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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 \textit{E. R. García Barroso} et al. [Math. Ann. 368, No. 3--4, 1359--1397 (2017; Zbl 1386.14018)], employ the Lagrange inversion formula to solve certain Newton-Puiseux equations when the solutions to the inverse problems are given. More precisely, for an irreducible \(f(x,y)\in K[[x,y]]\) over an algebraically closed field \(K\) of characteristic zero, they calculate the coefficients of \(\eta (x^{1/n})\) which would meet \(f(x,\eta (x^{1/n}))=0\) in terms of the coefficients of \(\xi (y^{1/m})\) that satisfy \(f(\xi (y^{1/m}),y)=0\). This article will present an alternative approach to solving the problem using diagonalizations on polynomial sequences of binomial-type. Along the way, a close relationship between binomial-type sequences and the Lagrange inversion formula will be observed. In addition, it will extend the result to give the coefficients of \(\eta (x^{1/n})\) directly in terms of the coefficients of \(f(x,y)\). As an application, an infinite series formula for the roots of complex polynomials will be obtained together with a sufficient condition for its convergence. Umbral calculus, Singularities of curves, local rings, Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral) Closed-form solutions to irreducible Newton-Puiseux equations by Lagrange inversion formula and diagonalization on polynomial sequences of binomial-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 In the article, the authors present an algorithm for computing the Betti numbers and torsion coefficients of closed semialgebraic sets.
Semialgebraic sets are subsets of \(\mathbb{R}^n\) defined by Boolean combinations (i.e., a sequence of unions, intersections, and complements) of polynomial equalities and inequalities. The class of these sets is closed under unions, intersections, complements, and projections as well as under taking images and preimages of polynomial maps. Semialgebraic sets play a distinguished role in several branches of mathematics including real algebraic geometry, complexity theory, mathematical programming, robotics, etc.
In this article, the authors focus on semialgebraic sets given by Boolean formulas without negations over lax polynomial inequalities i.e., they are defined by only unions and intersections of the atomic sets. In contrast to the doubly exponential complexity of previous algorithms for computing the sequence of homology groups of semialgebraic sets, the algorithm proposed in this article works in weak exponential time. This means that outside a subset of data having exponentially small measure, the cost of the algorithm is single exponential in the size of the data. homology groups; weak complexity; numerical algorithms; lax Boolean combinations Semialgebraic sets and related spaces, Numerical aspects of computer graphics, image analysis, and computational geometry, Complexity and performance of numerical algorithms, Analysis of algorithms and problem complexity Computing the homology of semialgebraic sets. I: Lax formulas | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We prove that the \textit{integral} cohomology modulo torsion of a rationally connected threefold comes from the integral cohomology of a smooth curve via the cylinder homomorphism associated to a family of \(1\)--cycles. Equivalently, it is of \textit{strong} coniveau \(1\). More generally, for a rationally connected manifold \(X\) of dimension \(n\), we show that the strong coniveau \(\widetilde{N}^{n-2}H^{2n-3}(X, \mathbb{Z})\) and coniveau \(N^{n-2} H^{2n-3}(X, \mathbb{Z})\) coincide for cohomology modulo torsion. coniveau filtration; strong coniveau filtration; integral cohomology; rationally connected manifold Algebraic cycles, Rationality questions in algebraic geometry, (Co)homology theory in algebraic geometry, \(3\)-folds, Rationally connected varieties On the coniveau of rationally connected threefolds | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We explain, in a non technical way, several general methods for constructing semi-universal deformations, especially by Kuranshi maps. Moreover, we give (standard) criteria of universality and smoothness of the semi-universal deformation, discuss the existence of operations on the base germ and describe intrinsically the Massey products. Kosarew, S.: Local moduli spaces and kuranishi maps, Manuscripta math. 110, No. 2, 237-249 (2003) Complex-analytic moduli problems, Local deformation theory, Artin approximation, etc. Local moduli spaces and Kuranishi maps | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(Q\) be an extended Dynkin quiver and \(Z(Q,d)\) the common zero locus of all non-constant homogeneous semi-invariant polynomial functions on the vector space parametrizing representations of \(Q\) of dimension vector \(d\). The main result of the paper is that \(Z(Q,d)\) is a complete intersection provided that the multiplicities of the summands in the so-called canonical decomposition of \(d\) are large enough (explicit bounds are given in the paper). This extends to arbitrary dimension vectors the earlier result of the same authors on the case of a prehomogeneous dimension vector [Comment. Math. Helv. 79, 350--361 (2004; Zbl 1063.14052)]. A notable new phenomenon in the case of a non-prehomogeneous dimension vector is that the number of irreducible components of \(Z(Q,d)\) can be arbitrarily large (except when \(Q\) is an oriented cycle). The proof builds on the description of algebras of semi-invariants of tame quivers by \textit{A. Skowronski} and \textit{J. Weyman} [Transform. Groups 5, No. 4, 361--402 (2000; Zbl 0986.16004)] and on the representation theoretic interpretation of semi-invariants of quivers due to \textit{A. Schofield} [J. Lond. Math. Soc. (2) 43, No. 3, 385--395 (1991; Zbl 0779.16005)]. semi-invariants; quivers; representations; cofree action; complete intersection Riedtmann, Ch., Zwara, G.: The zero set of semi-invariants for extended Dynkin quivers. Trans. Am. Math. Soc. \textbf{360}(12), 6251-6267 (2009i:14064) (2008) \textbf{(MR2434286)} Geometric invariant theory, Representations of quivers and partially ordered sets The zero set of semi-invariants for extended Dynkin quivers | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We prove that for a finite covering of curves the Clifford index of the source is at least that of the target. Coverings of curves, fundamental group, Special divisors on curves (gonality, Brill-Noether theory) An inequality of Clifford indices for a finite covering of curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(C\subset \mathbb {P}^{g-1}\) the canonical model of a smooth genus \(g\) curves. If \(g\ge 11\) the authors shows that the second syzygy scheme of \(C\) is \(\ne C\) if and only if \(C\) is a bielliptic curve. For lower \(g\) they classify the second syzygy scheme of the \(4\)-gonal curves. For the proof they first exclude the curves with Clifford index \(\ge 3\) and then analize the \(4\)-gonal curves. curves; syzygy; bielliptic curve; gonality; second syzygy scheme; tetragonal curve; Clifford index Special divisors on curves (gonality, Brill-Noether theory), Syzygies, resolutions, complexes and commutative rings A characterization of bielliptic curves via syzygy schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper, we present an explicit formula that connects the Kontsevich-Witten tau-function and the Hodge tau-function by differential operators belonging to the \(\widehat{GL(\infty)}\) group. Indeed, we show that the two tau-functions can be connected using Virasoro operators. This proves a conjecture posted by \textit{A. Alexandrov} [Lett. Math. Phys. 104, No. 1, 75--87 (2014; Zbl 1342.37066)]. Kontsevich-Witten tau-function; Hodge tau-function; Virasoro operators Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations, Virasoro and related algebras Connecting the Kontsevich-Witten and Hodge tau-functions by the \(\widehat{GL(\infty)}\) operators | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors identify certain Gromov-Witten invariants with Euler characteristics of moduli spaces of point configurations in projective spaces. They also discuss the similarities and differences between labeled floor diagrams and stable trees. Gromov-Witten; configuration space; Euler characteristic; moduli space; stable tree Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Moduli spaces of point configurations and plane curve counts | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 Grothendieck's \textit{dessins d}'\textit{enfants} in the context of the \( \mathcal{N}=2 \) supersymmetric gauge theories in (3 + 1) dimensions with product \(SU (2)\) gauge groups which have recently been considered by Gaiotto et al.. We identify the precise context in which dessins arise in these theories: they are the so-called ribbon graphs of such theories at certain isolated points in the moduli space. With this point in mind, we highlight connections to other work on trivalent dessins, gauge theories, and the modular group. supersymmetric gauge theory; gauge symmetry; differential and algebraic geometry He, Y-H; Read, J, Dessins denfants in \( {\mathcal{N}} =2 \) generalised quiver theories, JHEP, 08, 085, (2015) Supersymmetric field theories in quantum mechanics, Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations, Dessins d'enfants theory Dessins d'enfants in \( \mathcal{N}=2 \) generalised quiver theories | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We review the standard formulation of mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, and compare this construction to a description of mirror symmetry for \(K3\) surfaces which relies on a sublattice of the Picard lattice. We then show how to combine information about the Picard group of a toric ambient space with data about automorphisms of the toric variety to identify families of \(K3\) surfaces with high Picard rank. Mirror symmetry (algebro-geometric aspects), \(K3\) surfaces and Enriques surfaces, Special polytopes (linear programming, centrally symmetric, etc.), Toric varieties, Newton polyhedra, Okounkov bodies Reflexive polytopes and lattice-polarized \(K3\) surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We construct the normal forms of null-Kähler metrics: pseudo-Riemannian metrics admitting a compatible parallel nilpotent endomorphism of the tangent bundle. Such metrics are examples of non-Riemannian holonomy reduction, and (in the complexified setting) appear on the space of Bridgeland stability conditions on a Calabi-Yau threefold. Using twistor methods we show that, in dimension four -- where there is a connection with dispersionless integrability -- the cohomogeneity-one anti-self-dual null-Kähler metrics are generically characterised by solutions to Painlevé I or Painlevé II ODEs. non-Riemannian holonomy reduction; Bridgeland stability; Calabi-Yau threefold; using twistor methods General geometric structures on manifolds (almost complex, almost product structures, etc.), Global differential geometry of Hermitian and Kählerian manifolds, Issues of holonomy in differential geometry, Calabi-Yau manifolds (algebro-geometric aspects) Null Kähler geometry and isomonodromic deformations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The moduli space \(\overline{M}_{0,n}\) may be embedded into the product of projective spaces \(\mathbb{P}^1\times\mathbb{P}^2\times\cdots \times\mathbb{P}^{n-3}\), using a combination of the Kapranov map \(|\psi_n|:\overline{M}_{0,n}\to\mathbb{P}^{n-3}\) and the forgetful maps \(\pi_i:\overline{M}_{0,i}\to\overline{M}_{0,i-1}\). We give an explicit combinatorial formula for the multidegree of this embedding in terms of certain parking functions of height \(n-3\). We use this combinatorial interpretation to show that the total degree of the embedding (thought of as the projectivization of its cone in \(\mathbb{A}^2\times\mathbb{A}^3\cdots\times\mathbb{A}^{n-2})\) is equal to \((2(n-3)-1)!!=(2n-7)(2n-9)\cdots(5)(3)(1)\). As a consequence, we also obtain a new combinatorial interpretation for the odd double factorial.
Editorial remark: This is an extended version of [\textit{R. Cavalieri} et al., Sémin. Lothar. Comb. 84B, 84B.32, 12 p. (2020; Zbl 1447.05003)]. parking functions; moduli of curves; multidegree; intersection theory Permutations, words, matrices, Exact enumeration problems, generating functions, Families, moduli of curves (algebraic), Combinatorial aspects of algebraic geometry, Combinatorial aspects of representation theory Projective embeddings of \(\overline{M}_{0,n}\) and parking functions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We show that the Poincaré bundle gives a fully faithful functor from the derived category of a curve of sufficiently high genus into the derived category of its moduli space of bundles of rank \(r\) with fixed determinant of degree 1. This generalises results of \textit{M. S. Narasimhan} [J. Geom. Phys. 122, 53--58 (2017; Zbl 1390.14099)] and \textit{A. Fonarev} and \textit{A. Kuznetsov} [J. Lond. Math. Soc., II. Ser. 97, No. 1, 24--46 (2018; Zbl 1468.14037)] for the case of rank 2, and also gives an alternative proof of known results on deformations of universal bundles. Moreover we show that a twist of the embedding, together with 2 exceptional line bundles, gives the beginning of a semiorthogonal decomposition for the derived category of the moduli space. moduli of vector bundles; derived categories; semiorthogonal decompositions Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Vector bundles on curves and their moduli, Fano varieties Admissible subcategories in derived categories of moduli of vector bundles 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 In this article, the author shows an evaluation of the weighted average number of subgroups and that of cyclic subgroups of an elliptic curve \(E\) over the prime field \(\mathbb F_p\). Let \(C_\ell\) be the cyclic subgroup of order \(\ell\). Then the group \(E(\mathbb F_p)\simeq C_m\times C_n\), for \(m,n\in\mathbb Z^+, m|\operatorname{gcd}(n,p-1), |p+1-mn|\le 2\sqrt p\). Let \({\mathcal Ell}(p)\) be the set of rational isomorphic classes of elliptic curves defined over \(\mathbb F_p\). For a finite group \(G\), let \(s(G)\) (resp. \(c(G)\)) be its number of subgroups (resp. cyclic subgroups). The weighted average number of subgroups (resp. cyclic subgroups ) is defined as follows: \(h({\mathcal Ell}(p))=\frac 1p\sum_{E\in{\mathcal Ell}(p)}h(E(\mathbb F_p))/|\operatorname{Aut}(E)|\) with \(h=s\) (resp. \(h=c\)). Let \(h\in\{s,c\}\). Put \(P(m,n)=\frac 1p\sum_E\frac 1{|\operatorname{Aut}(E)|}\), where \(E\in {\mathcal Ell}(p)\) and \(E(\mathbb F_p)\simeq C_m\times C_n\). Then \(h(E(\mathbb F_p))=\sum_{m,n}h(C_m\times C_n)P(m,n)\), where \(m,n\in \mathbb Z^+, m|\operatorname{gcd}(n,p-1), |p+1-mn|\le 2\sqrt p\) and \(h(C_m\times C_n)=\sum_{d|\operatorname{gcd}(m,n)}F_h(d)\tau(m/d)\tau(n/d)\) with \(F_h(d)=\varphi(d)\) (resp. \(\mu*\phi(d)\)) when \(h=s\) (resp. \(c\)). Here \(\varphi\) is the Euler totient function and \(\mu\) is the Möbius function. In [\textit{C. David} et al., Math. Ann. 368, No. 1--2, 685--752 (2017; Zbl 1448.11116)], \(P(m,n)\) is given as a Euler product \(f_{\infty}(p+1-mn)\prod_{\ell:\text{prime }}f_\ell(m,n,p)\). See Propositions 2,2 and 2.3 in this article. Put \(w_{h,p}(m,n)=h(C_m\times C_n)f_\infty(p+1-mn)\). Then \(h({\mathcal Ell}(p))=\sum_{m,n}w_{h,p}(m,n)\prod_\ell f_\ell(m,n,p)\). To estimate \(h({\mathcal Ell}(p))\), the author uses the general results to evaluate sum of the form \(\sum_{a\in A}w_a\prod_\ell(1+\delta_\ell(a))\) given by [loc. cit.] and gives a good estimate for the divisor function in both short interval and arithmetic progression in Theorem 3.2.
As a main result, in Theorem 1.1, the author determines \(h({\mathcal Ell}(p))\) in an explicit but complicate form. By averaging this result over \(p\) the author obtains that \(\frac 1{\pi(x)}\sum_{p\le x}h({\mathcal Ell}(p))=(C_h+o(1))\log (x+1)\) as \(x\rightarrow \infty\), for some constant \(C_h\ge 1\), and further gives upper and lower bounds. As for upper bounds, for \(\varepsilon>0\), \(h({\mathcal Ell}(p))\ll_\varepsilon(\log p)^{1+e^\gamma+\varepsilon}\log\log p)\sum_{m|p-1}\tau(m^2)/m\) \(\ll_\varepsilon (\log p)^{1+e^\gamma+\varepsilon}\log\log p)\min((\log p)^4,\tau((p-1)^2)\sigma(p-1)/(p-1))\), where \(\sigma(n)\) is the sum of divisors of the positive integer \(n\) and \(\gamma=0.5772\cdots\) is the Euler-Mascheroni constant. For lower bounds, see Proposition 1.5. elliptic curve; subgroup Elliptic curves over global fields, Curves over finite and local fields, Elliptic curves, Asymptotic results on counting functions for algebraic and topological structures The average number of subgroups of elliptic curves over finite fields | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper under review is a detail survey of results obtained by the author with collaborators over the last two decades in the study of the classical theory of topological equisingularity and related topics. In the appendix, written by G.-M. Greuel and G. Pfister, some computational aspects of the theory are discussed. The bibliography contains 156 selected references and involves 16 items with the author's name.
The author begins with a reminder of basic notions, standard methods and known results concerning the topology of hypersurface singularities and its main objects such as the Milnor fibration, Morse theory, the polar and resolution methods, and others. Then he discusses some open problems closely related to Zariski's multiplicity conjecture [\textit{O. Zariski}, Bull. Am. Math. Soc. 77, 481--491 (1971; Zbl 0236.14002)], to questions on topological triviality for families of isolated hypersurfaces singularities and properties of \(\mu\)-constant strata and families, and so on. Further, the author explains his own contribution to the theory of topological triviality for families of nonisolated singularities [\textit{J. Fernández de Bobadilla}, Adv. Math. 248, 1199--1253 (2013; Zbl 1284.32018); \textit{J. Fernández De Bobadilla} and \textit{M. Marco-Buzunáriz}, Comment. Math. Helv. 88, No. 2, 253--304 (2013; Zbl 1271.14046)] and formulates several questions and conjectures. Among other things, some useful interactions of the theory of equisingularity with Floer homology of the Milnor fibration and Lipschitz geometry are mentioned also [\textit{L. Birbrair} et al., Proc. Am. Math. Soc. 144, No. 3, 983--987 (2016; Zbl 1338.14008); \textit{A. Némethi}, Publ. Res. Inst. Math. Sci. 44, No. 2, 507--543 (2008; Zbl 1149.14029)].
In the appendix you can find the following interesting remark: ``the failure to find a counter example to Zariski's conjecture was the most important reason for the development of SINGULAR as it is now'' (see [\textit{W. Decker} et al., SINGULAR 4-1-3 -- A computer algebra system for polynomial computations. (2020), \url{http://www.singular.uni-kl.de}]. nonisolated hypersurface singularities; Milnor fibration; morsification; equisingularity; Zariski's multiplicity conjecture; topological triviality; Floer homology; lattice homology; low dimensional topology; plumbing 3-manifolds; simultaneous resolutions; \(\mu\)-constant families; isolated surface singularities; topological triviality; Lipschitz equisingularity; motivic integration; arc spaces; vanishing cycles; monodromy; vanishing folds; cobordism theorem; computer algebra system ``Singular'' Singularities in algebraic geometry, Equisingularity (topological and analytic), Complex surface and hypersurface singularities, Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants Topological equisingularity: old problems from a new perspective (with an appendix by G.-M. Greuel and G. Pfister on \textsc{Singular}) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 exist many results showing that certain classes of functions are closed under integration; the main result of the present paper is also such a result, for functions on a finite extension \(K\) of the field \(\mathbb Q_p\) of \(p\)-adic numbers. Recall that there is a natural (Haar) measure on such fields \(K\), yielding a notion of integration of functions \(K^n \to \mathbb C\). More precisely, the authors specify certain classes \(\mathcal H\) of such functions which are ``base-stable under integration'', meaning that whenever a function \(f\colon K^{n+1} \to \mathbb C\) lies in \(\mathcal H\), then so does the function \(K^n \to \mathbb C, x \mapsto \int_{K} f(x, y)\,dy\). To be precise, the authors more generally consider functions on certain subsets of \(K^n \times \mathbb Z^m\), where \(\mathbb Z\) is equipped with the counting measure. In this review, I will for simplicity stick to functions on \(K^n\), and I will write \(\mathcal H(K^n)\) for those functions in \(\mathcal H\) which live on \(K^n\).
In many results of that kind (including the one in the present paper), one first fixes a certain first-order language \(\mathcal L\) on \(K\), yielding in particular a notion of \(\mathcal L\)-definable functions from \(K^n \to \mathbb Z\) (where \(\mathbb Z\) is considered as the value group of \(K\)). Then \(\mathcal H(K^n)\) is defined as some algebra generated by certain specific functions which are specified in terms of \(\mathcal L\)-definable functions.
A classical result of that kind is the one by \textit{J. Denef} [Prog. Math. None, 25--47 (1985; Zbl 0597.12021)], where \(\mathcal L\) is the valued field language, and where \(\mathcal H(K^n)\) is the \(\mathbb Q\)-algebra generated by functions of the form \(\alpha\colon K^n \to \mathbb Z\) and \(K^n \to \mathbb Q, x \mapsto q^{\alpha(x)}\), where \(\alpha\) is \(\mathcal L\)-definable. This has then been generalized in two directions: (a) allow other languages \(\mathcal L\); (b) add more generators to \(\mathcal H(K^n)\).
An important generalization of type (b) is to consider classes of functions containing additive characters \(\psi\colon (K, +) \to (\mathbb C^\times, \cdot)\), since those naturally appear in representation theory and in Fourier transforms. In direction (a), it is tempting to believe that one can take \(\mathcal L\) to be any P-minimal language on \(K\); P-minimality is an axiomatic condition implying that \(\mathcal L\)-definable objects are ``tame'' in some geometric sense. Up to recently, assuming P-minimality was not enough to prove the desired closedness under intergration; one additionally had to assume the existence of definable Skolem functions. This assumption has now been dropped. More precisely, in a previous paper, \textit{P. Cubides Kovacsics} and \textit{E. Leenknegt} [J. Symb. Log. 81, No. 3, 1124--1141 (2016; Zbl 1428.03060)] proved closedness under integration for any P-minimal \(\mathcal L\) when \(\mathcal H(K^n)\) is generated by the same kinds of functions as in the above-mentioned result by Denef. In the present paper, they now show that this result also extends to algebras \(\mathcal H(K^n)\) containing additive characters. In particular, the authors had to find the ``right'' algebras \(\mathcal H(K^n)\) (which are not a straight forward generalization of the ones when definable Skolem functions exist). \(P\)-minimality; \(p\)-adic integration; constructible functions; exponential-constructible functions Semialgebraic sets and related spaces, Model theory of ordered structures; o-minimality Exponential-constructible functions in \(P\)-minimal 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 We give a simplified proof (in characteristic zero) of the decomposition theorem for connected complex projective varieties with klt singularities and a numerically trivial canonical bundle. The proof mainly consists in reorganizing some of the partial results obtained by many authors and used in the previous proof but avoids those in positive characteristic by S. Druel. The single, to some extent new, contribution is an algebraicity and bimeromorphic splitting result for generically locally trivial fibrations with fibers without holomorphic vector fields. We first give the proof in the easier smooth case, following the same steps as in the general case, treated next. The last two words of the title are plagiarized from [\textit{F. Beukers} et al., Indag. Math., New Ser. 2, No. 1, 1--8 (1991; Zbl 0734.11039)]. Kähler-Einstein metrics; first Chern class; hyperkähler varieties; Calabi-Yau varieties; holonomy; algebraic foliations; fundamental group Calabi-Yau manifolds (algebro-geometric aspects), Holomorphic symplectic varieties, hyper-Kähler varieties, Transcendental methods of algebraic geometry (complex-analytic aspects), Kähler-Einstein manifolds, Calabi-Yau theory (complex-analytic aspects) The Bogomolov-Beauville-Yau decomposition for klt projective varieties with trivial first Chern class -- without tears | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 paper under review incorporate their approach (in the context of discrete differential geometry), which deals with so-called billiard systems within quadrics, playing the role of discrete analogues of geodesics on ellipsoids, into another one, so-called integrability or consistency conditions for quad-graphs.
The first approach has been developed by the authors of the paper under review in works [J. Math. Pures Appl. (9) 85, No. 6, 758--790 (2006; Zbl 1118.37029); Adv. Math. 219, No. 5, 1577--1607 (2008; Zbl 1154.37022); \textit{V. Dragović} and \textit{M. Radnović}, Poncelet porisms and beyond. Integrable billiards, hyperelliptic Jacobians and pencils of quadrics. Frontiers in Mathematics. Basel: Birkhäuser. (2011; Zbl 1225.37001)], and the second approach has been developed by other authors in [\textit{V. E. Adler, A. I. Bobenko} and \textit{Y. B. Suris}, Commun. Math. Phys. 233, No. 3, 513--543 (2003; Zbl 1075.37022); ``Discrete nonlinear hyperbolic equations: Classification of integrable cases'', Funct. Anal. Appl. 43, 3--17 (2009)] and [\textit{A. I. Bobenko} and\textit{Y. B. Suris}, Discrete differential geometry. Integrable structure. Graduate Studies in Mathematics 98. Providence, RI: American Mathematical Society (AMS). (2008, Zbl 1158.53001)].
The paper has 6 sections (including introduction). In the second and third sections the authors recall necessary notions from two approaches mentioned above, in section 4 they make a quad-graph interpretation of their earlier six-pointed star theorem. In section 5 they introduce a new notion, so called double reflection net, construct several examples of it and show that it induces a subclass of dual Darboux nets; the last nets are known to give examples of discrete integrable hierarchies. In section 6 they construct an example of Yang-Baxter map associated to confocal families of quadrics. ellipsoidal billiard; confocal quadrics; quad-graphs; integrability V. Dragović, Billiard algebra, integrable line congruences and DR-nets, J. Nonlinear Mathematical Physics, 19, (2012) Relationships between algebraic curves and integrable systems, Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Descriptive geometry, Arrangements of points, flats, hyperplanes (aspects of discrete geometry), Billiard algebra, integrable line congruences, and double reflection nets | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 develops some fundamental intersection theory of Deligne-Mumford stacks, building on earlier work of \textit{A. Vistoli} [Invent. Math. 97, No. 3, 613--670 (1989; Zbl 0694.14001)] and \textit{A. Kresch} [Invent. Math. 136, No. 3, 483--496 (1999; Zbl 0923.14003), by providing a more global framework which properly encodes global geometric information such as Chern classes and Chow ring structures. This machinery is applied, for instance, to certain weighted blow-ups, and more specifically, to the setting of moduli of stable maps, building on earlier work of the authors.
More precisely, the paper begins by developing machinery to in essence transform a proper local embedding of DM stacks into an actual embedding, via a fiber-product arising from a universally closed morphism that they construct. This construction provides the technical backbone of all intersection-theoretic work that follows, the main results of which are nicely described in the introduction. The first application is to Chern classes of weighted blow-ups. This is further applied to moduli of stable maps and intermediate weighted stable maps, which are spaces introduced in earlier work of the Mustatas as modular interpretations of the intermediate stages of an iterated blow-up occurring in a construction of the usual Kontsevich moduli space. Finally, the paper includes an appendix on the Euler sequence of a weighted projective bundle. Deligne-Mumford stack; local embedding; etale lift; Chern classes; weighted blow-ups; moduli space of stable maps A. M. Mustaţă and A. Mustaţă, The structure of a local embedding and Chern classes of weighted blow-ups , J. Eur. Math. Soc. 14 (2012), no. 6, 1739-1794. Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Algebraic cycles, Families, moduli of curves (algebraic) The structure of a local embedding and Chern classes of weighted blow-ups | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The topological zeta function and Igusa's local zeta function are respectively a geometrical invariant associated to a complex polynomial \(f\) and an arithmetical invariant associated to a polynomial \(f\) over a \(p\)-adic field. When \(f\) is a polynomial in two variables we prove a formula for both zeta functions in terms of the so-called log canonical model of \(f^{-1} \{0\}\) in \(\mathbb{A}^2\). This result yields moreover a conceptual explanation for a known cancellation property of candidate poles for these zeta functions. Also in the formula for Igusa's local zeta function appears a remarkable non-symmetric `\(q\)-deformation', of the intersection matrix of the minimal resolution of a Hirzebruch-Jung singularity. curve singularities; log canonical model; topological zeta function; Igusa's local zeta function Willem Veys.- Zeta functions for curves and log canonical models. Proceedings of the London Mathematical Society, 74 :360-378 (1997). Zbl0872.32022 MR1425327 Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants, Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.), Minimal model program (Mori theory, extremal rays), Local ground fields in algebraic geometry Zeta functions for curves and log canonical models | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper, we construct a new threshold-attribute-based signature (\(t\)-ABS) scheme that is significantly more efficient than all previous \(t\)-ABS schemes. The verification algorithm requires the computation of only 3 pairing operations, independently of the attribute set of the signature, and the size of the signature is also independent of the number of attributes. The security of all our schemes is reduced to the computational Diffie-Hellman problem. We also show how to achieve shorter public parameters based on the intractability of computational Diffie-Hellman assumption in the random oracle model. Authentication, digital signatures and secret sharing, Applications to coding theory and cryptography of arithmetic geometry Short pairing-efficient threshold-attribute-based signature | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This is a survey paper centered on a result of M. Emerton regarding the ordinary \(p\)-adic representations of \(\text{GL}_2({\mathbb Q}_p)\) and local-global compatibility.
Let \(E\) be a finite extension of \({\mathbb Q}_p\), whose ring of integers contains the Hecke eigenvalues of a proper parabolic modular form \(f\) of weight 2. Let also \(\rho_f\) be the \(p\)-adic representation of \(\text{Gal} (\bar{\mathbb Q}/{\mathbb Q})\) associated to \(f\) and, for a prime \(\ell\), \(\pi_l(\rho_f|_{\text{Gal} (\bar{\mathbb Q}_l/{\mathbb Q}_l)})\) the smooth representation of \(\text{GL}_2({\mathbb Q}_l)\) corresponding to the local Galois representation \(\rho_f|_{\text{Gal} (\bar{\mathbb Q}_l/{\mathbb Q}_l)}\) via the local Langlands correspondence. Then, by results of Langlands, Deligne and Carayol, \(\pi_l(\rho_f|_{\text{Gal} (\bar{\mathbb Q}_l/{\mathbb Q}_l)})\) essentially determines \(\rho_f|_{\text{Gal} (\bar{\mathbb Q}_l/{\mathbb Q}_l)}\) if \(\ell\neq p\); moreover, the local Langlands correspondence embeds into a global correspondence realized in a suitable cohomology space.
Emerton's result essentially extends this correspondence to the case \(\ell=p\). As a consequence, there is a compatibility between the classical local Langlands correspondence and the local \(p\)-adic Langlands correspondence for \(\text{GL}_2({\mathbb Q}_p)\).
Using Emerton's theorem, a number of other results were proven. Among them, the proof of the Fontaine-Mazur conjecture characterizing the representations of \(\text{Gal} (\bar{\mathbb Q}/{\mathbb Q})\) coming from classical modular forms and the proof of Kisin's conjecture charcaterizing the representations of \(\text{Gal} (\bar{\mathbb Q}/{\mathbb Q})\) coming from overconvergent modular forms. Langlands correspondence; local-global compatibility; modular forms Breuil, Christophe, Correspondance de Langlands \(p\)-adique, compatibilité local-global et applications [d'après Colmez, Emerton, Kisin, \(...\)], Astérisque, 348, Exp. No. 1031, viii, 119-147, (2012) Local ground fields in algebraic geometry, Langlands-Weil conjectures, nonabelian class field theory, Representation-theoretic methods; automorphic representations over local and global fields, Representations of Lie and linear algebraic groups over local fields, Geometric Langlands program: representation-theoretic aspects The \(p\)-adic Langlands correspondence, the local-global compatibility 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 These new \(a\)-ary subdivision schemes for curve design are derived easily from their corresponding two-scale scaling functions, a notion from the context of wavelets. [For part II, cf. Appl. Appl. Math. 3, No. 2, 176--187, electronic only (2008; Zbl 1175.65028)]. subdivision; curve design; stationary; refinable functions; \(a\)-ary Lian, J.-A., On \textit{a}-ary subdivision for curve design. I. 4-point and 6-point interpolatory schemes, \textit{Applications and Applied Mathematics}, 3, 1, 18-29, (2008) Computer-aided design (modeling of curves and surfaces), Computer science aspects of computer-aided design, Computational aspects of algebraic curves, Plane and space curves, Other \(n\)-ary compositions \((n \ge 3)\) On \(a\)-ary subdivision for curve design. I: 4-point and 6-point interpolatory 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 first family is ternary interpolatory and the second family is binary approximation. All these new schemes are circular-invariant, meaning that new vertices are generated from local circles formed by three consecutive old vertices. As consequences of the nonlinear schemes, two new families of linear subdivision schemes for curve design are established. The 3-point linear binary schemes, which are corner-cutting depending on the choices of the tension parameter, are natural extensions of the Lane-Riesenfeld schemes. The four families of both nonlinear and linear subdivision schemes are implemented extensively by a variety of examples. subdivision; curve design; computer-aided geometric design; refinable functions Computer-aided design (modeling of curves and surfaces), Computer science aspects of computer-aided design, Computational aspects of algebraic curves Circular nonlinear subdivision schemes for curve design | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 chapter covers the use of subdivision methods in algebraic geom- etry with an emphasis on intersection, self-intersection, and arrangement computation, for the case of semi-algebraic curves with either implicit or parametric representation. Special care is given to the genericity of the subdivision, which can be specified whatever the context is, and then specialized to meet the algorithm requirements. symbolic-numeric computation; topology; intersection; arrangement; polynomial solvers; mathematical software Mourrain, B., Wintz, J.: A subdivision method for arrangement computation of semi-algebraic curves. In: Nonlinear Computational Geometry, pp. 165-187, The IMA Volumes in Mathematics and its Applications, vol. 151, Springer, New York (2010) Symbolic computation and algebraic computation, Computer-aided design (modeling of curves and surfaces), Semialgebraic sets and related spaces A subdivision method for arrangement computation of semi-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 These new schemes further extend the family of the classical 4- and 6-point interpolatory schemes. [For part I, cf. Appl. Appl. Math. 3, No. 1, 18--29, electronic only (2008; Zbl 1175.65027)]. subdivision; curve design; stationary; refinable functions; \(a\)-ary Lian, J. A., On \(a\)-ary subdivision for curve design: II. 3-point and 5-point interpolatory schemes, Appl. Appl. Math., 3, 176-187, (2009) Computer-aided design (modeling of curves and surfaces), Computer science aspects of computer-aided design, Computational aspects of algebraic curves, Plane and space curves, Other \(n\)-ary compositions \((n \ge 3)\) On \(a\)-ary subdivision for curve design. II. 3-point and 5-point interpolatory 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 necessity to process data which live in nonlinear geometries (e.g. capture data, unit vectors, subspace, positive definite matrices) has led to some recent development in nonlinear multiscale representation and subdivision algorithms. The present paper analyzes convergence and \(C^1\)- and \(C^2\)-smoothness of subdivision schemes which operate in the matrix groups or general Lie groups, and which are defined by the so-called log-exponential analogy. It is shown that a large class of such schemes has essentially the same smoothness as the linear schemes derived from them.
The bibliography contains 17 sources. Lie group; matrix group; subdivision scheme; smoothness properties; nonlinear multiscale representation; convergence; log-exponential analogy P. Grohs and J. Wallner, \textit{Log-exponential analogues of univariate subdivision schemes in Lie groups and their smoothness properties}, in Approximation Theory XII, M. Neamtu and L. L. Schumaker, eds., Nashboro Press, Nashville, TN, 2008, pp. 181--190. Numerical aspects of computer graphics, image analysis, and computational geometry, Representations of Lie and linear algebraic groups over real fields: analytic methods, Group actions on varieties or schemes (quotients), Group actions on affine varieties, Associated Lie structures for groups, Analysis on real and complex Lie groups Log-exponential analogues of univariate subdivisional schemes in Lie groups and their smoothness properties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(S\) be a smooth projective surface with \(p_g=q=0\). We show how to use derived categorical methods to study the geometry of certain special iterated Hilbert schemes associated to \(S\) by showing that they contain a smooth connected component isomorphic to \(S\). Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, \(K3\) surfaces and Enriques surfaces, Surfaces of general type Smooth components on special iterated 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 We describe a subdivision method for handling algebraic implicit curves in 2d and 3d. We use the representation of polynomials in the Bernstein basis associated with a given box, to check if the topology of the curve is determined inside this box, from its points on the border of the box. Subdivision solvers are used for computing these points on the faces of the box, and segments joining these points are deduced to get a graph isotopic to the curve. Using envelop of polynomials, we show how this method allow to handle efficiently and accurately implicit curves with large coefficients.We report on implementation aspects and experimentations on 2d curves such as ridge curves or self intersection curves of parameterized surfaces, and on silhouette curves of implicit surfaces, showing the interesting practical behavior of this approach. Liang, C.; Mourrain, B.; Pavone, J., Subdivision methods for the topology of 2d and 3d implicit curves, (Jüttler, B.; Piene, R., Geometric Modeling and Algebraic Geometry, (2008), Springer Berlin Heidelberg), 199-214 Computational aspects of algebraic curves, Computer-aided design (modeling of curves and surfaces) Subdivision methods for the topology of 2d and 3d implicit 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 investigate both the \(2m\)-point a-ary for any \(a\geq 2\) and \((2m + 1)\)-point a-ary
for any odd \(a\geq 3\) interpolatory subdivision schemes for curve design. These schemes include
the extended family of the classical 4- and 6-point interpolatory a-ary schemes and the family
of the 3- and 5-point a-ary interpolatory schemes, both having been established in our previous
papers [ibid. 3, No. 1, 18--29 (2008; Zbl 1175.65027); ibid. 3, No. 2, 176--187 (2008; Zbl 1175.65028)]. Lian, J-A, On \(\alpha \)-ary subdivision for curve design. III. \(2m\)-point and \((2m+1)\)-point interpolatory schemes, Appl. Appl. Math., 4, 434-444, (2009) Computer-aided design (modeling of curves and surfaces), Computer science aspects of computer-aided design, Computational aspects of algebraic curves On \(a\)-ary subdivision for curve design. III: \(2m\)-point and \((2m + 1)\)-point interpolatory 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 an integral curve. Here we study the multiplication map \(H^0(X,L_1)\otimes\cdots\otimes H^0(X,L_s)\to H^0(X,L_1\otimes\cdots\otimes L_s)\) of global sections of \(s\geq 3\) line bundles, mainly when \(h^0(X,L_i)=2\) for all \(i\). multiplication map; integral curve; line bundle Special divisors on curves (gonality, Brill-Noether theory), Divisors, linear systems, invertible sheaves, Pencils, nets, webs in algebraic geometry On the iterated multiplication map for line bundles 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 The counting polynomial for a system of polynomial equations and inequalities in affine or projective space is defined via a decomposition of the solution set into subsets that can be defined by triangular-like systems, in the sense of \textit{J. M. Thomas} [Differential systems. AMS Coll. Publ. 21 (1937; JFM 63.0438.03)]. Its properties are similar to those of the Hilbert polynomial: its degree is the dimension of the solution set, and if the number of solution is finite, then the polynomial is the number of solutions. polynomial equations; polynomial inequations; solutions of polynomial systems; Thomas algorithm; counting polynomial; simple systems; triangular decomposition W. Plesken, ''Counting solutions of polynomial systems via iterated filtrations,'' Arch. Math., 92, 44--56 (2009). Enumerative problems (combinatorial problems) in algebraic geometry, Fibrations, degenerations in algebraic geometry, Computational aspects of field theory and polynomials, Computational aspects and applications of commutative rings, Computational aspects in algebraic geometry Counting solutions of polynomial systems via iterated fibrations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We present a purely numerical (i.e., non-algebraic) subdivision algorithm for computing an isotopic approximation of a simple arrangement of curves. The arrangement is ``simple'' in the sense that any three curves have no common intersection, any two curves intersect transversally, and each curve is non-singular. A curve is given as the zero set of an analytic function on the plane, along with effective interval forms of the function and its partial derivatives. Our solution generalizes the isotopic curve approximation algorithms of \textit{S. Plantinga} and \textit{G. Vegter} [``Isotopic approximation of implicit curves and surfaces'', in: Proceedings of the 2004 Eurographics/ACM SIGGRAPH symposium on geometry processing, SGP '04. New York, NY: Association for Computing Machinery (ACM). 245--254 (2004; \url{doi:10.1145/1057432.1057465})] and \textit{L. Lin} and \textit{C. Yap} [Discrete Comput. Geom. 45, No. 4, 760--795 (2011; Zbl 1221.65048)]. We use certified numerical primitives based on interval methods. Such algorithms have many favorable properties: they are practical, easy to implement, suffer no implementation gaps, integrate topological with geometric computation, and have adaptive as well as local complexity. A preliminary implementation is available in Core Library. isotopy; arrangement of curves; interval arithmetic; subdivision algorithms; marching-cube Geometric aspects of numerical algebraic geometry, Computational aspects of algebraic curves, Numerical aspects of computer graphics, image analysis, and computational geometry Isotopic arrangement of simple curves: an exact numerical approach based on subdivision | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems From author's abstract: We present an elementary construction of the multigraded Hilbert scheme of \(d\) points of \(\mathbb{A}^n_k=\text{Spec}(k[x_1,\dots,x_n])\), where \(k\) is an arbitrary commutative and unitary ring. This Hilbert scheme represents the functor from \(k\)-schemes to sets that associates to each \(k\)-scheme \(T\) the set of closed subschemes \(Z\subset T\times_k \mathbb{A}^n_k\) such that the direct image (via the first projection) of the structure sheaf of \(Z\) is locally free of rank \(d\) on \(T\). It is a special case of the general multigraded Hilbert scheme constructed by \textit{M. Haiman} and \textit{B. Sturmfels} [J. Algebr. Geom. 13, No. 4, 725--769 (2004; Zbl 1072.14007)]. Our construction proceeds by gluing together affine subschemes representing subfunctors that assign to \(T\) the subset of \(Z\) such that the direct image of the structure sheaf on \(T\) is free with a particular set of \(d\) monomials as basis. The coordinate rings of the subschemes representing the subfunctors are concretely described, yielding explicit local charts on the Hilbert scheme. Hilbert scheme of points; affine space; syzygies Huibregtse M.E.: An elementary construction of the multigraded Hilbert scheme of points. Pac. J. Math. 223, 269--315 (2006) Parametrization (Chow and Hilbert schemes), Syzygies, resolutions, complexes and commutative rings, Computational aspects and applications of commutative rings An elementary construction of the multigraded 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 Let \(X\) and \(Y\) be smooth varieties of dimension \(n-1\) and \(n\) over an arbitrary algebraically closed field, \(f:X\to Y\) a finite map that is birational onto its image. Suppose that \(f\) is curvilinear; that is, for all \(x\in X\), the Jacobian \(\partial f(x)\) has rank at least \(n-2\). For \(r\geq 1\), consider the subscheme \(N_r\) of \(Y\) defined by the \((r-1)\)th Fitting ideal of the \({\mathcal O}_Y\)-modulo \(f_*{\mathcal O}_X\), and set \(M_r: =f^{-1} N_r\). In this setting -- in fact, in a more general setting -- we prove the following statements, which show that \(M_r\) and \(N_r\) behave like reasonable schemes of source and target \(r\)-fold points of \(f\).
If each component of \(M_r\), or equivalently of \(N_r\), has the minimal possible dimension \(n-r\), then \(M_r\) and \(N_r\) are Cohen-Macaulay, and their fundamental cycles satisfy the relation, \(f_* [M_r] =r[N_r]\). Now, suppose that each component of \(M_s\), or of \(N_s\), has dimension \(n-s\) for \(s=1, \dots, r+1\). Then the blow-up \(\text{Bl}(N_r, N_{r+1})\) is equal to the Hilbert scheme \(\text{Hilb}^r_f\), and the blow-up \(\text{Bl}(M_r, M_{r+1})\) is equal to the universal subscheme \(\text{Univ}^r_f \times_YX\); moreover, \(\text{Hilb}^r_f\) and \(\text{Univ}^r_f\) are Gorenstein. In addition, the structure map \(h:\text{Hilb}^r_f\to Y\) is finite and birational onto its image; and its conductor is equal to the ideal \({\mathcal I}_r\) of \(N_{r+1}\) in \(N_r\), and is locally self-linked. Reciprocally, \(h_*{\mathcal O}_{\text{Hilb}^r_f}\) is equal to \(\Hom ({\mathcal I}_r, {\mathcal O}_{N_r})\). Moreover, \(h_*[h^{-1} N_{r+1}] =(r+1) [N_{r+1}]\). Similar assertions hold for the structure map \(h_1: \text{Univ}^r_f\to X\) if \(r\geq 2\). codimension one; multiple-point schemes; finite curvilinear map; blow-up; Hilbert scheme Kleiman S., Lipman J., Ulrich B.: The multiple-point schemes of a finite curvilinear map of codimension one. Ark. Mat. 34, 285--326 (1996) Singularities in algebraic geometry, Low codimension problems in algebraic geometry, Enumerative problems (combinatorial problems) in algebraic geometry, Local structure of morphisms in algebraic geometry: étale, flat, etc. The multiple-point schemes of a finite curvilinear map of codimension one | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors prove two non-intersecting extensions of result due to \textit{A. S. Gorodetski} et al. [Funct. Anal. Appl. 39, No. 1, 21--30 (2005); translation from Funkts. Anal. Prilozh. 39, No. 1, 27--38 (2005; Zbl 1134.37347)]. The first result states the existence of a \(C^2\)-open set of iterated function systems (IFS) with fully supported ergodic measures and only zero Lyapunov exponents. Such measures are constructed as weak-star limits of sequences of measures supported on periodic orbits of the appropriate skew-products. The second result asserts the existence of a \(C^1\)-open set of IFS's having not fully supported ergodic measures with only zero Lyapunov exponents and positive entropy. The proofs rely on the construction of non-hyperbolic measures for the induced IFS's on the flag manifold. Also some related questions are listed. non-hyperbolic measure; fibered Lyapunov exponents; Furstenberg vector; bootstrapping procedure; flag manifold Bochi, J.; Bonatti, C.; Díaz, LJ, Robust vanishing of all Lyapunov exponents for iterated function systems, Math. Z., 276, 469-503, (2014) Random dynamical systems aspects of multiplicative ergodic theory, Lyapunov exponents, Smooth ergodic theory, invariant measures for smooth dynamical systems, Grassmannians, Schubert varieties, flag manifolds Robust vanishing of all Lyapunov exponents for iterated function systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \( X \) be the Segre-Veronese embedding of \(\mathbb{P}^a\times\mathbb{P}^b\times\mathbb{P}^c\times \cdots\) with degree \( L=(l_1,l_2,l_3,\dots).\) Corresponding to it we have the multigraded ring \[ S[X]=\mathbb{C}[\alpha_0,\dots,\alpha_a,\beta_0,\dots,\beta_b,\gamma_0,\dots,\gamma_c,\dots] .\]
The ring \( S[X] \) has two dual interpretations. The first, more geometric, as ``functions'' on \(X\) and the second, more algebraic, in terms of derivations. We have then the well known apolarity lemma: \[F\in \langle \{ p_1,\dots,p_r \} \rangle \Leftrightarrow I(\{ p_1,\dots,p_r \}) \subset \mbox{Ann}(F)\] where \(F\in S[X], \ p_1,\dots,p_1\in X\) and \(\mbox{Ann}(F)\subset S[X]\) is the annihilator of \(F\) in the space of derivations.
This lemma gives a characterization of rank \(r\) tensors. In the present paper the authors introduce a result for border rank similar to the apolarity lemma. The main result goes as follows: Suppose a tensor or polynomial \(F\) has border rank at most \(r.\) Then there exists a multihomogeneous ideal \(I\subset S[X]\) such that \(I\subset \mbox{Ann}(F)\) and for each multidegree \(D\) the \(D\)th graded piece \(I_D\) of \(I\) has codimension equal to \(\min(r,\dim S[X]_D).\)
In fact, the result stated above is a consequence of more general results that they show for a smooth toric variety. They also have if and only if kind of results using more technical hypotheses.
Moreover, the results are applied to the cases \(\mathbb{P}^2\times \mathbb{P}^1 \times \cdots \times \mathbb{P}^1\) with arbitrary degree and \(\mathbb{P}^a\times \mathbb{P}^b\times \mathbb{P}^c\) with degree \((1,1,1)\) obtaining new bounds for the border rank of such Segre-Veronese varieties. apolarity; border rank; Cox ring; invariant ideals; multigraded Hilbert scheme; variety of sums of powers Parametrization (Chow and Hilbert schemes), Toric varieties, Newton polyhedra, Okounkov bodies, Multilinear algebra, tensor calculus, Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.) Apolarity, border rank, and multigraded Hilbert scheme | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a smooth complex quasiprojective variety. The Hilbert scheme \(\text{Hilb}^nX\) parametrizes zero-dimensional subschemes of length \(n\) of \(X\). If \(X\) is projective, \(\text{Hilb}^nX\) is a natural compactification of the parameter space of \(n\) distinct points of \(X\) and when \(X\) is a surface, it is, in fact, a smooth compactification. However, when the dimension of \(X\) is greater than two, the ``Hilbert scheme of points'' of length \(n\) on \(X\) is no longer smooth unless \(n\) is very small. Just as flag varieties are a natural generalization of grassmannians, it is natural to consider ``nested'' Hilbert schemes on \(X\) parametrizing nests \(Z_1 \subset Z_2 \subset \cdots \subset Z_m\) of zero-dimensional subschemes of \(X\) of given length.
By adapting the method of Ellingsrud and Strømme, we classify the nested Hilbert schemes on \(X\) which are smooth and obtain cellular decompositions for the smooth nested Hilbert schemes on affine and projective space as well as for the corresponding punctual nested Hilbert schemes. One deduces immediately that the Chow group and the (Borel-Moore) homology group of each of these spaces are isomorphic; odd homology vanishes and the \(2i\)-th homology group is the free abelian group generated by the classes of the closures of the \(i\)-cells. The number of cells of each dimension and hence the Betti numbers of the punctual nested Hilbert schemes are calculated here; the number of cells of each dimension of the smooth nested Hilbert schemes on affine and projective space can be read off formulae which are obtained in the sequels to this paper [see \textit{J. Cheah}, Math. Z. 227, No. 3, 479-504 (1998; Zbl 0890.14003) and J. Algebr. Geom. 5, No. 3, 479-511 (1996; Zbl 0889.14001)]. In fact, in the sequels, we use the cellular decompositions for the punctual nested Hilbert schemes obtained here to calculate the virtual Hodge polynomials of all the smooth nested Hilbert schemes on \(X\) (and of related varieties such as the universal families over some of these nested Hilbert schemes) in terms of the virtual Hodge polynomial of \(X\). Finally, following Göttsche, we study the Hilbert function strata of the punctual nested Hilbert schemes and cellular decompositions for these spaces. zero-dimensional subschemes; nested Hilbert schemes; virtual Holdge polynomials J. Cheah, ''Cellular Decompositions for Nested Hilbert Schemes of Points,'' Pac. J. Math. 183, 39--90 (1998). Parametrization (Chow and Hilbert schemes), Grassmannians, Schubert varieties, flag manifolds, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Cellular decompositions for nested Hilbert schemes of points | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(C\) be a smooth irreducible projective curve of genus \(g\) defined over the field \(\mathbb{C}\) of the complex numbers. Let \(J(C)\) be the Jacobian variety of \(C\). Fixing a base point \(P_0\) on \(C\) one defines naturals subschemes \(W^r_d\) di \(J(C)\). Those subschemes parametrize invertible sheafs \(L\) on \(C\) of degree \(d\) whose space of global sections have dimension at least \(r+1\). When \(C\) is a smooth curve embedded in some projective space \(\mathbb{P}^r\), using elementary constructions, such embeddings give rise to some naturally defined irreducible subset of some schemes \(W_e^1\).
In this article, the author finds sufficient conditions implying that such subset \(Z\) is a multiple irreducible component of \(W_e^1\). He shows that those sufficient conditions are satisfied for some natural types of embedded curves.
Those results are in sharp contrast to the known results in case \(C\) is a general curve. As a matter of fact, by the Brill-Noether Theory, in case \(C\) is a general curve of genus \(g\), many fundamental results concerning those schemes \(W_d^r\) are known. In particular, the scheme \(W_d^r\) is nonempty if and only if \(\rho_d^r(g)=g-(r+1)(g-d+r)\geq 0\). Moreover, in case \(\rho_d^r(g)\geq 0 \) then \(W_d^r\) is a reduced scheme of dimension \(\rho_d^r(g)\).
A lot of interesting curves do not satisfy the properties of general curves described by Brill-Noether Theory (e.g. smooth plane curves, complete intersection curves, double coverings of curves of low genus, and so on). Quite often by definition of the curve, they do not satisfy the nonexistence part. Nevertheless one could expect that the structure of those schemes satisfies nice properties such as reducedness for such ``natural'' curves.
It is important to observe that, in particular, the results of this article do indicate that one cannot expect that naturally occurring curves have always nice structures for their schemes \(W_d^r\).
An indication that such expectation could be wrong was already obtained by the author in a previous paper [J. Algebr. Geom. 4, No. 1, 1--15 (1995; Zbl 0842.14020)]. There, the author studied linear systems \(g_e^1\) on a smooth plane curve \(C\) of degree \(d\). As a corollary of the results one finds many schemes \(W_e^1\) that are not reduced. More concretely the following fact is proved. Let \(C\) be a smooth plane curve of degree \(d\). Let \(n, i, e\) be the integers satisfying \(1 \leq n \leq (d-2)/2\), \(0 \leq i \leq (n-1)(n-2)/2\), \(e=nd-n^2+i\). Then \(W_e^1\) has an irreducible component \(Z\) of dimension \(3n+i-2\) such that the Zariski tangent space to \(W^1_e\) at a general point of \(Z\) has dimension \(3n+2i-2\).
It should be remarked that, in general, for some curve and some integer \(e\), describing all linear systems \(g_e^1\) on that curve is a very difficult problem.
In order to prove the results of this article, the author cannot use a classical tangent dimension argument because there are multiple components. Instead the paper concerns with the use of some special arguments based on lower bounds on the dimension of the intersection of \(W_{e-1}^1+W_1^0\) and components \(Z\) of \(W_e^1\) not contained in \(W_{e-1}^1+W_1^0\). Using those bounds, the description of the Zariski tangent space to \(W^1_e\) of points on such intersection give rise to restriction on dim\((Z)\).
The main theorem is stated as follows. Theorem: Let \(C\subset \mathbb{P}^3\) be a smooth linearly normal and \(2-\)normal irreducible projective curve of degree \(d\) and genus \(g>2d-g\). Let \(x\in W^3_d\) be the point corresponding to \(C\subset \mathbb{P}^3\). Assume \(C\) is not contained in a quadric, and assume \(C\) has a \(3-\)secant line divisor \(E\) satisfying the generic condition. Then \(Z=x-W^0_2\) is a \(2-\)dimensional component of \(W^1_{d-2}\) satisfying tdim\((Z)=3\). In particular, \(x-W^0_2\) is a multiple component of \(W^1_{d-2}\).
Let \(g_d^r\) be a linear system on a curve \(C\), and let \(E\) be an effective divisor of degree \(e\) on \(C\). We say that \(E\) is an \(e-\)secant \(f-\)space divisor for \(g_d^r\) if \(\dim(g_d^r(-E))\geq r-f-1\). In case \(f=1\), then we talk about an \(e-\)secant line divisor for \(g_d^r\). \(E\) satisfies the generic condition if \(E\) is not contained in some \((e+1)-\)secant \(f-\)space divisor and \(2E\) imposes \(2e\) independent conditions on quadrics in \(\mathbb{P}^r\).
Since each smooth curve can be realized as a smooth space curve, not all smooth space curves have multiple components for some scheme \(W^1_e\) as it is the case for smooth plane curves of degree \(d \geq 8\). On the other hand, a smooth plane curve of degree \(d\) is always linearly normal, and in case \(d \geq 3\) then it is \(2-\)normal, and it is not contained in a conic. It is remarkable that in the case of space curves, linear sections already give rise to multiple components. Applying liaison to an arbitrary smooth space curve, the author obtains many examples of space curves satisfying the assumption of the main theorem, hence space curves of some degree \(d\) having a multiple component for \(W^1_{d-2}\).
Finally, the author gives a generalization of the main theorem to higher dimensional linear systems. In this generalization, we have to include some other types of conditions implying restrictions on its applicability. Therefore, this generalization is not considered as being the main result by the author. As a matter of fact, the author does expect the existence of much better generalizations. complete intersection curves; Jacobian; space curves; smooth curves; special divisor Special algebraic curves and curves of low genus, Plane and space curves, Special divisors on curves (gonality, Brill-Noether theory) Curves having schemes of special linear systems with multiple components | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(r\)-dimensional smooth complex quasiprojective variety and denote the Hilbert scheme parametrizing zero-dimensional subschemes of length \(n\) of \(X\) by \(\text{Hilb}^nX\). A nested Hilbert scheme on \(X\) is defined to be a scheme of the form
\[
Z_{\mathbf n}(X):=\{(Z_1, Z_2,\dots, Z_m) : Z_i \in \text{Hilb}^{n_i}X
\]
and \(Z_i\) is a subscheme of \(Z_j\) if \(i < j\},\) where the symbol \(\mathbf n\) is used as a shorthand for the \(m\)-tuple \((n_1, n_2,\dots,n_m)\). If \(({\mathcal U}_1, {\mathcal U}_2,\dots, {\mathcal U}_m)\) is the universal element over the nested Hilbert scheme \(Z_{\mathbf n}(X)\), we call the scheme \({\mathcal U}_1\times_{Z_{\mathbf n}(X)}{\mathcal U}_2 \times_{Z_{\mathbf n}(X)}\cdots \times_{Z_{\mathbf n}(X)} {\mathcal U}_m\) the universal family over \(Z_{\mathbf n}(X)\). \textit{J. Cheah} [J. Algebr. Geom. 5, No. 3, 479-511 (1996)], expressed the virtual Hodge polynomials of the smooth Hilbert schemes \(\text{Hilb}^n X\) in terms of that of \(X\). In the paper under review we indicate how the arguments of the cited paper can be modified to express the virtual Hodge polynomials of all the smooth nested Hilbert schemes (and those of their universal families when \(r\geq 2\)) in terms of that of \(X\). More generally, we obtain the virtual Hodge polynomials of the schemes
\[
\begin{cases} \text{Hilb}^nX,\\ Z_{n-1,n}(X),\\ {\mathcal F}_n(X), \\ {\mathcal F}_{n-1,n}(X),\\ \{(P,Z_1,Z_2)\in X\times \text{Hilb}^{n-1}X\times \text{Hilb}^n X: \\ \qquad P \text{ lies in the support of }Z_2, Z_1 \text{ is a subscheme of }Z_2\} \\ \{(P, Q, Z) \in X \times X\times \text{Hilb}^nX: P \text{ and }Q\text{ lie in the support of }Z\}\end{cases}\tag{1}
\]
in terms of the virtual Hodge polynomial of \(X\) and those of the reduced schemes
\[
\text{Hilb}^k (\mathbb{A}^r,0) = \{Z \in \text{Hilb}^k\mathbb{A}^r : Z\text{ is supported at the origin\}}
\]
and
\[
{\mathcal Z}_{k-1,k}(\mathbb{A}^r, 0)=\{(Z_1, Z_2) \in \text{Hilb}^{k-1}(\mathbb{A}^r,0)\times \text{Hilb}^k(\mathbb{A}^r,0): Z_1\text{ is a subscheme of }Z_2\}.
\]
When \(r= 2\) or \(n\geq 3\), the virtual Hodge polynomials of the spaces listed in (1) can be given purely in terms of that of \(X\). Note that when \(r\geq 2\), this includes all the smooth nested Hilbert schemes \(Z_{\mathbf n}(X)\) and their universal families. If \(r = 1\), the schemes \(Z_{\mathbf n}(X)\) are products of symmetric powers of \(X\) and their virtual Hodge polynomials are easily determined using the formula for the virtual Hodge polynomials of symmetric powers given in the paper cited above.
From the equations of virtual Hodge polynomials, we obtain for free analogous equations of virtual Poincaré polynomials and Euler characteristics. In fact, since the Euler characteristics of the spaces \(\text{Hilb}^k(\mathbb{A}^r,0)\) and \({\mathcal Z}_{k-1,k}(\mathbb{A}^r,0)\) can be expressed in terms of the numbers of certain higher dimensional partitions, the Euler characteristics of the schemes listed in (1) are expressible in terms of these numbers and the Euler characteristic of \(X\). If \(X\) is projective, we also obtain formulae giving the Hodge (resp. Poincaré) polynomials of the smooth nested Hilbert schemes in terms of that of \(X\). virtual Hodge polynomials; nested Hilbert schemes; Poincaré polynomials; Euler characteristic J. Cheah, ''The Virtual Hodge Polynomials of Nested Hilbert Schemes and Related Varieties,'' Math. Z. 227, 479--504 (1998). Parametrization (Chow and Hilbert schemes), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series The virtual Hodge polynomials of nested Hilbert schemes and related 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 chapter, we present an important result due to Drinfeld, Grinberg and Kazhdan (see [Zbl 1440.14071] in the present volume), which deals with the local structure of arc scheme at rational non-degenerate arcs, and various applications in the direction of singularity theory. Arcs and motivic integration, Formal neighborhoods in algebraic geometry The local structure of arc 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 \(H\) denote a multigraded Hilbert scheme (i.e., \(H\) parametrizes those ideals of a graded polynomial ring \(K[x_1,\dots,x_n]\) that have a fixed multi graded Hilbert function). The torus \(T = (K^*)^n\) acts naturally on \(H\) induced by its action on \(\mathbb{A}^n\). Then the \(T\)-graph of \(H\) is the graph whose vertices are zero-dimensional orbits of \(T\) while the edges are the one-dimensional orbits of \(T\). The \(T\)-graph encodes a great deal of structural information about \(H\). For example \(H\) is connected if and only if its \(T\)-graph is connected.
In general, the vertices of the \(T\)-graph will correspond to monomial ideals in \(H\) and an edge will exist between two such ideals \(I\) and \(J\) if and only if there exists a one-dimensional torus whose orbit closure contains both \(I\) and \(J\). As monomial ideals are essentially purely combinatorial objects one expects to be able to resolve questions about the \(T\)-graph by purely combinatorial means. In practice however this is exceedingly difficult.
In this paper, the authors find a necessary combinatorial condition for when two vertices in the \(T\)-graph are connected by an edge. They also apply their results to the interesting case of the Hilbert scheme of points in the plane which allows them to, in this case, resolve a question posed by Altman and Sturmfels by showing the \(T\)-graph depends on the characteristic of the ground field.
The paper is thorough and the authors illustrate the technique with some very interesting and explicit examples. It is an excellent resource for anyone interested in the multigraded Hilbert scheme. Hilbert schemes; torus actions; Gröbner bases Parametrization (Chow and Hilbert schemes), Polynomial rings and ideals; rings of integer-valued polynomials, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) The T-graph of a multigraded Hilbert scheme | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We work either in the algebraic geometric category over an algebraically closed field \(\mathbb{K}\) or in the complex analytic category. Here we consider the existence of families of data \((T,\{X_t\}_{t\in T}\), \(s:T\to T\), \(\{f_t\}_{t\in T})\) such that:
(i) \(T\) is a reduced and irreducible complex space;
(ii) \(\{X_t\}_{t\in T}\) is a smooth family of connected and compact complex manifolds;
(iii) \(s:T\to T\) is a set-theoretic map;
(iv) \(f_t:X_{s(t)}\to X_t\) is a locally invertible, but not globally invertible analytic map. In the algebraic category instead of (iv) we require that each \(f_t\) is étale. The main result is a classification in the case \(\dim X_t=2\). Moduli, classification: analytic theory; relations with modular forms, Singularities of curves, local rings, Compact complex surfaces Iterated covering maps in families of compact complex 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 is a very well written survey on the method of iterated Segre mappings for the studying of certain problems in CR geometry such as finite jet determination, algebraicity of holomorphic mappings between real algebraic manifolds, local stability groups and transversality. The author starts introducing the method of Segre mappings, discovered by the author herself, Baouendi and Ebenfelt and gives hints on its use for solving the above mentioned problems.
In the paper, there are several results stated which are obtained in the past decades and that can be (more or less explicitly) reduced to the method of Segre mappings.
The paper is pleasant to read and can be useful also for non-experts. Segre mappings; CR geometry; finite jet determination; algebraicity Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables, Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables, Proper holomorphic mappings, finiteness theorems, Real submanifolds in complex manifolds, Nash functions and manifolds, Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces Iterated Segre mappings of real submanifolds in complex space 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 a semisimple linear algebraic group over an algebraically closed field the parabolic subgroups have been classified. These subgroups are reduced schemes. But in positive characteristic there are also nonreduced parabolic subgroup-schemes. These have so far not been classified. This is done here, at least for characteristic greater than three. Also their structure is exhibited. semisimple linear algebraic group; parabolic subgroups; reduced schemes; parabolic subgroup-schemes Linear algebraic groups over arbitrary fields, Group schemes, Subgroup theorems; subgroup growth On the structure of nonreduced parabolic subgroup-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 \(X\subset \mathbb {P}^n\) be an integral and non-degenerate variety. For all \(e_1\geq \cdots \geq e_k>0\) consider the set of all curvilinear subschemes of \(X_{reg}\) with \(k\) connected components of degree \(e_1,\dots ,e_k\). For any \(P\in \mathbb {P}^n\) we define two X-ranks with respect to these curvilinear schemes. \(X\)-rank; curvilinear scheme Projective techniques in algebraic geometry, Computational aspects of algebraic curves Curvilinear schemes and the associated \(X\)-ranks | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 ramification sets of finite analytic mappings and applications of his results and methods to punctual Hilbert schemes and to finite analytic maps. He uses essentially the technique of ``chaining'' which consists in associating to a finite map a sequence (or chain) of sets, which are components of ramification loci of increasing multiplicity, and then in controlling their dimensions. For that purpose he uses a theorem of Grothendieck about the order of connexity of subgerms of an irreducible analytic germ, and also in the projective case the Fulton-Hansen theorem and a theorem of Deligne.
In {\S} 1, the author introduces three different notions of multiplicity. The topological multiplicity, the stable multiplicity and the algebraic one. In {\S} 2, he gives a fairly general lower bound to the dimension of the ramification set \(T^{d+1}(f)\), the set of points at which the multiplicity is at least \(d+1\). For the topological case he needs a hypothesis about f, called weak multitransversality which guarantees the additivity of multiplicity under deformation. This theorem is proved by a complicated induction involving multiproducts of ramification sets and the theorem of Grothendieck.
In {\S} 3, the author gives applications of {\S} 2, and of the chaining technique to the punctual Hilbert scheme \(Hilb'{\mathcal O}_{X,x}\) which parametrizes in \(Hilb'(X)\) the punctual schemes concentrated at \(x\in X\). The idea consists in identifying the germ of \(Hilb'({\mathcal O}_{X,x})\) at a smoothable element z with the ramification loci an appropriate map obtained by unfolding the equation of z. He thus obtains a lower bound for the local dimension at z of the open set U of smoothable points in X. This bound is (n-1)(\(\ell -1)\) with \(n=\dim (X)\) in the easiest case (X everywhere irreducible). Various, and more complicated results are obtained when we drop the irreducibility hypothesis or consider instead of U the open set of weakly smoothable (i.e. smoothable in a smooth ambient space) element.
Finally in {\S} 4, the author proves similar results for a finite projective morphism \(f:\quad X^ n\to P^ p.\) He generalizes a previous joint result of himself with Lazarsfeld (case \(n=p)\). This consists again in giving cases of non-emptiness for \(T^{d+1}(f)\) under some complicated numerical conditions. ramification sets of finite analytic mappings; punctual Hilbert schemes; ramification loci of increasing multiplicity T. Gaffney, ''Multiple points, chaining and Hilbert schemes,'' Amer. J. Math., vol. 110, iss. 4, pp. 595-628, 1988. Parametrization (Chow and Hilbert schemes), Singularities in algebraic geometry Multiple points, chaining and Hilbert schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A continuous map \(\mathbb R^m \to \mathbb R^N\) or \(\mathbb C^m \to \mathbb C^N\) is called \(k\)-regular if the images of any \(k\) points are linearly independent. Given integers \(m\) and \(k\) a problem going back to Chebyshev and Borsuk is to determine the minimal value of \(N\) for which such maps exist. The methods of algebraic topology provide lower bounds for \(N\), but there are very few results on the existence of such maps for particular values \(m\) and \(k\). Using methods of algebraic geometry we construct \(k\)-regular maps. We relate the upper bounds on \(N\) with the dimension of the locus of certain Gorenstein schemes in the punctual Hilbert scheme. The computations of the dimension of this family is explicit for \(k \leq 9\), and we provide explicit examples for \(k \leq 5\). We also provide upper bounds for arbitrary \(m\) and \(k\). \(k\)-regular embeddings; secants; punctual Hilbert scheme; finite Gorenstein schemes Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces, Immersions in differential topology, Parametrization (Chow and Hilbert schemes), Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) Constructions of \(k\)-regular maps using finite local 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 takes up ideas introduced by \textit{F.-O. Schreyer} [J . Reine Angew. Math. 421, 83-123 (1991; Zbl 0729.14021)]. Schreyer showed how a Gröbner basis for graded reverse lex can be immediately deduced from the Petri basis of the ideal of a canonical curve of arithmetical genus \(g\), and he proved conversely that a set of polynomials that looks like this Petri-Gröbner basis, and forms a Gröbner basis, generates the ideal of a (possibly reducible or singular) canonical curve. This motivates the definition in the present paper of the \textit{Petri scheme} \({\mathcal P}_g\) for each genus \(g\): Consider the set of polynomials in the Petri-Gröbner basis obtained in this way from a curve with genus \(g\) to have indeterminate coefficients at all non-leading terms, and take the scheme defined by setting all \(S\)-polynomial remainders equal to zero (Buchberger's criterion for Gröbner bases). The principal results of the paper under review are:
(1) The subset of \({\mathcal P}_6\) parametrizing canonical curves whose ideals are generated by quadrics is contained in one irreducible component of \({\mathcal P}_6\);
(2) For all sufficiently large \(g\), \({\mathcal P}_g\) has an irreducible component which is distinct from the irreducible component whose general point corresponds to a smooth Petri-general canonical curve. Gröbner basis; canonical curve; Petri scheme Computational aspects of algebraic curves, Special algebraic curves and curves of low genus, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) Canonical curves and the Petri 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 punctual Hilbert scheme has been known since the early days of algebraic geometry in EGA style. Indeed it is a very particular case of the Grothendieck's Hilbert scheme which classifies the subschemes of projective space. The general Hilbert scheme is a key object in many geometric constructions, especially in moduli problems. The punctual Hilbert scheme which classifies the 0-dimensional subschemes of fixed degree, roughly finite sets of fat points, while being pathological in most settings, enjoys many interesting properties especially in dimensions at most three. Most interestingly it has been observed in this last decade that the punctual Hilbert scheme, or one of its relatives, the \(G\)-Hilbert scheme of Itô-Nakamura, is a convenient tool in many hot topics, as singularities of algebraic varieties, e.g McKay correspondence, enumerative geometry versus Gromov-Witten invariants, combinatorics and symmetric polynomials as in Haiman's work, geometric representation theory (the subject of this school) and many others topics.
The goal of these lectures is to give a self-contained and elementary study of the foundational aspects around the punctual Hilbert scheme, and then to focus on a selected choice of applications motivated by the subject of the summer school, the punctual Hilbert scheme of the affine plane, and an equivariant version of the punctual Hilbert scheme in connection with the A-D-E singularities. As a consequence of our choice some important aspects are not treated in these notes, mainly the cohomology theory, or Nakajima's theory. for which beautiful surveys are already available in the current literature [\textit{V. Ginzburg}, Lectures on Nakajima's quiver varieties. Paris: Société Mathématique de France. Séminaires et Congrès 24, pt. 1, 145--219 (2012; Zbl 1305.16009); \textit{M. Lehn}, Lectures on Hilbert schemes. Providence, RI: American Mathematical Society (AMS). CRM Proceedings \& Lecture Notes 38, 1--30 (2004; Zbl 1076.14010); \textit{H. Kakajima}, Lectures on Hilbert schemes of points on surfaces. University Lecture Series. 18. Providence, RI: American Mathematical Society (AMS). xi, 132 p. (1999; Zbl 0949.14001)]. scheme; cluster; group; group action; matrix factorization; quotient scheme; singular point Bertin, J., The punctual Hilbert scheme: an introduction, Proceedings of the Summer School 'Geometric Methods in Representation Theory'. I, 1-102, (2012), Soc. Math. France, Paris Actions of groups on commutative rings; invariant theory, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), McKay correspondence, Cluster algebras The punctual Hilbert scheme: an introduction | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems How the nature of the base space affects the jet schemes is a natural question. This paper is motivated by a question: if a scheme is defined by a monomial ideal, then are its jet schemes also defined by monomial ideals? The answer is no. The paper gives a simple counterexample. But if one thinks of the reduced subschemes of the jet schemes instead of the jet schemes themselves, then the answer is yes. This paper proves it by describing the generators of the ideal of the reduced subschemes of the jet schemes concretely. Goward, Russell A. Jr. and Smith, Karen E. The jet scheme of a monomial scheme \textit{Comm. Algebra}34 (2006) 1591--1598 Math Reviews MR2229478 Computational aspects in algebraic geometry, Computational aspects and applications of commutative rings The jet scheme of a monomial 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 article is devoted to the description of an algorithm for subdivision of a plane into nonintersecting domains by a finite set of simple Jordan arcs. Each resultant domain is defined via a set of its boundary arcs and its indicator (a bounded or an unbounded domain) which determines the characteristic domain function. In addition, an algorithm is obtained for implementation of a regularized set operations on domains without cutoffs. It is based on subdividing a plane by common boundaries on subdomains and constructing on this base the result of the operation. For computing the intersection points of the boundary arcs, the Newton method is applied whose square convergence is proven for the case of convex and monotone curves. geometric intersection problem; curve intersection; algorithm; subdivision; Jordan arcs; Newton method; convergence Numerical aspects of computer graphics, image analysis, and computational geometry, Polyhedra and polytopes; regular figures, division of spaces, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Subdivision of a plane and set operations on domains | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We present a general framework for analyzing the complexity of subdivision-based algorithms whose tests are based on the sizes of regions and their distance to certain sets (often varieties) intrinsic to the problem under study. We call such tests diameter-distance tests. We illustrate that diameter-distance tests are common in the literature by proving that many interval arithmetic-based tests are, in fact, diameter-distance tests. For this class of algorithms, we provide both non-adaptive bounds for the complexity, based on separation bounds, as well as adaptive bounds, by applying the framework of continuous amortization.
Using this structure, we provide the first complexity analysis for the algorithm by Plantinga and Vegeter for approximating real implicit curves and surfaces. We present both adaptive and non-adaptive \textit{a priori} worst-case bounds on the complexity of this algorithm both in terms of the number of subregions constructed and in terms of the bit complexity for the construction. Finally, we construct families of hypersurfaces to prove that our bounds are tight. algorithmic complexity; worst case bit-complexity; subdivision algorithms; continuous amortization; adaptive complexity; separation bounds Computer graphics; computational geometry (digital and algorithmic aspects), Effectivity, complexity and computational aspects of algebraic geometry, Symbolic computation and algebraic computation, Analysis of algorithms The complexity of subdivision for diameter-distance tests | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The article studies Chen's iterated integrals from the point of view of synthetic differential geometry. The main result presented in the article is that Chen's iterated integrals produce a subcomplex of the de Rham complex on both the free path space as well as on based path spaces. synthetic differential geometry; path space; de Rham complex; Chen's iterated integrals; Hochschild complex; bar complex Synthetic differential geometry, Differential forms in global analysis, Category-theoretic methods and results in associative algebras (except as in 16D90), de Rham cohomology and algebraic geometry Synthetic differential geometry of Chen's iterated integrals | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We examine the slice spectral sequence for the cohomology of singular schemes with respect to various motivic \(T\)-spectra, especially the motivic cobordism spectrum. When the base field \(k\) admits resolution of singularities and \(X\) is a scheme of finite type over \(k\), we show that Voevodsky's slice filtration leads to a spectral sequence for \(\mathrm{MGL}_X\) whose terms are the motivic cohomology groups of \(X\) defined using the cdh-hypercohomology. As a consequence, we establish an isomorphism between certain geometric parts of the motivic cobordism and motivic cohomology of \(X\).
A similar spectral sequence for the connective \(K\)-theory leads to a cycle class map from the motivic cohomology to the homotopy invariant \(K\)-theory of \(X\). We show that this cycle class map is injective for a large class of projective schemes. We also deduce applications to the torsion in the motivic cohomology of singular schemes. algebraic cobordism; Milnor \(K\)-theory; motivic homotopy theory; motivic spectral sequence; \(K\)-theory; slice filtration; singular schemes Motivic cohomology; motivic homotopy theory, Algebraic cycles, Applications of methods of algebraic \(K\)-theory in algebraic geometry, \(K\)-theory of schemes, Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects) The slice spectral sequence for singular 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 Let \(V\) be a discrete valuation ring of mixed characteristic \(p\), with fraction field \(K\) and residue field \(k\). \(W = W(k)\) will denote the Witt ring of \(k\). For a scheme \(X\), let \(\alpha_X : M_X \to {\mathcal O}_X\) be a log structure. It is called constant if the associated sheaf of monoids \(M_X/{\mathcal O}^*_X\) is locally constant. A somewhat stronger condition is: for every \(x \in X\) and every non-unital local section \(m\) of \(M_{X,x}\), \(\alpha_X (m) = 0\). Such log structures are called hollow. For reduced \(X\) these notions coincide. For a scheme \(X\) with constant log structure \(\alpha_X : M_X \to {\mathcal O}_X\), one can define a unique hollow log structure \(\alpha^\natural_X : M_X \to {\mathcal O}_X\). The corresponding log scheme is called the hollowing out of \(X\). It is written \(X^\natural\). Write \(X \to \underline X\) for the canonical mapping from a log scheme \(X\) to the same scheme with the trivial log structure.
A (commutative) monoid is called saturated if any element \(y\) of the associated abelian group \(M^g\) such that \(ny \in M\) for some \(n \in \mathbb{N}\) already lies in \(M\). \(M\) is said to be fine if it is finitely generated. Sheafifying one has the analogous notions for log schemes. One defines an \(F^\infty\)-(iso)span \((E, \Phi, V)\) of width \(m\) on \(X/W\) as a sequence of (iso)crystals \(E^n\) and maps \(F^*_{X/W} E^{n + 1} @>\Phi_n>> E^n\) and \(E_n @>V_n>> F^*_{X/W} E^{n + 1}\) such that \(\Phi_n \circ V_n\) and \(V_n \circ\Phi_n\) are multiplication by \(p^m\). A morphism of level \(\ell\) between two \(F^\infty\)-spans is a sequence \(h_n\) of morphisms of the underlying crystals such that \(\Phi_n \circ F^*_{X/W} (h^{n + 1}) = p^\ell h_n \circ \Phi_n\) and \(F^*_{X/W} (h_{n + 1}) \circ V_n = p^\ell V_n \circ h_n\). For fine saturated log schemes \(X\) and \(Z\) of finite type over \(k\), and for a PD-enlargement \(T = (T, z_T)\) of \(Z\), one writes \(X_T\) for the pullback of \(X\) to \(Z_T\) and \(R\Gamma (X/T,E)\) for the crystalline cohomology \(R \Gamma (X_T/T, z^*_T E)\) of any crystal of \({\mathcal O}_{X/W}\)-modules. With these conventions one can now state the main comparison theorem:
Theorem 1. Let \(f : X \to Z\) be a perfectly smooth morphism of relative dimension \(d\) of fine saturated log schemes of finite type over \(k\) and let \((E, \Phi,V)\) be an \(F^\infty\)-span on \(X/W\) of width \(m\), flat over \({\mathcal O}_{Z/X} \). Suppose that \(T\) is an affine PD-enlargement of \(Z\) and that the log structure on \(Z\) is hollow. Then there is a family of isogenies
\[
\rho = : \bigl\{ \rho^n_T : R \Gamma (X/T, E^n) \otimes \mathbb{Q} \to R \Gamma (X/T^\natural, E^n) \otimes \mathbb{Q} \bigr\}.
\]
Furthermore \(\rho\) is compatible with the base change isomorphisms corresponding to PD-enlargements and with morphisms of \(F^\infty\)-spans (of arbitrary level).
The proof relies on a twisted inverse limit construction. One can give some variations and generalizations of the theorem. In particular, one obtains the Hyodo-Kato comparison theorem [\textit{O. Hyodo} and \textit{K. Kato}, ``Semi-stable reduction and crystalline cohomology with logarithmic poles'', in: Périodes \(p\)-adiques. Sémin. Bures-sur-Yvette 1988, Astérisque 223, 221-268 (1994)] in the context of semistable reduction. Let \(T = \text{Spec} (V)\), \(\eta = \text{Spec} (K)\) and \(\xi = \text{Spec} (k)\), all endowed with the trivial log structure. For any scheme \(Y/T\) let \(Y^\times\) denote the log scheme obtained by endowing \(Y\) with the direct image of the trivial log structure on \(Y_\eta\) via the open immersion \(Y_\eta \to Y\). Also, let \(\xi^\times_V\) denote the reduction of \(T^\times\) modulo the maximal ideal of \(V\). Hyodo and Kato constructed a hollow logarithmic formal scheme \(S^\natural_V\) in \(\text{Cris} (\xi^\times_V/W)\) via the Teichmüller mapping \(k^* \to W\). Write \(X^\times\) for the pullback of \(Y^\times/ \widehat T^\times\) to \(\xi^\times_V\). The choice of a uniformizer of \(V\) determines a morphism \(T^\natural \to S^\natural_V\). One can state:
Theorem (Hyodo-Kato). Suppose \(Y/T\) is proper and with semistable reduction. Then there is a canonical isomorphism \(R \Gamma (X^\times/S^\natural_V) \otimes K \cong R \Gamma (Y/K, \Omega^\bullet_{ Y_\eta/K})\).
Next, the Hyodo-Kato theorem is related to Christol's transfer theorem [\textit{G. Christol} in: Cohomologie \(p\)-adique, Astérisque 119-120, 151-168 (1984; Zbl 0553.12014)] in the theory of \(p\)-adic differential equations with regular singularities. For a fine saturated log scheme \(Z/k\) one introduces the notion of a transfer structure on a crystal \(E\) on \(Z/W\). The main results of above may be reformulated in the following way:
Theorem 2. Let \(f : X \to Z\) be a perfectly smooth morphism of fine saturated log schemes over \(k\) and assume the log structure on \(Z\) is constant. Then if \(E\) is an \(F^\infty\)-span on \(X/W\) and \(q \in \mathbb{N}\), there is a convergent \(F^\infty\)-isospan \(E^q\) of flat \(\mathbb{Q} \otimes {\mathcal O}_{Z/W}\)-modules such that for each \(p\)-adic enlargement \(T\) of \(Z/W\), \(E^q_T \cong \mathbb{Q} \otimes R^q f_{X/T^*}E\). -- Furthermore the value of \(E^q\) on each enlargement \((T,z_T)\) depends, up to canonical isomorphism, only on \((\underline T,M_T)\), not on \(\alpha_T\): \(M_T \to {\mathcal O}_T \). If \(\underline Z/k\) is smooth, the residue mapping of \(E^q\) is nilpotent.
The last section of the paper deals with the logarithmic construction of the crystalline analog of the Weil-Deligne group \({\mathbf W}_{\text{cris}} (\overline K)\). This \({\mathbf W}_{\text{cris}} (\overline K)\) can be thought of as the semi-direct product of the crystalline Weil group \(W_{\text{cris}} (\overline K)\), i.e. the group of all automorphisms of \(\overline K\) which act as some integral power of the Frobenius endomorphism of \(\overline k\), and the twisted form \(\overline K(1)\) of \(\overline K\). -- As a final result on obtains the following version of the Hyodo-Kato theorem:
Theorem 3. Suppose \(Y/V\) is proper and has semi-stable reduction and \(E\) is an \(F\)-crystal on the (logarithmic) special fiber \(X^\times/k^\times_V\). Then the de Rham cohomology \(H^q_{DR} ((E_Y, \nabla_Y)/K) \otimes \overline K\) admits a canonical action of \({\mathbf W}_{\text{cris}} (\overline K^\natural)\). In particular, \(H^q_{DR} (Y_K/K)\) admits such an action. Witt ring; log structure; hollow log structure; crystalline cohomology; comparison theorem; fine saturated log schemes; Weil-Deligne group Arthur Ogus, \?-crystals on schemes with constant log structure, Compositio Math. 97 (1995), no. 1-2, 187 -- 225. Special issue in honour of Frans Oort. \(p\)-adic cohomology, crystalline cohomology, (Co)homology theory in algebraic geometry, Witt vectors and related rings F-crystals on schemes with constant log structure | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 well-written paper, which is part I of II, the authors study degeneracy loci of morphisms of vector bundles on a smooth ambient space which, as they show, give rise to perfect obstruction theories and, hence, their associated virtual cycles. They manage to calculate these virtual cycles via the Thom-Porteous formula. The authors then give various applications of these results to punctual Hilbert schemes of \textit{nested} subschemes of a fixed projective surface \(S\), and discuss some implications for local Donaldson-Thomas theory.
More precisely, let \(E_{\bullet} = \{E_0 \overset{\sigma}{\rightarrow} E_1 \}\) be a two-term complex of vector bundles on a smooth ambient space \(A\), let \(n = \operatorname{dim} A\), and let \(r_i = \operatorname{rank}(E_i)\). The \(r\)-th degeneracy locus is defined set-theoretically as
\[Z_r = \bigl\{ x \in A \,\colon\, \operatorname{rank}(\sigma|_x) \leq r \bigr\}.\]
It is equipped with the scheme structure defined by the vanishing of the \((r+1) \times (r+1)\)-minors of \(\sigma\), i.e., by the vanishing of \(\Lambda^{r+1}\sigma \colon \Lambda^{r+1}E_0 \to \Lambda^{r+1}E_1\). In this paper, the authors restrict their attention to the smallest \(r\) for which \(Z = Z_r\) is non-empty; the general case is treated in the sequel [\textit{A. Gholampour} and \textit{R. P. Thomas}, Compos. Math. 156, No. 8, 1623--1663 (2020; Zbl 1454.14028)]. Their first main result, Theorem 3.6, yields a perfect obstruction theory on \(Z\) and provides an expression for its associated virtual cycle \(\iota_*[Z]^{\textrm{vir}}\) where \(\iota \colon Z \hookrightarrow A\) is the embedding.
Let \(S\) be a fixed projective surface, let \(n_1,n_2\) be non-negative integers, and let \(S^{[n_i]}\) denote the Hilbert scheme of zero-dimensional length \(n_i\) closed subschemes of \(S\). The easiest case is the 2-step nested punctual Hilbert scheme,
\[S^{[n_1,n_2]} = \bigl\{ I_1 \subseteq I_2 \subseteq \mathcal{O}_S \,\colon\, \operatorname{length}(\mathcal{O}_S / I_i) = n_i \bigr\}.\] It lies in the ambient space \(S^{[n_1]} \times S^{[n_2]}\) as the locus of points \((I_1,I_2)\) for which there is exists a non-zero map \(\mathrm{Hom}_S(I_1,I_2) \neq 0\). Let \(\pi \colon S^{[n_1]} \times S^{[n_2]} \times S \to S^{[n_1]} \times S^{[n_2]}\) be the natural projection, let \(\mathcal{Z}_1, \mathcal{Z}_2\) denote the universal closed subschemes, and let \(\mathcal{I}_1, \mathcal{I}_2\) denote the corresponding universal ideal sheaves. Then \(S^{[n_1,n_2]}\) arises as the degeneracy locus of the complex of vector bundles
\[R\mathcal{H}om_{\pi}(\mathcal{I}_1,\mathcal{I}_2) = R\pi_{*}R\mathcal{H}om(\mathcal{I}_1,\mathcal{I}_2)\] over \(S^{[n_1]} \times S^{[n_2]}\).
The second main result, Theorem 6.3, states that for any smooth projective surface \(S\), the \(2\)-step nested Hilbert scheme \(S^{[n_1,n_2]}\) carries a natural perfect obstruction theory and virtual cycle
\[\bigl[ S^{[n_1,n_2]} \bigr]^{\textrm{vir}} \in A_{n_1+n_2}\bigl( S^{[n_1,n_2]} \bigr)\]
whose push-forward to \(S^{[n_1]} \times S^{[,n_2]}\) equals \(c_{n_1+n_2}\bigl( R\mathcal{H}om_{\pi}(\mathcal{I}_1,\mathcal{I}_2[1] \bigr)\)
Since Theorem 6.3 expresses \([S^{[n_1,n_2]}]^{\text{vir}}\) as a degeneracy locus, the authors are able to use the splitting principle from topology to give a proof of the following vanishing result of Carlsson and Okounkov [\textit{E. Carlsson} and \textit{A. Okounkov}, Duke Math. J. 161, No. 9, 1797--1815 (2012; Zbl 1256.14010)]: For any smooth projective surface \(S\) we have the vanishing
\[c_{n_1+n_2+i}\bigl( R\mathcal{H}om_{\pi}(\mathcal{I}_1,\mathcal{I}_2[1]\bigr) = 0, \qquad i >0, \]
over \(S^{[n_1]} \times S^{[n_2]}\). The methods in this work allow the authors to generalise this result in Theorem 8.3, yielding the vanishing
\[c_{n_1+n_2+i}\bigl( R\pi_{*}\mathcal{L} - R\mathcal{H}om_{\pi}(\mathcal{I}_1,\mathcal{I}_2 \otimes \mathcal{L}) \bigr) = 0, \qquad i >0, \]
over \(S^{[n_1]} \times S^{[n_2]} \times \operatorname{Pic}_{\beta}(S)\). Here \(\beta \in H_2(S,\mathbb{Z})\) is a curve class and \(\mathcal{L} \to S \times \operatorname{Pic}_{\beta}(S)\) is a Poincaré line bundle. The proof relies on a more general notion of nested Hilbert scheme, denoted \(S_{\beta}^{[n_1,n_2]}\) of subschemes \(S \supset Z_1 \supseteq Z_2\) where \(Z_1\) is allowed to be of dimension \(\leq 1\). In the sequel to this paper [\textit{A. Gholampour} and \textit{R. P. Thomas}, Compos. Math. 156, No. 8, 1623--1663 (2020; Zbl 1454.14028)], the authors construct a natural perfect obstruction theory and virtual cycle on \(S_{\beta}^{[n_1,n_2]}\) for any curve class \(\beta\).
Throughout the paper, attention is paid to surfaces for which either or both of \(h^{0,1}(S)\) and \(h^{0,2}(S)\) are non-zero, either of which necessitate further consideration. The final section of the paper provides an alternative, more geometric, approach to the virtual cycle construction by incorporating the Jacobian \(\operatorname{Jac}(S)\) directly in case \(h^{0,1}(S) \neq 0\). degeneracy locus; nested Hilbert scheme; Thom-Porteous formula; local Donaldson-Thomas theory; Vafa-Witten invariants Algebraic moduli problems, moduli of vector bundles, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Parametrization (Chow and Hilbert schemes), Applications of global analysis to structures on manifolds Degeneracy loci, virtual cycles and nested Hilbert schemes. 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 Let \(n,q>0\) be integers, and let \(IMM_q^n\) be the polynomial on \(n\)-tuples of \(q\times q\) matrices whose value on \((X_1,\dots,X_n)\) is \(\text{tr}(X_n\cdots X_1)\). Then \(IMM_q^n\) is a homogeneous polynomial in \(nq^2\) variables; if we let \(V\) be the complex vector space of \(n\)-tuples of \(q\times q\) matrices, then we write \(IMM_q^n\in S^{n}V\). In the work under review, some geometric properties of \(IMM_q^n\) are given. The tools used are representation theory and algebraic geometry.
Geometric Complexity Theory seeks to study polynomials which are characterized by their symmetry group. Recall that \(\text{GL}(V)\) acts on \(S^nV\); the symmetry group \(\mathcal{S}\) of \(IMM_q^n\in S^nV\) is the stabilizer of \(IMM_q^n\) under this action. A main result in this paper is that \(\mathcal{S}\cong \mathcal{S}_0\rtimes D_n\), where \(\mathcal{S}_0\) is the connected component of the identity of \(\mathcal{S}\) and \(D_n\) is the group of symmetries of a regular \(n\)-gon. Alternatively, any polynomial in \(S^nV\) which is stabilized by \(\mathcal{S}_0\) is shown to be a complex multiple of \(IMM_q^n\), hence \(IMM_q^n\) is characterized by its stabilizer.
The remainder of the results concerns the hypersurface determined by the polynomial. Let \(\mathcal{I}mm_q^n\) be the hypersurface \(V(IMM_q^n)\subseteq\mathbb{P}V^*\). It is shown that its dual variety \((\mathcal{I}mm_q^n)^{\vee}\) is also a hypersurface in \(\mathbb{P}V^*\). Let \(\mathcal{S}ing_q^n\) denote the singular locus of the affine cone in \(V^*\) over \(V(IMM_q^n)\subseteq\mathbb{P}V^*\). Then a description of the irreducible components of \(\mathcal{S}ing_q^n\) in terms of certain nilpotent representations of the Euclidean equioriented quiver and dimensions are computed; examples are provided for \(n\leq 3\). Finally, let \(\mathcal{W}=V(J)\) where \(J\) is the ideal of \(IMM_q^n\) generated by all \(n-2\) order partial derivatives. The irreducible components of \(\mathcal{W}\) are given, and it is shown that \(\text{dim} \mathcal{W}=\lfloor (5/4)q^2 \rfloor\). iterated matrix multiplication; symmetry group of a polynomial; Jacobian loci of hypersurfaces F. Gesmundo, \textit{Geometric aspects of iterated matrix multiplication}, J. Algebra, 461 (2016), pp. 42--64. Other algebraic groups (geometric aspects), Linear preserver problems, Representations of quivers and partially ordered sets Geometric aspects of iterated matrix multiplication | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A 4-point nonlinear corner-cutting subdivision scheme is established. It is induced by a special \(C\)-shaped biarc structure. The scheme is circular-invariant and can be effectively applied to sets of 2-dimensional (2D) data that are convex. The scheme is also extended adaptively to non-convex data. Explicit examples are demonstrated. biarc; circular-invariant; computer-aided geometric design; nonlinear; subdivision Computer-aided design (modeling of curves and surfaces), Plane and space curves, Computer science aspects of computer-aided design A new four point circular-invariant corner-cutting subdivision for curve design | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Towers of function fields meeting the Drinfeld-Vlăduţ bound with equality were recently constructed by Garcia-Stichtenoth. Modular uniformizations thereof were pointed out by Elkies. We connect this fact to hypergeometric analogs of the arithmetic-geometric mean and related theta series identities discovered by Borweins and Garvan. algebraic-geometric codes; lattices; towers of function fields; hypergeometric analogs; theta series identities Solé, P.: Towers of function fields and iterated means. IEEE trans. Inform. theory 46, 1532-1535 (2000) Geometric methods (including applications of algebraic geometry) applied to coding theory, Applications to coding theory and cryptography of arithmetic geometry Towers of function fields and iterated means | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 determine the real counting function for the hypothetical ``curve'' \(C\) over \({\mathbb{F}}_1\), whose corresponding zeta function is the complete Riemann zeta function. Such a counting function exists as a distribution, is positive on \((1, \infty)\) and takes the value \(- \infty\) at \(q=1\) as expected from the infinite genus of \(C\).
As an application, the authors apply their functorial theory in order to interpret conceptually the spectral realization of zeros of L-functions. absolute point; counting functions; zeta functions Connes, A.; Constani, C., Schemes over \(\mathbf{F}_1\) and zeta functions, Compositio Mathematica, 146, 6, 1383-1415, (2010) Schemes and morphisms, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture Schemes over \(\mathbb F_{1}\) and zeta functions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The article continues a series of papers of the author, resp. joint work with L. Alonso Tarrio and A. Jeremias Lopez. They are aimed to develop an infinitesimal theory of locally noetherian formal schemes ([\textit{L. Alonso Tarrío} et al., Commun. Algebra 35, No. 4, 1341--1367 (2007; Zbl 1124.14006); J. Pure Appl. Algebra 213, No. 7, 1373--1398 (2009; Zbl 1206.14012)], [\textit{M. Pérez Rodríguez}, J. Pure Appl. Algebra 212, No. 11, 2381--2388 (2008; Zbl 1151.14007)]). Here the author introduces a suitable definition of a cotangent complex and provides a deformation theory in this category. Let \(f:{ \mathcal X }\to \mathcal Y\) be a morphism of locally noetherian formal schemes. The author defines a cotangent complex \(\hat{\mathcal L}_{{\mathcal X} / {\mathcal Y}}\) in the derived category \(D^{-} ( {\mathcal X} )\) and proves basic properties like the existence of an augmentation map \(\hat{\mathcal L}_{{\mathcal X} / {\mathcal Y}} \to \hat{\Omega}^1_{{\mathcal X} / {\mathcal Y}}\), functoriality, localization, coherence, the existence of distinguished triangels, flat base change. Results on deformations provide a characterization of smooth and etale morphisms.
The paper concludes with a discussion of the cotangent complex and infinitesimal conditions (etaleness, regular closed immersions). formal scheme; cotangent complex; lifting; deformation; local homology Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Generalizations (algebraic spaces, stacks), Infinitesimal methods in algebraic geometry, Formal neighborhoods in algebraic geometry, Local structure of morphisms in algebraic geometry: étale, flat, etc., Formal methods and deformations in algebraic geometry Deformation of formal schemes through local homology | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In the paper under review the author reconstructs a reduced noetherian scheme from the derived category of perfect complexes over the scheme together with the tensor product structure on the derived category.
The main tool is a result of Thomason which gives an inclusion preserving correspondence between certain subcategories \({\mathcal C}\) of the category of perfect complexes over a noetherian scheme \(X\) and closed subsets \(Y\) of \(X\) so that whenever \(y\) is in \(Y\) also its closure is in \(Y\). The author associates to a triangulated category \({\mathcal C}\) which is equipped with some sort of a tensor product \(\otimes\) a topological space \(Spc ({\mathcal C},\otimes)\) in some very natural way. Then, a presheaf \(K\) of triangulated categories is constructed on this topological space and finally a ringed space given basically by the endomorphisms of the identity functor.
Earlier, Bondal and Orlov got a completely different way to reconstruct the original variety out of its derived category by considering the group of self-equivalences of the derived category. isomorphism problem for derived categories P. Balmer, Presheaves of triangulated categories and reconstruction of schemes, Math. Ann. 324 (2002), no. 3, 557-580. Derived categories, triangulated categories, Schemes and morphisms Presheaves of triangulated categories and reconstruction 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 Let R be a Noetherian ring. An R-sequence \(x_ 1,...,x_ n\) is built by always choosing \(x_ i\) to be not in the union of the primes in Ass R/(x\({}_ 1,...,x_{i-1})R\). In the terminology of this paper, \(A(I)=Ass R/I\) is a grade scheme and its associated grade function is \(f(I)=the\) (classical) grade of I. If instead, we take \(A(I)=\{P| P\quad is\quad an\) essential prime (respectively, asymptotic prime) of \(1\}\), and construct a sequence in the same way, we get the essential (respectively asymptotic) grade function. This paper studies the abstract nature of such grade schemes and their associated grade functions, in particular, finding which properties of the above examples are abstract in nature, and which are indigeneous to the given example.
The main theorem says that if f is a function from the set of all ideals in all localizations of R, to the set of nonnegative integers, then f is the grade function of some grade scheme A(I) if and only if f satisfies (i) f(I\({}_ S)=\min \{f(P_ P)| P_ S\) is
\[
a prime
\]
containing \(I/_ S\}\), (ii) f(P\({}_ P)\leq height P\) for all \(P\in Spec R\), and \((iii)\quad if\quad (Q,U)\) is a conforming pair in R, and if \(f(P_ P)\leq n\) for all \(P\in U\), then \(f(Q_ Q)\leq n-1\). - Here, (Q,U) is a conforming pair if \(Q\in Spec R\) and U is an infinite set of primes, each properly containing Q, such that if W is any infinite subset of U, then \(\cap \{P\in W\}=Q\). asymptotic grade; essential grade; Noetherian ring; grade function; grade scheme Commutative ring extensions and related topics, Commutative Noetherian rings and modules, Relevant commutative algebra Grade schemes and grade functions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a smooth complex algebraic surface. The Hilbert scheme \(X^{[n]}\) of \(n\) points on \(X\) is a smooth variety of dimension \(2n\). \textit{E. Carlsson} studied the generating series for the intersection pairings between the total Chern class of the tangent bundle and the Chern classes of tautological bundles on \((\mathbb C^2)^{[n]}\), proving that the reduced series \(\langle \text{ch}_{k_1} \dots \text{ch}_{k_N} \rangle^\prime\) is a quasi-modular form [Adv. Math. 229, 2888--2907 (2012; Zbl 1255.14005)]. \textit{A. Okounkov} conjectured that these reduced series are multiple \(q\)-zeta values [Funct. Anal. Appl. 48, 138--144 (2014; Zbl 1327.14026)]. \textit{Z. Qin} and \textit{F. Yu} [Int. Math. Res. Not. 2018, 321--361 (2018; Zbl 1435.14007)] proved the conjecture modulo lower weight terms via the reduced series
\[
\overline F^{\alpha_1,\dots,\alpha_N}_{k_1,\dots,k_N} (q) = (q;q)_\infty^{\chi (X)} \cdot \sum_n q^n \int_{X^{[n]}} (\Pi_{i=1}^N G_{k_i} (\alpha_i,n)) c(T_{X^{[n]}})
\]
where \(0 \leq k_i \in \mathbb Z\), \(\alpha_i \in H^* (X), (q;q)_\infty = \Pi_{n=1}^\infty (1-q^n)\) and \(G_{k_i}(\alpha_i, n) \in H^* (X^{[n]})\) are classes which play a role in the study of the geometry of \(X^{[n]}\) (see work of \textit{Z. Qin} [Hilbert schemes of points and infinite dimensional Lie algebras. Providence, RI: American Mathematical Society (AMS) (2018; Zbl 1403.14003)]).
In the paper under review, the authors further study the series \(\overline F^{\alpha_1,\dots,\alpha_N}_{k_1,\dots,k_N} (q)\). Defining functions \(\Theta_k^\alpha (q)\) depending on \(\alpha \in H^* (X)\) and \(k \geq 0\), they fix \(0 \leq k_1, \dots, k_N \in \mathbb Z\) and \(\alpha_1, \dots, \alpha_N \in H^* (X, \mathbb Q)\) and prove the following:
(1) If \(\langle K_X^2,\alpha_i \rangle =0\) and \(2|k_i\) for each \(i\), then the leading term \(\Pi_{i=1}^N \Theta_k^\alpha (q)\) of \(\overline F^{\alpha_1,\dots,\alpha_N}_{k_1,\dots,k_N} (q)\) is either \(0\) or a quasi-modular form of weight \(\sum (k_i+2)\).
(2) Suppose \(|\alpha_i|=4\) for each \(i\). If \(2|k_i\) for each \(i\), then \(\overline F^{\alpha_1,\dots,\alpha_N}_{k_1,\dots,k_N} (q)\) is a quasi-modular form of weight \(\sum (k_i+2)\). if \(2 \not |k_i\) for some \(i\), then \(\overline F^{\alpha_1,\dots,\alpha_N}_{k_1,\dots,k_N} (q)=0\).
These results are proved by relating the leading term of \(\overline F^{\alpha_1,\dots,\alpha_N}_{k_1,\dots,k_N} (q)\) for \(X\) to the leading term of \(\langle \text{ch}_{k_1} \dots \text{ch}_{k_N} \rangle^\prime\) for \(\mathbb C^2\) studied by Carlsson [loc. cit.]. Hilbert schemes of points on a surface; quasi-modular forms; multiple zeta value; generalized partition Parametrization (Chow and Hilbert schemes), Binomial coefficients; factorials; \(q\)-identities, Vertex operators; vertex operator algebras and related structures Hilbert schemes of points and quasi-modularity | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 construction of virtual classes on nested Hilbert scheme of points and curves of smooth projective surfaces. Let \(S\) be a smooth projective surface, \(r>0\), \(\boldsymbol{n}=n_1,\dots,n_r\in \mathbb{Z}_{\geq 0}\) and \(\boldsymbol{\beta}=\beta_1,\dots, \beta_{r-1}\in H_2(S,\mathbb{Z})\). The closed points of the nested Hilbert scheme \(S_{\boldsymbol{\beta}}^{[\boldsymbol{n}]}\) consist of tuples of subschemes of \(S\)
\[Z_1,\dots,Z_r,\quad C_1,\dots,C_{r-1} \]
where \(Z_i\) is 0-dimensional of length \(n_i\), \(C_i\) is a divisor of class \(\beta_i\) and the the following nesting condition holds
\[ I_{Z_i}(-C_i)\subset I_{Z_{i+1}}, \]
where \(I_{Z_i}\) is the ideal sheaf of \(Z_i\).
For \(S\) a smooth projective surface and \(r\geq 2\), the authors construct a virtual fundamental class \[
[S_{\boldsymbol{\beta}}^{[\boldsymbol{n}]}]^{\text{vir}}\in A_d(S_{\boldsymbol{\beta}}^{[\boldsymbol{n}]}), \quad d=n_1+n_r+\frac{1}{2}\sum_{i=0}^{r-1}\beta_i\cdot(\beta_i-c_1(\omega_S))
\]
and, under suitable hypotheses on \(S\), a reduced virtual fundamental class \([S_{\boldsymbol{\beta}}^{[\boldsymbol{n}]}]^{\text{vir}}_{\text{red}}\in A_*(S_{\boldsymbol{\beta}}^{[\boldsymbol{n}]})\).
This class, constructed in great generality, recovers various virtual classes already present in the literature. The main examples are for the Hilbert scheme of points \(S^{[n]}\), the Hilbert scheme of divisors \(S_\beta\), and the moduli space of stable pairs \(S^{[0,n]}_\beta\).
As an application, the authors compute a closed formula for an operator introduced by Carlsson-Okounkov on Hilbert schemes of points. Define a \(K\)-theory class on \( S^{[n_1]}\times S^{[n_2]}\)
\[
\mathrm{E}^{n_1,n_2}_M:=[\mathbf{R}\pi_*p^*M]-[\mathbf{R}\mathcal{H}om_\pi(\mathcal{I}^{[n_1]},\mathcal{I}^{[n_2]}\otimes p^*M)],
\]
where \(M\) is a line bundle on \(S\). Then
\[ \sum_{n_1\geq n_2\geq 0}(-1)^{n_1+n_2}\int_{S^[n_1\geq n_2]^{\text{vir}}}i^*c(\mathrm{E}^{n_1,n_2}_M)q_1^{n_1}q_2^{n_2}=\\
\prod_{n>0}\left(1-q_2^{n-1}q_1^n\right)^{\langle K_S, K_S-M\rangle}\left(1-q_2^{n}q_1^n\right)^{\langle K_S-M, M\rangle-e(S)},
\]
where \(\langle-,-\rangle\) is the Poincaré pairing and \(i:S^{[n_1\geq n_2]}\hookrightarrow S^{[n_1]}\times S^{[n_2]}
\). nested Hilbert scheme; projective surfaces; virtual fundamental class Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Parametrization (Chow and Hilbert schemes) Nested Hilbert schemes on surfaces: virtual fundamental class | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 investigates flat families over discrete valuation rings describing the degeneration of projective toric varieties into reducible schemes with toric components. The combinatorial counterpart is given by a certain kind of non-compact polyhedra; the irreducible components of the special fibers correspond to the compact faces of the corresponding polytope. The whole construction is similar to perturbing the equation of a reducible toric divisor.
As an application, the author obtains a formula for the number of solutions with prescribed order of a sufficiently general system of Laurent polynomials. This result generalizes the well-known formula of Kouchnirenko. discrete valuation rings; degeneration of projective toric varieties; non-compact polyhedra; toric divisor; number of solutions of Laurent polynomials A. L. Smirnov, Torus schemes over a discrete valuation ring, Algebra i Analiz 8 (1996), no. 4, 161 -- 172 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 8 (1997), no. 4, 651 -- 659. Toric varieties, Newton polyhedra, Okounkov bodies, Formal power series rings, Families, fibrations in algebraic geometry, Valuation rings Torus schemes over a discrete valuation ring | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a ringed space defined over an algebraically closed field \(k\) of uncountable cardinality. The inductive limit over a family of closed subschemes of \(X\) directed by inclusion in ringed spaces is called an ind-scheme. For \(X=k^{{\mathbb{N}}}\) an example of an ind-scheme is an ind- affine scheme. Ind-affine schemes form a Cartesian closed category denoted ind-\(\underline{\text{aff}}\). It is shown (section~1) that ind-\(\underline{\text{aff}}\) is closed under suitable translation by a \(k\)-valued point. It is proved (section~2) that a basic open set of an ind-affine scheme is isomorphic to some closed subset of some ind-affine scheme with respect to the induced ringed space structures. In section~3 using the definitions given there it is proved that a full closed subset of an ind-affine scheme respecting its inductive limit structure with its induced ringed space structure is isomorphic (in ringed spaces) to some ind-affine scheme. Using results from section~1 and the notion of \(\infty\)-tuple embedding from section~3, the countable coproduct of ind-affine schemes in ringed spaces is shown to be isomorphic to an ind-affine scheme (in ringed spaces). It is shown (section~5) that if \(X\) is separated and satisfies a soberness condition, the intersection of two open ind-affine subschemes of \(X\) is again an open ind-affine subscheme of \(X\). It is proved that if \(X\) is a locally ind-affine ringed space which patches properly (using notions of section~6) and two open ind-affine subschemes of \(X\) meet \(Z\)-affinely, then \(X\) is an ind-scheme. In section~7 it is shown how to obtain an ind-affine scheme whose \(k\)-points correspond to the elements of \(\Hom_ S(X,Y)\) in a natural way (here \(\Hom_ S(X,Y)\) denotes the collection of morphisms from a scheme \(X\) to a scheme \(Y\)). In section~8 the autor enlarges ind-\(\underline{\text{aff}}\), adding ringed spaces which are the inductive limits of certain diagrams in ind-\(\underline{\text{aff}}\). ind-affine scheme; ringed spaces 4 P. Cherenack , Extending schemes to a Cartesian closed category , Quaestiones Math. 9 ( 1986 ), 95 - 133 . MR 862296 | Zbl 0653.14001 Schemes and morphisms, Generalizations (algebraic spaces, stacks), Closed categories (closed monoidal and Cartesian closed categories, etc.), Applications of methods of algebraic \(K\)-theory in algebraic geometry Extending schemes to a Cartesian closed 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 Let \(K\) be a number field and let \(\phi(x)\) be a rational function of degree greater than 1 defined over \(K\). Let \(\Phi_n(x,t)=\phi^{(n)}(x)-t\) where \(\phi^{(n)}\) is the \(n\)th iterate of \(\phi\). The authors give a formula for the discriminant of the numerator of \(\Phi_n(x,t)\). If \(\phi\) is \textit{postcritically finite}, i.e. if the forward orbit of the critical points of \(\phi\) under all iterations is a finite set, then the authors also prove that for each specialization of \(t\) to \(t_0 \in K\), then there exists a finite set of primes of \(K\) containing the prime divisors of the discriminant of \(\Phi_n\) for all \(n\). Cullinan, John; Hajir, Farshid, Ramification in iterated towers for rational functions, Manuscripta Math., 137, 3-4, 273-286, (2012) Algebraic functions and function fields in algebraic geometry Ramification in iterated towers for rational functions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The Coleman integral is a \(p\)-adic line integral. Double Coleman integrals on elliptic curves appear in \textit{M. Kim}'s nonabelian Chabauty method [J. Am. Math. Soc. 23, No. 3, 725--747 (2010; Zbl 1225.11077)], the first numerical examples of which were given by the author, \textit{K.S. Kedlaya}, and \textit{M. Kim} [J. Am. Math. Soc. 24, No. 1, 281--291 (2011; Zbl 1285.11082)]. This paper describes the algorithms used to produce those examples, as well as techniques to compute higher iterated integrals on hyperelliptic curves, building on previous joint work with \textit{R.W. Bradshaw} and \textit{K.S. Kedlaya} [Lect. Notes Comput. Sci. 6197, 16--31 (2010; Zbl 1261.14011)]. Coleman integration; \(p\)-adic integration; iterated Coleman integration; hyperelliptic curves; nonabelian Chabauty; integral points Balakrishnan J.\ S., Iterated Coleman integration for hyperelliptic curves, ANTS-X: Proceedings of the tenth algorithmic number theory symposium, Open Book Series 1, Mathematical Sciences Publishers (2013), 41-61. Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.), Curves of arbitrary genus or genus \(\ne 1\) over global fields, Analytic computations, \(p\)-adic cohomology, crystalline cohomology Iterated Coleman integration for hyperelliptic 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 obtain a theorem which allows to prove compact generation of derived categories of Grothendieck categories, based upon certain coverings by localizations. This theorem follows from an application of Rouquier's cocovering theorem in the triangulated context, and it implies Neeman's result on compact generation of quasi-compact separated schemes. We prove an application of our theorem to non-commutative deformations of such schemes, based upon a change from Koszul complexes to Chevalley-Eilenberg complexes. Grothendieck category; derived category; compact generation; deformation; surface Lowen, W.; Bergh, M., On compact generation of deformed schemes, Adv. Math., 244, 441-464, (2013) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Derived categories, triangulated categories, Formal methods and deformations in algebraic geometry On compact generation of deformed 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 motivation of introducing the pro-étale topology is as follows: For a scheme \(X\), there are the étale cohomology groups \(\mathrm{H}^i(X_{\mathrm{et}},\mathbb{Q}_\ell)\) defined indirectly as \(\varprojlim_n\mathrm{H}^i(X_{\mathrm{et}},\mathbb{Z}/\ell^n) \otimes_{\mathbb{Z}_\ell} \mathbb{Q}_\ell\). The naive definition as \(\mathrm{H}^i(X_{\mathrm{et}},\mathbb{Q}_\ell)\) with \(\mathbb{Q}_\ell\) considered as a constant sheaf does not give the correct result, since e.\,g.\ for \(X\) a smooth projective connected curve over an algebraically closed field, one would have \(\mathrm{H}^i(X_{\mathrm{et}},\mathbb{Q}_\ell) = \mathbb{Q}_\ell\) for \(i = 0\) and \(= 0\) for \(i > 0\), which is not a good Weil cohomology theory.
The authors introduce a new Grothendieck topology \(X_{\mathrm{proet}}\) for a scheme \(X\) whose underlying topological space is Noetherian. The proétale site \(X_{\mathrm{proet}}\) is the site of weakly étale \(X\)-schemes, with covers given by fpqc covers. A morphism \(f: Y \to X\) is called weakly étale if \(f\) is flat and \(\Delta_f: Y \to Y \times_X Y\) is flat. (Definition 1.2) A morphism of rings is étale iff it is weakly étale and finitely presented, and an ind-étale morphism of rings is weakly étale. (Theorem 1.3) For a ring \(A\), the site of weakly étale \(A\)-algebras is equivalent to the site of ind-étale \(A\)-algebras.
A sheaf \(L\) of \(\bar{\mathbb{Q}}_\ell\)-modules on \(X_{\mathrm{proet}}\) is lisse if it is locally free of finite rank. A sheaf \(C\) of \(\bar{\mathbb{Q}}_\ell\)-modules on \(X_{\mathrm{proet}}\) is constructible if there is a finite stratification \(\{X_i \to X\}\) into locally closed subsets \(X_i \subseteq X\) such that \(C|_{X_i}\) is lisse. (Definition 1.1) An object \(K \in \mathrm{D}(X_{\mathrm{proet}}, \bar{\mathbb{Q}}_\ell)\) is constructible if it is bounded and all cohomology sheaves are constructible. Let \(\mathrm{D}_{\mathrm{cons}}(X_{\mathrm{proet}}, \mathbb{Q}_\ell)\) be the corresponding full triangulated subcategory.
Let \(x\) be a geometric point of a locally topologically Noetherian connected scheme \(X\) with \(\mathrm{ev}_x: \mathrm{Loc}_X \to \mathrm{Set}\) the associated functor \(F \mapsto F_x\), with \(\mathrm{Loc}_X\) the full subcategory of locally constant sheaves \(F \in \mathrm{Shv}(X_{\mathrm{proet}})\), i.\,e.\ such that there exists a cover \(\{X_i \to X\}\) in \(X_{\mathrm{proet}}\) with \(F|_{X_i}\) constant. The pro-étale fundamental group \(\pi_1^{\mathrm{proet}}(X,x)\) is defined as \(\mathrm{Aut}(\mathrm{ev}_x)\) using the compact-open topology on \(\mathrm{Aut}(S)\) for any set \(S\). (Definition 7.4.2) The category \(\mathrm{Loc}_X\) is equivalent to the category \(\mathrm{Cov}_X\) of étale \(X\)-schemes which satisfy the valuative criterion of properness. (Theorem 1.10)
Then: \(\mathrm{H}^i(X_{\mathrm{et}},\mathbb{Q}_\ell) = \mathrm{H}^i(X_{\mathrm{proet}},\mathbb{Q}_\ell)\). The full triangulated subcategory \(\mathrm{D}_{\mathrm{cons}}(X_{\mathrm{proet}}, \mathbb{Q}_\ell) \subseteq \mathrm{D}(X_{\mathrm{proet}}, \mathbb{Q}_\ell)\) of constructible objects is stable under Grothendieck's six operations, and there is a natural equivalence between \(\mathrm{D}_{\mathrm{cons}}(X_{\mathrm{proet}}, \mathbb{Q}_\ell)\) and \(\mathrm{D}^{\mathrm b}_{\mathrm c}(X,\mathbb{Q}_\ell)\). There is an equivalence between the category of finite dimensional continuous representations of \(\pi_1^{\mathrm{proet}}(X,x)\) and that of lisse \(\mathbb{Q}_\ell\)-sheaves on a locally topologically Noetherian connected scheme \(X\). If \(X\) is geometrically unibranch, \(\pi_1^{\mathrm{proet}}(X,x) \cong \pi_1^{\mathrm{et}}(X,x)\). (Lemma 7.4.10)
Most of the things above work for any local field \(E/\mathbb{Q}_\ell\) instead of \(\mathbb{Q}_\ell\). étale cohomology; higher regulators; zeta and L-functions; Grothendieck topologies; coverings; fundamental group Bhatt, Bhargav; Scholze, Peter, The pro-étale topology for schemes, Astérisque, 369, 99-201, (2015) Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects), Grothendieck topologies and Grothendieck topoi, Coverings of curves, fundamental group The pro-étale topology for schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Alexander schemes form the firm foundation for a study of the Chow group. The author defines a notion of Alexander scheme equivalent to that of A. Vistoli on connected components. Furthermore, it becomes clear from the definition that Alexander schemes are the most general natural class of schemes which behave like smooth schemes for intersection theory. A practical criterion for deciding whether a given scheme is Alexander is given. A natural extension from Alexander scheme to Alexander morphism is made. Finally, using the tools developed, conditions defining whether a cone over a smooth projective variety is Alexander are found. Alexander schemes; Chow group; Alexander morphism; cone Parametrization (Chow and Hilbert schemes), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Schemes and morphisms On the characterization of Alexander 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 prove the existence of many zero-dimensional schemes \(Z\subset\mathbb P^n\) with a small number of connected components, all of them defined over \(\mathbb F_q\), and with good postulation (even if \(\text{length}(Z)\gg \sharp(\mathbb P^n(\mathbb F_q)))\). Projective techniques in algebraic geometry, Finite ground fields in algebraic geometry Unreduced zero-dimensional schemes defined over \(\mathbb F_q\) with good postulation | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This book grew out of a course on iterated integrals and cycles on an algebraic manifold the author held in Fall 2001 at the Nankai Mathematical Institute. This book certainly is the first self contained introduction to this subject which is also adapted for non experts and graduate students. The book is divided into three chapters: 1.) Iterated Integrals, Chen's flat connection and \(\pi_1\) 2.) Iterated Integrals on Compact Riemann Surfaces 3.) The generalized Linking Pairing and the Heat Kernel.
Let us describe the content of these chapters more detailed: Chapter one starts with first examples of iterated integrals which, loosely spoken, are integrals over simplices. Then it introduces Lie algebras, the universal enveloping algebra and gives Chen's construction for a Lie algebra \(\mathfrak{g}\) and a connection on \(\mathfrak{g}\) associated to a connected manifold \(X\). This construction also is explained in a broader context which goes back to Quillen and Lazard. The next chapter can be seen as an application of this theory to Riemann surfaces and the Jacobian manifold of Riemann surfaces. It culminates in a proof of the fact that for a general Riemann surface the cycle \(X-X^-\) in the Jacobian Jac\((X)\) is homologous to zero but not algebraically equivalent to zero (\(X^-\) denotes the image of the cycle \(X \to \) Jac\((X)\) under the inversion morphism on Jac\((X)\)). To obtain this result, it is shown that the iterated integrals of harmonic one forms on \(X\) give the Abel-Jacobi map. The conclusion is that \(X- X^-\) can algebraically be equivalent to zero only if all these iterated integrals are integers. This allows for the Fermat quartic curve \(X = V(X^4+Y^4-Z^4) \subset {\mathbb P}^2\) with a rough integral estimate to show that \(X-X^-\) is not equivalent to zero. The last chapter generalizes this approach to Kähler manifolds of higher dimension. It introduces a generalized linking number to cycles \(C_1,C_2, \ldots C_k\) satisfying that any \(k-1\) have empty intersection and the codimensions add up to dim\((X)+1\). This linking number can be computed by the heat kernel and by an iterated integral as well. It is shown that the symmetric group on \(k\) letters operates by changing the sign on these linking numbers. Furthermore, the situation for complex analytic cycles \(C_i\) is studied, too. On behalf of the author, let me give two slight corrections in the main theorems of this chapter: In theorem 3.2 on page 85, the words ``i.e. either side of the equation is a function of the homology classes of the \(C^i\)'' should be removed. Similarly, in the statement of theorem 3.3 on page 92, the words ``depends only on the homology classes of the \(C^i\)'' should be replaced by ``is unchanged if any \(C_i\) is replaced by a homologous complex cycle \(C_i'\) satisfying the same intersection conditions.'' Chen's connection; algebraic cycles; Abel-Jacobi map Harris (B.).- Iterated Integrals and Cycles on Algebraic Manifolds, Nankai Tracts in Mathematics, vol. 7, World Scientific (2004). Zbl1063.14010 MR2063961 Algebraic cycles, Knots and links (in high dimensions) [For the low-dimensional case, see 57M25], Heat kernels in several complex variables, Differential forms in global analysis, Integration on analytic sets and spaces, currents Iterated integrals and cycles on algebraic 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 article studies seminormal schemes which were discussed earlier by \textit{C.~Traverso} [Ann. Sc. Norm. Sup. Pisa 24 585--595 (1970; Zbl 0205.50501), \textit{S. Greco} and \textit{C.~Traverso} [Compos. Math. 40, 325--365 (1980; Zbl 0412.14024)].
Following Greco-Traverso [loc. cit.], a ring \(A\) is said to be Mori if it is reduced and its integral closure \(A^\nu\) in \(Q(A)\) is finite over \(A\). In this case, the largest subring \(A^{sn}\) such that
(1) \(A\subseteq A^{sn} \subseteq A^\nu\),
(2) \({\text{ Spec}} A^{sn} \to {\text{ Spec}} A\) is bijective,
(3) all maps on the residue fields are isomorphisms
is said to be the semi-normalization of \(A\).
The following is shown:
Given a Noetherian and Mori ring \(A\), the semi-normalization of \(A\) is exactly the subring of \(\prod_{ {\mathbf p} \in {\text{ Spec}} A} \kappa({ p})\) consisting of pointwise functions which vary algebraically along DVRs.
As a corollary, a characterization of morphisms of a seminormal scheme is deduced which relates them to certain compatible set maps of points over DVRs.
Another application is a theorem on the \(a^2, a^3 \Rightarrow a\in A\) characterization.
Furthermore, the article gives a direct proof of the equivalence of the pointwise property and the simplicial characterization of the semi-normalization, as described by \textit{M. Saito} [Math. Ann. 316, No. 2, 283--331 (2000; Zbl 0976.14011)]. The final section gives an application to characterize the Chow variety using the pointwise characterization of semi-normalization. seminormal scheme; Mori ring; semi-normalization A. Röllin and N. Ross. A probabilistic approach to local limit theorems with applications to random graphs. , 2010. Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Local theory in algebraic geometry A characterization of seminormal 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 his fundamental paper [``Techniques de construction et théorèmes d'existence en géométrie algébrique. IV: Les schemes de Hilbert'', Sémin. Bourbaki, exp. 221 (1961; Zbl 0236.14003)], \textit{A. Grothendieck} introduced the so called Hilbert scheme, which is an object which parametrizes all projective subschemes of the projective space with fixed Hilbert polynomial.
In the paper under review, the authors present a more general object, called the multigraded Hilbert scheme, parametrizing all homogeneous ideals with fixed Hilbert function in a graded polynomial ring \(S\).
As in the case of Hilbert schemes, the multigraded Hilbert scheme is a projective scheme (quasi-projective if the grading of \(S\) is not positive), and, when the ground ring is a field, its tangent space at a point corresponding to an ideal \(I\) has a simple description: it is canonically isomorphic to the degree \(0\) piece of \(\Hom(I,S/I)\).
The construction of the multigraded Hilbert scheme is obtained in a great generality, and it enables the authors to prove a conjecture from \textit{D. Bayer}'s thesis [The division algorithm and the Hilbert scheme, Ph.D. thesis, Harvard University (1982)] on equations defining the Hilbert scheme, and to construct a natural morphism from the toric Hilbert scheme to the toric Chow variety, resolving Problem 6.4 appearing in the paper of \textit{B. Sturmfels} [The geometry of A-graded algebras, preprint, \texttt{http://arXiv.org/abs/math.AG/9410032}]. graded polynomial ring; Hilbert function; Chow morphism M. Haiman - B. Sturmfels, Multigraded Hilbert schemes. J. Algebraic Geom., 13 (4) (2004, pp. 725-769. Zbl1072.14007 MR2073194 Parametrization (Chow and Hilbert schemes), Toric varieties, Newton polyhedra, Okounkov bodies, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Grassmannians, Schubert varieties, flag manifolds Multigraded 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 In two previous papers [in: Geometric and combinatorial aspects of commutative algebra. Papers based on lectures delivered at the international conference on commutative algebra and algebraic geometry, Messina, Italy, June 16--20, 1999. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 217, 21--41 (2001; Zbl 1059.14059); J. Pure Appl. Algebra 165, No. 3, 235--253 (2001; Zbl 1094.13519)], we defined, for every projective, \(0\)-dimensional, reduced scheme \(X\), a set of numerical sequences, which turns out to be a refinement of the Hilbert function of \(X\). Here, we extend that definition to the case of a scheme \(X\) not necessarily reduced; the aim is reached by replacing a point by its corresponding ``separating ideal'' in its coordinate ring. The numerical sequences are obtained by taking the degrees of the elements appearing in suitable sequences of separating ideals. These latter sequences are themselves a good tool in the search for subschemes of \(X\) not in general position. Beccari, G.; Massaza, C.: Separating sequences of 0-dimensional schemes, Matematiche 61, 37-68 (2006) Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Graded rings, Computational aspects in algebraic geometry Separating sequences of \(0\)-dimensional schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We introduce symmetrizing operators of the polynomial ring \(A[x]\) in the variable \(x\) over a ring \(A\). When \(A\) is an algebra over a field \(k\) these operators are used to characterize the monic polynomials \(F(x)\) of degree \(n\) in \(A[x]\) such that \(A\otimes_k k[x]_{(x)}/(F(x))\) is a free \(A\)-module of rank \(n\). We use the characterization to determine the Hilbert scheme parameterizing subschemes of length \(n\) of \(k[x]_{(x)}\). local rings; symmetrizing operators; free quotient algebras; monic polynomials Laksov, D. andSkjelnes, R. M., The Hilbert scheme parameterizing finite length subschemes of the line with support at the origin,Compositio Math. 126 (2001), 323--334. Parametrization (Chow and Hilbert schemes), Singularities of curves, local rings, Regular local rings The Hilbert scheme parameterizing finite length subschemes of the line with support at the origin. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 iterated images of a Jacobian pair \(f: \mathbb{C}^2\to\mathbb{C}^2\) stabilize; that is, all the sets \(f^k(\mathbb{C}^2)\) are equal for \(k\) sufficiently large. More generally, let \(X\) be a closed algebraic subset of CN, and let \(f: X\to X\) be an open polynomial map with \(X- f(X)\) a finite set. We show that the sets \(f^k(X)\) stabilize, and for any cofinite subset \(\Omega\subset X\) with \(f\Omega)\subseteq \Omega\), the sets \(f^k(\Omega)\) stabilize. We apply these results to obtain a new characterization of the two dimensional complex Jacobian conjecture related to questions of surjectivity. stable image; polynomial map; étale; Jacobian conjecture Peretz Nguyen Van Chau, R., Campbell L.A., Gutierrez, C.: Iterated images and the plane Jacobian conjecture. In: Discret and Continuous Dynamical Systems, vol. 16, No. 2, pp. 455--461 (2006) Jacobian problem, Rational and birational maps, Birational automorphisms, Cremona group and generalizations Iterated images and the plane Jacobian conjecture | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 a new recollement for complexes on schemes. Let \(U\) be an open subscheme of a scheme \(X\) and \(Z\) the closed complement of \(U\) in \(X\). Let \(D(X)\) be the derived category of complexes of \(O_X\)-modules with quasi-coherent cohomology and \(D_Z(X)\) the full subcategory consisting of complexes with quasi-coherent cohomology supported on \(Z\). \textit{R.W. Thomason} and \textit{T. Trobaugh}, [Higher algebraic K-theory of schemes and of derived categories. The Grothendieck Festschrift, Vol. III, Prog. Math. 88, 247--435 (1990; Zbl 0731.14001)] described a localization sequence of triangulated categories
\[
D_Z(X) \to D(X) \to D(U).
\]
\textit{A. Neeman} [Ann. Sci. Éc. Norm. Supér. (4) 25, No. 5, 547--566 (1992; Zbl 0868.19001)] showed that this can be generalized replacing \(D(X)\) with a compactly generated triangulated category \(T\) and \(D_Z(X)\) with a subcategory \(K\) generated by a set of compact objects:
\[
K^{\perp} \to T \to K.
\]
The author shows that the localization sequences of Thomason and Trobaugh and of Neeman can be enhanced into a recollement of triangulated categories, in the sense described in [\textit{A. A. Beilinson, J. Bernstein, P. Deligne}, ''Faisceaux pervers,'' Astérisque 100, 172 p. (1982; Zbl 0536.14011)]. Concretely one needs to construct adjoint functors with good properties completing the diagrams above. triangulated categories; derived categories; localization of triangulated categories; recollement of triangulated categories Derived categories, triangulated categories, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] A new recollement for schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems If one starts with a simply laced Dynkin diagram, then a quiver representation has finitely many orbits. Any fixed such orbit determines its equivariant fundamental class, or, its quiver polynomial. These quiver polynomials are universal polynomials representing degeneracy loci. They generalize several important polynomials in algebraic combinatorics (e.g., Giambelli-Thom-Porteous formulas, Schur and Schubert polynomials of Schubert calculus, and the quantum and universal Schubert polynomials). They have several nice structure properties (stability, positivity). The article provides a nonconventional generating description of these polynomials. quivers; quiver varieties; quiver representations, quiver polynomials; Thom polynomials; simply laced Dynkin graphs; degeneracy loci; Schur polynomials Rimányi, R., Quiver polynomials in iterated residue form, J. Algebraic Combin., 40, 2, 527-542, (2014) Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Representations of quivers and partially ordered sets, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Global theory of complex singularities; cohomological properties, Grassmannians, Schubert varieties, flag manifolds Quiver polynomials in iterated residue form | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 fine mixed subdivision of a \((d-1)\)-simplex \(T\) of size \(n\) gives rise to a system of \(d\atopwithdelims ()2\) permutations of \([n]\) on the edges of \(T\), and to a collection of \(n\) unit \((d-1)\)-simplices inside \(T\). Which systems of permutations and which collections of simplices arise in this way? The spread out simplices conjecture of \textit{F. Ardila} and \textit{S. Billey} [Adv. Math. 214, No. 2, 495--524 (2007; Zbl 1194.14078)] proposes an answer to the second question.
We propose and give evidence for an answer to the first question, the acyclic system conjecture. We prove that the system of permutations of \(T\) determines the collection of simplices of \(T\). This establishes the acyclic system conjecture as a first step towards proving the spread out simplices conjecture. We use this approach to prove both conjectures for \(n=3\) in arbitrary dimension. fine mixed subdivisions; triangulations; product of simplices; tropical geometry Ardila, Federico; Ceballos, Cesar: Acyclic systems of permutations and fine mixed subdivisions of simplices, Discrete comput. Geom. 49, No. 3, 485-510 (2013) Combinatorial aspects of simplicial complexes, , Classical problems, Schubert calculus Acyclic systems of permutations and fine mixed subdivisions of simplices | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Fix a field \(k\) and consider the class of all varieties which are smooth over \(k\). If \(W @>\pi>> \text{Spec} (k)\) is within this class, we can attach to it a relative tangent bundle or, equivalently, a locally free sheaf \(\Omega^1_{W/k}\) of relative differentials. This tangent bundle, intrinsic to \(\pi\), plays a central role in algebraic geometry. For instance
(a) in studying the birational class of \(W\),
(b) in analyzing the singular locus of a closed embedded subscheme of \(W\) (e.g. jacobian ideals).
Essential for the development of some of these problems is the fact that the class is closed by blowing up regular centers, namely has the following property:
(P) Let \(W\) be smooth over \(k\), \(C\) a regular closed subscheme of \(W\), and \(W\leftarrow W_1\) the blow-up at \(C\). Then \(W_1\) is also in the class (is also smooth over the field \(k)\).
However, this property fails to hold if we consider now the class of smooth schemes over \(\mathbb{Z}\). In this work we replace \(\text{Spec} (k)\) by a Dedekind scheme of characteristic zero (for instance \(Y=\text{Spec} (\mathbb{Z}))\), and define a class of schemes over \(Y\) such that:
(1) the class includes the smooth schemes over \(Y\),
(2) to any \(W @>\pi>> Y\) in the class there is an intrinsically defined tangent bundle,
(3) the class is closed by blowing up convenient regular centers.
As an application, in \(\S 4\), we analyze the behaviour of jacobian ideals of embedded arithmetic schemes. sheaf of relative differentials; tangent bundle; birational class; singular locus; blowing up; smooth schemes; jacobian ideals of embedded arithmetic schemes Villamayor, O.: On smoothness and blowing ups of arithmetical schemes. Math. Z. 225, 317-332 (1997) Schemes and morphisms, Global theory and resolution of singularities (algebro-geometric aspects), Arithmetic problems in algebraic geometry; Diophantine geometry On smoothness and blowing ups of 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 A curvilinear subscheme of \(\mathbb{P}^r\) is a subscheme of finite length \(l\) of the form \({\mathcal O}_{C,p}/m^l_p\) for some smooth point \(p\) on a reduced curve \(C\). We present an elementary proof of the following theorem:
A general collection of curvilinear subschemes \(S_1, \dots, S_k\) of \(\mathbb{P}^r\) of lengths \(l_1,\dots,l_k\), respectively, impose independent conditions on hypersurfaces of degree \(d\) whenever \(\Sigma_i l_i \leq{d+r\choose r}\). curvilinear subscheme; hypersurfaces C. Ciliberto and R. Miranda: ''Interpolations on curvilinear schemes'', J. Algebra, Vol. 203, (1998), pp. 677--678. Projective techniques in algebraic geometry, Arithmetic ground fields for surfaces or higher-dimensional varieties Interpolation on curvilinear 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 study the stability of coherent systems defined over \(\mathbb{R}\) on genus 0 and genus 1 smooth real projective curves. Vector bundles on curves and their moduli, Topology of real algebraic varieties, Real algebraic and real-analytic geometry Coherent systems on real low genus 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 study relationships between the Nisnevich topology on smooth schemes and certain Grothendieck topologies on proper and not necessarily proper modulus pairs, which were introduced in previous papers. Our results play an important role in the theory of sheaves with transfers on proper modulus pairs. Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects), Motivic cohomology; motivic homotopy theory, Higher symbols, Milnor \(K\)-theory, Symbols and arithmetic (\(K\)-theoretic aspects) Topologies on schemes and modulus pairs | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 define and analyze various generalizations of the punctual Hilbert scheme of the plane, associated to complex or real Lie algebras. Out of these, we construct new geometric structures on surfaces whose moduli spaces share multiple properties with Hitchin components, and which are conjecturally homeomorphic to them. For simple complex Lie algebras, this generalizes the higher complex structure. For real Lie algebras, this should give an alternative description of the Hitchin-Kostant-Rallis section. higher Teichmüller theory; Hitchin components; geometric structures; punctual Hilbert schemes Teichmüller theory for Riemann surfaces, General geometric structures on manifolds (almost complex, almost product structures, etc.), Parametrization (Chow and Hilbert schemes) Generalized punctual Hilbert schemes and \(\mathfrak{g}\)-complex 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 A projective scheme \(X\) is \textit{determinantal} if its homogeneous ideal is the ideal of \(r \times r\) minors of a homogeneous \(p \times q\) matrix, and the codimension of \(X\) is the expected value, namely \(c = (p-r+1) (q-r+1)\). If \(r = \min \{ p,q \}\) then \(X\) is \textit{standard determinantal}. If \(X\) is, in addition, a generic complete intersection then it is called \textit{good determinantal}. The degrees of the entries of the matrix can be determined by a \(p\)-tuple and a \(q\)-tuple of integers \(a_1 \leq \dots \leq a_p\) and \(b_1 \leq \dots \leq b_q\); we denote by \(W(\underline{b},\underline{a})\) the stratum in the appropriate Hilbert scheme corresponding to good determinantal schemes, and by \(W_s(\underline{b}, \underline{a})\) the stratum corresponding to standard determinantal schemes. One of the author's results is that \(W_s(\underline{b}, \underline{a})\) is irreducible, and \(W(\underline{b},\underline{a}) \neq \emptyset\) if and only if \(W_s(\underline{b},\underline{a}) \neq \emptyset\). This paper considers mainly the case of zero-dimensional schemes, and focuses on the following problems: (1) Determine when the closure of \(W(\underline{b}, \underline{a})\) is an irreducible component of the Hilbert scheme; (2) Find the codimension of \(W(\underline{b},\underline{a})\) in the Hilbert scheme if its closure is not a component; (c) Determine when the component of the Hilbert scheme is generically smooth along \(W(\underline{b},\underline{a})\). determinantal scheme; standard determinantal scheme; good determinantal scheme; Hilbert scheme; unobstructed J.O. Kleppe, Families of low dimensional determinantal schemes. J. Pure Appl. Alg., online 9.11.2010, DOI: 101016/j.jpaa.2010.10.007. Determinantal varieties, Parametrization (Chow and Hilbert schemes), Deformations and infinitesimal methods in commutative ring theory, Families, moduli of curves (algebraic), Families, moduli, classification: algebraic theory Families of low dimensional determinantal schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For a smooth projective surface \(S\) and a linear system \(|L|\) let \(N_r\) be the degree of the locus of \(r\)-nodal curves. It is known from the Göttsche conjecture that \(N_r\) is a polynomial of the four relevant numbers of the situation. An earlier paper of the author proves that \(r!N_r=P_r(a_1,\ldots,a_r)\) where \(P_r\) is the \(r\)th complete exponential Bell polynomial and \(a_i\) are universal integer coefficient linear polynomials of the four numbers.
In this paper, the author gives an intersection theoretic interpretation of the \(a_i\) linear forms. In addition the relation of the results with \textit{M. È. Kazaryan}'s [Russ. Math. Surv. 58, No. 4, 665--724 (2003); translation from Usp. Mat. Nauk 58, No. 4, 29--88 (2003; Zbl 1062.58039)] multi-singularity formulas is discussed. Göttsche formulas; nodal curve locus; multisingularity formulas Qviller, N., Structure of node polynomials for curves on surfaces, Math. Nachr., 287, 1394-1420, (2014) Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Partitions of sets Structure of 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 Borel-fixed ideals play a key role in the study of Hilbert schemes. Indeed each component and each intersection of components of a Hilbert scheme contains at least one Borel-fixed point, i.e. a point corresponding to a subscheme defined by a Borel-fixed ideal. Moreover Borel-fixed ideals have good combinatorial properties, which make them very interesting in an algorithmic perspective. In this paper, we propose an implementation of the algorithm computing all the saturated Borel-fixed ideals with number of variables and Hilbert polynomial assigned, introduced from a theoretical point of view in [\textit{F. Cioffi} et al., Discrete Math. 311, No. 20, 2238--2252 (2011; Zbl 1243.14007)]. Hilbert polynomial; Hilbert scheme; Borel-fixed ideals Lella, P., An efficient implementation of the algorithm computing the Borel-fixed points of a Hilbert scheme, (Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation, (2012), ACM), 242-248 Symbolic computation and algebraic computation, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Parametrization (Chow and Hilbert schemes) An efficient implementation of the algorithm computing the Borel-fixed points of a Hilbert scheme | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The Picard scheme of a smooth curve and a smooth complex variety is reduced. In this note we discuss which classes of surfaces in terms of the Enriques-Kodaira classification can have non-reduced Picard schemes and whether there are restrictions on the characteristic of the ground field. It turns out that non-reduced Picard schemes are uncommon in Kodaira dimension \(\kappa \leq 0\), that this phenomenon can be bounded for \(\kappa =2\) (general type) and that it is as bad as can be for \(\kappa =1\). C. Liedtke. A note on non-reduced Picard schemes. J. Pure Appl. Algebra, 213:737--741, 2009. Families, moduli, classification: algebraic theory, Picard schemes, higher Jacobians, Divisors, linear systems, invertible sheaves A note on non-reduced 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 The paper under review studies the Segre classes of subschemes of projective space defined by monomial ideals. The Segre class is shown to be computable by a formula involving the log canonical thresholds of certain extensions of the ideal. The author poses the question to what extent his result might hold for non-monomial schemes.
The proof relies on a result of \textit{J. A. Howald} describing the multiplier ideal of a monomial ideal [Trans. Am. Math. Soc. 353, No. 7, 2665--2671 (2001; Zbl 0979.13026)] and a more recent result of the author [``Segre classes as integrals over polytopes'', to appear in J. Eur. Math. Soc., \url{arXiv:1307.0830}]. log canonical threshold; Segre class; multiplier ideal; Howald's theorem Paolo Aluffi, Log canonical threshold and Segre classes of monomial schemes, Manuscripta Math. 146 (2015), no. 1-2, 1 -- 6. Singularities in algebraic geometry, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Log canonical threshold and Segre classes of monomial schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Given an algorithm for resolution of singularities that satisfies certain conditions (``a good algorithm''), natural notions of simultaneous algorithmic resolution, and of equi-resolution, for families of embedded schemes (parametrized by a reduced scheme \(T\)) are defined. It is proved that these notions are equivalent. Something similar is done for families of sheaves of ideals, where the goal is algorithmic simultaneous principalization. A consequence is that given a family of embedded schemes over a reduced \(T\), this parameter scheme can be naturally expressed as a disjoint union of locally closed sets \(T_j\), such that the induced family on each part \(T_j\) is equi-resolvable. In particular, this can be applied to the Hilbert scheme of a smooth projective variety; in fact, our result shows that, in characteristic zero, the underlying topological space of any Hilbert scheme parametrizing embedded schemes can be naturally stratified in equi-resolvable families. resolution of singularities; algorithmic resolution; simultaneous resolution; Hilbert schemes Encinas, S., Nobile, A. and Villamayor, O.: On algorithmic equi-resolution and stratification of Hilbert schemes. Proc. London Math. Soc. 86 (2003), no. 3, 607-648. Global theory and resolution of singularities (algebro-geometric aspects) On algorithmic equi-resolution and stratification of Hilbert schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\Gamma \subset \mathbb{P}^c_K\) denote a nondegenerate finite subscheme of length \(d\). Related to \(\Gamma\) the authors investigate the following numerical invariants:\begin{itemize} \item[(1)] \(\operatorname{reg} \Gamma\), the Castelnuovo-Mumford regularity. \item[(2)] \(m(\Gamma)\) the maximal degree in a minimal generating set of \(I_{\Gamma}\), the saturated defining ideal of \(\Gamma\). \item[(3)] \(\ell(\Gamma)\) the largest integer \(\ell\) such that \(\Gamma\) admits a proper \(\ell\)-secant line. Then the following inequalities hold \( (\star) : \ell(\Gamma) \leq m(\Gamma) \leq \operatorname{reg}(\Gamma)\).\end{itemize} In their main result, the authors prove that equalities in \((\star)\) hold provided \((\#) : \operatorname{reg} \Gamma \geq \frac{d-c+5}{2}\). In case the last bound \((\#)\) is not fulfilled, then there is no strong relation between \(\operatorname{reg} \Gamma\) and \(\ell(\Gamma)\) as shown by examples. Moreover, if \((\#)\) is satisfied the authors study the Betti numbers of \(\Gamma\). Let \(\mathbb{L}\) denote a \(\operatorname{reg} \Gamma\)-secant line of \(\Gamma\) and let \(X\) denote the scheme-theoretic union \(\Gamma \cap \mathbb{L}\). It is shown that the Betti table of \(\Gamma\) is determined by those of \(X\) and \(\mathbb{L}\). This turns out because \(\operatorname{reg} X\) is strictly smaller than \(\operatorname{reg} \Gamma\). finite schemes; Castelnuovo-Mumford regularity; secant line Projective techniques in algebraic geometry, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Questions of classical algebraic geometry Regularity and multisecant lines of finite 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 the entire collection see Zbl 0635.00006.]
The author studies the moduli problem for basic surfaces of Picard \(number\quad n,\) i.e. for surfaces obtained as ordered blowings-up of \({\mathbb{P}}^ 2\) at n-1 points. He constructs:
(1) a versal family \(P=\{\pi_ n: X_{n+1}\to X_ n\}\), such that any family locally for the étale topology ``comes'' from P by fibre- products.
(2) a scheme \(I_ n\) representing the functor \(Isom_ n\) defined as follows: for any scheme U and any pair of morphisms \(f,g: U\to X_ n,\) \(Isom_ n(U)\) is the set of U-isomorphisms \(U\times_ fX_{n+1}\cong U\times_ gX_{n+1},\)
(3) natural morphisms s,t: \(I_ n\to X_ n.\)
Since in this case the moduli functor is not representable, the existence of such a family is the best result one can hope to find.
Then the author studies in detail the Picard functor and the scheme structure of \(I_ n\). moduli problem for basic surfaces of Picard \(number\quad n,\); Picard functor Harbourne, B.: Iterated blow-ups and moduli for rational surfaces. Lecture notes in math. 1311, 101-117 (1988) Families, moduli, classification: algebraic theory, Rational and birational maps, Picard groups, Rational and unirational varieties Iterated blow-ups and moduli for rational surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Given a real function \(f(X,Y)\), a box region \(B\) and \(\varepsilon>0\), we want to compute an \(\varepsilon\)-isotopic polygonal approximation to the curve \(C: f(X,Y)=0\) within \(B\). We focus on subdivision algorithms because of their adaptive complexity. \textit{S. Plantinga} and \textit{G. Vegter} [``Isotopic approximation of implicit curves and surfaces'', in: Proc. Eurographics. Symposium on Geometry Processing. New York: ACM Press. 245--254 (2004)] gave a numerical subdivision algorithm that is exact when the curve \(C\) is non-singular. They used a computational model that relies only on function evaluation and interval arithmetic. We generalize their algorithm to any (possibly non-simply connected) region \(B\) that does not contain singularities of \(C\). With this generalization as subroutine, we provide a method to detect isolated algebraic singularities and their branching degree. This appears to be the first complete numerical method to treat implicit algebraic curves with isolated singularities.
For Part I, see [\textit{C. K. Yap}, in: Proceedings of the 22nd annual symposium on computational geometry, SCG'06, Sedona, Arizona, USA, June 5--7, 2006. New York, NY: Association for Computing Machinery (ACM). 217--226 (2006; Zbl 1153.65324)]. Numerical aspects of computer graphics, image analysis, and computational geometry, Computer graphics; computational geometry (digital and algorithmic aspects), Computational aspects of algebraic curves, Singularities of curves, local rings Complete subdivision algorithms. II: Isotopic meshing of singular 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 In this paper, the authors study degeneracy loci of maps of vector bundles on smooth ambient spaces, virtual fundamental cycles,
and their applications to nested Hilbert schemes.
Let \(X\) be a smooth complex quasi-projective variety and \(\sigma: E_0 \to E_1\) be a map of vector bundles of ranks \(e_0\) and \(e_1\) over \(X\).
For a positive integer \(r \le e_0\), the \(r\)-th degeneracy locus of \(\sigma\) is defined to be \(D_r(\sigma) = \{x \in X| \dim \ker (\sigma_x) \ge r\}\), i.e., the \((e_1 - e_0 + r - 1)\)-th Fitting scheme of the first cohomology sheaf \(h^1(E_*) =\mathrm{coker}(\sigma)\) of the complex \(E_*\).
The authors prove that the virtual resolution \(\widetilde D_r(\sigma)\) of \(D_r(\sigma)\) admits a perfect obstruction theory which depends only on the quasi-isomorphism class of the complex \(E_*\), and that the resulting virtual fundamental cycle \([\widetilde D_r(E_*)]^{\mathrm{vir}}\), when pushed forward to \(X\), is given by the Thom-Porteous formula.
When \(\widetilde D_r(\sigma)\) can be naturally embedded into the Grassmannian bundle \(\mathrm{Gr}(r,B)\) for some vector bundle
\(B\) over \(X\), the push-forward of \([\widetilde D_r(E_*)]^{\mathrm{vir}}\) to the Chow group \(A_*(\mathrm{Gr}(r,B))\) is also expressed
via the Thom-Porteous formula.
These results are applied to the nested Hilbert schemes \(S_\beta^{[n_1, n_2]} = \{I_1(-D) \subset I_2 \subset \mathcal O_S|[D] = \beta, \mathrm{length}(\mathcal O_S/I_i) = n_i \}\) of points and curves on a smooth projective surface \(S\) where \(\beta \in H^2(S, \mathbb Z)\) and \(n_1\) and \(n_2\) are non-negative integers.
The key observation is that these nested Hilbert schemes is identified with the virtual resolution \(\widetilde D_1(E_*)\) of the degeneracy locus \(D_1(E_*)\) for some \(2\)-term complex \(E_*\) over the product \(S^{[n_1]} \times S^{[n_2]}\) of two Hilbert schemes of points.
The earlier result describes the push-forward of \([S_\beta^{[n_1, n_2]}]^{\mathrm{vir}}\) to \(S^{[n_1]} \times S^{[n_2]} \times \mathbb P(H^0(X, \mathcal O_X(D)))\).
Similar result holds for the reduced class \([S_\beta^{[n_1, n_2]}]^{\mathrm{red}}\) as well. Moreover, a comparison theorem formulates \([S_\beta^{[n_1, n_2]}]^{\mathrm{vir}}\) and \([S_\beta^{[n_1, n_2]}]^{\mathrm{red}}\) in terms of the Carlsson-Okounkov K-theory class \(\mathrm{CO}_\beta^{[n_1, n_2]}\) on \(S^{[n_1]} \times S^{[n_2]} \times S_\beta\) where \(S_\beta\) denotes \(S_\beta^{[0, 0]}\). It follows that many integrals from the Vafa-Witten invariants of \(S\) can be evaluated in terms of the Seiberg-Witten invariants of \(S\) and
certain tautological virtual bundles over \(S^{[n_1]} \times S^{[n_2]} \times\mathrm{Pic}_\beta(S)\). Hilbert scheme; degeneracy locus; Thom-Porteous formula; local Donaldson-Thomas theory; Vafa-Witten invariants Algebraic moduli problems, moduli of vector bundles, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Parametrization (Chow and Hilbert schemes) Degeneracy loci, virtual cycles and nested Hilbert schemes. II | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \( E_{/\mathbb{Q}}\) be an elliptic curve of conductor \(N\) and let \(f\) be the weight 2 newform on \(\Gamma_0(N)\) associated to it by modularity. Building on an idea of S.~Zhang, an article by Darmon, Rotger, and Sols [\textit{H. Darmon} et al., Publ. Math. Besançon. Algèbre et Théorie des Nombres 2012/2, 19-46 (2012; Zbl 1332.11054)] describes the construction of so-called \textit{Chow-Heegner points}, \(P_{T,f}\in E({\bar {\mathbb{Q}}})\), indexed by algebraic correspondences \(T\subset X_0(N)\times X_0(N)\). It also gives an analytic formula, depending only on the image of \(T\) in cohomology under the complex cycle class map, for calculating \(P_{T,f}\) numerically via Chen's theory of iterated integrals. The present work describes an algorithm based on this formula for computing the Chow-Heegner points to arbitrarily high complex accuracy, carries out the computation for all elliptic curves of rank 1 and conductor \(N< 100\) when the cycles \(T\) arise from Hecke correspondences, and discusses several important variants of the basic construction. Deninger, C., Scholl, A.: The Beilinson conjectures. In: ''\(L\)-functions and arithmetic.'' (Durham, 1989), pp. 173-209. London Math. Soc. Lecture Note Ser., vol. 153. Cambridge Univ. Press, Cambridge (1991) Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols, Elliptic curves over global fields, Number-theoretic algorithms; complexity, (Equivariant) Chow groups and rings; motives Algorithms for Chow-Heegner points via iterated integrals | 0 |
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