Grothendieck}, ``Technique de descente et théorèmes d'existence en géometrie algébrique. I-IV'', Sémin. Bourbaki 12, Exp. No. 190 and 195 (1960; Zbl 0229.14007 and Zbl 0234.14007)]. Let \(R\) be a discrete valuation ring with field of fractions \(K\). In the final section, the results from \textit{M. Raynaud}'s paper [Bull. Soc. Math. Fr. 102(1974), 241-280 (1975; Zbl 0325.14020)] are duplicated, where he describes prolongations of \(K\)-group schemes to \(R\). abelian varieties; isogeny; flat group scheme; Hopf algebra Tate, J., \textit{finite flat group schemes}, Modular forms and Fermat's last theorem (Boston, MA, 1995), 121-154, (1997), Springer, New York Group schemes, Isogeny, Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act Finite flat group schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Recently there has been growing interest in the description of involutions of minimal surfaces of general type with geometric genus 0, following results of \textit{A. Calabri} et al. [Trans. Am. Math. Soc. 359, No. 4, 1605--1632 (2007; Zbl 1124.14036)]. We recall that a numerical Campedelli (resp. numerical Godeaux) surface is a minimal surface of general type with geometric genus 0 and canonical degree 2 (resp. 1).
The authors show the existence of a simply connected numerical Campedelli surface with an involution whose quotient is birational to a simply connected numerical Godeaux surface.
It was indeed already known that one could have an involution on a numerical Campedelli surface with \(4\) isolated fixed points, thus yielding a numerical Godeaux surface at the quotient as in [\textit{D. Frapporti}, Collect. Math. 64, No. 3, 293--311 (2013; Zbl 1303.14045)]. Still, the simple connectedness here is an important property, since even the existence of simply connected numerical Godeaux and Campedelli surfaces has been a challenging problem for a long time, and we are still far from classifying all of them.
The strategy combines double covering and \(\mathbb{Q}\)-Gorenstein smoothing techniques, and both surfaces are obtained by smoothing a (different) singular rational elliptic surface. To show the existence of the smoothing, the authors need to show that there is no local-to-global obstruction. This involves a new technique which generalizes a result of \textit{D. M. Burns jun.} and \textit{J. M. Wahl} [Invent. Math. 26, 67--88 (1974; Zbl 0288.14010)] describing the space of first-order deformations of a singular complex surface with only rational double points. Park, H.; Shin, D.; Urzúa, G., A simply connected numerical Campedelli surface with an involution, Math. Ann., 357, 31-49, (2013) Surfaces of general type, Families, moduli, classification: algebraic theory, Singularities of surfaces or higher-dimensional varieties, Symplectic manifolds (general theory) A simply connected numerical Campedelli surface with an involution | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors construct division polynomials for binary Edwards curves, which are symmetry polynomials in \(x\) and \(y\). The main property of the division polynomials is that they characterize \(n\)-torsion points of a binary Edwards curve for a positive integer \(n\). Then they generalize the same results on Edwards curves over non-binary fields and also construct an Edwards curve with a point of order 12. Edwards curve; division polynomial; elliptic curve cryptosystem Curves over finite and local fields, Elliptic curves, Algebraic coding theory; cryptography (number-theoretic aspects), Cryptography Division polynomials for binary Edwards 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 algebraic geometry Algebraic geometry Sur les variétés algébriques à trois dimensions de genres un contenant des involutions cycliques | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Infinitesimal deformations are commonly used in real algebraic geometric algorithms. The author uses homotopy continuation to model the deformation, and presents an algorithm for computing a finite set of real roots of a polynomial system containing a point on each connected component. The algorithm computes a finite superset of the isolated roots over the real numbers. Some numerical examples are presented to demonstrate the computational steps of the algorithm. real algebraic geometry; infinitesimal deformation; numerical algebraic geometry; polynomial system; homotopy continuation; algorithm; real roots; numerical examples Hauenstein, JD, Numerically computing real points on algebraic sets, Acta applicandae mathematicae, 125, 105-119, (2013) Numerical computation of solutions to systems of equations, Polynomials, factorization in commutative rings, Global methods, including homotopy approaches to the numerical solution of nonlinear equations, Computational aspects of higher-dimensional varieties Numerically computing real points on algebraic sets | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \({\mathbb P}^{d}\) denote projective space over an algebraically closed field \(k\). Let \(K_{0}( {\mathbb P}^{d} )\) denote the Grothendieck group of locally free sheaves on \({\mathbb P}^{d}\). For \(x \in K_{0}( {\mathbb P}^{d} )\) with Chern classes \(c_{i}(x)\) the Chern polynomial is defined by
\[
C_{x}(t) = 1 + c_{1}(x)t + c_{2}(x)t^{2} + \dots + c_{d}(x)t^{d}
\]
lying in \({\mathbb Z}[t]/(t^{d+1})\). On the other hand a graded, finitely generated module \(M\) over the graded ring \(k[x_{0}, \dots , x_{d}]\) gives a coherent sheaf on \({\mathbb P}^{d}\) and a class in the Grothendieck group \(G_{0}( {\mathbb P}^{d} ) \cong K_{0}( {\mathbb P}^{d} )\). The Hilbert polynomial \(P_{M}(t)\) is related to \(C_{M}(t)\) by the Hirzebruch-Riemann-Roch theorem [see \textit{D. Eisenbud}, ``Commutative algebra. With a view towards algebraic geometry'', Grad. Texts Math. 150 (1995; Zbl 0819.13001)].
The author shows that the homomorphism
\[
\xi : K_{0}( {\mathbb P}^{d} ) \rightarrow ({\mathbb Z}[t]/(t^{d+1}))^{*} \times {\mathbb Z}
\]
given by \(\xi(M) = (C_{M}(t) , \text{rank}M))\) is injective. As an application she shows that the classes of \(M\) and \(N\) in \(K_{0}({\mathbb P}^{d})\) are equal if and only if \(C_{M}(t) = C_{N}(t)\) and \( \text{rank}M) = \text{rank}N)\) or if and only if \(P_{M}(t) = P_{N}(t)\). The paper concludes with a section detailing the precise relation between Chern and Hilbert polynomials, which appears in [\textit{D. Eisenbud}, loc. cit., Exercise 19.18]. Chan, C. -Y.J.: A correspondence between Hilbert polynomials and Chern polynomials over projective spaces, Illinois J. Math. 48, No. 2, 451-462 (2004) Applications of methods of algebraic \(K\)-theory in algebraic geometry, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry A correspondence between {H}ilbert polynomials and {C}hern polynomials over projective spaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems If \(X\) is a complex algebraic variety, the virtual Poincaré polynomial \(P_X(t)\) is uniquely determined by the properties:
(1) \(P_X(t)= P_{X-Y} (t)+P_Y(t)\), for every closed subvariety \(Y\).
(2) If \(X\) is smooth and complete, then \(P_X(t)\) is the usual Poincaré polynomial of \(X\).
The authors consider the case \(X=G/H\), where \(G\) is a complex connected linear algebraic group and \(H\) is its closed subgroup.
The main result: \(P_{G/H}(t) =t^{2u} (t^2-1)^rQ_{G/H} (t^2)\), where \(Q_{G/H} (t^2)\) is a polynomial with non-negative integer coefficients. For a regular embedding \(X\) of \(G/H\) [\textit{E. Bifet}, \textit{C. De Concini} and \textit{C. Procesi}, Adv. Math. 82, 1-34 (1990; Zbl 0743.14018)] it is proved (provided \(G\) is complete and \(H\) is connected) that \(P_X(t)= Q_{G/H}(t^2) R_X(t^2)\), where \(R_X(t^2)\) is a polynomial with non-negative integer coefficients.
Reviewer's remark. The authors' assertion that ``the usual Poincaré polynomials for homogeneous spaces are generally unknown'' is properly not quite correct. As soon as the cohomology of homogeneous spaces can be described by the H. Cartan algebra, the Poincaré polynomial can be written using its algebraic characteristics (Gröbner bases, for instance) [see \textit{I. Z. Rozenknop}, Usp. Mat. Nauk 25, No. 5, (155), 245-246 (1970; Zbl 0204.06001)]. virtual Poincaré polynomial Brion, Michel; Peyre, Emmanuel, The virtual Poincaré polynomials of homogeneous spaces, Compositio Math., 0010-437X, 134, 3, 319-335, (2002) Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Homogeneous spaces and generalizations The virtual Poincaré polynomials of homogeneous spaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The classification of multigerms with respect to \( \mathcal A \)-equivalence is studied. A classification of \( r \)-multigerms from \( {\mathbb{R}}^2 \) to \( {\mathbb{R}}^3 \) of codimension not greater than \( 6-r \) is carried out, and the bifurcation geometry of such singularities is analysed. This classification yields, in particular, a full list of simple multigerm singularities from \( {\mathbb{R}}^2 \) to \( {\mathbb{R}}^3 \). The authors use the standard philosophy of reducing such questions to computations in jet spaces. The problem is then one of Lie groups acting on affine spaces and the fundamental techniques of \textit{J. N. Mather} [Publ. Math., Inst. Hautes Étud. Sci. 35, 127-156 (1968; Zbl 0159.25001)] can be applied, together with results of \textit{J. W. Bruce, N. P. Kirk} and \textit{A. A. du Plessis} on so-called complete transversals [Nonlinearity 10, 253-275 (1997; Zbl 0929.58019)]. An important amount of computer algebra is also involved. classification of multigerms; \(\mathcal A\)-equivalence; bifurcation Hobbs, C. A. and Kirk, N. P., On the classification and bifurcation of multigerms of maps from surfaces to 3-space, Math. Scand. 89 (2001), 57--96. Classification; finite determinacy of map germs, Complex surface and hypersurface singularities, Singularities of surfaces or higher-dimensional varieties On the classification and bifurcation of multigerms of maps from surfaces to 3-space | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We give a noncommutative geometric description of the internal Hom dg-category in the homotopy category of dg-categories between two noncommutative projective schemes in the style of \textit{M. Artin} and \textit{J. J. Zhang} [Adv. Math. 109, No. 2, 228--287 (1994; Zbl 0833.14002)]. As an immediate application, we give a noncommutative projective derived Morita statement along the lines of Rickard and Orlov. noncommutative algebra; noncommutative projective schemes; derived categories; Fourier-Mukai transforms Noncommutative algebraic geometry, Fundamental constructions in algebraic geometry involving higher and derived categories (homotopical algebraic geometry, derived algebraic geometry, etc.), Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Derived categories and associative algebras Kernels for noncommutative projective 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 I give a conjectural generating function for the numbers of \(\delta\)-nodal curves in a linear system of dimension \(\delta\) on an algebraic surface. It reproduces the results of \textit{I. Vainsencher} [J. Algebr. Geom. 4, No. 3, 503-526 (1995; Zbl 0928.14035)] for the case \(\delta\leq 6\) and \textit{S. Kleiman} and \textit{R. Piene} for the case \(\delta\leq 8\). The numbers of curves are expressed in terms of five universal power series, three of which I give explicitly as quasimodular forms. This gives in particular the numbers of curves of arbitrary genus on a K3 surface and an abelian surface in terms of quasimodular forms, generalizing the formula of \textit{S.-T. Yau} and \textit{E. Zaslow} [Nucl. Phys., B 471, No. 3, 503-512 (1996)] for rational curves on K3 surfaces. The coefficients of the other two power series can be determined by comparing with the recursive formulas of \textit{L. Caporaso} and \textit{J. Harris} [Invent. Math. 131, No. 2, 345-392 (1998; Zbl 0934.14038)] for the Severi degrees in \(\mathbb{P}_2\). We verify the conjecture for genus 2 curves on an abelian surface. We also discuss a link of this problem with Hilbert schemes of points. numbers of nodal curves; numbers of curves on a K3 surface; genus 2 curves an an abelian surface; Hilbert schemes of points Göttsche, L., A conjectural generating function for numbers of curves on surfaces, Commun. Math. Phys., 196, 523-533, (1998) Enumerative problems (combinatorial problems) in algebraic geometry, Divisors, linear systems, invertible sheaves, Plane and space curves, \(K3\) surfaces and Enriques surfaces A conjectural generating function for numbers of curves on surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be an irreducible smooth projective variety of dimension \(n\) over \({\mathbb Z}\) which has an elusive model \({\mathcal X}\) over the field \({\mathbb F}_1\) with one element. The zeta function of \({\mathcal X}\) is \(\zeta_{\mathcal X}(s)=\prod_{i=0}^n(s-i)^{-b_{2i}}\), where \(b_0,\ldots,b_{2n}\) are the Betti numbers of \(X\). The author proves the functional equation
\[
\zeta_{\mathcal X}(n-s)=(-1)^\chi \zeta_{\mathcal X}(s),
\]
where \(\chi=\sum_{i=0}^n b_{2i}\) is the Euler characteristic of \(X_{\mathbb C}\). If \(G\) is a split reductive group scheme of rank \(r\) with \(N\) positive roots, then
\[
\zeta_{\mathcal G}(r+N-s)=(-1)^\chi(\zeta_{\mathcal G}(s))^{(-1)^r}.
\]
\({\mathbb F}_1\)-scheme; zeta function; functional equation Lorscheid, O, Functional equations for zeta functions of \(\mathbb{F}_1\)-schemes, C. R. Math. Acad. Sci. Paris, 348, 1143-1146, (2010) Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Varieties over finite and local fields Functional equations for zeta functions of \(\mathbb F_1\)-schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\mathcal O\) be the complete local ring of an irreducible curve singularity over an algebraically closed field of characteristic zero. The authors describe the Hilbert schemes of length \(r\) closed subschemes of \(\text{Spec } {\mathcal O}\) as constructed by \textit{G. Pfister} and \textit{J. H. M. Steenbrink} [J. Pure Appl. Algebra 77, No. 1, 103--116 (1992; Zbl 0752.14007)]. Recalling their work, let \(\overline {\mathcal O}\) be the normalization and set \(\delta = \dim_k ({\overline {\mathcal O}}/{\mathcal O})\). Letting \(I(2 \delta) \subset {\overline {\mathcal O}}\) denote the ideal of elements of order at least \(2 \delta\), the subset \(M \subset \mathbb G = \text{Gr} (\delta, {\overline {\mathcal O}}/I(2 \delta))\) consisting of subspaces \(V\) which are \({\mathcal O}\)-submodules maps onto a variety \(\mathcal M = \psi(M)\) via the Plücker embedding \(\psi: \mathbb G \hookrightarrow \mathbb P^N\). Further, if \({\mathcal I}_r\) is the set of ideals \(I \subset {\mathcal O}\) with \(\dim_k {\mathcal O}/I = r\), the map \(\varphi_r: {\mathcal I}_r \to \mathbb G\) given by \(I \mapsto t^{-r}I / I(2 \delta)\) is injective for all \(r\) with Zariski closed image \({\mathcal M}_r \subset \mathcal M\), defining the Hilbert schemes. Moreover \({\mathcal M}_r = \mathcal M\) for \(r \geq 2 \delta\).
The authors consider the case \(\mathcal O = k[[t^2, t^{2d+1}]] \subset {\overline {\mathcal O}} = k[[t]]\) for fixed \(d\): denote the Hilbert schemes above by \({\mathcal A}_d = {\mathcal M}\) and \({\mathcal A}_{d,r} = {\mathcal M}_r\). Pfister and Steenbrink [loc. cit.] had already shown that \({\mathcal A}_{d,r} = {\mathcal A}_d\) is a rational variety of dimension \(d\) for \(r \geq 2d\), so they focus on the case \(1 \leq r \leq 2d\). Using their previous work [\textit{Y. Sōma} and \textit{M. Watari}, J. Singul. 8, 135--145 (2014; Zbl 1312.14016)] and semi-groups generated by orders of elements to study the form of the ideals, the authors prove that \({\mathcal A}_{d,2s} \cong {\mathcal A}_{d,2s+1}\) if \(0 < s < d\) and \({\mathcal A}_{d,r} \cong {\mathcal A}_{e,r}\) for \(0 < r < \min\{2d,2e\}\). Applying these results, they show that for \(1 \leq r \leq 2d\) and \(s = [ r/2 ]\), \({\mathcal A}_{d,r}\) is a rational projective variety of dimension \(s\) which is isomorphic to \({\mathcal A}_s\) if \(r \geq 2\). Moreover \({\mathcal A}_{d,r}\) consists of a single point for \(r=1\), is isomorphic to \(\mathbb P^1\) for \(r=2,3\) and is a singular rational projective variety for \(4 \leq r \leq 2d\) whose singular locus is described. punctual Hilbert schemes Parametrization (Chow and Hilbert schemes), Singularities of curves, local rings The punctual Hilbert schemes for the curve singularities of type \(A_{2d}\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 computation of the multiplicity and the approximation of isolated multiple roots of polynomial systems is a difficult problem. In recent years, there has been an increase of activity in this area. Our goal is to translate the theoretical background developed in the last century on the theory of singularities in terms of computation and complexity.
This paper presents several different views that are relevant to address the following issues: predict the multiplicity of a root and/or determine the number of roots in a ball, approximate fast a multiple root and give complexity results for such problems.
Finally, we propose a new method to determine a regular system, called equivalent but deflated, i.e., admitting the same root as the initial singular one. Giusti, M; Yakoubsohn, JC, Multipliity hunting and approximating multiple roots of polynomials systems, Contemp. Math., 604, 105c-128, (2013) Solving polynomial systems; resultants, Effectivity, complexity and computational aspects of algebraic geometry, Numerical computation of solutions to systems of equations, Symbolic computation and algebraic computation Multiplicity hunting and approximating multiple roots of polynomial systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The modified Korteweg-de Vries hierarchy (mKdV) is derived by imposing isometry and isoenergy conditions on a moduli space of plane loops. The conditions are compared to the constraints that define Euler's elastica. Moreover, the conditions are shown to be constraints on the curvature and other invariants of the loops which appear as coefficients of the generating function for the Faber polynomials.{
\copyright 2016 American Institute of Physics} Korteweg-de Vries hierarchy; isometry/isoenergy conditions; Euler's elastica Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), KdV equations (Korteweg-de Vries equations), Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials), Special sequences and polynomials From Euler's elastica to the mKdV hierarchy, through the Faber polynomials | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Any complex abelian variety \(A\) of dimension \(g\) can be represented as a complex torus \({\mathbb{C}}^g/{\mathcal L}\simeq ({\mathcal L}\otimes_{\mathbb{Z}}{\mathbb{R}})/{\mathcal L}\) and every point \(\xi\) of \(A\) may be identified by its \(2g\) real coordinates in a mesh of the lattice \({\mathcal L}\). When \((A,\xi)\) moves in an algebraic family, one gets the Betti map \(\beta\) from the parameter space \(S\) to \({\mathbb{R}}^{2g}\), which is a multivalued analytic map. This is a convenient tool for the study of the distribution of torsion values. The authors list a sample of classical or recent occurrences of torsion value problems and/or Betti maps, and undertake a systematic study of the Betti map. They relate the derivative of the Betti map to the Kodaira-Spencer map. Under natural hypotheses, they show that the Betti map of every section not contained in an algebraic subgroup is a submersion, and deduce the density of torsion values for every section.
Let \(A\to S\) be an abelian scheme of relative dimension \(g\) over a smooth complex algebraic variety, \(\xi:S\to A\) a section, \(\widetilde{S}\) the universal covering of \(S({\mathbb{C}})\), \(\widetilde{\beta}:\widetilde{S}\to {\mathbb{R}}^{2g}\) the associated Betti map. The authors give formulae for the generic rank \({\mathrm{rk}}\beta\) of \(\beta\), namely the maximal value of the rank of the derivative \(d\beta(\tilde{s})\) when \(\tilde{s}\) runs through \(\tilde{S}\). The authors determine this rank in relative dimension \(\le 3\) and investigate in detail the case of jacobians of families of hyperelliptic curves, both in the real and complex case. The main application is obtained in collaboration with Z.~Gao. Let
\(A \to S\) be a principally polarized abelian scheme of relative dimension \(g\) which has no non-trivial endomorphism (on any finite covering). Assume that the image of \(S\) in the moduli space \(A_g\) has dimension at least \(g\). Then the Betti map of any non-torsion section \(\xi\) is generically a submersion, so that \(\xi^{-1} A_{\mathrm {tors}}\) is dense in \(S({\mathbb C})\). The proof involves an application of the pure Ax-Schanuel theorem written by Z.~Gao in an Appendix to the paper under review (see [\textit{N. Mok} et al., Ann. Math. (2) 189, No. 3, 945--978 (2019; Zbl 1481.14048)]). abelian varieties; torsion; Betti coordinates; Kodaira-Spencer map; Manin theorem of kernel; Ax-Schanuel theorem Abelian varieties of dimension \(> 1\), Algebraic moduli of abelian varieties, classification, Transcendence theory of elliptic and abelian functions, Special varieties The Betti map associated to a section of an abelian 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 \textit{L. Godeaux} [Atti Accad. Naz. Lincei, Rend., VI. Ser. 14, 479--481 (1931; Zbl 0004.01703; JFM 57.0838.02)] gave the first example of a minimal surface of general type with geometric genus \(p_g=0\) by considering a quotient of a quintic surface in the complex projective space \(\mathbb P^3\) by a freely acting cyclic group of order \(5\) of projective transformations. For this surface the self-intersection of the canonical divisor is \(K^2=1\). Nowadays a surface of general type with the same values of the invariants is called a \textit{numerical Godeaux surface}. Among these surfaces, the ones admitting an involution have been classified by the work of \textit{J. Keum} and \textit{Y. Lee} [Math. Proc. Camb. Philos. Soc. 129, No. 2, 205--216 (2000; Zbl 1024.14019)] and \textit{A. Calabri} et al. [Trans. Am. Math. Soc. 359, No. 4, 1605--1632 (2007; Zbl 1124.14036)]. They show that such surfaces are birationally equivalent either to double covers of Enriques surfaces or to double planes, describing the two possibilities for their plane models. A numerical Godeaux surface over \(\mathbb C\) has \(p_g=q=0\), but \(p_g=q=1\) can occur in characteristic \(p=2,3\) or \(5\). In the paper under review it is shown that if the characteristic is \(p\geq 5\), then the situation is analogous to the complex case: the order of the torsion group is at most \(5\) and the quotient by an involution is either birational to an Enriques surface or is a rational surface. Moreover, there are given examples in characteristic \(5\) of quintic surfaces in \(\mathbb P^3\) with a free action of a group \(G\cong\mathbb Z/5\mathbb Z\) and having extra symmetry by \(\mathrm{Aut }G\cong\mathbb Z/4\mathbb Z\). Godeaux surface; involution Surfaces of general type Numerical Godeaux surfaces with an involution 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 Semiorthogonal decompositions give important informations on the structure of the bounded derived category \(D(X)\) of (quasi-)coherent sheaves of a projective variety \(X\), and are strictly related to the birational geometry of \(X\) [see for example \textit{A. I. Bondal} and \textit{D. O. Orlov}, Semiorthogonal decompositions for algebraic varieties, \url{arXiv:alg-geom/9506012}]. In general, if \(D\) is a triangulated category defined over a field \(k\), a semiorthogonal decomposition of \(D\) is a sequence of admissible triangulated subcategories \(\sigma=(D_{1},\dots,D_{n})\) (i. e. \(D_{i}\) is a full triangulated subcategory of \(D\) whose inclusion functor admits a right adjoint for every \(i=1,\dots,n\)) such that \(\sigma\) generates the category \(D\), and for every \(i>j\) we have \(D_{j}\subseteq D_{i}^{\perp}\) (i. e. \(\Hom_{D}(B,A)=0\) for every \(B\in D_{i}\) and every \(A\in D_{j}\)).
\textit{D. O. Orlov} [Russ. Acad. Sci., Izv., Math. 41, No. 1, 133--141 (1993; Zbl 0798.14007)] proved a semiorthogonal decomposition for the bounded derived category \(D(X)\) of coherent sheaves, where \(X=\mathbb{P}(E)\) is the projective bundle associated to a vector bundle \(E\) of rank \(r+1\) on a smooth projective variety \(S\). It is of the form \((D(S)_{0},\dots,D(S)_{r})\), where \(D(S)_{i}\) is the full triangulated subcategory of \(D(X)\) of objects of the form \(p^{*}A\otimes\mathcal{O}_{X}(i)\) (here \(p:X\longrightarrow S\) is the canonical map, \(A\in D(S)\) and \(\mathcal{O}_{X}(i)=\mathcal{O}_{X}(1)^{\otimes i}\), where \(\mathcal{O}_{X}(1)\) is the tautological line bundle of \(X\)).
In the paper under review the author generalizes this result to any Brauer-Severi scheme \(f:X\longrightarrow S\), where \(S\) is a locally noetherian scheme. A Brauer-Severi scheme on \(S\) is a locally noetherian scheme \(X\) together with a flat and proper morphism \(f:X\longrightarrow S\) whose geometric fibers are isomorphic to \(\mathbb{P}^{r}\). They can be viewed as projective bundles on \(S\): if \(f:X\longrightarrow S\) is a Brauer-Severi scheme, then there is \(\alpha\) in the cohomological Brauer group \(Br(S)=H^{2}_{et}(S,\mathbb{G}_{m})\) of \(S\), and a locally free \(\alpha-\)twisted sheaf \(E\) of rank \(r+1\) such that \(X\simeq\mathbb{P}(E)\). In section 2 the author recalls some basic facts about Brauer-Severi schemes and twisted sheaves.
The main result of the paper, which is shown in section 4, is that if \(D(X)\) is the category of perfect complexes of coherent sheaves on a Brauer-Severi scheme \(f:X\longrightarrow S\) (i. e. complexes of coherent sheaves on \(X\) which are locally isomorphic to bounded complexes of vector bundles), then \(D(X)\) admits a semiorthogonal decomposition of the form \((D(S,X)_{0},\dots,D(S,X)_{r})\), where \(r\) is the relative dimension of \(f\), and \(D(S,X)_{i}\) is equivalent to the category \(D(S,\alpha^{-i})\) of perfect complexes of coherent sheaves on \(S\) twisted by \(\alpha^{-i}\). If \(S\) and \(X\) are smooth projective varieties, we get the generalization to Brauer-Severi varieties of the previous result by Orlov. derived categories; twisted sheaves; Brauer group; Brauer-Severi schemes Bernardara, M.: A semiorthogonal decomposition for Brauer Severi schemes. Math. Nachr. \textbf{282}, 1406-1413 (2009). arXiv:math.AG/0511497 Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Derived categories, triangulated categories, Brauer groups of schemes A semiorthogonal decomposition for Brauer-Severi 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 extremely nicely written paper, the authors introduce a new notion of asymptotic behavior of line bundles on irregular varieties. First of all, the authors introduce the eventual map which associates a map to a line bundle on a variety of maximal Albanese dimension, and the continuous rank function which is a continuous function defined on a line in the space of \(\mathbb{R}\)-divisor classes that has properties similar to those of the volume function. Let \(X\) be a normal complex projective variety, \(T \subseteq X\) a subvariety of dimension \(m\), and consider \(a : X \rightarrow A\) a morphism to an abelian variety such that \(\mathrm{Pic}^{0}(A)\) injects into \(\mathrm{Pic}^{0}(T)\). Let \(L\) be a line bundle on \(X\) and \(\alpha \in\mathrm{Pic}^{0}(A)\) a general element. For any integer \(d\geq 1\), we consider the connected variety \(X^{(d)}\) defined by a map \(\tilde{\mu}_{d}\) which comes from the multiplication map by \(d\) on \(A\), i.e., \(\mu_{d} : A \rightarrow A\), namely \(\tilde{\mu}_{d} : X^{(d)} \rightarrow X\). For \(L \in\mathrm{Pic}(X)\), set \(L^{(d)} = \tilde{\mu}_{d}^{*}(L)\) and we are going to look at the linear system \(|L^{(d)} \otimes \alpha|_{|_{T^{(d)}}}\), where \(T^{(d)}:=\tilde{\mu}_{d}^{*}(T)\), \(\alpha \in\mathrm{Pic}^{0}(A)\) general, and \(d\) sufficiently large and divisible.
We start with the continuous rank of \(L\) with respect to \(a\) as the integer
\[h_{a}^{0}(X,L):=\min\{h^{0}(X, L\otimes \alpha) \, : \, \alpha \in\mathrm{Pic}^{0}(A)\}.\]
The continuous rank measures positivity of \(L\), for instance if \(h^{0}_{a}(X,L) > 0\), then \(L\) is big.
The first result of the paper tells us that when \(h^{0}_{a}(X,L)>0\) then there exists a generically finite dominant rational map, the eventual map \(\phi: X \rightarrow Z\), such that \(a\) is composed with \(\phi\), and for large \(d\) and divisible enough and \(\alpha \in\mathrm{Pic}^{0}(A)\) general the map given by \(|L^{(d)} \otimes \alpha|\) is obtained from \(\phi\) by base change with the \(d\)-th multiplication map.
Secondly, the authors introduce the so-called continuous continuous rank function by
\[x \mapsto h^{0}_{a}(X|_{T}, L + x\cdot M),\]
where \(M = a^{*}H\) with \(H\) being ample on \(A\) and \(x\) is rational. It turns out that this function extends to a continuous convex function of \(\mathbb{R}\) which is differentiable except possibly countable many points. It is worth emphasizing that this rank function is rather a subtle invariant that is not easy to compute.
As an application, the authors provide quick short proofs of a wide range of new Clifford-Severi inequalities, i.e., geographical bounds of the following form
\[\mathrm{vol}_{X|T}(L) \geq C(m) h_{a}^{0}(X_{|_{T}},L),\]
where \(C(m) = \mathcal{O}(m!)\) depends on several geometrical properties of \(X,L\), or \(a\). For instance, if \(K_{X} - L\) is pseufoeffecitve and \(a\) is birational onto its image, then
\[\mathrm{vol}(L) \geq \frac{5}{2}n! \cdot h^{0}_{a}(X,L),\]
where \(n\) is the dimension of \(X\). irregular variety; variety of maximal Albanese dimension; eventual map; continuous rank function; Clifford-Severi inequalities Divisors, linear systems, invertible sheaves, Surfaces of general type, \(3\)-folds, \(4\)-folds, \(n\)-folds (\(n>4\)) Linear systems on irregular 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 From the summary: ``For any weight vector \(\chi\) of positive integers, the weighted projective space \(\mathbf{P}(\chi)\) is a projective toric variety, and has orbifold singularities in every case other than standard projective space. Our principal aim is to study the algebraic topology of \(\mathbf{P}(\chi)\), paying particular attention to its localisation at individual primes \(p\). We identify certain \(p\)-primary weight vectors \(\pi\) for which \(\mathbf{P}(\pi)\) is homeomorphic to an iterated Thom space, and discuss how any weighted projective space may be reassembled from its \(p\)-primary parts. The resulting Thom isomorphisms provide an alternative to Kawasaki's calculation of the cohomology ring of \(\mathbf{P}(\chi)\), and allow us to recover Al Amrani's extension to complex \(K\)-theory. Our methods generalise to arbitrary complex oriented cohomology algebras and their dual homology coalgebras, as we demonstrate for complex cobordism theory, the universal example. In particular, we describe a fundamental class that belongs to the complex bordism coalgebra of \(\mathbf{P}(\chi)\), and may be interpreted as a resolution of singularities.'' weighted projective space; Thom space; orbifold Topology and geometry of orbifolds, Toric varieties, Newton polyhedra, Okounkov bodies, Bordism and cobordism theories and formal group laws in algebraic topology Weighted projective spaces and iterated Thom spaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(S\) be a scheme over a perfect field \(k\) of characteristic \(p>0\) and let \(W(k)\) be the ring of Witt vectors. Let \(T\) be a \(p\)-torsion free formal \(W(k)\)-scheme (for the \(p\)-adic topology). Let \(S\to T\) be a closed \(W(k)\)-immersion defined by an ideal equipped with divided powers. This note presents a classification of the liftings to \(T\), of commutative and locally free \(S\)-group-schemes with rank a power of \(p\), called finite \(p\)-groups. formal scheme; \(p\)-adic topology; liftings of group-schemes; Witt vectors Badra, A.: Deformations of finite p-group schemes to a formal scheme, comptes rendus de l' academie des sciences. Serie I-mathematique 325, 177-181 (1997) Formal groups, \(p\)-divisible groups, Local ground fields in algebraic geometry, Algebraic theory of quadratic forms; Witt groups and rings, Group schemes Deformations of finite \(p\)-group schemes to a formal 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 \(R\) be a commutative ring with identity and \(M\) be a unitary \(R\)-module. In this paper, we obtain a scheme \((\mathcal{X}(M),\mathbb{O}_{\mathcal{X}(M)})\) over the primary-like spectrum \(\mathcal{X}(M)\) of \(M\) and prove that \((\mathcal{X}(M),\mathbb{O}_{\mathcal{X}(M)})\) is a Noetherian scheme when \(R\) is a Noetherian ring. Zariski topology; sheaf of rings; scheme Schemes and morphisms, Theory of modules and ideals in commutative rings Primary-like submodules and a scheme over the primary-like spectrum of modules | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors of the article continue their study of representation growth and rational singularities of moduli spaces of local systems
[\textit{A. Aizenbud} and \textit{N. Avni}, Invent. Math. 204, No. 1, 245--316 (2016; Zbl 1401.14057)], now from a more general point of view.
They relate algebraic geometry over finite rings with representation theory, and obtain two main results concerning estimates (bounds) for the number of points of schemes over finite rings (Theorem A) and estimates (bounds) for the number of irreducible representations of arithmetic lattices of algebraic group schemes (Theorem B).
Let $X$ be a scheme of finite type over ${\mathbb Z}$ with the reduced, absolutely irreducible generic fiber
$X_{\mathbb Q} := X \times_{\operatorname{Spec}{\mathbb Z}} \operatorname{Spec}{\mathbb Q}$ of $X$ which is a local complete intersection.
Let $\#X(R)$ (the authors use the notation $|X(R)|$) be the number of points of the scheme $X$ over the finite ring $R$.
Theorem A states, in particular, that the next conditions are equivalent:
For any $m$, $\lim_{p \to \infty} \frac{\#X({\mathbb Z}/{p^m})}{p^{m \cdot \dim X_{\mathbb Q}}} = 1$; $X_{\mathbb Q}$ has rational singularities.
Theorem B states that for any algebraic group scheme $G$, whose generic fiber $G_{\mathbb Q}$ is simple, connected, simply connected, and of ${\mathbb Q}$-rank at least 2, and for every $C > 40$, the number of irreducible representations of $G({\mathbb Z})$ of dimension $n$ is equal to $ o(n^C)$.
The main tools in proving these theorems and their generalizations are Poincare series by Borevich-Shafarevich, Igusa zeta functions and other $p$-adic integrals, Lang-Weil bounds, deformation schemes and the theorem of Frobenius.
``For a topological group $\Gamma$, let $r_n(\Gamma)$ be the number of isomorphism classes of irreducible, $n$-dimensional, complex, continuous representations of $\Gamma$'', and let $\zeta_{\Gamma}(s)$ be the representation zeta function of $\Gamma$.
Let $k$ be a global field and let $T$ be a finite set of places of $k$ containing all Archimedean places.
By ${\mathcal O}_{k,T}$ authors denote the ring of $T$-integers of $k$ and by $\widehat{{\mathcal O}_{k,T}}$ the profinite completion of ${\mathcal O}_{k,T}$.
Let $\alpha(\Gamma)$ be the abscissa of convergence of $\zeta_{\Gamma}(s)$.
The second section of the article deals with preliminaries, which include (along with the above) elements of singularities and theorems by \textit{J. Denef} [Am. J. Math. 109, 991--1008 (1987; Zbl 0659.14017)], and by \textit{M. Mustaţă} [Invent. Math. 145, No. 3, 397--424 (2001; Zbl 1091.14004)].
In the next sections authors of the article under review ``study the number of points of schemes over finite rings'' and prove
(a generalization) of Theorem A.
The article closes with results on representation zeta functions of compact $p$-adic groups, of adelic groups and of arithmetic groups.
In the forth section the authors prove Theorem II on the abscissa of convergence $\alpha(G({\mathcal O}_{k,T}))$, Theorem III on abscissa of convergence $\alpha(G({\widehat{{\mathcal O}_{k,T}}}))$ and Theorem V on estimates of representation zeta function values at integer points $2n - 2, \; n \ge 2$.
Reviewer's remark: It is interesting to relay results of this article with results of the paper by \textit{B. Frankel} [J. Algebra 510, 393--412 (2018; Zbl 1436.14041)]. representation growth; Igusa zeta function; points of schemes over finite ring; complete intersection; rational singularities; representation zeta function Rational points, Singularities in algebraic geometry, Asymptotic properties of groups, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Representation theory for linear algebraic groups, Linear algebraic groups over global fields and their integers Counting points of schemes over finite rings and counting representations of arithmetic lattices | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Während im allgemeinen die Jacobischen von Kurven vom Geschlecht 2 und 3 in Produkte von elliptischen Kurven zerfallen, wird hier die Existenz von Kurven vom Geschlecht 2 und 3 (über \({\mathbb{C}})\) mit sogenannten elementaren Jacobischen gezeigt. Eine abelsche Varietät wird hier elementar genannt, wenn sie keine nicht-trivialen abelschen Untervarietäten enthält. Genauer: Sei R der Ring der ganzen Zahlen in \({\bar {\mathbb{Q}}}_ p\). Es gibt Kurven C über R vom Geschlecht 2 und 3, derart, daß \(Pic^ 0_ R(C)\) unendlich viele R-Automorphismen hat und die Fasern von \(Pic^ 0_ R(C)\) über Spec R elementare abelsche Varietäten sind. \(Pic^ 0_ R(C)\) besitzt unendlich viele kanonische Polarisierungen. Da es Einbettungen \(R\hookrightarrow {\mathbb{C}}\) gibt, erhält man Kurven der gewünschten Art über \({\mathbb{C}}\). Jacobian; elementary abelian varieties; canonical polarization Jacobians, Prym varieties, Picard schemes, higher Jacobians Canonical polarizations of Picard schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(C\) be a reduced Gorenstein projective curve. A generalized linear system on \(C\) is a datum \((\mathcal {F},e)\), where \(\mathcal {F}\) is a rank 1 torsion free sheaf on \(C\) and \(e: V \to H^0(C,\mathcal {F})\) a linear map of vector spaces; it is called strongly non-degenerate if for each irreducible component \(C_i\) of \(C\) the composition of \(e\) with the map \(H^0(C, \mathcal {F}) \to H^0(C_i,\mathcal {F}| C_i)\) is injective. This is a strong restriction for reducible curves, essential for almost everything here, but satisfied in many very interesting cases (not by \(H^0(C,\omega _C)\) for most reducible curves). The authors associate to \((\mathcal {F},e)\) two intrinsic objects: a zero-dimensional subscheme \(Z(\mathcal {F},e)\subset C\) and a \(0\)-cycle \(R(\mathcal {F},e)\). Now assume that \((C,(\mathcal {F},e))\) is a one-dimensional limit of a family \((C_t,L_t)\) of true linear systems. They prove that the Weierstrass divisors of \((C_t,L_t)\) parametrizing weighted Weierstrass points of \(L_t\) converge to a subscheme of \(C\) containing \(Z(\mathcal {F},e)\) and with \(R(\mathcal {F},e)\) as its associated \(0\)-cicle (the limit may depends on the family). The proofs use Wronskians for families of curves ([\textit{E. Esteves}, Ann. Sci. Éc. Norm. Supér. (4) 29, No. 1, 107--134 (1996; Zbl 0872.14025)]) and limits of Cartier divisors ([\textit{E. Esteves}, J. Pure Appl. Algebra 214, No. 10, 1718--1728 (2010; Zbl 1184.14011)]). A key part of the paper is a very useful comparison of all concepts with respect to a partial normalization \(C' \to C\). The results are proved in characteristic zero and it is observed that it works if the characteristic is \(> \dim (V)\), where \(V\) is the vector space of the given generalized linear system. curves; degenerations; linear systems; Weierstrass points; ramification point; Gorenstein curve Riemann surfaces; Weierstrass points; gap sequences, Families, moduli of curves (algebraic), Fibrations, degenerations in algebraic geometry Generalized linear systems on curves and their Weierstrass points | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors study the relation between the scaling 1-hook property of the coloured Alexander polynomial $\mathcal {A}_R^{\mathcal{K}}\left( q \right)$ and the KP hierarchy. The Alexander polynomial arises as a special case of the HOMFLY polynomial $\mathcal {H}_R^{\mathcal{K}}\left(q,a \right)$ of the knot $\mathcal{K}$ coloured with representation $R$ [\textit{E. Witten}, Commun. Math. Phys. 121, No. 3, 351--399 (1989; Zbl 0667.57005); \textit{ D. Bar-Natan}, J. Knot Theory Ramifications 4, No. 4, 503--547 (1995; Zbl 0861.57009)]:
\[{\mathcal {H}}_R^{\mathcal {K}}\left( {q,a} \right) = \frac{1} {Z}\int {DA{e^{ - \frac{i} {\hbar }{S_{CS}}\left[ A \right]}}{W_R}\left( {K,A} \right)}, \]
where the Wilson loop is
\[{W_R}\left( {K,A} \right) = {\text{t}}{{\text{r}}_R}P \exp \left( {\oint {A_\mu ^a\left( {\text{x}} \right){T^a}d{{\text{x}}^\mu }} } \right),\]
and the Chern-Simons action is
\[{S_{CS}}\left[ A \right] = \frac{\kappa } {{4\pi }}\int\limits_M {\text{Tr} \left( {A \wedge dA + \frac{2} {3}A \wedge A \wedge A} \right)}, \]
with $q = {e^\hbar },a = N\hbar $ and $\hbar = \frac{{2\pi i}} {{\kappa + N}}.$
The limiting case $\hbar \to 0,N \to \infty $ such that $N\hbar$ remains fixed, i.e., $q=1$, of the HOMFLY polynomials gives the special polynomials ${\mathcal {H}}_R^{\mathcal {K}}\left( {q,a} \right) = \sigma _R^{\mathcal {K}}\left( a \right)$, whose $R$ dependence makes them expressible in the form $\sigma _R^{\mathcal {K}}\left( a \right) = {\left( {\sigma _{\left[ 1 \right]}^{\mathcal {K}}\left( a \right)} \right)^{\left| R \right|}}$ for the Young diagram $R = \left\{ {{R_i}} \right\},{R_1} \geqslant {R_2} \geqslant \ldots \geqslant {R_{l\left( R \right)}},\left| R \right|: = \sum\nolimits_i {{R_i}} $. This provides the construction of a KP $\tau$-function (see for example [\textit{P. Dunin-Barkowski} et al., J. High Energy Phys. 2013, No. 3, Paper No. 021, 85 p. (2013; Zbl 1342.57004)]).
\par The dual limit as $a \to 1$ of the HOMFLY polynomials, i.e., ${\mathcal {H}}_R^{\mathcal {K}}\left( {q,1} \right)$, for the fundamental representation $R$ gives the Alexander polynomial, the coloured version of which exhibits a dual property with respect to $R$, viz., ${\mathcal {A}}_R^{\mathcal {K}}\left( q \right) = {\mathcal {A}}_{\left[ 1 \right]}^{\mathcal {K}}\left( {{q^{\left| R \right|}}} \right)$, which holds only for the representations corresponding to 1-hook Young diagrams ${R = \left[ {r,{1^L}} \right]}.$
\par In this paper the authors study this property perturbatively and claim that while the special polynomials provide solutions to the KP hierarchy, the Alexander polynomials induce the equations of the KP hierarchy.
\par The main result of the paper is stated in Section 5, where, by considering the generating function of the KP hierarchy, replacing the Hirota operators with the Casimir eigenvalues and symmetrizing the identity, the authors find that Hirota KP bilinear equations are satisfied if and only if
\({\mathcal {A}}_R^{\mathcal {K}}\left( q \right) = {\mathcal {A}}_{\left[ 1 \right]}^{\mathcal {K}}\left( {{q^{\left| R \right|}}} \right)\).
The authors give only the first half of the proof of this result. \par The paper explores interesting
connections between the KP hierarchy and the coloured Alexander polynomials. Chern-Simons theory; knot invariant; Kontsevich integral; Vassiliev invariants; Hirota bilinear identities; KP hierarchy; Young diagrams; Gromov-Witten theory; Schur polynomial Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Knots and links in the 3-sphere, Invariants of knots and \(3\)-manifolds, Knots and links (in high dimensions) [For the low-dimensional case, see 57M25], Eta-invariants, Chern-Simons invariants, Casimir effect in quantum field theory, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Coloured Alexander polynomials and KP hierarchy | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors construct a differential graded Lie algebra \(\mathrm{Der}_k(R,R)\) controlling infinitesimal flat deformations of a separated \(k\)-scheme \(X\) where \(k\) is a field of characteristic \(0\). This is an application of the homotopy-theoretic deformation theory they develop in [\textit{M. Manetti}: ``Formal deformation theory in left-proper model categories'', Preprint, \url{arXiv:1802.06707}] and translates Palamodov's resolvent of a complex space into the algebraic category. Bases of deformations are non-positive local Artin dg algebras \(A \in \mathbf{DGArt}^{\leq 0}_k\), a derived analogue of classical local Artin algebras which are recovered as dg algebras concentrated in degree \(0\). In case \(X = \mathrm{Spec}\ S\) is affine (treated in the above mentioned work), cofibrant deformations \(A \to S_A\) of \(k \to S\) up to weak equivalence in the model category \(\mathbf{CDGA}^{\leq 0}_k\) form a (derived) deformation functor
\[
\mathrm{Def}_S: \mathbf{DGArt}^{\leq 0}_k \to \mathbf{Set}
\]
which generalizes flat deformations of \(S\). Taking a cofibrant resolution \(R \to S\), the deformation functor \(\mathrm{Def}_S\) is controlled by the dgla \(\mathrm{Der_k(R,R)}\) of derivations of \(R\). The authors call \(R \to S\) a (local) \textit{Tate-Quillen resolution} as it is the dg analogue of Quillen's simplicial resolution of rings and a generalization of the Koszul-Tate resolution. The dgla \(\mathrm{Der}_k(R,R)\) corresponds to the tangent complex in complex geometry.
The paper under review treats the general, non-affine case. Here the model category \(\mathbf{CDGA}^{\leq 0}_k\) is replaced by the (Reedy) model category
\[
\mathbf{M} := \mathrm{Fun}(\mathcal{N}, \mathbf{CDGA}^{\leq 0}_k)
\]
of diagrams of shape \(\mathcal{N}\) where \(\mathcal{N}\) is the nerve of an affine covering \(\{\mathcal{U}_\alpha\}\) of \(X\). The algebra \(S\) is replaced by the diagram \(S_\bullet \in \mathbf{M}\) of sections \(S_\alpha = \Gamma(U_\alpha,\mathcal{O}_X)\) on the opens from which \(X\) can be recovered by gluing. For a dg Artin algebra \(A \in \mathbf{DGArt}^{\leq 0}_k\), cofibrant deformations \(\Delta A \to S_{A\bullet}\) over the constant diagram \(\Delta A\) up to weak equivalence yield a deformation functor
\[
\mathrm{Def}_{S_\bullet}: \mathbf{DGArt}^{\leq 0}_k \to \mathbf{Set}
\]
which generalizes flat deformations of \(X\). Again a cofibrant resolution \(R_\bullet \to S_\bullet\) in \(\mathbf{M}\) provides a dgla controlling the deformation functor. The authors call such a cofibrant resolution a \textit{Reedy-Palamodov resolvent}. Whereas the first name refers to the use of the direct Reedy category \(\mathcal{N}\) and the Reedy model structure on \(\mathbf{M}\), the second honours Palamodov's globalization of Tyurina's semifree resolution of an analytic algebra by means of polyhedral coverings which yields the tangent complex and the tangent cohomology of a complex space. model categories; deformation theory; differential graded algebras; algebraic schemes; cotangent complex Formal methods and deformations in algebraic geometry, Graded rings and modules (associative rings and algebras), Homotopical algebra, Quillen model categories, derivators, Deformations of complex structures Deformations of algebraic schemes via Reedy-Palamodov cofibrant resolutions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We present an algorithm that reduces the problem of calculating a numerical approximation to the action of absolute Frobenius on the middle-dimensional rigid cohomology of a smooth projective variety over a finite field, to that of performing the same calculation for a smooth hyperplane section. When combined with standard geometric techniques, this yields a method for computing zeta functions which proceeds `by induction on the dimension'. The `inductive step' combines previous work of the author on the deformation of Frobenius with a higher rank generalisation of Kedlaya's algorithm. The analysis of the loss of precision during the algorithm uses a deep theorem of \textit{G. Christol} and \textit{G. Dwork} [Duke Math. J. 62, 689--720 (1991; Zbl 0762.12004)] on \(p\)-adic solutions to differential systems at regular singular points. We apply our algorithm to compute the zeta functions of compactifications of certain surfaces which are double covers of the affine plane. Lauder A.G.B.: A recursive method for computing zeta functions of varieties. LMS. J. Comp. Math. 9, 222--269 (2006) Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Zeta and \(L\)-functions in characteristic \(p\), Analytic computations A recursive method for computing zeta functions of 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 braid arrangement \({\mathcal A}_{\ell-1}\) consists of the hyperplanes in \({\mathbb C}^\ell\) defined by the equations \(x_i=x_j\), for \(1\leq i<j\leq \ell\). The complement \({\mathbb C}^\ell - \bigcup {\mathcal A}_{\ell-1}\) is homeomorphic to the space of configurations of \(\ell\) distinct labelled points in the plane. \textit{C. DeConcini} and \textit{C. Procesi} [Sel. Math., New Ser. 1, No. 3, 459-494 (1995; Zbl 0842.14038)] constructed a nonsingular algebraic variety \({\mathcal Y}_{{\mathcal A}_{\ell-1}}\) and an algebraic map \(p: {\mathcal Y}_{{\mathcal A}_{\ell-1}} \to {\mathbb C}^\ell\) such that the preimage of \(\bigcup {\mathcal A}_{\ell-1}\) is a divisor with normal crossings, and \(p\) restricts to an isomorphism on the complement of this divisor. Furthermore, \({\mathcal Y}_{{\mathcal A}_{\ell-1}}\) is homeomorphic to the moduli space of stable \((\ell+1)\)-pointed curves of genus zero.
The natural action of \(\Sigma_\ell\) on \({\mathbb C}^\ell\) can be lifted to an action on \({\mathcal Y}_{{\mathcal A}_{\ell-1}}\), yielding a representation of \(\Sigma_\ell\) on the cohomology \(H^*({\mathcal Y}_{{\mathcal A}_{\ell-1}})\). The paper under review gives a complete description of (the character of) this representation. More specifically, the author derives a formula for the generalized Poincaré polynomial
\[
P_w(q)=\sum \text{tr} w|_{H^{2i}({\mathcal Y}_{{\mathcal A}_{\ell-1}})}q^i
\]
for arbitrary \(w\in \Sigma_\ell\).
These traces are calculated directly by a combinatorial argument, using a description due to \textit{S. Yuzvinsky} [Invent. Math. 127, No. 2, 319-335 (1997)] of an integral basis for \(H^*({\mathcal Y}_{{\mathcal A}_{\ell-1}})\). Basis elements are indexed by (graph-theoretic) forests with \(\ell\) leaves labelled by the integers \(1, \ldots, \ell\), rooted components, and internal vertices labelled subject to certain valence conditions. The natural \(\Sigma_\ell\) action on these forests, by permutation of the leaf labels, corresponds to the \(\Sigma_\ell\) representation on \(H^*({\mathcal Y}_{{\mathcal A}_{\ell-1}})\). Then the trace of \(w \in \Sigma_\ell\) on \(H^{2i}({\mathcal Y}_{{\mathcal A}_{\ell-1}})\) can be obtained by a counting argument.
The count is actually not so easy to carry out, and the resulting formula for \(P_w(q)\) is fairly complicated. In the last part of the paper, the author reformulates the main result in terms of a more general series \(\mathcal H\) in formal variables, which can be described in a more compact form. The Poincaré polynomial \(P_w(q)\) is obtained from \(\mathcal H\) by a certain specialization process involving the cycle structure of \(w\).
The expression for \(\mathcal H\) contains certain sums over trees. The author derives closed forms for these sums that may be of independent interest. braid arrangement; DeConcini-Procesi model; representation; symmetric group; cohomology ring; Poincaré series Arrangements of points, flats, hyperplanes (aspects of discrete geometry), Families, moduli of curves (algebraic) Generalized Poincaré series for models of the braid arrangements | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Deformation theory in algebraic geometry is a basic tool that is just as old as the idea of classifying algebraic varieties. In his famous memoir on abelian functions from 1857, Riemann initiated the study of deformations of the complex structure of compact one-dimensional complex manifolds of given topological genus \(g\), that is of compact Riemann surfaces (or non-singular complex projective curves) of genus \(g\). The deformation of complex algebraic surfaces seems to have been considered first by Max Noether in 1888. However, a systematic deformation theory for higher-dimensional complex manifolds could be developed only as late as in 1958 and in the years thereafter, thus even more than 100 years after Riemann's pioneering memoir, and with the modern conceptual framework of sheaf theory, sheaf cohomology, and Hodge theory as a crucial ingredient [cf.: \textit{K. Kodaira}, ``Complex manifolds and deformation of complex structures.'' Grundlehren der Mathematischen Wissenschaften, 283. New York etc.: Springer-Verlag. X, 465 (1986; Zbl 0581.32012)].
This modern approach to complex-analytic deformation theory is well-known as Kodaira-Nirenberg-Spencer-Kuranishi theory, and its immediate effect on the development of classification theory in algebraic geometry has been equally revolutionary. In fact, A. Grothendieck immediately realized the functorial character of the Kodaira-Spencer theory of infinitesimal deformations of complex-analytic manifolds, and he outlined an analogous approach within his newly created foundational framework of algebraic schemes and formal algebraic geometry right away. These fundamental ideas, which eventually led to what is now called algebraic deformation theory, were published in a series of Bourbaki seminar expositions collected in his celebrated ``Fondements de la Géométric Algébrique'' [\textit{A. Grothendieck} ``FGA'', Secretariat Math. Paris (1962; Zbl 0239.14002)]. In the sequel, algebraic deformation theory has rapidly grown into a vast and central topic in modern abstract algebraic geometry, due to its crucial importance in regard to variational problems, including local properties of moduli spaces of varieties, vector bundles, and singularities. In its present state of art, algebraic deformation theory is highly formalized, both conceptually and methodically rather involved, widely ramified within algebraic geometry, and therefore not easily accessible to non-experts in the field. Although being of such importance and ubiquity in variational algebraic geometry, and while still developing rapidly, algebraic deformation theory had so far not yet found an adequate reflection in the relevant textbook literature. This fact made it rather difficult for non-specialists to find a solid orientation in this vast field of contemporary mathematical research, all the more so as numerous subtle technicalities and allegedly well-known results are scattered in the huge literature as ``folklore'', without rigorous and detailed proofs. The book under review is an attempt to partially fill this bothersome gap in the literature, and to provide a largely self-contained and comprehensive account of deformation theory in classical algebraic geometry, with complete proofs of those results and techniques that are needed to fully understand the local deformation theory of algebraic schemes over an algebraically closed field. This includes the careful explanation of those basic tools that are indispensable, for example, in the local study Hilbert schemes, Quot schemes, Picard schemes, and other classifying objects in this context. In this vein, and for the first time in the literature, the author compiles some of the many folklore results scattered in the literature, with detailed and systematic proofs, which must be seen as a just as valuable as rewarding contribution towards a solid foundation of algebraic deformation theory, and as an utmost useful service to the mathematical community likewise. As for the contents, the present book consists of four chapters, each of which is divided into several sections and subsections. In addition, and for the convenience of the non-expert reader, there are five appendices devoted to some basic facts from commutative algebra and algebraic geometry which are used throughout the text.
After a brief introduction, in which an outline of the complex-analytic infinitesimal deformation theory à la Kodaira-Nirenberg-Spencer-Kuranishi serves as an explanation of the logical structure of algebraic (and functorial) deformation theory, Chapter 1 introduces the reader to infinitesimal deformations in an elementary fashion. The first section discusses extensions of rings and algebras, together with their generalizations to schemes over a base scheme, whereas the second section explains locally trivial deformations of schemes. This includes infinitesimal deformations of non-singular affine schemes, automorphisms of deformations and their extendability properties, first-oder locally trivial deformations, higher-order deformations, and the related (cohomological) obstruction theory for deformations. Chapter 2 provides the foundations of formal deformation theory. Starting with the concepts of formal smoothness and relative obstruction spaces for ring extensions, the author reconsiders infinitesimal deformations of algebraic schemes via M. Schlessinger's theory of functors of Artin rings, culminating in Schlessinger's famous theorem on the existence of (semi-) universal formal deformations [cf.: \textit{M. Schlessinger}, Trans. Am. Math. Soc. 130, 208--222 (1968; Zbl 0167.49503)]. This is followed by a discussion of deformation functors and local moduli functors in general, including their obstruction spaces as well as concrete applications to the deformation theory of algebraic surfaces in characteristic zero. In this regard, the author provides a largely self-contained treatment of formal deformation theory along the classical approach, that is without introducing cotangent complexes (à la L. Illusie) or methods of differential graded Lie algebras as more recent tools in deformation theory.
Chapter 3 illustrates the general theory of deformation functors by several concrete examples. With the single exception of deformations of algebraic vector bundles, which have already been exhaustively treated in several recent research monographs like the book of \textit{D. Huybrechts} and \textit{M. Lehn} [``The geometry of moduli spaces of sheaves''. Aspects Math. E 31 (1997; Zbl 0872.14002)], the author describes in great detail the most important deformation functors in current algebraic geometry, mainly by carefully verifying Schlessinger's conditions for the existence of (semi-)universal deformation spaces in the respective cases. Moreover, the first-order deformations, i.e., the tangent spaces of these functors, as well as the corresponding obstruction spaces are thoroughly analyzed, thereby revealing a similar pattern in almost all of these exemplary cases. As for the concrete examples treated in this chapter, the author discusses the deformation functor of an affine scheme with at most quotient singularities, the local (relative) Hilbert functor of closed subschemes of a scheme, the local Picard functor of a scheme, deformations of sections of an invertible sheaf on a scheme, deformations of various types of morphisms of schemes, deformations of a closed embedding, and (co-)stable subschemes.
Chapter 4 gives a thorough introduction to Hilbert schemes, Quot schemes, and flag Hilbert schemes. As the author points out, these objects are needed to construct important examples of global deformations, and to study their local behaviour in the framework of the main theme of the present text. Besides, until now it was rather difficult to give precise references for many results on the geometry of these classifying objects, just like in local deformation theory, and this very fact has been another reason for the author to include that chapter in his book. Together with the very recent monograph ``Fundamental Algebraic Geometry: Grothendieck's FGA Explained'' by \textit{B. Fantechi} et al. [Math. Surv. Monogr. 123 (2005; Zbl 1085.14001)], this chapter in the book under review provides the only (and overdue) reasonably comprehensive exposition on Grothendieck's Hilbert and Quot schemes. Apart from a general introduction to Castelnuovo-Mumford regularity, flattening stratifications, Hilbert schemes, Quot schemes, flag Hilbert schemes, and Grassmannians, together with applications to families of projective schemes, there is a concluding section on plane curves, their equisingular infinitesimal deformations, and their so-called Severi varieties. The author's approach to the proof of existence of nodal curves with an arbitrary number of singularities is very original and apparently new. Based on the use of multiple point schemes, this example strikingly illustrates the power of algebraic deformation theory even in classical curve theory.
The five appendices at the end of the book collect some basic standard topics and are titled as follows: A. Flatness; B. Differentials; C. Smoothness; D. Complete intersections; E. Functorial language. Most of the results presented here come with full proofs, which further strengthens the already high degree of self-containedness of the book.
There are no explicit exercises or working problems accompanying the text, but there is a wealth of illustrating concrete examples, additional remarks, historical annotations, and hints to further reading. The bibliography includes 190 references, ranging from the very classical up to the most recent articles and books, and both a very carefully compiled list of used symbols and a just as thorough alphabetical index considerably enhance the value of the entire treatise.
Without any doubt, this is a masterly book on a highly advanced topic in algebraic geometry. The author's style of writing captivates by its high degree of comprehensiveness, completeness, rigour, sytematical exposition, creative originality, lucidity, and user-friendliness in a like manner. The entire text is kept at a level that makes it suitable for graduate students with a solid background in commutative algebra, homological algebra, and basic algebraic geometry. But even for experts and active researchers in algebraic geometry, this unique book on algebraic deformation theory offers a great deal of inspiration and new insights, too, and its future role as a standard source and reference book in the field can surely be taken for granted from now on. textbook; formal methods; local deformation theory; local moduli; Hilbert schemes; Picard functors E. Sernesi, \textit{Deformations of algebraic schemes}. Springer 2006. MR2247603 Zbl 1102.14001 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Formal methods and deformations in algebraic geometry, Local deformation theory, Artin approximation, etc., Infinitesimal methods in algebraic geometry, Algebraic moduli problems, moduli of vector bundles, Parametrization (Chow and Hilbert schemes) Deformations of algebraic 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 From the fact that a foliation by curves of degree greater than one, with isolated singularities, in the complex projective plane \(\mathbb P^2\) is uniquely determined by its subscheme of singular points (the singular subscheme of the foliation), we pose the problem of existence of proper closed subschemes \(Z\) of the singular subscheme which still determine the foliation in a unique way. We prove the existence of such subschemes \(Z\) for foliations with reduced singular subscheme. Unlike the degree \(\deg Z\) of such subschemes is not sharp for the posed problem, we show that it is so in the sense that \(Z\) defines the so-called polar net of the foliation. Campillo, A., Olivares, J.: Special subschemes of the scheme of singularities of a plane foliation. Comptes Rendus de l'Académie de Sciences - Série I - Mathématiques 344(9), 581-585 (2007) Singularities of holomorphic vector fields and foliations, Cycles and subschemes Special subschemes of the scheme of singularities of a plane foliation | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author proves that, if the chosen projection avoids the bitangents and the inflections to the small enough level curves of a real bivariate polynomial function near a strict local minimum at the origin, then the asymptotic Poincaré-Reeb tree becomes complete binary and its vertices become totally ordered. This projection direction is called generic. She proves that for any such asymptotic family of level curves, there are finitely many intervals on the real projective line outside of which all the directions are generic with respect to all the curves in the family. If the projection is generic, then the local shape of the curves can be encoded in terms of alternating permutations, called snakes. Snakes offer an effective description of the local geometry and topology, well-suited for computations. generic projection; real algebraic curve; Poincaré-Reeb tree; permutation; snake; polar curve; bitangent; inflection; dual curve; strict local minimum Topology of real algebraic varieties, Real algebraic sets, Singularities of curves, local rings, Singularities in algebraic geometry, Trees Permutations encoding the local shape of level curves of real polynomials via generic projections | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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{M.-H. Shih} and \textit{J.-L. Dong} [Adv. Appl. Math. 34, No. 1, 30--46 (2005; Zbl 1060.05505)] have proved a boolean analogue of the Jacobian problem: if a map from \(\{0,1\}^n\) to itself has the property that all the boolean eigenvalues of the discrete Jacobian matrix of each element of \(\{0,1\}^n\) are zero, then it has a unique fixed point. In this note, this result is extended to any map \(F\) from the product \(X\) of \(n\) finite intervals of integers to itself. Our method of proof reveals an interesting property of the asynchronous state graph of \(F\) used to model the qualitative behavior of genetic regulatory networks. discrete dynamical systems; asynchronous automata networks; Jacobian conjecture; discrete Jacobian matrix; Boolean eigenvalue; fixed point; genetic regulatory networks Richard, A., An extension of a combinatorial fixed point theorem of shih and dong, Advances in Applied Mathematics, 41, 4, 620-627, (2008) Combinatorial aspects of matrices (incidence, Hadamard, etc.), Cellular automata (computational aspects), Topological dynamics, Jacobian problem, Boolean functions, Combinatorics in computer science, Genetics and population dynamics, An extension of a combinatorial fixed point theorem of Shih and Dong | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(W^ 2_ 6\) be a scheme parametrizing pairs (L,C), where C is a smooth algebraic curve of genus 10 and L is a line bundle of degree 6 such that \(h^ 0(C,L)\geq 3\). It is shown that one of the irreducible components of this scheme is nonreduced. A general point of this component consists of a smooth hyperelliptic curve C and a line bundle L of the form \(L={\mathcal O}(2g^ 1_ 2+Q+R)\), where Q and R are general points on C. linear system; algebraic curve Divisors, linear systems, invertible sheaves, Curves in algebraic geometry On a family of special linear systems on algebraic curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper treats a generalization of the Briançon-Skoda theorem about integral closures of ideals for graded systems of ideals satisfying a certain geometric condition. This work divides in two sections, the first contains some definitions and basic properties of stable graded systems of ideals. The main result of section 2 is the following:
Let \(a_{\bullet}=\{a_n\}_{n\in \mathbb N}\) be a stable graded system of ideals. Then there exists an integer \(C\) so that for all \(n\gg 0\), \(\mathcal{J}(Cn.{a}_{\bullet})\subseteq {a}_{n},\) where \(\mathcal{J}(Cn.a_{\bullet})\) denotes the level \(n\) asymptotic multiplier ideal attached to graded system of ideal \(a_{\bullet}\).
As asymptotic multiplier ideals are integrally closed, the fact that \(\overline{a}_m \subseteq \mathcal{J}(m.a_{\bullet})\) then implies the following generalization of the aforementioned result of Briançon and Skoda.
Let \(a_{\bullet}=\{a_n\}_{n\in \mathbb N}\) be a stable graded system of ideals. Then there exists a positive integer \(C\), such that for all \(n\gg 0\), \({\overline{a}}_{Cn}\subseteq a_n\). asymptotic order of vanishing; ideal sheaf; level-\(n\) asympotic multiplier ideal; graded system of ideals; smooth complex variety; symbolic power of an ideal Ideals and multiplicative ideal theory in commutative rings, Integral closure of commutative rings and ideals, Structure, classification theorems for modules and ideals in commutative rings, Singularities in algebraic geometry, Divisors, linear systems, invertible sheaves, Vanishing theorems in algebraic geometry A Briançon-Skoda type theorem for graded systems of ideals | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Given a smooth curve on a smooth surface, the Hilbert scheme of points on the surface is stratified according to the length of the intersection with the curve. The strata are highly singular. We show that this stratification admits a natural log-resolution, namely the stratified blowup. As a consequence, the induced Poisson structure on the Hilbert scheme of a Poisson surface has unobstructed deformations. complex surface; Hilbert scheme; stratification; normal crossings; Poisson structure Ran, Z, Incidence stratifications on Hilbert schemes of smooth surfaces, and an application to Poisson structures, Int. J. Math., 27, 1650006, (2016) Parametrization (Chow and Hilbert schemes), Surfaces and higher-dimensional varieties, Compact complex surfaces, Poisson manifolds; Poisson groupoids and algebroids Incidence stratifications on Hilbert schemes of smooth surfaces, and an application to Poisson structures | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author proves that if \(S\) is a Poisson surface, i.e. a smooth projective surface with a Poisson structure, then the Hilbert scheme of points of \(S\) has a natural Poisson structure, induced by the one of \(S\). This generalizes previous results obtained by \textit{A. Beauville} [J. Differ. Geom. 18, 755-782 (1983; Zbl 0537.53056)] and \textit{S. Mukai} [Invent. Math. 77, 101-116 (1984; Zbl 0565.14002)] in the symplectic case, i.e. when \(S\) is an Abelian or K3 surface. Finally, the author applies his results to give some examples of integrable Hamiltonian systems naturally defined on these Hilbert schemes. In the case \(S= \mathbb{P}^2\) he obtains a large class of integrable systems, which includes the ones studied by \textit{P. Vanhaecke} [Prog. Math. 145, 187-212 (1997; Zbl 0873.58038)]. Poisson surface; Hilbert scheme; Poisson structure; integrable Hamiltonian systems; Hilbert schemes F. BOTTACIN, Poisson structures on Hilbert schemes of points of a surface and integrable systems, Manuscripta Math., 97 (1998), pp. 517-527. Zbl0945.53049 MR1660136 Poisson manifolds; Poisson groupoids and algebroids, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Parametrization (Chow and Hilbert schemes), Special surfaces Poisson structures on Hilbert schemes of points of a surface and 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 If \(X\) is a set of distinct points in \(\mathbb{P}^2\) with given graded Betti numbers, we produce a new set of points \(Y\) with the same graded Betti numbers as \(X\) which admits all possible conductor degrees according to the graded Betti numbers. Moreover, for such schemes we can compute the conductor degree for each point.
We conclude by generalizing the construction of these schemes, obtaining again the same results. Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) On 0-dimensional schemes with all permissible conductor degrees | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Based on the Lenard recursion equations and the stationary zero-curvature equation, we derive the coupled Sasa-Satsuma hierarchy, in which a typical number is the coupled Sasa-Satsuma equation. The properties of the associated trigonal curve and the meromorphic functions are studied, which naturally give the essential singularities and divisors of the meromorphic functions. By comparing the asymptotic expansions for the Baker-Akhiezer function and its Riemann theta function representation, we arrive at the finite genus solutions of the whole coupled Sasa-Satsuma hierarchy in terms of the Riemann theta function. coupled Sasa-Satsuma hierarchy; trigonal curve; Baker-Akhiezer function; finite-genus solution Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions, Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems, Theta functions and curves; Schottky problem, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Relationships between algebraic curves and integrable systems The coupled Sasa-Satsuma hierarchy: trigonal curve and finite genus solutions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For the entire collection see Zbl 0695.00010.]
This paper is a sequel to a previous paper by the author [Acta Math. 147, 13-49 (1981; Zbl 0479.14004)], where the method of iteration was used to derive r-fold-point formulas for a proper map \(f:X\to Y\) [see \textit{S. Katz} in Algebraic geometry, Proc. Conf., Sundance/Utah 1986, Lect. Notes Math. 1311, 147-155 (1988; Zbl 0657.14035)]. - In the present paper a defect of the method of iteration is pointed out in the case of torsion and new formulas are derived using the Hilbert scheme. More precisely the new formulas are expressed in terms of effectively computable polynomials in the Chern classes of f with integer coefficients and in terms of three multiple-point cycles \(t_ r, u_ r\) and \(v_ r\) where \(t_ r\) enumerates the points of Y whose fibers contain length r subschemes, \(u_ r\) enumerates the points of X which are parts of length r subschemes of the fibers and \(v_ r\) enumerates the points x in X such that there exists a length r\(+1\) subscheme of the fiber \(f^{-1}fx\) that is an extension of some length r subscheme by x. Finally, the author raises several unanswered questions that could be investigated in the future. iteration; r-fold-point; Hilbert scheme; multiple-point cycles Steven L. Kleiman, Multiple-point formulas. II. The Hilbert scheme, Enumerative geometry (Sitges, 1987) Lecture Notes in Math., vol. 1436, Springer, Berlin, 1990, pp. 101 -- 138. Enumerative problems (combinatorial problems) in algebraic geometry, Parametrization (Chow and Hilbert schemes), Algebraic cycles Multiple-point formulas. II: The Hilbert scheme | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For a commuative field \(F\), the ring \(T\) of upper triangular \(2\times 2\) matrices over \(F\) ist called the ring of ternions. It is shown that the set of all free cyclic submodules of \(T^2\) admits a point model in \(\mathrm{PG}(7,F)\) which is a smooth algebraic variety \(\mathcal{X}\cup \mathcal{Y}\), where \(\mathcal X\) corresponds to the unimodular submodules and \(\mathcal Y\) (corresponding to the non-unimodular ones) is a line. ternions; projective line; cyclic submodules; point model; smooth variety Havlicek, H., Kosiorek, J., Odehnal, B.: A point model for the free cyclic submodules over ternions. Results Math. 63, 1071-1078 (2013) Nonlinear incidence geometry, Ring geometry (Hjelmslev, Barbilian, etc.), Rational and ruled surfaces, Free, projective, and flat modules and ideals in associative algebras A point model for the free cyclic submodules over ternions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 develops techniques to construct tropicalization maps associated to log schemes.
A log scheme $X$ is a scheme with additional data, given by sheaves of monoids, which typically records information on infinitesimal deformations of a family, where $X$ appears as a general fiber [\textit{K. Kato}, Proc. JAMI Inaugur. Conf., Baltimore/MD (USA) 1988, 191--224 (1989; Zbl 0776.14004)]. Tropicalizing $X$ provides a combinatorially effective framework to study enumerative problems in log geometry [\textit{M. Gross} and \textit{B. Siebert}, J. Am. Math. Soc. 26, No. 2, 451--510 (2013; Zbl 1281.14044)].
In this article, techniques from Berkovich analytic spaces are used to construct a natural tropicalization map. The author shows that this map is functorial with respect to logarithmic morphisms. The approach developed in the article, in particular, generalises Thuillier's retraction map onto the non-Archimedean skeleton [\textit{A. Thuillier}, Manuscr. Math. 123, No. 4, 381--451 (2007; Zbl 1134.14018)].
As an application of the techniqes developed, a criterion for a Schön Variety [\textit{J. Tevelev}, Am. J. Math. 129, No. 4, 1087--1104 (2007; Zbl 1154.14039)] to admit a faithful tropicalization is given. tropicalization; Berkovich analytic geometry; log geometry , Rigid analytic geometry, Commutative semigroups Functorial tropicalization of logarithmic schemes: the case of constant 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 \textit{R. Hirota} and \textit{K. Kimura} [J. Phys. Soc. Japan 69, No. 3, 627--630 (2000; Zbl 1058.70504)] discovered integrable discretizations of the Euler and the Lagrange tops, given by birational maps. Their method is a specialization to the integrable context of a general discretization scheme introduced by W. Kahan and applicable to any vector field with a quadratic dependence on phase variables. According to a proposal by T. Ratiu, discretizations of Hirota-Kimura type can be considered for numerous integrable systems of classical mechanics. Due to a remarkable and not well understood mechanism, such discretizations seem to inherit the integrability for all algebraically completely integrable systems. We introduce an experimental method for a rigorous study of integrability of such discretizations.
Application of this method to the Hirota-Kimura-type discretization of the Clebsch system leads to the discovery of four functionally independent integrals of motion of this discrete-time system, which turn out to be much more complicated than the integrals of the continuous-time system. Further, we prove that every orbit of the discrete-time Clebsch system lies in an intersection of four quadrics in the six-dimensional phase space. Analogous results hold for the Hirota--Kimura-type discretizations for all commuting flows of the Clebsch system, as well as for the \(so(4)\) Euler top. integrable discretization; computer-assisted proof; birational dynamics; Clebsch system; integrable tops Petrera, M., Pfadler, A., and Suris, Yu. B., On Integrability of Hirota-Kimura-Type Discretizations: Experimental Study of the Discrete Clebsch System, Experiment. Math., 2009, vol. 18, no. 2, pp. 223--247. Rational and birational maps, Relationships between algebraic curves and integrable systems, Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems, Discrete version of topics in analysis, Integrable cases of motion in rigid body dynamics On integrability of Hirota-Kimura-type discretizations: experimental study of the discrete Clebsch system | 0 |
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