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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 Based on Pieri's formula on Schubert varieties [see \textit{F. Sottile}, Can. J. Math. 49, 1281-1298 (1997; Zbl 0933.14031)], the Pieri homotopy algorithm was first proposed by \textit{B. Huber, F. Sottile}, and \textit{B. Sturmfels} [J. Symb. Comput. 26, 767-788 (1998; Zbl 1064.14508)]\ for numerical Schubert calculus to enumerate all \(p\)-planes in \({\mathbb C}^{m+p}\) that meet \(n\) given planes in general position. The algorithm has been improved by \textit{B. Huber} and \textit{J. Verschelde} [SIAM J. Control Optim. 38, 1265-1287 (2000; Zbl 0955.14038)]\ to be more intuitive and more suitable for computer implementations.
A different approach of employing the Pieri homotopy algorithm for numerical Schubert calculus is presented in this paper. A major advantage of our method is that the polynomial equations in the process are all square systems admitting the same number of equations and unknowns. Moreover, the degree of each polynomial equation is always 2, which warrants much better numerical stability when the solutions are being solved. Numerical results for a big variety of examples illustrate that a considerable advance in speed as well as much smaller storage requirements have been achieved by the resulting algorithm. enumerative geometry; Schubert variety; Pieri formula; Pieri homotopy algorithm; Pieri poset; algorithm Li D, Qi L, Zhou S (2002) Descent directions of quasi-Newton methods for symmetric nonlinear equations. SIAM J Numer Anal 40(5): 1763--1774 Grassmannians, Schubert varieties, flag manifolds, Computational aspects of higher-dimensional varieties, Numerical computation of solutions to systems of equations, Symbolic computation and algebraic computation, Enumerative problems (combinatorial problems) in algebraic geometry Numerical Schubert calculus by the Pieri homotopy algorithm | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Sublinear circuits are generalizations of the affine circuits in matroid theory, and they arise as the convex-combinatorial core underlying constrained non-negativity certificates of exponential sums and of polynomials based on the arithmetic-geometric inequality. Here, we study the polyhedral combinatorics of sublinear circuits for polyhedral constraint sets. We give results on the relation between the sublinear circuits and their supports and provide necessary as well as sufficient criteria for sublinear circuits. Based on these characterizations, we provide some explicit results and enumerations for two prominent polyhedral cases, namely the non-negative orthant and the cube \([- 1,1]^n\). positive function; sublinear circuit; sums of arithmetic-geometric exponentials; non-negativity certificate; polyhedron Combinatorial aspects of matroids and geometric lattices, Real algebraic sets, Convex sets in \(n\) dimensions (including convex hypersurfaces), Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.), Nonlinear programming, Semidefinite programming, Convex programming Sublinear circuits for polyhedral sets | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper considers multipliers of periodic orbits for complex polynomial dynamical systems in one variable, addressing in particular the behavior of such multipliers under perturbation of a polynomial map. The main results show that the multipliers of a set of distinct periodic orbits of a polynomial map on \(\mathbb{C}\) will in general vary independently if the coefficients of the polynomial are perturbed.
Given a monic polynomial \(g(z)\) of degree \(n\) and points \(\beta_1,\dots,\beta_l\) in \(\mathbb{C}\) with exact periods \(r_1,\dots,r_l\) under \(g(z)\), the behavior of the orbit of each \(\beta_i\) is understood via the map \(\text{Mult}_{\beta_i}: f(z) \mapsto (f^{\circ r_i})'(\beta_i(f))\) whose domain is a neighborhood (in the space of monic polynomials of degree \(n\)) of \(g(z)\) defined so that each \(\beta_i\) can be viewed as a holomorphic function on the neighborhood such that \(\beta_i(f)\) has exact period \(r_i\) under \(f(z)\). So the multipliers of the orbits of \(\beta_1,\dots,\beta_l\) vary independently at \(g(z)\) if the gradients of \(\text{Mult}_{\beta_1},\dots,\text{Mult}_{\beta_l}\) at \(g(z)\) are linearly independent in \(\mathbb{C}^n\). The paper concludes that independence is achieved for a generic choice of \(g(z)\) if \(n \geq 3\), \(r_1,\dots,r_l < n\) (with equality allowed if no \(\beta_i\) is a fixed point), and the orbits of \(\beta_1,\dots,\beta_l\) are pairwise disjoint. The conclusion follows from computations of the gradients of \(\text{Mult}_{\beta_1},\dots,\text{Mult}_{\beta_l}\) which show that maps of the form \(z\mapsto z^n\) achieve independence under the given assumptions, combined with an argument that the set of monic polynomial maps achieving linear independence must be Zariski open. complex polynomial dynamics in one variable; multipliers of periodic points DOI: 10.1070/IM2013v077n04ABEH002657 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets, Polynomials and rational functions of one complex variable, Elementary questions in algebraic geometry One-dimensional polynomial maps, periodic points and multipliers | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a quasi-compact separated scheme, and let \(X = \bigcup X_\alpha\) be its open covering. Suppose that a closed subscheme \(Z\) is proregular embedded in \(X.\) Thus, the defining ideal of \(Z\) is generated by a proregular sequence of sections from \(\bigcup \Gamma(X_\alpha, {\mathcal O}_{X_\alpha})\) [see \textit{J. P. C. Greenless} and \textit{J. P. May}, J. Algebra 149, No. 2, 438-453 (1992; Zbl 0774.18007)]. In the case where \(X\) is noetherian any closed subscheme is proregular embedded in \(X.\) For the pair \((X,Z)\) the authors describe a universal functorial duality expressed in terms of the right and left derived of the homomorphism and completion functors, respectively, which gives a sort of adjointness between the local cohomology and local homology supported in \(Z.\) In fact, using these results the authors generalize GM-duality (loc.cit.), the Peskine-Szpiro duality sequence [\textit{C. Peskine} and \textit{L. Szpiro}, Publ. Math., Inst. Hautes Étud. Sci. 42 (1972), 47-119 (1973; Zbl 0268.13008)], affine and formal duality theorems of Hartshorne [\textit{R. Hartshorne}, ``Residues and duality'', Lect. Notes Math. 20 (1996; Zbl 0212.26101)], and others. quasi-compact scheme; formal scheme; Koszul complex; proregular sequence; local cohomology; local homology; adjointness; derived functor; Bousfield localization; Matlis duality; Grothendieck duality; Warwick duality theorem; Peskine-Szpiro duality; functorial duality Tarrío, L. A.; López, A. J.; Lipman, J., Local homology and cohomology on schemes, Ann. Sci. Éc. Norm. Supér. (4), 30, 1, 1-39, (1997) Local cohomology and algebraic geometry, Duality theorems for analytic spaces, Formal neighborhoods in algebraic geometry, Étale and other Grothendieck topologies and (co)homologies, Derived categories, triangulated categories Local homology and cohomology on 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 article presents a theory of equisingularity for families of \(0\)-dimensional sheaves of ideals on smooth algebraic surfaces in an arithmetic context. The value of this work is precisely this: It is pioneering in setting the formalisms in order that we may speak of equisingularity in the arithmetic case. It deals with families of \(0\)-dimensional schemes on regular surfaces, which is an interesting case to start with, since they appear both in the geometric theory of Enriques with the notion of proximity, and in the theory of Zariski of complete ideals in a \(2\)-dimensional regular local ring. As the authors say in the introduction, some of the results in the paper have been used by \textit{A. Nobile} and \textit{O. E. Villamayor} [Commun. Algebra 26, 2669-2688 (1998; Zbl 0938.14001)] to develop a theory analogous to Zariski's one for sheaves of ideals on an arithmetic \(3\)-fold. More precisely, the authors consider a Dedekind scheme \(T\) with the condition that, for all \(t \in T\), the residue field \(k(t)\) is a perfect field, and a smooth morphism \(\pi: X \rightarrow T\), where \(X\) is a \(3\)-dimensional scheme. By an arithmetic family of \(0\)-dimensional ideals on the fibers of \(X\) they mean a \(1\)-dimensional coherent sheaf of \({\mathcal O}_X\)-ideals \(I\) such that \(\pi\) induces a flat morphism \(V(I) \rightarrow T\), where \(V(I)\) is the subscheme defined by \(\sqrt I\). Then, at each point \(t \in T\), a geometric fiber of \(\pi\) at \(t\) is a smooth surface, and \(I\) induces a \(0\)-dimensional sheaf of ideals on it. The authors follow the notions appearing in a paper by \textit{J. J. Risler} [Bull. Soc. Math. Fr. 101, 3-16 (1973; Zbl 0256.14006)] in the context of local analytic geometry. The first condition they introduce, called (a) in the paper and generalizing (a) in the paper by Risler cited above, is the following:
The morphism \(V(I) \rightarrow T\) is étale, if we consider the blowing up \(X_1 \rightarrow X\) at \(V(I)\) and the proper transform \(I_1\) of \(I\), then \(V(I_1) \rightarrow T\) is étale, and so on.
The authors prove that this is equivalent to say that, for all geometric points \(\overline t\), the ideals \(I_{\overline t}\) have the same associated forests with the same weights given by the orders of the propers transforms (this is called condition (b)).
A third approach is to add the proximity relations to these weighted forests (called then condition (c)) and to relate it to the equisingularity of \(T\)-curves \(C\) belonging to \(I\): To have equivalence the curves need to be ``general enough''. equisingularity; schemes over Dedekind rings; \(0\)-dimensional ideals; smooth surfaces; arithmetic \(3\)-fold [NV2]---, Arithmetic families of smooth surfaces and equisingularity of embedded schemes.Manuscripta Math., 100 (1999), 173--196. Global theory and resolution of singularities (algebro-geometric aspects), Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Dedekind, Prüfer, Krull and Mori rings and their generalizations Arithmetic families of smooth surfaces and equisingularity of embedded 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 provide a framework for working with Gröbner bases over arbitrary rings \(k\) with a prescribed finite standard set \(\Delta\). We show that the functor associating to a \(k\)-algebra \(B\) the set of all reduced Gröbner bases with standard set \(\Delta\) is representable and that the representing scheme is a locally closed stratum in the Hilbert scheme of points. We cover the Hilbert scheme of points by open affine subschemes which represent the functor associating to a \(k\)-algebra \(B\) the set of all border bases with standard set \(\Delta\) and give reasonably small sets of equations defining these schemes. We show that the schemes parametrizing Gröbner bases are connected; give a connectedness criterion for the schemes parametrizing border bases; and prove that the decomposition of the Hilbert scheme of points into the locally closed strata parametrizing Gröbner bases is not a stratification. Hilbert scheme of points; Gröbner bases; standard sets; locally closed strata Mathias Lederer, Gröbner strata in the Hilbert scheme of points , J. Comm. Alg. 3 (2011), 349-404. 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) Gröbner strata in the Hilbert scheme of points | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The set of multisecant spaces to a given projective variety is a classic subject of study (especially in the case of curves), and also in recent times it has been considered and worked upon in many papers for different reasons (enumerative geometry point of view, study of special divisors).
This paper deals with the problem of determining the connectedness of this set, and the main result gives numerical conditions on \(a,e,t,n\) in order to have that the set of \(e\)-secant \(a\)-planes to a curve \(C\) in \(\mathbb{P}^n\) is connected when it is at most \(t\)-dimensional. From this the author also proves the following: Let \(C \subseteq \mathbb{P}^3_k\), \(k = \overline k\), be a smooth connected algebraic curve, with degree \(d\) and genus \(g\); let \(S\) be the scheme of its trisecant lines. Then in any of the following cases \(S\) is connected:
(1) \(h^1 (C, {\mathcal O}_C(1)) = 0\);
(2) \(\text{char} (k) = 0\), \(d \geq g + 4\), and \(g \geq 8\) if \(d = g + 4\);
(3) \(\text{char} (k) = 0\), \(d = g + 3\), \(g \geq 14\) and \(C\) is a normal point of \(\text{Hilb}_{d,g} (\mathbb{P}^3)\).
The main ingredients in the proof of these results are the Schwartzenberger's rk\(e\)-vector bundle on the \(e\)-th symmetric product of \(C\), the connectedness theorem of \textit{W. Fulton} and \textit{R. Lazarsfeld} [Acta Math. 146, 271-283 (1981; Zbl 0469.14018)], and some recent work by \textit{F. Laytimi} [Math. Ann. 294, No. 3, 459-462 (1992; Zbl 0757.14019)]. connectedness of multisecant spaces E. Ballico, ''On the connectedness of the scheme of multisecants to a projective curve,''Geometriae Dedicata,53, 327--332 (1994). Plane and space curves, Connected and locally connected spaces (general aspects), Topological properties in algebraic geometry, Enumerative problems (combinatorial problems) in algebraic geometry On the connectedness of the scheme of multisecants to a projective curve | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper, we study length categories using iterated extensions. We fix a field \(k\), and for any family \(\mathsf{S}\) of orthogonal \(k\)-rational points in an Abelian \(k\)-category \(\mathcal{A} \), we consider the category \(\mathbf{Ext}(\mathsf{S})\) of iterated extensions of \(\mathsf{S}\) in \(\mathcal{A} \), equipped with the natural forgetful functor \(\mathbf{Ext}(\mathsf{S}) \to \mathcal{A}(\mathsf{S})\) into the length category \(\mathcal{A}(\mathsf{S})\). There is a necessary and sufficient condition for a length category to be uniserial, due to Gabriel, expressed in terms of the Gabriel quiver (or Ext-quiver) of the length category. Using Gabriel's criterion, we give a complete classification of the indecomposable objects in \(\mathcal{A}(\mathsf{S})\) when it is a uniserial length category. In particular, we prove that there is an obstruction for a path in the Gabriel quiver to give rise to an indecomposable object. The obstruction vanishes in the hereditary case, and can in general be expressed using matric Massey products. We discuss the close connection between this obstruction, and the noncommutative deformations of the family \(\mathsf{S}\) in \(\mathcal{A}\). As an application, we classify all graded holonomic \(D\)-modules on a monomial curve over the complex numbers, obtaining the most explicit results over the affine line, when \(D\) is the first Weyl algebra. We also give a non-hereditary example, where we compute the obstructions and show that they do not vanish. finite length categories; uniserial categories; iterated extensions; noncommutative deformations Abelian categories, Grothendieck categories, Representation theory of associative rings and algebras, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials Iterated extensions and uniserial length categories | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems It is known that the variety parametrizing pairs of commuting nilpotent matrices is irreducible and that this provides a proof of the irreducibility of the punctual Hilbert scheme in the plane. We extend this link to the nilpotent commuting variety of some parabolic subalgebras of \(M_{n}(\Bbbk)\) and to the punctual nested Hilbert scheme. By this method, we obtain a lower bound on the dimension of these moduli spaces. We characterize the cases where they are irreducible. In some reducible cases, we describe the irreducible components and their dimensions. Hilbert scheme; commuting variety; GIT; parabolic algebra; nilpotent orbit [4] Michael Bulois &aLaurent Evain, &Nested punctual Hilbert schemes and commuting varieties of parabolic subalgebras&#xhttp://arxiv.org/abs/1306.4838 Parametrization (Chow and Hilbert schemes), Group actions on varieties or schemes (quotients), Geometric invariant theory, Coadjoint orbits; nilpotent varieties Nested punctual Hilbert schemes and commuting varieties of parabolic subalgebras | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 there exists a scheme which represents the functor of line modules over a graded algebra, and call it the `line scheme' of the algebra. We study its properties and its relationship to the point scheme. If the line scheme of a quadratic, Auslander-regular algebra of global dimension four has dimension one, then it determines the defining relations of the algebra.
Moreover, we prove the following counter-intuitive result: if the zero locus of the defining relations of a quadratic (not necessarily regular) algebra on four generators with six defining relations is finite, then it determines the defining relations of the algebra. Although this result is non-commutative in nature, its proof uses only commutative theory.
We also use the structure of the line scheme and the point scheme of a four-dimensional regular algebra to determine basic incidence relations between line modules and point modules. regular algebras; quadratic algebras; linear modules; line modules; point schemes; line schemes Shelton, Brad; Vancliff, Michaela, Schemes of line modules. I, J. London Math. Soc. (2), 65, 3, 575-590, (2002) Rings arising from noncommutative algebraic geometry, Noncommutative algebraic geometry, Graded rings and modules (associative rings and algebras), Quadratic and Koszul algebras Schemes of line modules. I | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In the famous article [Compos. Math. 110, No. 1, 65--126 (1998; Zbl 0894.18005)] \textit{E. Getzler} and \textit{M. Kapranov} studied operadic type structures related to the moduli space of algebraic curve. Intuitively, algebraic curves with marked points can be glued along the marked points generating operations on the moduli spaces. However, when considering arbitrary genuses of curves, the classical operadic picture, in which operations are labeled by trees, is replaced by operation labeled instead by graphs. They call this operadic structure modular. Moreover, moduli of curves with marked points do have typically (for instance when considering genus \(0\) curves) an extra cyclic symmetry obtained by permuting the punctures. In the same paper, they show how to construct a Feynman transform on the category of dg-modular operads and how to compute its Euler characteristic in terms of the Wick's theorem, hence highlighting the relation of this operad with mathematical physics.
In this paper the authors present a generalization of these results for curves with marked points \(k-\log\) canonically embedded, meaning admitting a projective embedding by a complete linear system. The study of log canonical models for curves has been central in the study of moduli spaces of curves and for its relationships to the Minimal Model program. There are three results presented: first, they show that for \(k\geq 5\) the moduli spaces of \(k-\log\) canonically embedded curves assemble together in a modular operad in Deligne-Mumford stacks. Second, they show that for \(k\geq 1\) the moduli spaces of \(k-\log\) canonically embedded curves of genus \(0\) assemble together in a cyclic operad in schemes. Third, they show that for \(k\geq 2\) the moduli spaces of \(k-\log\) canonically embedded curves assemble together in a stable cyclic operad in Deligne-Mumford stacks. In order to prove these results, they construct morphisms on these moduli spaces corresponding to the gluing of two embedded curves and to the gluing of two points together on the same embedded curve.
The proofs of these statements appear correct. Would be interesting, as a follow up work, to understand weather the construction of Getzler and Kapranov of the Feynman transform could be generalized to this setting. modular operad; log-canonical Hilbert scheme Families, moduli of curves (algebraic), , Parametrization (Chow and Hilbert schemes) Modular operads of embedded 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 prove that the generating series of the Poincaré polynomials of quasihomogeneous Hilbert schemes of points in the plane has a beautiful decomposition into an infinite product. We also compute the generating series of the numbers of quasihomogeneous components in a moduli space of sheaves on the projective plane. The answer is given in terms of characters of the affine Lie algebra \(\widehat{sl}_m\). Parametrization (Chow and Hilbert schemes), Combinatorial aspects of representation theory, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Algebraic moduli problems, moduli of vector bundles Generating series of the Poincaré polynomials of quasihomogeneous 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 \(C\) be a curve in \(\mathbb P^3\) over an algebraically closed field \(k\) and let \(R = k[x_0,x_1,x_2,x_3]\). Denote by \(I(C)\) its homogeneous ideal, and by \(\mathcal I_C\) its ideal sheaf. The \textit{Rao module} (sometimes \textit{Hartshorne-Rao module} or \textit{deficiency module}) of \(C\) is the graded \(R\)-module \(M = H_*^1(\mathcal I_C) := \bigoplus_{t \in \mathbb Z} H^1(\mathbb P^3, \mathcal I_C (t))\). The curve \(C\) is said to be \textit{Buchsbaum} if \(M\) is annihilated by the irrelevant ideal of \(R\). This is true in particular when \(M\) has only one non-zero component, and such curves are the central object of study of this paper. Assume from now on that \(C\) is such a curve. We consider all the components, \(V\), of the Hilbert scheme \(H(d,g)\) that contain \(C\). The author determines \(V\) from the point of view of describing the graded Betti numbers of the generic curve of \(V\) in terms of the graded Betti numbers of \(C\). This deep and careful study of the behavior of the Betti numbers of curves in these families, combined with earlier work of the author, gives us a good understanding of the Hilbert scheme of \(C\), and its singular locus. Hilbert scheme; space curve; Buchsbaum curve; graded Betti numbers; ghost term; linkage Kleppe, J.O., The Hilbert scheme of Buchsbaum space curves, Ann. inst. Fourier, 62, 6, 2099-2130, (2012) Parametrization (Chow and Hilbert schemes), Plane and space curves, Linkage, Syzygies, resolutions, complexes and commutative rings, Linkage, complete intersections and determinantal ideals The Hilbert scheme of Buchsbaum space curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A locally recoverable code is a code over a finite alphabet such that the value of any single coordinate of a codeword can be recovered from the values of a small subset of other coordinates. Building on work of \textit{A. Barg} et al. [IEEE Trans. Inf. Theory 63, No. 8, 4928--4939 (2017; Zbl 1372.94480)], we present several constructions of locally recoverable codes from algebraic curves and surfaces. locally recoverable code Geometric methods (including applications of algebraic geometry) applied to coding theory, Applications to coding theory and cryptography of arithmetic geometry Locally recoverable codes from algebraic curves and surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a smooth quasi-projective surface. It is well known by work of \textit{J. Fogarty} that the Hilbert scheme \(\mathrm{Hilb}^n (X)\) parametrizing closed subschemes of length \(n\) is a smooth irreducible variety of dimension \(2n\) [Amer. J. Math. 90, 511--521 (1968; Zbl 0176.18401)]. The corresponding universal family \(Z^n \subset \mathrm{Hilb}^n (X) \times X\) is smooth for \(n=2\) (it is isomorphic to the blow-up of \(X \times X\) along the diagonal), but singular for \(n>2\). \textit{J. Fogarty} showed that \(Z^n\) is irreducible, normal, Cohen-Macaulay and satisfies Serre's condition \(R_3\) [Am. J. Math. 95, 660--687 (1973; Zbl 0299.14020)]
In this note the author gives more detailed information about the local structure of \(Z^n\), proving that its singularities are rational but not \(\mathbb Q\)-Gorenstein. He also proves that at a closed point \(\zeta = (\xi,p) \in Z^n\), the Samuel multiplicity \(\mu\) is given by \(\binom{b_2+1}{2}\), where \(b_2\) is the dimension of the socle of the local ring \({\mathcal O}_{\xi,p}\). He also proves a sharp upper bound on \(b_2\), namely \(\displaystyle b_2 \leq \lfloor \frac{\sqrt{1+8n}-1}{2} \rfloor\). This implies that the Samuel multiplicity satisfies \(\mu \leq n\), corroborating a result of \textit{M. Haiman} [J. Amer. Math. Soc. 14, No. 4, 841--1006 (2001; Zbl 1009.14001)]. Hilbert scheme of points on a surface; universal family; rational singularities; Samuel multiplicity Parametrization (Chow and Hilbert schemes) On the universal family of Hilbert schemes of points on a surface | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This note is an announcement of two results about ideals of fat points in the plane. The first one is about the dimension of linear systems of plane curves with given multiplicities at a set of generic points \(P_1,\ldots , P_s\) in \({\mathbb P}^2\). Let \({\mathcal L}(t,m_1,\ldots ,m_s)\) be the linear system of curves of degree \(t\) with multiplicity at least \(m_i\) at each \(P_i\).
Theorem A in the paper states that, if \(t\geq m_1+m_2\), \(m_1\geq m_2 \geq \ldots \geq m_s \geq 1\) and the expected dimension of the system is such that:
\[
{t+2 \choose 2}- \sum_{i=1}^s {m_i+1 \choose 2} \geq {1\over 2}(sm_4^2-7m_4+2),
\]
then the system is non-special (i.e. its dimension is the expected one).
Theorem B states that for \(P_1,\ldots , P_s\) as above, and \(I= I(P_1)^{m_1}\cap \ldots \cap I(P_s)^{m_s}\), with \(s\geq 9\), we have that if either
1) at least \(s-1\) of the \(m_i\)'s are equal,
or
2) \(m_1\geq m_2 \geq \ldots \geq m_s \geq {m_1 \over 2}\) ,
then \(I^{(2r)} \subset M^rI^r\), where \(M=(x_0,x_1,x_2)\) is the irrelevant ideal.
The two theorems are related to two conjectures; the first (which goes back to B. Segre in 1961) states that \({\mathcal L}(t,m_1,\ldots ,m_s)\) always has the expected dimension unless its base locus contains a \((-1)\)-curve; the second (by Harbourne and Huneke) states that any ideal made as \(I\) in Theorem B (in every \({\mathbb P}^n\)) is such that \(I^{(nr)} \subset M^{r(n-1)}I^r\), \(\forall r\geq 1\).
The proofs of the theorems can be found in [\textit{M. Dumnicki, T. Szemberg} and \textit{H. Tutaj-Gasińska},``A vanishing theorem and symbolic powers of planar point ideals'', \url{arXiv:1302.0871}]. fat points; symbolic powers; postulation Divisors, linear systems, invertible sheaves, Vanishing theorems in algebraic geometry New results on fat points schemes in \(\mathbb{P}^2\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A famous integrable case of the rigid body motion around a fixed point is the Kowalevski top. There is a large literature dedicated to understanding Kowalevski's original integration procedure. In a recent paper by one of the authors, a new approach to the Kowalevski integration procedure has been suggested. Its novelty lies in the introduction of the new notion of discriminantly separable polynomials. Here, it is shown that by using discriminantly separable polynomials of degree two in each of three base variables of the problem, it is possible to construct a class of integrable dynamical systems so that all main steps of the Kowalevski's integration procedure follow as easy and transparent logical consequences. Some new examples are discussed. motion of a rigid body with a fixed point; Kowalevski top; integrable cases Dragović, V; Kukić, K, Systems of the Kowalevski type and discriminantly separable polynomials, Regul. Chaotic Dyn., 19, 162-184, (2014) Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Motion of a rigid body with a fixed point, Integrable cases of motion in rigid body dynamics, Relationships between algebraic curves and integrable systems Systems of Kowalevski type and discriminantly separable polynomials | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We study the role of the mirabolic subgroup \(P\) of \(G=\mathrm{GL}_n(F)\) (\(F\) a \(p\)-adic field) for smooth irreducible representations of \(G\) that are distinguished relative to a subgroup of the form \(H_{k} = \mathrm{GL}_k(F)\times \mathrm{GL}_{n-k}(F)\). We show that if a non-zero \(H_1\)-invariant linear form exists on a representation, then the a priori larger space of \(P\cap H_1\)-invariant forms is one-dimensional. When \(k>1\), we give a reduction of the same problem to a question about invariant distributions on the nilpotent cone tangent to the symmetric space \(G/H_k\). Some new distributional methods for non-reductive groups are developed. distinguished representations; \(p\)-adic symmetric spaces; mirabolic subgroup; invariant distributions Representations of Lie and linear algebraic groups over local fields, Analysis on \(p\)-adic Lie groups, Local ground fields in algebraic geometry, Other algebraic groups (geometric aspects) A distributional treatment of relative mirabolic multiplicity 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 Let \(k\) be a field of characteristic zero. Let \(f\in k[x,y]\) be an irreducible polynomial. In this chapter, we link the differential operators of \(k[x,y]\) that appear in the Bernstein functional equation for \(f\) to the nilpotent elements of the \(k\)-algebra of the arc scheme associated with the affine plane curve defined by the datum of \(f\). Arcs and motivic integration, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Singularities of curves, local rings Arc scheme and Bernstein 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 consider some cases of equivariant Hilbert schemes for cyclic group actions on the plane. The main theorem of the paper establishes generating series for some of these.
The authors include a discussion of the equivariant Hilbert scheme more generally. Furthermore, beyond the cases where generating series are established by the authors' main theorem, they also include a conjecture in one other case. The paper is economical and contains some useful examples. Hilbert schemes of points; equivariant Hilbert schemes; generating series Gusein-Zade, SM; Luengo, I; Melle-Hernández, A, On generating series of classes of equivariant Hilbert schemes of fat points, Mosc. Math. J., 10, 593-602, (2010) Parametrization (Chow and Hilbert schemes), Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) On generating series of classes of equivariant Hilbert schemes of fat points | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems More than 50 years ago \textit{D. Mumford} gave an example of a generically non-reduced irreducible component of the Hilbert scheme of smooth irreducible space curves [Am. J. Math. 84, 642--648 (1962; Zbl 0114.13106)]. Expanding on Mumford's example, \textit{J. O. Kleppe} used Hilbert flag schemes to show that if a general curve \(C\) in a Hilbert scheme component \(V\) lies on a smooth cubic surface \(X\) and \(H^1({\mathcal O}_X (-C)(3)) \neq 0\), then \(V\) is generically non-reduced [Lect. Notes Math. 1266, 181--207 (1985; Zbl 0631.14022)]. Recently \textit{J. O. Kleppe} and \textit{J. C. Ottem} extended these ideas to produce examples in which the general curve \(C\) lies on a quartic surface [Int. J. Math. 26, Article ID 1550017, 30 p. (2015; Zbl 1323.14005)], but their method fails for curves lying on smooth surfaces \(X\) of degree \(d \geq 5\) or having Picard number \(\rho (X) > 2\).
Here the author combines Hilbert flag scheme methods and the theory of Hodge loci to construct more examples of non-reduced Hilbert schemes whose general curve \(C\) lies on a smooth surface \(X\) of degree \(d \geq 5\) (and on no quartic) satisfying \(\rho (X) > 2\). He starts with a smooth surface \(X\) of degree \(d \geq 5\) containing two coplanar lines \(L_1\) and \(L_2\), noting that \(\rho (X) > 2\). The Hilbert scheme of extremal curves of the form \(D=2L_1 + L_2 \subset X\) is generically non-reduced by work of \textit{M. Martin-Deschamps} and \textit{D. Perrin} [Ann. Scient. École Norm. Sup. 29, 757--785 (1996; Zbl 0892.14005)]. Letting \(\gamma \in H^{1,1}(X, \mathbb Z)\) denote the cohomology class of of \(D\), the Zariski closure of the associated Hodge locus \(\text{NL}(\gamma)\) in the open set \(U \subset |{\mathcal O}_{\mathbb P^3} (d)|\) of smooth surfaces is irreducible and generically non-reduced. With arguments involving exact sequences and Mumford regularity, he shows that the general member \(C \in |{\mathcal O}_X (D) (m)|\) is a smooth connected curve for \(m \geq 2d-2\). Letting \(X\) and \(C\) vary and taking the closure in the Hilbert scheme gives an irreducible component which is generically non-reduced. The arguments are clearly written and easy to follow. Hilbert scheme of space curves; Hilbert flag scheme; Hodge locus; non-reduced components Parametrization (Chow and Hilbert schemes), Transcendental methods of algebraic geometry (complex-analytic aspects), Variation of Hodge structures (algebro-geometric aspects) On generically non-reduced components of Hilbert schemes of smooth curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X \to Y\) be a \(Y\)-scheme; a proper \(Y\)-scheme \(Z@>p>>Y\) is called a compactification of the \(Y\)-scheme \(X\) if there exist an open dense immersion \(X@>i>>Z\) over \(Y\). -- \textit{M. Nagata} proved earlier (1962) that if \(Y\) is a noetherian scheme and \(X\to Y\) is separated and of finite type, then it admits a \(Y\)-compactification. In this article, this result is proved by alternative methods. In the first part of the paper are given some results closed to blowing-ups and compactification problem on extensions of morphisms. The result of Nagata is a consequence of these preliminary results. The paper also contains many results useful in birational geometry. open immersion; blowing-ups; compactification; extensions of morphisms; birational geometry Lütkebohmert, W., On compactification of schemes, Manuscripta Math., 80, 95-111, (1993) Rational and birational maps, Global theory and resolution of singularities (algebro-geometric aspects), Schemes and morphisms On compactification 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 \(Z \subset \mathbb{P}^2\) be a zero-dimensional scheme. Fix \(t \in \mathbb N\). In this paper we study the following question: find assumptions on \(Z\) and \(t\) such that \(h^1 (\mathcal{I}_A (t)) < h^1 (\mathcal{I}_Z (t))\) for all \(A\subsetneq Z\) and check if \(t\) does not exist for a certain class of schemes \(Z\). zero-dimensional scheme; plane curve; Hilbert function Projective techniques in algebraic geometry Zero-dimensional schemes in the plane | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We study multiview moduli problems that arise in computer vision. We show that these moduli spaces are always smooth and irreducible, in both the calibrated and uncalibrated cases, for any number of views. We also show that these moduli spaces always admit open immersions into Hilbert schemes for more than two views, extending and refining work of \textit{C. Aholt} et al. [Can. J. Math. 65, No. 5, 961--988 (2013; Zbl 1284.13035)]. We use these moduli spaces to study and extend the classical twisted pair covering of the essential variety. multiview geometry; Hilbert scheme; computer vision; moduli theory; deformation theory Stacks and moduli problems, Computational aspects of higher-dimensional varieties, Parametrization (Chow and Hilbert schemes), Infinitesimal methods in algebraic geometry, Machine vision and scene understanding Two Hilbert schemes in computer vision | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(Y\) be a compact complex space of general type and \(X\) a Moishezon space. Then \textit{S. Kobayashi} and \textit{T. Ochiai} [Invent. Math. 31, 7--16 (1975; Zbl 0331.32020)] have shown that the set of dominant meromorphic maps from \(X\) to \(Y\) is finite. This was later generalized by \textit{M. Deschamps} and \textit{R. L. Menegaux} [Bull. Soc. Math. Fr. 106, 279--286 (1978; Zbl 0417.14007)] who showed that if \(X\) and \(Y\) are smooth projective varieties defined over a field \(k\) of any characteristic and \(Y\) is of general type, then the set of separable dominant rational maps from \(X\) to \(Y\) is finite. Furthermore, \textit{R. Tsushima} [Proc. Japan Acad., Ser. A 55, 95--100 (1979; Zbl 0443.14006)] showed finiteness in the case of open varieties.
The paper under review generalizes the previous results in the category of log schemes. In particular, the following is shown. Let \(X\) and \(Y\) be proper varieties defined over an algebraically closed field \(k\), with at most normal crossing singularities. Let \(M_X\), \(M_Y\) and \(M_k\) be fine log structures on \(X\), \(Y\) and \(\mathrm{Spec}(k)\), respectively, such that there are log smooth integral maps \((X,M_X) \rightarrow (\mathrm{Spec}(k), M_k)\) and \((Y,M_Y) \rightarrow (\mathrm{Spec}(k), M_k)\). Moreover, assume that \((Y,M_Y)\) is of log general type over \(\mathrm{Spec}(k)\). Then the set of all log rational maps \((\phi, h) \colon (X,M_X) \dasharrow (Y,M_Y)\), over \((\mathrm{Spec}(k),M_k)\), that satisfy the following two properties, is finite: \begin{itemize}\item[1.] \(\phi \colon X \dasharrow Y\) is a rational map defined over a dense open set \(U\) such that \(\mathrm{codim}(X-U,X) \geq 2\); \item[2.] For any irreducible component \(X^{\prime}\) of \(X\), there is an irreducible component \(Y^{\prime}\) of \(Y\) such that \(\phi(X^{\prime}) \subset Y^{\prime}\) and the induced rational map \(\phi^{\prime} \colon X^{\prime} \dasharrow Y^{\prime}\) is dominant and separable.
The main advantage of this result compared to the previous ones is that it allows \(X\) and \(Y\) to have normal crossing singularities.\end{itemize} rational maps; log schemes; log general type I. Iwanari and A. Moriwaki, Dominant rational maps in the category of log schemes, to appear in Tohoku Math. J. 59 (2007). Rational and birational maps, Rational points Dominant rational maps in the category of log 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 \(\Phi_0\) be a Lie algebra law on the vector space \(k^n\) \((k\) is an algebraically closed field of characteristic zero). The author works on the category of deformations of \(\Phi_0\) parametrized by a local ring \(A\), understood as morphisms \({\mathcal O}\to A\) where \({\mathcal O}\) is the local ring at the point \(\Phi_0\) in the scheme defined by antisymmetric and Jacobi identities. The definition contains deformations in the sense of Gerstenhaber [\textit{R. Carles}, C.R. Acad. Sci., Paris, Sér. I 312, 671-674 (1991; Zbl 0734.17008)]. If \(A\) is complete, the author shows that each deformation -- up to an equivalence -- has certain structure constants fixed and constant values. Deformations expressed with parameters which are running over orbits (under the canonical action of \(GL(n,k))\) distinct from the orbit of \(\Phi_0\) are studied. In particular, the number of the essential parameters is calculated. Finally, the author examines the case of the complex Lie algebra \(sl(2,\mathbb{C})\otimes\mathbb{C}^n\). cohomology of Lie algebras; Gerstenhaber deformations; deformations Cohomology of Lie (super)algebras, Formal methods and deformations in algebraic geometry Deformations in the schemes defined by Jacobi identities | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \({\mathbb {F}}\) be any field. The Grassmannian \({\mathrm {Gr}}(m,n)\) is the set of \(m\)-dimensional subspaces in \({\mathbb {F}}^n\), and the general linear group \({\mathrm{GL}}_n({\mathbb {F}})\) acts transitively on it. The Schubert cells of \({\mathrm {Gr}}(m,n)\) are the orbits of the Borel subgroup \({\mathcal {B}} \subset {\mathrm {GL}}_n({\mathbb {F}})\) on \({\mathrm {Gr}}(m,n)\). We consider the association scheme on each Schubert cell defined by the \({\mathcal {B}}\)-action and show it is symmetric and it is the generalized wreath product of one-class association schemes, which was introduced by \textit{R. A. Bailey} [Eur. J. Comb. 27, No. 3, 428--435 (2006; Zbl 1111.05100)]. subspace lattice; Schubert cell; Borel subgroup; association scheme; generalized wreath product Association schemes, strongly regular graphs, Combinatorial structures in finite projective spaces, Grassmannians, Schubert varieties, flag manifolds Association schemes on the Schubert cells of a Grassmannian | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Hilbert schemes \(\mathrm{Hilb}^{p(t)} (\mathbb P^m)\) parametrizing closed subschemes \(X \subset \mathbb P^m\) with Hilbert polynomial \(p(t)\) have received much attention from algebraic geometers since their construction in the early 1960s. Although connected [\textit{R. Hartshorne}, Publ. Math., Inst. Hautes Étud. Sci. 29, 5--48 (1966; Zbl 0171.41502)], \(\mathrm{Hilb}^{p(t)} (\mathbb P^m)\) exhibits many bad behaviors, for example it can have non-reduced components [\textit{D. Mumford}, Am. J. Math. 84, 642--648 (1962; Zbl 0114.13106)] and every singularity type appears in some Hilbert scheme [\textit{R. Vakil}, Invent. Math. 164, No. 3, 569--590 (2006; Zbl 1095.14006)]. In the paper under review, the authors classify all smooth Hilbert schemes.
Recall that a Hilbert polynomial of a closed subscheme \(X \subset \mathbb P^m\) can be uniquely written in the form \(p(t) = \sum_{i=1}^r \binom{t+\lambda_i-i}{\lambda_i-1}\) where \(\lambda = (\lambda_1, \dots, \lambda_r)\) is a partition of integers satisfying \(\lambda_1 \geq \dots \geq \lambda_r \geq 1\) [\textit{G. Gotzmann}, Math. Z. 158, 61--70 (1978; Zbl 0352.13009)]. The authors prove that the partitions corresponding to a smooth Hilbert scheme are precisely those in the following list:
(1) \(m = 2 \geq \lambda_1\).
(2) \(m \geq \lambda_1\) and \(\lambda_r \geq 2\).
(3) \(\lambda = (1)\) or \(\lambda = (m^{r-2}, \Lambda_{r-1},1)\), where \(r \geq 2\) and \(m \geq \lambda_{r-1} \geq 1\).
(4) \(\lambda = (m^{r-s-3}, \lambda_{r-s-s}^{s+2},1)\), where \(r-3 \geq s \geq 0\) and \(m-1 \geq \lambda_{r-s-2} \geq 3\).
(5) \(\lambda = (m^{r-s-5}, 2^{s+4},1)\), where \(r-5 \geq s \geq 0\).
(6) \(\lambda = (m^{r-3},1^3)\), where \(r \geq 3\).
(7) \(\lambda = (m+1)\) or \(r=0\).
Moreover, the authors describe the schemes parametrized by each family listed. For example, the general member of family (3) is a union of a hypersurface of degree \(r-2\), a linear subspace of dimension \(\lambda_{r-1}\) and a point while the general member of family (5) is a union of a hypersurface of degree \(r-s-3\), a hypersurface of degree \(s+2\) of a linear subspace of dimension \(\lambda_{r-s-2}\) and a point. The families in (7) correspond to the one-point Hilbert scheme parametrizing \(\mathbb P^m\) and the empty scheme. All families in the theorem were previously known to be smooth: smoothness for families (1) and (6) follows from work of \textit{J. Fogarty} [Am. J. Math. 90, 511--521 (1968; Zbl 0176.18401)], families (2) and (3) were shown smooth by
\textit{A. P. Staal} [Math. Z. 296, No. 3--4, 1593--1611 (2020; Zbl 1451.14010)] and families (4) and (5) were shown smooth by \textit{R. Ramkumar} [J. Algebra 617, 17--47 (2023; Zbl 1503.13007)] in his work on Hilbert schemes with at most two Borel-fixed ideals. The main contribution here is that this list is complete.
As to the strategy of the proof, it is known from work of \textit{A. Reeves} and \textit{M. Stillman} that the lexicographical point is always smooth on the Hilbert scheme and determines a unique irreducible component of \(\mathrm{Hilb}^{p(t)} (\mathbb P^m)\) of computable dimension [J. Algebr. Geom. 6, No. 2, 235--246 (1997; Zbl 0924.14004)]. In theory one could attempt to show smoothness by computing the dimension of the Zariski tangent space at the other Borel-fixed points, but this is unwieldy. Instead, the authors construct families of subschemes corresponding to points on Hilbert schemes that necessarily singular. To describe these points, define a \textit{residual inclusion} \(X \subset Y \subset \mathbb P^m\) to be a closed immersion such that there is a linear subspace \(\Lambda \subset \mathbb P^m\) containing \(X\) and a hypersurface \(D \subset \Lambda\) with \(Y\) the residual scheme of \(D \subset X\) in \(\Lambda\). A \textit{residual flag} is a flag \(\emptyset \subset X_e \subset X_{e-1} \subset \dots \subset X_1\) of residual inclusions. For most Hilbert polynomials not in the list, the authors produce such an \(X_1\) near the lexicographic point which corresponds to a singular point on the Hilbert scheme. The others not on the list are handled with three other singular families. The proof is valid over \(\mathrm{Spec}\,\mathbb Z\). The authors credit \texttt{Macaulay2} [\url{http://www.math.uiuc.edu/Macaulay2/}] for many experimental computations that were indispensable in discovering their results. Hilbert schemes; Borel fixed ideals; partial flag varieties Parametrization (Chow and Hilbert schemes), Fibrations, degenerations in algebraic geometry, Geometric invariant theory, Actions of groups on commutative rings; invariant theory Smooth Hilbert schemes: their classification and geometry | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems What is now called a Fourier-Mukai functor was first studied by \textit{S. Mukai} [Nagoya Math. J. 81, 153--175 (1981; Zbl 0417.14036)]. Such a functor sends an object \(F\) of a derived category on \(X\) to \(Rq_{\ast}(K \otimes p^{\ast} F)\), where the so-called kernel \(K\) is an object of a derived category on \(X\times Y\) and \(p: X\times Y \rightarrow X\) and \(q: X\times Y\rightarrow Y\) are projections in the category of schemes. Such functors were shown to be a useful tool in the study of moduli spaces of coherent sheaves and to understand the structure of the derived category of coherent sheaves on a smooth variety. More recently, attempts were made to use derived categories and Fourier-Mukai functors to tackle problems in higher dimensional birational geometry.
The relevance of Fourier-Mukai functors in the smooth case is highlighted by Orlov's Theorem which says that any exact equivalence between bounded derived categories of coherent sheaves on smooth projective varieties is a Fourier-Mukai functor. Fourier-Mukai functors on singular varieties have not yet been studied in depth. The paper under review is one of the first attempts to generalise results to singular varieties which were previously known in the smooth case only. One of the main results of this article is a characterisation of those kernels \(K\) which give fully faithful functors (resp. equivalences) between bounded derived categories on Gorenstein schemes. This generalises a result of \textit{A. Bondal} and \textit{D. Orlov} [Semiorthogonal decomposition for algebraic varieties, preprint MPIM 95/15 (1995), see also \url{arXiv:math/9506012}]. An interesting example in characteristic \(p>0\) is provided which shows that the given criterion does not hold in the case of positive characteristic. As an application of their characterisation of Fourier-Mukai equivalences, the authors show that the dimension and the order of the canonical line bundle are Fourier-Mukai invariants of projective Gorenstein schemes.
In the final section of this paper, relative Fourier-Mukai transforms are studied. As their main application, they give a new proof of a result of \textit{I. Burban} and the reviewer [Manuscr. Math. 120, No. 3, 283--306 (2006; Zbl 1105.18011)] which says that the Fourier-Mukai functor, given by the relative Poincaré bundle of an elliptic fibration with irreducible fibres, is an equivalence. Mukai pair; spanning class; elliptic fibration; strongly simple; fully faithful integral transform; D-equivalence; derived category Hernández Ruipérez, D.; López Martín, A.C.; Sancho de Salas, F., Fourier-Mukai transforms for Gorenstein schemes, Adv. math., 211, 2, 594-620, (2007) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Derived categories, triangulated categories, Elliptic surfaces, elliptic or Calabi-Yau fibrations, Minimal model program (Mori theory, extremal rays), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Fourier--Mukai transforms for Gorenstein 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 \(W=m_1P_1+\cdots+m_sP_s\) be a scheme of fat points in \(\mathbb P^n_K\), where \(K\) is a field of characteristic zero.
The authors deal with the problem of computing the Hilbert polynomial of the modules of the Kähler differential \(k\)-forms of the coordinate ring of \(W\), \(\Omega^{k}_{R_{W/K}}\). After introducing notation and basic facts in Section 2, they show in Theorem 3.7 that the Hilbert polynomial of \(\Omega^{n+1}_{R_{W/K}}\) is \(\sum_j\binom{m_j+n-2}{n},\) i.e., it is the Hilbert polynomial of the coordinate ring of the fat points scheme \((m_1-1)P_1+\cdots+(m_s-1)P_s\).
This answers positively to a conjecture the authors stated in a previous paper, see Conjecture 5.7 in [\textit{M. Kreuzer} et al., J. Algebra 501, 255--284 (2018; Zbl 1388.13051)].
Moreover, making use of Theorem 3.7, the authors compute the Hilbert polynomial of the modules of the Kähler differential of a scheme of fat points in \(\mathbb P^2\), see Proposition 4.1. Hilbert function; fat point scheme; regularity index; Kähler differential module Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Modules of differentials, Cycles and subschemes Hilbert polynomials of Kähler differential modules for fat point 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 first construct and give basic properties of a fibered coproduct in the category of ringed spaces (which is just a particular type of colimit). We then look at some special cases where this actually gives a fibred coproduct
in the category of schemes. Intuitively this is gluing a collection of schemes along some collection of other schemes (possibly subschemes). We then use this to construct a scheme without closed points. Schwede, K., Gluing schemes and a scheme without closed points, No. 386, 157-172, (2005), Providence Schemes and morphisms Gluing schemes and a scheme without closed 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 We provide an overview of the \textit{VersalDeformations} package for \textit{Macaulay2} which computes versal deformations of isolated singularities and local multigraded Hilbert schemes. Ilten N.\ O., Versal deformations and local Hilbert schemes, J. Software Algebra Geom. 3 (2012), 12-16. Formal methods and deformations in algebraic geometry, Parametrization (Chow and Hilbert schemes), Software, source code, etc. for problems pertaining to algebraic geometry Versal deformations and local 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 \(k\) be a perfect field of characteristic \(p \geq 3.\) \textit{J.-M. Fontaine} classified \(p\)-divisible and finite flat commutative \(p\)-groups over the Witt vectors \(W(k)\) in terms of filtered modules. The paper extends these classifications replacing \(W(k)\) by an arbitrary complete discrete valuation ring of unequal characteristic with residue field \(k\). ``Generalized'' filtered modules are used. finite group schemes; characteristic \(p\); \(p\)-divisible group; generalized filtered modules Christophe Breuil, Schémas en groupe et modules filtrés, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), no. 2, 93 -- 97 (French, with English and French summaries). Group schemes, Formal groups, \(p\)-divisible groups, Finite ground fields in algebraic geometry Group schemes and filtered modules | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A natural question in the theory of Tannakian categories is: What if you don't remember Forget? Working over an arbitrary commutative ring \(R\), we prove that an answer to this question is given by the functor represented by the étale fundamental groupoid \(\pi _{1}\)(spec\((R)\)), i.e. the separable absolute Galois group of \(R\) when it is a field. This gives a new definition for étale \(\pi _{1}\)(spec\((R)\)) in terms of the category of \(R\)-modules rather than the category of étale covers. More generally, we introduce a new notion of ``commutative 2-ring'' that includes both Grothendieck topoi and symmetric monoidal categories of modules, and define a notion of \(\pi _{1}\) for the corresponding ``affine 2-schemes.'' These results help to simplify and clarify some of the peculiarities of the étale fundamental group. For example, étale fundamental groups are not ``true'' groups but only profinite groups, and one cannot hope to recover more: the ``Tannakian'' functor represented by the étale fundamental group of a scheme preserves finite products but not all products. higher category theory; presentable categories; fundamental groupoids; Galois theory; categorification; affine 2-schemes; Tannakian reconstruction Chirvasitu, A; Johnson-Freyd, T, The fundamental pro-groupoid of an affine 2-scheme, Appl. Categ. Struct., 21, 469-522, (2013) Double categories, \(2\)-categories, bicategories, hypercategories, Monoidal categories (= multiplicative categories) [See also 19D23], Homotopy theory and fundamental groups in algebraic geometry The fundamental pro-groupoid of an affine 2-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 prove a closed formula for the integrals of the top Segre classes of tautological bundles over the Hilbert schemes of points of a \(K3\) surface \(X\). We derive relations among the Segre classes via equivariant localization of the virtual fundamental classes of Quot schemes on \(X\). The resulting recursions are then solved explicitly. The formula proves the \(K\)-trivial case of a conjecture of \textit{M. Lehn} [Invent. Math. 136, No. 1, 157--207 (1999; Zbl 0919.14001)].
The relations determining the Segre classes fit into a much wider theory. By localizing the virtual classes of certain relative Quot schemes on surfaces, we obtain new systems of relations among tautological classes on moduli spaces of surfaces and their relative Hilbert schemes of points. For the moduli of \(K3\) surfaces, we produce relations intertwining the \(\kappa\) classes and the Noether-Lefschetz loci. Conjectures are proposed. Hilbert schemes of points; Quot schemes; moduli of \(K3\) surfaces; tautological classes Parametrization (Chow and Hilbert schemes), Families, moduli, classification: algebraic theory, \(K3\) surfaces and Enriques surfaces, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Segre classes and Hilbert schemes of points | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For the entire collection see Zbl 0509.00008.]
Author's abstract: ''I will give a simple description of how to locally represent the punctual Hilbert scheme \(Hilb^ n{\mathbb{C}}^ 2\) as a local flattener of some unfolding of a finite map-germ and how to derive information on the geometry of the Hilbert scheme \(Hilb^ n{\mathbb{C}}\{x,y\}\) from the study of the ramification loci of that unfolding; and conversely I will also describe how to obtain information on the topological classification of finite stable map-germs of type \(\Sigma_ 2\) using properties of \(Hilb^ n{\mathbb{C}}\{x,y\}\), following \textit{M. Granger}, ''Géométrie des schémas de Hilbert ponctuels'' (Thèse, Nice 1980); see also Mém. Soc. Math. Fr., Nouv. Sér. 8 (1983; Zbl 0534.14002), and \textit{J. Briançon, M. Granger}, and \textit{J. P. Speder}, Ann. Sci. Éc. Norm. Super., IV. Sér. 14, 1-25 (1981; Zbl 0463.14001) and \textit{J. Damon} and the author, Invent. Math. 32, 103- 132 (1976; Zbl 0333.57017).''
The paper is short and contains sketches of proofs. punctual Hilbert scheme; local flattener; unfolding of a finite map-germ; topological classification of finite stable map-germs Galligo, A.: Hilbert scheme as flattener. Proc. sympos. Pure math. 40, 449-452 (1983) Parametrization (Chow and Hilbert schemes), Formal methods and deformations in algebraic geometry, Deformations of submanifolds and subspaces, Germs of analytic sets, local parametrization Hilbert scheme as flattener | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Many authors have studied the Hilbert scheme \(\text{Hilb}^d (X)\) parametrizing closed subschemes \(Z \subset X\) of length \(d\) on a variety \(X\). \textit{J. Fogarty} showed that \(\text{Hilb}^d (X)\) is smooth and irreducible when \(X\) is a smooth connected surface [Am. J. Math. 90, 511--521 (1968; Zbl 0176.18401)], but for \(n>2\) and \(d \gg 0\) \textit{A. Iarrobino} [Invent. Math. 15, 72--77 (1972; Zbl 0227.14006)] proved that \(\text{Hilb}^d (\mathbb A^n)\) is reducible by constructing \textit{elementary} components \(Z \subset \text{Hilb}^d (\mathbb A^n)\), those parametrizing subschemes supported at a single point.
The author proves that if \(R \subset \mathbb A^n\) is supported at the origin, then the corresponding point \([R] \in \text{Hilb}^d (\mathbb A^n)\) lies on an elementary component if \(R\) has \textit{trivial negative tangents}, meaning that the tangent map \(\langle \partial_1, \dots, \partial_n \rangle \to \text{Hom}(I_R,{\mathcal O}_R)_{<0}\) of the orbit of \([R]\) under translation is surjective. Conversely, if \(Z \subset \text{Hilb}^d (\mathbb A^n)\) is a generically reduced elementary component and char \(k =0\), then the general point \([R] \in Z\) has trivial negative tangents (it is unknown whether there exist generically non-reduced components).
The main tools in the proof are obstruction theory and an extension of the decomposition theorem of \textit{A. Bialynicki-Birula} [Ann. Math. (2) 98, 480--497 (1973; Zbl 0275.14007)] from smooth proper varieties to the singular non-proper Hilbert schemes \(\text{Hilb}^d (\mathbb A^n)\).
The author uses his criterion to construct infinitely many smooth points of distinct elementary components of \(\text{Hilb}^d (\mathbb A^4)\); these examples show that the Gröbner fan need not distinguish components of \(\text{Hilb}^d (\mathbb A^n)\). He also reduces the question of whether \(\text{Hilb}^d (\mathbb A^n)\) is reduced to a testable conjecture involving the trivial negative tangents condition. Hilbert scheme of points; elementary components Parametrization (Chow and Hilbert schemes), Group actions on varieties or schemes (quotients), Deformations and infinitesimal methods in commutative ring theory Elementary components of 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 A hierarchy of Hamiltonian systems including the Garnier-Rosochatius system as the first member is introduced. This hierarchy of Hamiltonian systems is proved to be completely integrable, and the corresponding flows are linearized on the Jacobi variety of the associated hyperelliptic curve. In addition, a relation between these flows and the KdV equation is found, and the connection with finite gap solution of KdV equation is shown.{
\copyright 2011 American Institute of Physics} DOI: 10.1063/1.3597231 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions, Relationships between algebraic curves and integrable systems A hierarchy of Garnier-Rosochatius 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 Complements of linear hyperplane arrangements have been studied for a long time. The authors study duality properties of more general arrangements. More precisely, of arrangements of smooth hypersurfaces, these are collections of smooth, irreducible, codimension 1 subvarieties, embedded in a smooth, connected, complex projective algebraic variety, that intersect locally like hyperplanes.
The general hypotheses are: Let \(U\) be a connected, smooth, complex quasiprojective variety of dimension \(n\). Assume \(U\) has a smooth compactification \(Y\) such that: {\parindent=6mm \begin{itemize}\item[(1)] The components of the boundary \(D=Y\setminus U\) form a nonempty arrangement of hypersurfaces \(\mathcal{A}\). \item[(2)] For each submanifold \(X\) in the intersection poset \(L(\mathcal{A})\), the complement of the restriction of \(\mathcal{A}\) to \(X\) is either empty or a Stein manifold.
\end{itemize}} Let \(X\) be a connected \(CW\)-complex of finite type and fundamental group \(G\). The space \(X\) is called a duality space of dimension \(n\) if \(H^q(X;\mathbb{Z}[G])=0\) for \(q\not=n\) and \(H^n(X;\mathbb{Z}[G])\) is nontrivial and torsion-free. The space \(X\) is called an abelian duality space if the analogous property holds for the coefficient \(G\)-module \(\mathbb{Z}[G^{ab}]\). The first result is that under the hypotheses (1) and (2), it follows that \(U\) is a duality space and an abelian duality space. Further analysis gives similar results in the context of \(\ell^{2}\) cohomology, nonempty elliptic arrangements, toric arrangements and orbit configuration spaces. duality spaces; arrangements of hypersurfaces Denham, G.; Suciu, A., Local systems on arrangements of smooth, complex algebraic hypersurfaces, Forum Math. Sigma, 6, (2018) Homology with local coefficients, equivariant cohomology, Compactifications; symmetric and spherical varieties, Homological methods in group theory, Stein spaces, Relations with arrangements of hyperplanes, Rational homotopy theory, Discriminantal varieties and configuration spaces in algebraic topology, Duality in applied homological algebra and category theory (aspects of algebraic topology), Topological methods in group theory Local systems on complements of arrangements of smooth, complex algebraic hypersurfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The main emphasis in these notes is on the Fourier-Mukai transforms as equivalence of derived categories of coherent sheaves on algebraic varieties. For this reason, the first section is devoted to a basic (but we hope, understandable) introduction to derived categories. In the second section we develop the basic theory of Fourier-Mukai transforms. Another aim of our lectures was to outline the relations between Fourier-Mukai and Nahm transforms. This is the topic of section 3. Finally, section 4 is devoted to the application of the theory of Fourier-Mukai transforms to the study of coherent systems. Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Derived categories, triangulated categories Introduction to Fourier-Mukai and Nahm transforms with an application to coherent systems on elliptic curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be an elliptic curve defined over \(\mathbb{F}_q\). Here we study the existence of geometrically polystable vector bundles \(E\) defined over \(\mathbb{F}_q\) and with further geometric properties and the existence of \(\alpha\)-stable coherent systems \((E,V)\) of type \((n,d,k)\), \(k\in \{n-1,n,n+1\}\), defined over \(\mathbb{F}_q\). Vector bundles on curves and their moduli, Elliptic curves, Finite ground fields in algebraic geometry Coherent systems on smooth elliptic curves over a finite field | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The main theorem states that the cohomology of \(\text{ Hilb}^{[n]}\), the Hilbert scheme of \(n\)-points on a \(K3\)-surface, is the \(S_n\)-invariant part of the \(S_n\)-Frobenius algebra associated to the symmetric product of the cohomology of the surface twisted by a discrete torsion. Here \(S_n\) denotes the symmetric group of \(n\) letters. The main idea is to use the notion of \(G\)-Frobenius algebras for a finite group \(G\), which arise from the stringy study of objects with a global \(G\)-action.
In section one, the author presents the general functorial setup for extending functors to Frobenius algebras to those with values in \(G\)-Frobenius algebras. Section two contains the basic definitions of \(G\)-Frobenius algebras. Section three introduces intersection Frobenius algebras which are adapted to the situation in which one can take successive intersections of fixed point sets. Section four reviews the analysis of discrete torsion. In section five, the author recalls the results on the structure of \(S_n\)-Frobenius algebras. Section six assembles these results in the case of any \(S_n\)-Frobenius algebra twisted by a specific discrete torsion. The result applied to the situation of the Hilbert scheme yields the above main theorem. Kaufmann, R. M.: Discrete torsion, symmetric products and the Hilbert scheme. Frobenius manifolds, quantum cohomology, and singularities (2004) Noncommutative algebraic geometry, Parametrization (Chow and Hilbert schemes) Discrete torsion, symmetric products and the Hilbert scheme | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The existence of a desingularization of quasi-excellent schemes as conjectured by \textit{A. Grothendieck} [Publ. Math., Inst. Hautes Étud. Sci. 20, 101--355 (1964; Zbl 0136.15901); ibid. 24, 1--231 (1965; Zbl 0135.39701); ibid. 28, 1--255 (1966; Zbl 0144.19904); ibid. 32, 1--361 (1967; Zbl 0153.22301)] was shown in the author's work [Adv. Math. 219, No. 2, 488--522 (2008; Zbl 1146.14009)] in 2008. Compared to the analogue for varieties, that result had the following disadvantages: Centers of the necessary blowups in the resolution procedure could be non-regular, and functoriality was not satisfied for the given construction.
The article under review is the first of two papers strengthening the results previously obtained: A desingularization is given by only blowing up regular centers and such that the resulting sequence of blowing ups gives an object which is functorial for regular morphisms. The case treated here is the non-embedded desingularization, whereas for the embedded one the author refers to his forthcoming paper [``Functorial desingularization over \(\mathbb{Q}\): boundaries and the embedded case'', \url{arXiv:0912.2570}].
Main result of the article is
Theorem 1.2.1: For any Noetherian quasi-excellent generically reduced scheme \(X=X_0\) over \(\text{Spec} (\mathbb{Q})\) there exists a blow-up sequence \({\mathcal F} (X): X_n \dashrightarrow X_0\) such that the following conditions are satisfied: {\parindent=8mm \begin{itemize}\item[(i)] the centers of the blowups are disjoint from the preimages of the regular locus \(X_{\mathrm{reg}}\); \item[(ii)] the centers of the blowups are regular; \item[(iii)] \(X_n\) is regular; \item[(iv)] the blow-up sequence \({\mathcal F} (X)\) is functorial with respect to all regular morphisms \(X' \to X\), in the sense that \({\mathcal F} (X')\) is obtained from \({\mathcal F} (X)\times_X X'\) by omitting all empty blowups.
\end{itemize}}
The Construction of \({\mathcal F} \) is done starting with any algorithm \({\mathcal F}_{\mathrm{Var}}\) giving desingularizations for varieties in characteristic 0 and which is functorial for regular morphisms in the sense of (i), (iii) and (iv). Furthermore, \({\mathcal F} \) will be found to satisfy the above condition (ii) if this is the case for the algorithm \({\mathcal F}_{\mathrm{Var}}\). This algorithm is extended to pairs \((X,Z)\) of quasi-excellent schemes \(X\) and Cartier divisors \(Z\) in \(X\) containing the singular locus and isomorphic to a disjoint union of varieties, such that \({\mathcal F}_{\mathrm{Var}} (X,Z) \) desingularizes \(X\). Now the formal completion \({\mathcal X} := \hat{X}_Z\) is algebraized by some \(X'\), and \({\mathcal F}_{\mathrm{Var}} (X')\) gives rise to desingularizations on \(\mathcal X\) (and on \(X\)). The main work remaining now is to show that \({\mathcal F}_{\mathrm{Var}} (\mathcal X) = \widehat{{\mathcal F}_{\mathrm{Var}} (X')}\) is canonically defined by \(X_n\), where \(X_n\subseteq \mathcal X\) is some sufficiently large nilpotent neighborhood of the closed fibre. Algebraization is done using the classical approximation results of \textit{R. Elkik} [Ann. Sci. Éc. Norm. Supér. (4) 6, 553--603 (1973; Zbl 0327.14001)].
From the author's abstract: ``As a main application, we deduce that any reduced formal variety of characteristic zero admits a strong functorial desingularization. Also, we show that as an easy formal consequence of our main result one obtains strong functorial desingularization for many other spaces of characteristic zero including quasi-excellent stacks, formal schemes, and complex or nonarchimedean analytic spaces. Moreover, these functors easily generalize to noncompact settings by use of generalized convergent blow-up sequences with regular centers.'' desingularization of quasi-excellent schemes; nonembedded desingularization; functorial desingularization Temkin, M., Functorial desingularization of quasi-excellent schemes in characteristic zero: the nonembedded case, Duke Mathematical Journal, 161, 2207-2254, (2012) Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Valuation rings Functorial desingularization of quasi-excellent schemes in characteristic zero: the nonembedded case | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a smooth complex projective curve of genus \(g\). A coherent system of type \((n,d,k)\) is a pair \((E,V)\) where \(E\) is a rank \(n\) vector bundle on \(X\) with degree \(d\) and \(V\) is a \(k\)-dimensional linear subspace of \(H^0(E)\). For any fixed real number \(\alpha\) there is a notion of stability and semistability of coherent systems on \(X\) (\(\alpha\)-stability and \(\alpha\)-semistability). There are moduli spaces of \(\alpha\)-semistable vector bundles on \(X\) of type \((n,d,k)\). Coherent systems were introduced (for arbitrary varieties) by \textit{J. Le Potier} [in: N. J. Hitchin, P. E. Newstead, W. M. Oxbury eds., Vector bundles in Algebraic Geometry. London: Cambridge University Press. London Mathematical Society Lecture Notes Series, Vol. 208, 179--239 (1995; Zbl 0847.14005)] and \textit{N. Raghavendra} and \textit{P. A. Vishwanath} [Tôhoku Math. J. 46, No. 3, 321--340 (1994; Zbl 0828.14005)]. They were extensively used to study moduli spaces of stable vector bundles and the Brill-Noether theory of stable and semistable vector bundles, e.g., [\textit{S. B. Bradlow} and \textit{O. García-Prada}, J. Lond. Math. Soc. (2) 60, No. 1, 155--170 (1999; Zbl 0954.32014)]; [\textit{H. Lange} and \textit{P. E. Newstead}, Int. J. Math. 15, No. 9, 409--424 (2004; Zbl 1152.14018), Int. J. Math. 16, No. 7, 787--805 (2005; Zbl 1078.14045), Int. J. Math. 18, No. 4, 363--393 (2007; Zbl 1114.14022), Int. J. Math. 19, No. 9, 1103--1119 (2008; Zbl 1152.14018)]. In the paper under review \(k<n\) and there are studied the ones coming from a BGN extension
\[
0 \to V\otimes \mathcal {O}_X \to E \to F\to 0
\]
with strictly stable \(F\) (the hardest case, the case not handled in previous references). In this case he describes a stratification of the moduli space of coherent systems. Each strata is also described as the complement of a determinantal variety. Each strata is proved to be smooth and irreducible. In some cases this stratification allows the author to compute the Hodge polynomials of these moduli spaces. Explicit computations are given for them and the usual Poincaré polynomials when \((n,d,k) = (3,d,1)\) with \(d\) even. For earlier work on vector bundles (not coherent systems) see [\textit{C. González-Martínez}, Manuscr. Math. 137, No. 1--2, 19--55 (2012; Zbl 1245.14034)]. coherent systems; moduli spaces of vector bundles; vector bundles on curves; moduli spaces; Hodge polynomials Vector bundles on curves and their moduli, Algebraic moduli problems, moduli of vector bundles, Topological properties in algebraic geometry Hodge polynomials of some moduli spaces of coherent 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 Fix an algebraically closed field \(k\) of characteristic zero and let \(H^d (\mathbb A^n_k)\) denote the Hilbert scheme parametrizing zero dimensional closed subschemes of length \(d\). The \textit{principal component} of \(H^d (\mathbb A^n_k)\) consists of flat limits of \(d\) distinct points, but sometimes there are other components, the first examples being due to \textit{A. Iarrobino} [Invent. Math. 15, 72--77 (1972; Zbl 0227.14006)]. An \textit{elementary component} of \(H^d (\mathbb A^n_k)\) is one whose general member corresponds to a closed subscheme supported at a single point and is generically smooth. \textit{A. Iarrobino} and \textit{J. Emsalem} constructed an explicit example of an elementary component some 40 years ago [Compos. Math. 36, 145--188 (1978; Zbl 0393.14002)]. After describing their example carefully, the author uses the theory of border basis schemes due to \textit{M. Kreuzer} and \textit{L. Robbiano} [Collect. Math. 59, No. 3, 275--297 (2008; Zbl 1190.13022)]; J. Pure Appl. Algebra 215, No. 8, 2005--2018 (2011; Zbl 1216.13018)] to extend the Iarrobino-Ensalem construction, obtaining more examples of elementary components. Hilbert scheme of points; elementary component Parametrization (Chow and Hilbert schemes) Some elementary components of the 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 Verf. giebt eine neue Darstellung seiner früheren Resultate (Math. Ann. XLV; F. d. M. XXV. 1893/94. 690, JFM 25.0690.01) und bringt sie dabei in Uebereinstimmung mit denen des Hrn. Hensel (J. f. M. CIX; F. d. M. XXIII. 1891. 432, JFM 23.0432.02). Algebraic functions of one variable Algebraic functions and function fields in algebraic geometry On fundamental systems for algebraic functions. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors are interested in mixed splines in a \(d\)-dimensional polyhedral complex, which are splines where the order of smoothness may change on the adjacency points of various \((d-1)\)-faces. The problem of finding the dimension of the vector space of this type of functions is studied, giving a complete answer. The authors use the technique developed in 2009 by \textit{T. McDonald} and \textit{H. Schenck} [Adv. Appl. Math. 42, No. 1, 82--93 (2009; Zbl 1178.41008)], extending it in the framework of mixed splines. Their technique is used in few well chosen examples that conclude the paper. mixed splines; polyhedral complex; dimension formula; Hilbert polynomial Geometric aspects of numerical algebraic geometry, Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Spline approximation, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Numerical computation using splines Piecewise polynomials with different smoothness degrees on polyhedral complexes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 genus \(g\) curve, \(n,d,k\) integers with \(n>0\), \(k \geq 0\) and \(\alpha \in \mathbb R\). A coherent system of type \((n,d,k)\) on \(X\) is a pair \((E,V)\), where \(E\) is a degree \(d\) and rank \(n\) vector bundle on \(X\) and \(V\) is a \(k\)-dimensional linear subspace of \(H^0(X,E)\). Set \(\mu _\alpha (E,V) = d/n + \alpha \cdot k\) (the \(\alpha\)-slope) and use the \(\alpha\)-slope to define the \(\alpha\)-stability of a coherent system. Coherent systems give nice moduli spaces [\textit{S. B. Bradlow}, \textit{O. García-Prada}, \textit{V. Munoz} and \textit{P. E. Newstead}, Int. J. Math. 14, No. 7, 683--733 (2003; Zbl 1057.14041)]. Here the authors give a very precise description of these moduli spaces when \(X\) is an elliptic curve (non-emptyness, irreduciblity, smoothness). For weaker results in the genus \(0\) case, see \textit{H. Lange} and \textit{P. E. Newstead} [Int. J. Math. 15, No. 4 409--424 (2004; Zbl 1072.14039)]. semistable vector bundle; vector bundles on elliptic curves Lange H., Newstead P.E.: Coherent systems on elliptic curves. Int. J. Math. 16(7), 787--805 (2005) Vector bundles on curves and their moduli, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Sheaves and cohomology of sections of holomorphic vector bundles, general results Coherent systems on elliptic 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 the paper under review a birational morphism of the Faltings compactified moduli space \(A_ 2(3)\) (principally polarized abelian surfaces with level 3 structure) onto a singular quartic in \({\mathbb{P}}^ 4\) is constructed in explicit form. 45 surfaces, which are isomorphic to \({\mathbb{P}}^ 1\times {\mathbb{P}}^ 1\) and correspond to abelian surfaces decomposable as a product (and nothing more) are contracted to points by this morphism. birational morphism of compactified moduli space G. van der Geer, ''Note on abelian schemes of level three,''Math. Ann.,278, Nos. 1--4, 401--408 (1987). Algebraic moduli of abelian varieties, classification, Rational and birational maps Note on abelian schemes of level three | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 principal specialization \(\nu_w=\mathfrak{S}_w(1,\dots,1)\) of the Schubert polynomial at \(w\), which equals the degree of the matrix Schubert variety corresponding to \(w\), has attracted a lot of attention in recent years. In this paper, we show that \(\nu_w\) is bounded below by \(1+p_{132}(w)+p_{1432}(w)\) where \(p_u(w)\) is the number of occurrences of the pattern \(u\) in \(w\), strengthening a previous result by \textit{A. E. Weigandt} [Algebr. Comb. 1, No. 4, 415--423 (2018; Zbl 1397.05205)]. We then make a conjecture relating the principal specialization of Schubert polynomials to pattern containment. Finally, we characterize permutations \(w\) whose RC-graphs are connected by simple ladder moves via pattern avoidance. pattern containment; permutation patterns Combinatorial aspects of algebraic geometry, Classical problems, Schubert calculus Principal specializations of Schubert polynomials and pattern containment | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(R\) be a commutative ring containing \(\mathbb Q\). The polynomials \(f_1,\ldots,f_m\in R[X_1,\ldots,X_n]\) form a partial coordinate system if there exist polynomials \(f_{m+1},\ldots,f_n\) such that \(R[f_1,\ldots,f_n]=R[X_1,\ldots,X_n]\). For an element \(a\in R\) the system \(f_1,\ldots,f_m\in R[X_1,\ldots,X_n]\) is called an \(a\)-strongly partial residual coordinate system if the images of \(f_1,\ldots,f_m\) in \((R/aR)[X_1,\ldots,X_n]\) and \(R_a[X_1,\ldots,X_n]\) form a partial coordinate system. One of the results in [\textit{J. Berson} et al., J. Pure Appl. Algebra 184, No. 2--3, 165--174 (2003; Zbl 1028.13005)] states that if \(a\in R\) is not a zero-divisor and \(f_1,\ldots,f_{n-1}\in R[X_1,\ldots,X_n]\) is an \(a\)-strongly partial residual coordinate system, then it is also a partial coordinate system for \(R[X_1,\ldots,X_n]\). The paper contains the conjecture that the same holds for an arbitrary \(a\in R\). The main result of the paper under review confirms this conjecture and establishes the above stated result for an arbitrary \(a\in R\). The proof uses some results from [\textit{P. Das} and \textit{A. K. Dutta}, J. Pure Appl. Algebra 218, No. 10, 1792--1799 (2014; Zbl 1291.14090)]. polynomial algebra; coordinate; residual coordinate Polynomials over commutative rings, Affine fibrations A note on partial coordinate system in a polynomial ring | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The aim of this paper is to show that certain types of schemes can be recovered from categories associated to them.
The author starts by studying the case of locally Noetherian schemes. To such a scheme \(X\), he associates the category \(\text{Sch}(X)\) of morphisms of finite type from Noetherian schemes to \(X\). He then shows step by step how to reconstruct \(X\) by purely categorical techniques from the category \(\text{Sch}(X)\).
In the second part of the paper, this is extended to fine saturated log schemes whose underlying scheme is locally Noetherian. Here the appropriate category is the category of morphisms from fine saturated log schemes whose underlying scheme is Noetherian and which have the property that the underlying morphism of schemes is of finite type. anabelian geometry Mochizuki, Shinichi: Categorical representation of locally noetherian log schemes, Adv. math. 188, No. 1, 222-246 (2004) Schemes and morphisms, Generalizations (algebraic spaces, stacks), Categories in geometry and topology Categorical representation of locally Noetherian log 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 present new results on standard basis computations of a 0-dimensional ideal \(I\) in a power series ring or in the localization of a polynomial ring over a computable field \(K\). We prove the semicontinuity of the ``highest corner'' in a family of ideals, parametrized by the spectrum of a Noetherian domain \(A\). This semicontinuity is used to design a new modular algorithm for computing a standard basis of \(I\) if \(K\) is the quotient field of \(A\). It uses the computation over the residue field of a ``good'' prime ideal of \(A\) to truncate high order terms in the subsequent computation over \(K\). We prove that almost all prime ideals are good, so a random choice is very likely to be good, and whether it is good is detected a posteriori by the algorithm. The algorithm yields a significant speed advantage over the non-modular version and works for arbitrary Noetherian domains. The most important special cases are perhaps \(A={\mathbb{Z}}\) and \(A=k[t], k\) any field and \(t\) a set of parameters. Besides its generality, the method differs substantially from previously known modular algorithms for \(A={\mathbb{Z}} \), since it does not manipulate the coefficients. It is also usually faster and can be combined with other modular methods for computations in local rings. The algorithm is implemented in the computer algebra system Singular and we present several examples illustrating its power. standard bases; algorithm for zero-dimensional ideals; semicontinuity; highest corner Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Singularities in algebraic geometry, Effectivity, complexity and computational aspects of algebraic geometry Using semicontinuity for standard bases computations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 reproduces, with the addition of an introduction and some notes to make it more readable and up to date, a 1986 letter from the author to \textit{W. Messing}, in which he proves, as an application of the comparison theorem between crystalline cohomology and \(p\)-adic étale cohomology [cf. \textit{J.-M. Fontaine} and \textit{W. Messing}, in: Current trends in arithmetical algebraic geometry, Proc. Summer Res. Conf., Arcata, Contemp. Math. 67, 179-207 (1987; Zbl 0632.14016)], the following:
Theorem 1. Let \({\mathcal X}\) be a proper and smooth scheme over \(\mathbb{Z}\) and let \(X = {\mathcal X} \otimes \mathbb{Q}\) (or, slightly more generally, let \(X\) be a proper and smooth scheme over \(\mathbb{Q}\) with good reduction everywhere). Then \(H^j (X, \Omega^i_X) = 0\) whenever \(i,j \in \mathbb{N}\) satisfy \(i \neq j\) and \(i + j \leq 3\).
This theorem generalizes a previous result of the author on the non- existence of abelian varieties over \(\mathbb{Z}\) to higher dimensions [\textit{J.-M. Fontaine}, Invent. Math. 81, No. 3, 515-538 (1985; Zbl 0612.14043)], and, as was the case in that paper, the author begins by giving a bound for the different of certain field extensions. He starts with \(K\), a field of characteristic 0, complete for a discrete valuation, with perfect residue field of characteristic \(p > 0\) and absolutely unramified. If we take \(\overline K\) to be an algebraic closure of \(K\) and \(G_K = \text{Gal} (\overline K/K)\), he proves the following.
Theorem 2. Let \(V\) be a crystalline \(p\)-adic representation of \(G_K\) whose Hodge-Tate weights are in \([0,r]\), where \(r\) is an integer, \(0 < r < p - 1\). Let \(U\) be a subquotient of \(V\), stable under \(G_K\) and killed by \(p\). Let \(H\) be the kernel of the action of \(G_K\) on \(U\) and \(L = \overline K^H\). If \(v_0\) denotes the valuation on \(L\) normalized so that \(v_0 (p) = 1\), and if \({\mathcal D}_{L/K}\) is the different of the extension \(L/K\), then \(v_0 ({\mathcal D}_{L/K}) \leq 1 + r/(p - 1)\).
In the proof of this theorem a key role is played by the comparison functor between filtered Dieudonné modules and crystalline Galois representations. -- The author uses this bound, together with the general lower bounds for the \(n\)-th root of the discriminant of a field extension of degree \(n\) found by Diaz y Diaz, using the method of Odlyzko-Poitou- Serre, to study the category of 7-adic finite-dimensional representations of \(G =\text{Gal} (\overline \mathbb{Q}/ \mathbb{Q})\), unramified outside 7 and such that, when seen as representations of \(\text{Gal} (\overline \mathbb{Q}_7/ \mathbb{Q}_7)\), they are crystalline with Hodge-Tate weights between 0 and 3. He proves in particular the following proposition: Let \(V\) be such a representation and \(\chi\) be the cyclotomic character. -- Put \(V_4 = 0\) and for \(i = 0,1,2,3\), \(V_i = \{v \in V |gv - \chi^i (g)v \in V_{i + 1}\) for all \(g \in G\}\). Then \(V_0 = V\).
Theorem 1 follows from this proposition, applied to the representation dual to the one given by the action of \(G\) on \(H^m_{\text{ét}} (X \otimes \overline \mathbb{Q}, \mathbb{Q}_7)\), \(m \in \{1,2,3\}\), which is explicitly related to de Rham cohomology through the comparison theorem of Fontaine-Messing. Note that, as we have pointed out, the relation between filtered modules and Galois representations figures prominently in the proofs of the two main results of the paper. -- The author mentions that results similar to those in this paper, and in particular a generalization of theorem 2, have been obtained independently by \textit{V. A. Abrashkin} [see Invent. Math. 101, No. 3, 631-640 (1990; Zbl 0761.14006) and references therein]. comparison theorem between crystalline cohomology and \(p\)-adic étale cohomology; vanishing theorem; 7-adic finite-dimensional representations Jean-Marc Fontaine, Schémas propres et lisses sur \({\mathbf Z}\), Proceedings of the Indo-French Conference on Geometry (Bombay, 1989), Hindustan Book Agency, Delhi, 1993, pp. 43-56 (French). \(p\)-adic cohomology, crystalline cohomology, Étale and other Grothendieck topologies and (co)homologies, Local ground fields in algebraic geometry Schemes which are proper and smooth over \(\mathbb{Z}\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 establishes the basic theory on piecewise polynomial functions that are \(C^r\) with different smoothness degree \(r\) over the whole domain, and presents the theory and methods for solving zero-dimensional parametric piecewise polynomial systems. It shows that solving such a system amounts to solving \(m\) parametric semi-algebraic systems, which is then reduced to the computation of \(m\) discriminant varieties. Here, \(m\) is the number of \(n\)-dimensional cells in the hereditary partition of the domain. The parametric piecewise polynomial system is ultimately solved utilizing the critical points method and the Collins partial cylindrical algebraic decomposition method.
The paper also proposes a classification method and its algorithm to address whether there exists an open set in the parameter domain such that, for each point in the open set, the corresponding zero-dimensional non-parametric piecewise polynomial system (a realization of the parametric piecewise polynomial system) has exactly the given number of torsion-free real zeros in the \(m\) cells respectively. piecewise polynomial; parametric piecewise polynomial system; parametric semi-algebraic systems; discriminant variety; number of real zeros; cylindrical algebraic decomposition method; algorithm Lai Y S, Wang R H, Wu J M. Solving parametric piecewise polynomial systems. J Comput Appl Math, 2011, 236: 924--936 Numerical computation of solutions to systems of equations, Computational aspects of algebraic surfaces, Numerical computation of solutions to single equations, Solving polynomial systems; resultants Solving parametric piecewise polynomial systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The theory of DAHA-Jones polynomials is extended from torus knots to iterated torus knot, for any reduced root systems and weights. This is inspired by \textit{P. Samuelson}'s construction for the \(\mathfrak{sl}_2\) case [``Iterated torus knots and double affine Hecke algebras'', Preprint, \url{arXiv:1408.0483}]. The paper proves polynomiality, duality and other properties, and computes several examples. They conjecture that these polynomials specialize to Khovanov-Rozansky polynomials, which was since proven by \textit{H. Morton} and \textit{P. Samuelson} [Duke Math. J. 166, No. 5, 801--854 (2017; Zbl 1369.16034)].
The same authors have since extended the DAHA-Jones polynomials to iterated torus links [\textit{A. Beliakova} (ed.) and \textit{A. D. Lauda} (ed.), Categorification in geometry, topology, and physics. Providence, RI: American Mathematical Society (AMS) (2017; Zbl 1362.81007)]. double affine Hecke algebra; Jones polynomials; HOMFLY-PT polynomial; Khovanov-Rozansky homology; iterated torus knot; cabling; MacDonald polynomial; plane curve singularity; generalized Jacobian; Betti numbers; Puiseux expansion Cherednik, I.; Danilenko, I., DAHA and iterated torus knots, Algebr. Geom. Topol., 16, 843-898, (2016) Singularities of curves, local rings, Knots and links in the 3-sphere, Hecke algebras and their representations, Braid groups; Artin groups, Compact Riemann surfaces and uniformization, Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.), Lie algebras of linear algebraic groups, Singular homology and cohomology theory DAHA and iterated torus knots | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The present paper develops new algorithms for computing zeta functions of algebraic varieties over finite fields. It builds on previous work of the author for hyperelliptic curves and improves substantially on previously known algorithms if it comes to complexity, both on the time- and space-scale. Quite surprisingly, the algorithms do not require very sophisticated machinery, such as \(p\)-adic or \(\ell\)-adic cohomology theories. There are also applications of zeta functions of schemes over \(\mathbb Z\). zeta function; algorithm; arithmetic scheme; hypersurface Harvey, David, Computing zeta functions of arithmetic schemes, Proc. Lond. Math. Soc. (3), 111, 6, 1379-1401, (2015) \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture, Zeta and \(L\)-functions in characteristic \(p\), Number-theoretic algorithms; complexity, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Computing zeta functions 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 Let \(\Delta\) be the partition of a region in affine space into a finite number of polyhedral cells. A \textit{multivariate spline} with smoothness \(\mu\) over \(\Delta\) is a \(C^\mu\)-function whose restriction to each cell of \(\Delta\) is a polynomial. A \(C^\mu\) \textit{piecewise algebraic variety} is the zero set of a collection of multivariate splines with smoothness \(\mu\). The article serves primarily as a review of the algebraic structure of multivariate spline spaces and of the concepts from algebraic geometry that translate to the more general setting of multivariate splines, namely the correspondence between ideals and varieties. In addition, although the Hilbert Nullstellensatz does not hold in general for splines, the authors provide some instances in which it does.
The article targets a general mathematical audience familiar with the basic concepts of algebraic geometry. piecewise algebraic varieties; algebraic varieties; multivariate splines; ideals Numerical computation using splines, Spline approximation, Polynomial rings and ideals; rings of integer-valued polynomials, Computational aspects of algebraic curves The correspondence between multivariate spline ideals and piecewise algebraic varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Quantum \(\mathbb P^n\)s are regular algebras of global dimension \(n+1\), and are considered to be noncommutative analogs of the polynomial ring in \(n+1\) variables. The quantum \(\mathbb P^2\)s have been classified by \textit{M. Artin} et al. [Prog. Math. 86, 33--85 (1990; Zbl 0744.14024)] by means of point schemes. However, the classification of quantum \(\mathbb P^3\)s is still unknown. Motivated by the classifcation of quantum \(\mathbb P^2\)s, the authors propose that the identification of the point schemes and line schemes that arise from quantum \(\mathbb P^3\)s should be the first step toward completing the classification of quantum \(\mathbb P^3\)s. Note that most regular algebras of global dimension \(4\) are quadratic, so attention is restricted to quadratic quantum \(\mathbb P^3\)s, ie. quadratic, Noetherian, AS-regular algebras with Hilbert series \((1-t)^{-4}\).
The current article computes the line schemes of a specific family of quadratic quantum \(\mathbb P^3\)s in order to supplement a lack of examples in the current literature and to validate certain predictions on the generic structure of quadratic quantum \(\mathbb P^3\)s. Van den Bergh has proved that generically quadratic quantum \(\mathbb P^3\)s have a point scheme consisting of twenty distinct points and a one-dimensional line scheme. Furthermore, explicit calculation of the line schemes of a certain family of quadratic quantum \(\mathbb P^3\)s lead \textit{R. G. Chandler} and the second author [J. Algebra 439, 316--333 (2015; Zbl 1348.14005)] to conjecture that the line scheme of a generic quadratic quantum \(\mathbb P^3\) consists of a union of two degree-four spatial elliptic curves and four planar elliptic curves. New in this article, the authors provide support for this conjecture by computing the line schemes of a \(1\)-parameter family of algebras \(\mathcal A(\alpha)\) which are quadratic quantum \(\mathbb P^3\)s. While the line schemes obtained from this family are not unions of elliptic curves as conjectured, they are in fact degenerations of the conjectured collections of planar and spatial elliptic curves. As such, the \(\mathcal A(\alpha)\) may reasonably be expected to be not generic but rather limits of families of generic curves. In this way, the conjecture is supported by the results of the paper. However, further calculation of certain line schemes of other families of quadratic quantum \(\mathbb P^3\)s have led the authors to modify the conjecture to include the possibility of four degree-four spatial elliptic curves and two nonsingular conics. The updated conjecture appears as Conjecture 4.3.
The algebras \(\mathcal A(\alpha)\) are specific cases of the generalized graded Clifford algebras constructed by Cassidy and Vancliff as possible generic quantum \(\mathbb P^3\)s. \(\mathcal A(\alpha)\) is presented explicitly as a quotient of the free algebra in four variables by six quadratic relations. The point scheme of \(\mathcal A(\alpha)\) is computed following the method of Artin et al. [loc. cit.] and shown to consist of twenty distinct points. The line scheme is computed following the method originally introduced by \textit{B. Shelton} and the second author [Commun. Algebra 30, No. 5, 2535--2552 (2002; Zbl 1056.14002)], which identifies it with the zero set of fourty-five quartic polynomials and one quadratic polynomial in six variables. Mathematica is used to compute these polynomials along with a Gröbner basis. Using this, the authors identify the line scheme with a union of eight irreducible curves in \(\mathbb P^5\): two lines, two nonsingular conics, two planar elliptic curves, one spatial elliptic curve, and one spatial rational curve with a singular point. The singular rational curve is viewed as a degeneration of a spatial elliptic curve, while each line + conic pair is viewed as a degeneration of a planar elliptic curve. Additionally, the authors investigate the intersection points of the irreducible components of the line scheme as well as the lines in the line scheme which contains points of the point scheme. In particular, they show that four points of the point scheme lie on infinitely many points of the line scheme, while the remaining \(16\) points of the point scheme lie on exactly six distinct lines of the line scheme, counting multiplicity. line scheme; point scheme; elliptic curve; regular algebra; Plücker coordinates Tomlin, D., Vancliff, M.: The one-dimensional line scheme of a family of quadratic quantum \({\mathbb{P}}^{3}\)s (2017) \textbf{(preprint)}. arXiv:1705.10426 Noncommutative algebraic geometry, Quadratic and Koszul algebras, Rings arising from noncommutative algebraic geometry The one-dimensional line scheme of a family of quadratic quantum \(\mathbb{P}^3\)s | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 Riemann surface and \(E\to X\) be a fixed smooth complex bundle. A coherent system on \(E\) is a pair \(({\mathcal E},V)\), where \({\mathcal E}\) is a holomorphic bundle isomorphic to \(E\), and \(V\) is a linear subspace of \(H^0(X, {\mathcal E})\).
After recalling the notions of \(\alpha\)-stability for \(({\mathcal E},V)\) \((\alpha>0)\) and the orthogonal vortex equations depending on \(\alpha >0\) and \(\tau\in R\), the authors prove the following theorem, which is in a sense the converse to the authors' previous result [the authors and \textit{G. Daskalopoulos} and \textit{R. Wentworth}, Lond. Math. Soc. Lect. Note Ser. 208, 15-67 (1995; Zbl 0827.14010)]: Let \(({\mathcal E},V)\) be an \(\alpha\)-stable coherent system for some \(\alpha>0\). Then there is a smooth solution to the orthonormal vortex equations on \(({\mathcal E},V)\) for a suitable \(\tau\). \(\alpha\)-stability; coherent system DOI: 10.1112/S002461079900767X Holomorphic bundles and generalizations, Algebraic moduli problems, moduli of vector bundles, Kähler-Einstein manifolds A Hitchin-Kobayashi correspondence for coherent systems on Riemann surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For a smooth projective surface \(V\), let \(\mathrm{Hilb}^m_V\) denote the scheme parametrizing effective divisors \(D\) in \(V\) such that its first Chern class \(c_1({\mathcal O}_V(D))\) equals the fixed class \(m\in H^2(V,{\mathbb Z})\). In an earlier paper [Topology, 46, No. 3, 225--294 (2007; Zbl 1120.14034)] the authors constructed the virtual class \([[\mathrm{Hilb}^m_V]]\), which in the present paper is considered as a homology class by the cycle map.
Let \(C\subset V\) be an integral curve, and let \(c\) denote its first Chern class. By adding the curve to the divisors one gets a closed embedding \(\mathrm{Hilb}^{m}_V \to \mathrm{Hilb}^{m+c}_V\). The main result of the article relates the classes \([[\mathrm{Hilb}^{m-c}_V]]\) and \([[\mathrm{Hilb}^m_V]]\) when \(m\cdot c <0\), and the classes \([[ \mathrm{Hilb}^m_V]]\) and \([[\mathrm{Hilb}^{m+c}_V]]\) when \((k-m)\cdot c<0\), where \(k\) is the first Chern class of the tautological line bundle of the surface. Hilbert scheme; virtual fundamental class Parametrization (Chow and Hilbert schemes), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Relations for virtual fundamental classes of Hilbert schemes of curves on surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(K\) be an algebraically closed field of characteristic \(0,\) \(S=K[x_0,\dots,x_n]\) and \({\mathbb P}^n_K=\mathrm{Proj} S.\) Let us consider on the set of the terms of \(S\) a degreverse order. The authors prove that in the Hilbert scheme of points in \({\mathbb P}^n_K,\) the point corresponding to a segment ideal, with respect to a degreverse order, is singular. Unfortunately this result cannot be generalized to Hilbert schemes with a Hilbert polynomial of a positive degree.
Moreover they provide an algorithm for computing all the saturated Borel ideals with a given Hilbert polynomial. Hilbert scheme of points; Borel ideal; segment ideal; Gröbner stratum; degrevlex term order; Gotzmann number Cioffi, F.; Lella, P.; Marinari, M. G.; Roggero, M., Segments and Hilbert schemes of points, Discrete Math., 311, 20, 2238-2252, (2011) Parametrization (Chow and Hilbert schemes), Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) Segments and Hilbert schemes of points | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let X be a variety defined over an algebraically closed field k, \({\mathcal F}\) a coherent sheaf on X. For each \(x\in X\) and Schubert symbol a the local Schubert multiplicity \(e_ x({\mathcal F},a)\) is defined. If \({\mathcal F}=\Omega^ 1\!_ X\), one can express the local Euler obstruction of X at x as an alternating sum of these multiplicities. A formula is given, which yields in a special case one of \textit{Lê Dung Trang} and \textit{B. Teissier} [Ann. Math., II. Ser. 114, 457-491 (1981; Zbl 0488.32004)], thus generalizing this formula to varieties of arbitrary characteristic. local Euler obstruction; blowing-up; Borel-Moore cohomology; Grassmannian; coherent sheaf; local Schubert multiplicity Grassmannians, Schubert varieties, flag manifolds, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Singularities in algebraic geometry Sur les multiplicités de Schubert locales des faisceaux algébriques cohérents | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 and denote with \(S^{[n]}\) the Hilbert scheme that parameterizes length \(n\) subschemes of \(S\). There exists a natural map from \(S^{[n]}\) to the symmetric power \(S^{(n)}\), whose fibers over multiplicity \(n\) cycles define the \textit{punctual Hilbert scheme} \(P_n\). In other words, \(P_n\) parameterizes subschemes of length \(n\) supported at one point. The authors study the tangent space \(T_\xi\) to \(P_n\) at a scheme \(\xi\in P_n\). The dimension \(\dim(T_\xi) \) is bounded below by the corank of the normal map \(\alpha_{n,\xi}\) which sends \(H^0(S, T_{S|\xi})\) to Hom\((\mathcal I_\xi,\mathcal O_\xi)\). The authors prove that when the ideal \(I_\xi\) of \(\xi\) is a monomial ideal, then \(\dim(T_\xi) \) is indeed equal to the corank of \(\alpha_{n,\xi}\). They also show how, for schemes \(\xi\) defined by monomials, the corank of \(\alpha_{n,\xi}\) can be computed from the Young diagram associated to \(I_\xi\). Hilbert scheme Parametrization (Chow and Hilbert schemes) The tangent space of the punctual Hilbert scheme | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors develop the Littlewood-Richardson homotopy algorithm. For this purpose, they use numerical continuation to compute solutions to Schubert problems on Grassmannians and it is based on the geometric Littlewood-Richardson rule. The main idea of the algorithm is the new optimal formulation of Schubert problems in local Stiefel coordinates as systems of equations. The implementation presented can solve problem instances with tens of thousands of solutions. Schubert calculus; Grassmannian; Littlewood-Richardson rule; numerical homotopy continuation Classical problems, Schubert calculus, Numerical computation of solutions to systems of equations, Geometric aspects of numerical algebraic geometry Numerical Schubert calculus via the Littlewood-Richardson homotopy algorithm | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Maps between \(k\)-algebras \(f:B\rightarrow C\) are considered to be [\textit{first-order infinitesimal}] \textit{neighbours} if \((f(a)-g(a))(f(b)-g(g))=0\) for all \(a,b\). If \(2\in k\) is invertible, this is equivalent to \((f(a)-g(a))^2\) being zero. This paper gives an equivalent formulation of this property in terms of ideals, and uses this to study the neighbour property where \(B\) is a polynomial algebra.
A \((p+1)\)-tuple of mutually neighbouring algebra maps is defined to be an \textit{infinitesimal \(p\)-simplex}. It is shown that affine combinations of such algebra maps are themselves algebra maps (which is not true in the general case), and that two such affine combinations are themselves neighbours. Results are extended to vectors with entries from \(C\), and to vector spaces over a ring of scalars. first neighbourhood of the diagonal; neighbour points; affine schemes; affine combinations Infinitesimal methods in algebraic geometry, Formal neighborhoods in algebraic geometry, Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem), Synthetic differential geometry Affine combinations in affine schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We show that the weights on a tropical variety can be recovered from the tropical scheme structure proposed in [GG16], so there is a well-defined Hilbert-Chow morphism from a tropical scheme to the underlying tropical cycle. For a subscheme of projective space given by a homogeneous ideal \(I\) we show that the Giansiracusa tropical scheme structure contains the same information as the set of valuated matroids of the vector spaces \(I_d\) for \(d \geq 0\). We also give a combinatorial criterion to determine whether a given relation is in the congruence defining the tropical scheme structure. tropical scheme; valuated matroid Geometric aspects of tropical varieties, Combinatorial aspects of matroids and geometric lattices, Foundations of tropical geometry and relations with algebra, Combinatorial aspects of tropical varieties Tropical schemes, tropical cycles, and valuated matroids | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \({\mathcal H}(d,g,r)\) denote the Hilbert scheme of smooth curves of degree \(d\) and genus \(g\) in the complex projective space \(\mathbb{P}^r\). The authors are primarily interested in the Hilbert scheme of curves of low gonality and its geometric properties. In particular, they examine the so-called Brill-Noether-Petri properties for the Hilbert scheme \({\mathcal H}(2g-8,g,g-5)\) in section 2. In section 3, they look for the components of Hilbert schemes which violates the Brill-Noether-Petri properties. In fact they exhibit extra components of Hilbert schemes with an open dense subset considering of \(k\)-gonal curves. Finally, they look at the Hilbert scheme consisting of nearly extremal curves; they give an example of a Hilbert scheme with two components consisting of smooth curves which achieve nearly maximal genus with respect to the degree in projective space \(\mathbb{P}^r\). Brill-Noether-Petri properties Special divisors on curves (gonality, Brill-Noether theory), Parametrization (Chow and Hilbert schemes) On the Hilbert scheme of trigonal curves and nearly extremal 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 construct a descent-of-scalars criterion for \(K\)-linear abelian categories. Using advances in the Langlands correspondence due to \textit{T. Abe} [J. Am. Math. Soc. 31, No. 4, 921--1057 (2018; Zbl 1420.14044)], we build a correspondence between certain rank 2 local systems and certain Barsotti-Tate groups on complete curves over a finite field. We conjecture that such Barsotti-Tate groups ``come from'' a family of fake elliptic curves. As an application of these ideas, we provide a criterion for being a Shimura curve over \(\mathbb{F}_{q}\). Along the way we formulate a conjecture on the field-of-coefficients of certain compatible systems. Shimura curves; Barsotti-Tate groups; local systems Finite ground fields in algebraic geometry, Modular and Shimura varieties, Arithmetic ground fields for curves Rank 2 local systems, Barsotti-Tate groups, and Shimura 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 border basis schemes provide a nice open cover of the Hilbert schemes parametrising zero-dimensional schemes in an affine space over a field \(K\). Hence, such Hilbert schemes can be investigated by means of this open cover. Since border basis schemes can be explicitly computed, this approach gives a computational perspective to this kind of investigation.
The authors of this paper face the interesting problem of determining the defining equations of particular loci in a border basis scheme. They deal with the loci consisting of the \(K\)-rational points that are characterized by one of the following properties: locally Gorenstein, strict Gorenstein, strict complete intersection, Cayley-Bacharach, and strictly Cayley-Bacharach. Concerning the Cayley-Bacharach property, a generalization is considered of the classical definition for zero-dimensional reduced schemes to every zero-dimensional scheme. Explicit algorithms that compute the defining ideals of these loci are exhibited. All these loci turn out to be constructible subsets.
More precisely, except for the locally Gorenstein property (see Section 4), the authors compute the considered loci in a stratification of a border basis scheme, in which every stratum consists of schemes with a specific affine Hilbert function (see Sections 5 and 10). In the case the affine Hilbert function is the same as the affine Hilbert function of the order ideal of the border basis scheme, the stratum is characterized by a suitable shape of the border bases of its schemes (see Proposition 5.3). The authors also highlight that this stratum is the positive Bialinicki-Birula decomposition of the border basis scheme. Several useful and explicit examples are given throughout the paper. zero-dimensional ideal; border basis; border basis scheme; Cayley-Bacharach property; Gorenstein ring; strict complete intersection Linkage, complete intersections and determinantal ideals, Complete intersections, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Parametrization (Chow and Hilbert schemes), Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) Computing subschemes of the border basis 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 study the structure of the relative Hilbert scheme for a family of nodal (or smooth) curves via its natural cycle map to the relative symmetric product. We show that the cycle map is the blowing up of the discriminant locus, which consists of cycles with multiple points. We discuss some applications and connections, notably with birational geometry and intersection theory on Hilbert schemes of smooth surfaces. Revised version corrects some minor errors. Z. Ran, \textit{Cycle map on Hilbert schemes of nodal curves}, in \textit{Projective Varieties with Unexpected Properties}, Walter De Gruyter, Berlin, 2005, pp. 363-380. Families, moduli of curves (algebraic), Parametrization (Chow and Hilbert schemes) Cycle map on Hilbert schemes of nodal 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 From the Abstract. ``This article contains a survey of a new algebro-geometric approach for working with iterated algebraic loop groups associated with iterated Laurent series over arbitrary commutative rings and its applications to the study of the higher-dimensional Contou-Carrère symbol. In addition to the survey, the article also contains new results related to this symbol.''
The detailed survey starts with several motivations for studying Contou-Carrère symbols. There are relations with tame symbols, Weil reciprocity, higher dimensional class field theory, deformations of a flag of subvarities. (An example of a flag would be a point lying on a curve lying on a surface inside a given variety.)
For a commutative ring \(A\) let \(\mathcal L A=A((t))=A[[t]][t^{-1}]\) be the ring of Laurent series over \(A\) with its usual ``\(t\)-adic'' topology. More generally, let \(\mathcal L^n A=A((t_1))\cdots ((t_n))\), with the appropriate topology. Let \(n\geq1\). (In the original Contou-Carrère symbol \(n\) equals one.) The \(n\)-dimensional Contou-Carrère symbol \(\mathrm{CC}_n\) associates to \(f_0\), \dots, \(f_n\) in \((\mathcal L^n A)^*\) an element of \(A^*=\mathbb G_m(A)\). There are several constructions of \(\mathrm{CC}_n\). The symbol is multilinear and antisymmetric. It satisfies the Steinberg relations: If \(f_i+f_{i+1}=1\) for some \(i\), then \(\mathrm{CC}_n(f_0,\dots,f_n)=1\). Thus it factors through the Milnor \(K\)-group \(K_{n+1}^M(\mathcal L^n A)\), where the Milnor \(K\)-group is defined as if \(\mathcal L^n A\) were a field. Such a definition is reasonable because \(\mathcal L^n A\) has enough units.
If \(F\) is a functor defined on the category of commutative \(R\)-algebras over some commutative ring \(R\), then \(L^nF\) denotes the \(n\)-iterated loop functor \(F\circ \mathcal L^n\). So the \(n\)-dimensional Contou-Carrère symbol may be viewed as describing a morphism of functors from \(L^nK_{n+1}^M\) to \(\mathbb G_m\). Instead of \(\mathbb G_m\) one may take another group scheme over a ring \(R\), but it is shown that any Contou-Carrère like symbol factors through \(\mathbb G_m\) under mild hypotheses on the target group scheme.
Higher algebraic \(K\)-theory yields a boundary map \(\partial:L^nK_{n+1}\to L^{n-1}K_n\) in the appropriate localisation sequence. Iterating this, we get a map \(L^nK_{n+1}\to K_1\). Now \(\mathrm{CC}_n\) may be defined as the composite
\[
L^nK_{n+1}^M\to L^nK_{n+1}\to K_1\overset{\det}\to\mathbb G_m.
\]
Thus if \(a,b\in A^*\), then \(\mathrm{CC}_n(a,t_1,\cdots,t_n)=a\) and \(\mathrm{CC}_n(a,b,f_2\cdots,f_n)=1\). Inspired by residues in complex analysis one may define \(\mathrm{res}:\Omega^n_{\mathcal L^n A}\to A\). The residue of an absolute Kähler differential \(n\)-form \(\omega\in\Omega^n_{\mathcal L^n A}\) is ``the coefficient of \(\mathop\mathrm{dlog} t_1\wedge\cdots\wedge \mathop\mathrm{dlog}t_n\)''. Both \(\mathrm{CC}_n\) and the residue are invariant under continuous \(A\)-linear automorphisms of \({\mathcal L^n A}\).
If \(A\) is a \(\mathbb Q\)-algebra, and \(f_0\in A[[t_1]]\cdots[[t_n]]\subset \mathcal L^n A\) has constant term \(1\), then
\[
\mathrm{CC}_n(f_0,\dots,f_n)= \exp \mathrm{res}\left(\log f_0 \frac{df_1}{f_1}\wedge \cdots \wedge\frac{df_n}{f_n}\right).
\]
\par Expanding this formula in terms of the nonzero coefficients of the \(f_i\) one encounters no denominators and no infinite sums. The resulting expression is still valid when \(A\) is no \(\mathbb Q\)-algebra. This is one of the main results proved in the paper. There is a more elaborate formulation that is relevant to the case that \(A\) has nilpotents or nontrivial idempotents.
The authors develop a theory that can deal with \((\mathcal L^n A)^*\) despite the fact that it does not seem to be described by an ind-flat scheme. They introduce so-called thick ind-cones, with coordinate rings that embed into power series rings in infinitely many variables. As an application of the theory an invertibility criterion is given for continuous endomorphisms of \(\mathcal L^n A\). When 2 is invertible in \(A\) they show that the residue is the tangent to the map \(\mathrm{CC}_n\). group schemes; iterated Laurent series over rings; higher-dimensional Contou-Carrère symbol; higher-dimensional Witt pairing; Milnor \(K\)-group of ring Higher symbols, Milnor \(K\)-theory, Power series rings, Group schemes, Generalized class field theory (\(K\)-theoretic aspects), Formal power series rings Iterated Laurent series over rings and the Contou-Carrère symbol | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(X/\mathbb{Q}\) be a \(K3\) surface. In a previous paper the authors indentified two consequences of real and of complex multiplication for the zeta function of the reductions of \(X\) modulo primes of good reduction. In this paper the authors aim to check these consequence for particular examples. In order to do this they need an efficient algorithm to determine the zeta function of a \(K3\) surface over a finite field.
More concretely, the authors present a list of possible families of \(K3\) surfaces with real or with complex mulitplication. Then for various members of these families they calculate the zeta function and check that the surface satisfy the above mentioned consequences. The authors use a variant of a method by Harvey's to calculate the zeta function. K3 surfaces; real multiplication; complex multiplication; zeta function \(K3\) surfaces and Enriques surfaces, Complex multiplication and moduli of abelian varieties, Number-theoretic algorithms; complexity, Zeta and \(L\)-functions in characteristic \(p\) Point counting on \(K3\) surfaces and an application concerning real and complex 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 An efficient algorithm for computing the branching structure of an algebraic curve is given. To describe the branching structure one has to specify which sheets of the covering are connected in which way at a given branch point, i.e. one has to identify the monodromy.
Generators of the fundamental group of the base of the ramified covering punctured at the discriminant points of the curve are constructed via a minimal spanning tree of the discriminant points. This leads to paths of minimal length between the points. The branching structure is obtained by analytically continuing the roots of the equation defining the curve along the previously constructed generators of the fundamental group. Riemann surfaces; algebraic curves; monodromy; foundamental group Frauendiener J, Klein C and Shramchenko V 2011 Efficient computation of the branching structure of an algebraic curve \textit{Comput. Methods Funct. Theory}11 527--46 Computational aspects of algebraic curves, Riemann surfaces; Weierstrass points; gap sequences Efficient computation of the branching structure of an algebraic curve | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This title introduces the theory of arc schemes in algebraic geometry and singularity theory, with special emphasis on recent developments around the Nash problem for surfaces. The main challenges are to understand the global and local structure of arc schemes, and how they relate to the nature of the singularities on the variety. Since the arc scheme is an infinite dimensional object, new tools need to be developed to give a precise meaning to the notion of a singular point of the arc scheme.
Other related topics are also explored, including motivic integration and dual intersection complexes of resolutions of singularities. Written by leading international experts, it offers a broad overview of different applications of arc schemes in algebraic geometry, singularity theory and representation theory.
The articles of this volume will be reviewed individually. Research exposition (monographs, survey articles) pertaining to algebraic geometry, Arcs and motivic integration, Local theory in algebraic geometry, Collections of articles of miscellaneous specific interest Arc schemes and singularities | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We determine the irregular Hodge filtration, as introduced by Sabbah, for the purely irregular hypergeometric \(\mathcal{D}\)-modules. We obtain, in particular, a formula for the irregular Hodge numbers of these systems. We use the reduction of hypergeometric systems from GKZ-systems as well as comparison results to Gauss-Manin systems of Laurent polynomials via Fourier-Laplace and Radon transformations. D-modules; irregular Hodge filtration; hypergeometric systems; twistor D-modules Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Sheaves of differential operators and their modules, \(D\)-modules Irregular Hodge filtration of some confluent hypergeometric 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 By results of \textit{M. Kashiwara} [Lect. Notes Math. 1016, 134--142 (1983; Zbl 0566.32022)] and \textit{B. Malgrange} [Astérisque 101--102, 243--267 (1983; Zbl 0528.32007)] the roots of the Bernstein-Sato polynomial of a hypersurface give the monodromy eigenvalues on the Milnor fibre. This paper describes a generalisation to the case of a collection of polynomials. The author conjectures that the Bernstein-Sato ideal is generated by products of linear polynomials of the form \(\alpha_1 s_1+\dots+\alpha_r s_r +\alpha\), with \(\alpha_i\in \mathbb{Q}_{\geq0}\) and \(\alpha\in \mathbb{Q}_{>0}\). He first arrived at it from computing examples in \textsc{Singular}, and the paper evolved out of the effort to understand this behaviour.
Let \(F=(f_1,\dots,f_r)\) be a collection of polynomials on \(X=\mathbb{C}^n\), to be considered locally at a point \(x\in X\), with zero set \(D=\bigcup V(f_j)\). In this situation the generalisation of Deligne's nearby cycle functor is the Sabbah specialisation functor \(\psi_F\). It is shown that the uniform support \(\text{Supp}_x^{\text{unif}}(\psi_F\mathbb{C}_X)\) is a finite union of torsion translated subtori of \((\mathbb{C}^*)^r\). The conjecture for the zero set of the Bernstein-sato ideal \(B_{F,x}\) is now that \(\exp(V( B_{F,x} ))=\bigcup_{y\in D \text{ near } x} \text{Supp}_y^{\text{unif}}(\psi_F\mathbb{C}_X)\). One direction is proved and a significant step towards the converse is made. The support of the Sabbah specialisation complex is related to the cohomology support locus of the complement of \(D\) in a small open ball around \(x\), involving the local systems from the title.
The author also discusses a multi-variable version of the monodromy conjecture, which is shown to follow from the usual single-variable one, and proves it in the case of hyperplane arrangements. Bernstein-Sato ideal; local systems; cohomology jump loci; characteristic variety; Sabbah specialization; Alexander module; Milnor fiber; Monodromy Conjecture; hyperplane arrangements Budur, N, Bernstein-Sato ideals and local systems, Ann. Inst. Fourier (Grenoble), 65, 549-603, (2015) Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Relations with arrangements of hyperplanes Bernstein-Sato ideals and local systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Consider a linear block code \(C\subset \mathrm{GF}(q)^n\) of length \(n\), dimension \(k\), and minimum distance \(d\) over a finite field \(\mathrm{GF}(q)\), where \(q\) is a prime power. Coordinates of a codeword \(c\in C\) are denoted \(c=(c_1,c_2,\ldots,c_n)\). We say \(i\in \{1,2,\ldots,n\}\) is \textit{locally recoverable with locality \(r\)} if there is a set \(R_i\subset \{1,2,\ldots,n\}-\{i\}\) of cardinality \(r\) with the property: for all codewords \(u,v\in C\), \(pr_i(u)=pr_i(v)\) implies \(u_i=v_i\) (where \(pr_i\) is the projection on the coordinates of \(R_i\)). For example, if \(C\) has a generator matrix in standard form then for any coordinate \(i>k\) is determined by the information bits in the first \(k\) coordinates, so we may take \(R_i=\{1,2,\dots,k\}\). The code \(C\) is called \textit{locally recoverable with locality \(r\)} if every coordinate is locally recoverable with locality at most \(r\). Locally recoverable codes (LCRs) were introduced relatively recently due to their application to cloud storage systems.
The paper under review considers the question: which linear codes \(C\) are LCR codes with ``small'' \(r\)? Along with the upper bound \(r\leq k\), there is a lower bound provided by the Singleton-like inequality
\[ k+d+\left\lceil \frac{k}{r}\right\rceil \leq n+2. \]
MDS codes have \(r=k\). A. Barg and co-authors have introduced LRC analogues of RS codes and AG codes. This paper under review studies LRC AG codes arising from ``separated varibles'' curves defined by polynomial equations over \(\mathrm{GF}(q)\) with separated variables, \(A(y)=B(x)\). For example, an elliptic curve can be modeled by such an equation.
We recall the AG code construction: Let \(X\) be a (projective, non-singular, absolutely irreducible, algebraic) curve of genus \(g\) defined over the field \(\mathrm{GF}(q)\). Let \(P = \{P_1 , \dots , P_n \} \subset X(\mathrm{GF}(q))\) be a set of \(n\) rational distinct points, \(D = P_1 + \dots + P_n\), and let \(G\) be a rational divisor with support disjoint from \(P\). The AG code \(C(P, L(G))\) is defined as \(C(P, L(G)) = ev_P (L(G))\), where \(L(G)\) is the Riemann-Roch space associated to \(G\) and \(ev_P\) is the evaluation at \(P\) map, \(ev_P (f ) = (f (P_1 ), \dots , f (P_n ))\).
We can construct LRC AG codes from algebraic curves as follows: let \(X\), \(Y\) be two algebraic curves over \(\mathrm{GF}(q)\) and let \(\phi\colon X \to Y\) be a rational separable morphism of degree \(r + 1\). Take a set \(U\subset Y(\mathrm{GF}(q))\) of rational points with totally split fibres and let \(P = \phi^{-1}(U)\). By the separability of \(\phi\) there exists \(x \in \mathrm{GF}(q) (X )\) satisfying \(\mathrm{GF}(q) (X ) = \mathrm{GF}(q) (Y)(x)\). Let
\[ V = \left\{ \sum_{i=1}^{r-1}\sum_{j=1}^m a_{ij} f_j x^i \mid a_{ij} \in \mathrm{GF}(q) \right\} \]
where \(\{f_1 , \ldots , f_m \}\) is a basis of \(L(E)\). The LRC AG code \(C\) is defined as \(C = ev_P (V ) \subset \mathrm{GF}(q)^n\), with \(n = |P|\).
The following result was previously known from the work of Barg et al.
Theorem (Barg et al.): Let \(P\), \(V\) and \(r\) as above. If \(ev_P\) is injective on \(V\) then \(C \subset \mathrm{GF}(q)^n\) is a linear \([n, k, d]\) LRC code with parameters \(n = s(r + 1)\), \(k = r(E)\), \(d \geq n - \deg(E)(r + 1) - (r - 1) \deg(x)\) and locality \(r\).
Theorem 3.2, one of the main results of the article under review, is a very similar theorem, but valid for the class of ``separated variables'' curves studied in this paper. However, it is much too technical to state here. There are other interesting bounds on the parameters proven in the paper. For example, the author's Theorem 5.1 provides a generalization of the Singleton-like inequality stated above. We refer to the paper itself for more precise statements.
The remainder of the paper deals with decoding/recovery methods for such codes and numerous examples. See the paper itself for more precise statements of these results. error-correcting code; locally recoverable code; algebraic-geometric code Geometric methods (including applications of algebraic geometry) applied to coding theory, Curves over finite and local fields, Algebraic coding theory; cryptography (number-theoretic aspects), Applications to coding theory and cryptography of arithmetic geometry, Linear codes (general theory) Locally recoverable codes from algebraic curves with separated variables | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 a smooth projective curve of genus \(g\) can be reconstructed from its polarized Jacobian \((X, \Theta)\) as a certain locus in the Hilbert scheme \(\mathrm{Hilb}^{d}(X)\), for \(d = 3\) and for \(d = g+2\), defined by geometric conditions in terms of the polarization \(\Theta.\) The result is an application of the Gunning-Welters trisecant criterion and the Castelnuovo-Schottky theorem by Pareschi-Popa and Grushevsky, and its scheme theoretic extension by the authors. Schottky problem; Hilbert scheme; Jacobian; theta duality; trisecant criterion Theta functions and curves; Schottky problem, Parametrization (Chow and Hilbert schemes) Schottky via the punctual Hilbert scheme | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This article surveys the author's work on iterated integrals, Archimedean height pairings and their connection to algebraic cycles and the Abel--Jacobi map. First, he discusses iterated integrals. To do so, he considers three disjoint closed curves \(C_1\), \(C_2\), and \(C_3\) on a Riemann surface \(S\) with Jacobian \(J(S)\). Denote their Poincaré dual harmonic forms by \(\alpha_i\). Since the integral \(\int_S (\alpha_i \wedge \alpha_j) =0\), we have 1--forms \(\eta_{ij}\) with \(\alpha_i \wedge \alpha_j = d \eta_{ij}\). These forms \(\eta_{ij}\) become unique when we require that they are coexact (orthogonal to closed forms). He defines the iterated integral homomorphism to be
\[
I(\alpha_1,\alpha_2,\alpha_3):= \int_{C_3} [(\alpha_1,\alpha_2) - \eta_{12}],
\]
where \((\alpha_1,\alpha_2)\) is the iterated integral along \(C_3\). This yields a mapping from the primitive part \(PH^3(J(S),{\mathbb Z})\) of \(H^3(J(S),{\mathbb Z})\) to \({\mathbb R}/{\mathbb Z}\). We regard \(S\) as a cycle in \(J(S)\) and let \(S^-\) be the \((-1)\) involution applied to this cycle.
The author shows that the image of the cycle \(\nu(S-S^-)\) under the Abel--Jacobi map is (up to a factor 2) given by the iterated integral \(I\). Using integral estimates, this was used by the author to show that \((S-S^-)\) is not algebraically equivalent to zero. Next, the Archimedean height pairing is explained. Here he takes two disjoint cycles \(A\) and \(B\) on a Riemann manifold \(X\), such that dim\((A) + \) dim\((B)\) = dim\((X)-1\). The heat kernel \(p_t \in A^n(X \times X)\) tends to \(H\), which is the harmonic projection, when \(t \to \infty\). There exists a coexact \(\gamma_t \in A^{n-1}(X \times X)\) with \(d\gamma_t=p_t-H\). This allows him to define the Archimedean height pairing \((A,B)_X\) to be
\[
(A,B)_X : = \lim_{t \to 0} \int _{A \times B} \gamma_t \,.
\]
The connection to the iterated integrals is given by the formula
\[
(\Delta_S,C_1 \times C_2 \times C_3)_{S \times S \times S} = I(\alpha_1,\alpha_2,\alpha_3)\,,
\]
where \(\Delta_S\) is the diagonal in \(S \times S \times S\).
These results are generalized to higher dimensional Riemann manifolds \(S\) obtaining a generalized pairing \((S,C_1 \times \ldots \times C_k)_{S^k}\) for transversally intersecting cycles \(C_i \subset S\) such that the intersection of any \(k-1\) of them is empty. Again, the author gives an expression of this cycle pairing in terms of iterated integrals. The article closes with mentioning the case of complex cycles on compact Kähler manifolds where a similiar result holds. Indeed, here we can find \(\Gamma_t \in A^{n-1,n-1}(X \times X)\) such that \(dd^c \Gamma_t = p_t - H\). If all the cycles \(C_i\) are divisors, we obtain Deligne's determinant of cohomology. iterated integrals; Abel-Jacobi map; height pairing; Kähler manifold; Riemann manifold; cycles Algebraic cycles, Integration on analytic sets and spaces, currents, Kähler manifolds Chen's iterated integrals and algebraic cycles. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For a separated map \(\pi: S \to B\) of schemes, \textit{S. L. Kleiman} iteratively constructed schemes \(X_r\) which naturally parametrize ordered clusters of \(r\) points (possibly infinitely near) in the fibers of \(\pi\) [Acta Math. 147, 13--49 (1981; Zbl 0479.14004)]. One can think of an \(r\)-cluster as a sequence \((\sigma_r, \sigma_{r-1}, \dots, \sigma_0)\) of blow ups centered at points over a common image \(b \in B\), as seen in work of \textit{B. Harbourne} [Lect. Notes Math. 1311, 101--117 (1988; Zbl 0661.14030)]. To extend these ideas to the relative case, the author starts with a flat separated surjective family \(\pi: S \to B\) of finite type with \(B\) irreducible and generically irreducible fibers. For such a family, the relative notion of a 1-cluster is a continuous choice of points in each fiber of \(\pi\), in other words a section \(\sigma_0\) to \(\pi\); the notion of a 2-cluster (allowing infinitely near points) consists of a section \(\sigma_1\) of the composite \(\tilde S \to S \to B\), where \(\tilde S \to S\) is the blow up at \(\sigma_0 (B)\), and so on for higher ordered clusters. The author makes formal definitions to this effect, showing that under good conditions there exist schemes \(\text{Cl}_r\) which naturally parametrize \(r\)-relative clusters of \(\pi\), that is the corresponding moduli functor is represented by a scheme. In particular recovers Kleiman's schemes \(X_r\) in the absolute case when \(B = \operatorname{Spec}k\), where \(k\) is a field.
When \(B\) is smooth, the author gives a recursive construction, exhibiting \(\text{Cl}_{r+1}\) as a blow up of \(\text{Cl}_r^2 = \text{Cl}_r \times_{\text{Cl}_{r-1}} \text{Cl}_r\) along a closed subscheme which fails to be Cartier only along the diagonal. Some new phenomena arise in this relative construction, for example it is shown that there is an open subset \(V \subset \text{Cl}_{r+1}\) corresponding to relative clusters for which the last section is not infinitely near to the previous section. A flattening stratification of \(\text{Cl}_r^2\) is given in which the open set \(V\) is described as the union of certain admissible strata. The author closes with some examples of surfaces in which the \(\text{Cl}_r\) behave differently from Kleiman's iterated blow ups. Iterated blowups; infinitely near points; universal family of sections Fibrations, degenerations in algebraic geometry, Rational and birational maps On the universal scheme of \(r\)-relative clusters of a family | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For any quasi-projective variety \(X\) over \(\mathbb C\), \textit{P. Deligne} defined [Actes Congr. Internat. Math. 1970, 1, 425--430 (1971; Zbl 0219.14006); Publ. Math., Inst. Hautes Étud. Sci. 40, 5--57 (1971; Zbl 0219.14007); and Publ. Math., Inst. Hautes Étud. Sci. 44, 5--77 (1974; Zbl 0237.14003)] a mixed Hodge structure on the cohomology groups \(H^k_c(X,{\mathbb C})\) with associated \textit{Hodge polynomial} \({\epsilon}_X(u,v)\). When \(X\) is a smooth projective variety, \({\epsilon}_X(u,v) = \sum_{p.q}h^{p,q}(X)u^pv^q\).
A \textit{coherent system} of type \((n,d,k)\) on a smooth projective curve \(C\) is a pair \((E,V)\) consisting of a vector bundle \(E\) of rank \(n\) and degree \(d\) over \(C\) and a \(k\)-dimensional vector subspace \(V \subseteq H^0(E)\). For \(\alpha \in {\mathbb R}\), the \(\alpha\)-slope of a coherent system is defined by:
\[
{\mu}_{\alpha}(E,V) := \frac{d}{n} + \alpha \frac{k}{n}\, .
\]
Using the \(\alpha\)-slope on can define the notion of \(\alpha\)-\textit{stability} for coherent systems and one can show that there exists a quasi-projective variety \(G(\alpha ;n,d,k)\) which is a moduli space for the \(\alpha\)-stable coherent systems of type \((n,d,k)\) on \(C\).
Assume, from now on, that \(C\) is an \textit{elliptic curve}. In their previous work [Int. J. Math. 16, No. 7, 787--805 (2005; Zbl 1078.14045)], the authors of the paper under review showed that \(G(\alpha ;n,d,k)\) is smooth and irreducible of the expected dimension \(k(d-k) + 1\) (if non-empty). In the paper under review, they compute the Hodge polynomials and determine the birational types of the spaces \(G(\alpha ;n,d,k)\), in some cases.
More precisely, the authors show that if \(\text{gcd}(n,d) = 1\) and \(\alpha\) is a small positive real number then \(G(\alpha ;n,d,k)\) is a \(\text{Grass}(k,d)\)-bundle over \(C\) and compute its Hodge polynomial. They also show that if \(\text{gcd}(n,d) = 2\), \(k = 1\) and \(\alpha\) is arbitrary, \(G(\alpha ;n,d,1)\) is birational to \({\mathbb P}^{d-1}\times C\). Moreover, the Fourier-Mukai transform \({\Phi}_a\) defined by \textit{D. Hernández Ruipérez} and \textit{C. Tejero Prieto} [J. Lond. Math. Soc., II Ser. 77, 15--32 (2008; Zbl 1133.14012)] induces an isomorphsm of moduli spaces \(G(\alpha ;2,d,1) \simeq G(\alpha ;2+ad,d,1)\). Finally, if \(\text{gcd}(n,d) = h > 1\) and \(k < d\) then \(G(\alpha ;n,d,k)\) is birational to a variety fibred over \(\text{Symm}^hC\) with general fibre unirational. moduli space; vector bundle; projective curve; coherent system; birational type; Hodge polynomial Lange, Hodge polynomials and birational types of moduli spaces of coherent systems on elliptic curves, Manuscr. Math. 130 (1) pp 1-- (2009) Vector bundles on curves and their moduli, Transcendental methods, Hodge theory (algebro-geometric aspects), Birational geometry Hodge polynomials and birational types of moduli spaces of coherent systems on elliptic 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 Hilbert scheme \(S^{[n]}\) of points on an algebraic surface \(S\) is a simple example of a moduli space and also a nice (crepant) resolution of singularities of the symmetric power \(S^{(n)}\). For many phenomena expected for moduli spaces and nice resolutions of singular varieties it is a model case. Hilbert schemes of points have connections to several fields of mathematics, including moduli spaces of sheaves, Donaldson invariants, enumerative geometry of curves, infinite dimensional Lie algebras and vertex algebras and also to theoretical physics. This talk will try to give an overview over these connections. moduli spaces; vertex algebras; orbifolds; resolution of singularities; Donaldson invariants L Göttsche, Hilbert schemes of points on surfaces (editor T T Li), Higher Ed. Press (2002) 483 Parametrization (Chow and Hilbert schemes), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants) Hilbert schemes of points on surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author surveys a series of conjectures in combinatorics using new results on the geometry of Hilbert schemes. The combinatorial results include the positivity conjecture for the Macdonald symmetric functions, and the ``\(n!\)'' and ``\((n+1)^{n-1}\)'' conjectures which relate Macdonald polynomials to the characters of doubly graded \(S_n\)-modules.
In 1987 Macdonald unified the theory of Hall-Littlewood symmetric functions with that of spherical functions on symmetric spaces, introducing a class of symmetric functions, now known as Macdonald polynomials, with coefficients depending on two parameters \(q\) and \(t\). There are bivariate analogues of the Kostka-Foulkes polynomials, and Macdonald conjectured that these more general Kostka-Foulkes polynomials should have positive integer coefficients. In 1993 Garsia and the author introduced some bigraded \(S_n\)-module and conjectured that its dimension is \(n!\) and this is known as the \(n!\) conjecture. It is known that the \(n!\) conjecture implies the Macdonald positivity conjecture. The spaces figuring in the \(n!\) conjecture are quotients of the ring of coinvariants for the diagonal action of \(S_n\) on \({\mathbb C}^n\oplus{\mathbb C}^n\), and it was natural to investigate the characters of the full coinvariant ring. The space of coinvariants has dimension \((n+1)^{n-1}\). This involved \(q\)-analogues of this number and the Catalan numbers \(C_n\) in the data. A menagerie of things studied earlier by combinatorialists for their own sake turned up unexpectedly in this new context. Further, Procesi suggested that the diagonal coinvariants might be interpreted as sections of a vector bundle on the Hilbert scheme \(H_n\) of points in the plane. Understanding this geometric context has led the author to the proofs of the \(n!\) and \((n+1)^{n-1}\) conjectures. The full explanation depends on properties of the Hilbert scheme which were not known before, and had to be established from scratch in order to complete the picture. One might say, that the main results are not the \(n!\) and \((n+1)^{n-1}\) theorems, but new theorems in algebraic geometry.
In order to make the exposition self-contained, the author includes background from combinatorics, theory of symmetric functions, representation theory and geometry. At the end of the paper he discusses future directions, new conjectures, and related work of other mathematicians. Macdonald polynomials; symmetric polynomials; partitions; Hilbert scheme; \(n!\) conjecture Mark Haiman, Combinatorics, symmetric functions, and Hilbert schemes, Current developments in mathematics, 2002, Int. Press, Somerville, MA, 2003, pp. 39 -- 111. Symmetric functions and generalizations, Parametrization (Chow and Hilbert schemes) Combinatorics, symmetric functions, 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 [For the entire collection see Zbl 0561.00002.]
The author specializes results of Demazure and Bernstein-Gelfand about the intersection theory of quotients of reductive algebraic groups by Borel subgroups to obtain a description of the intersection theory (the basis theorem, Giambelli's (the determinantal) formula and Pieri's formula) for Grassmann manifolds. The reviewer recommends the reader to consult the long row of articles on the subject by \textit{A. Lascoux} and \textit{M.-C. Schützenberger} [e.g. Astérisque 87-88, 249-266 (1981; Zbl 0504.20007); C. R. Acad. Sci., Paris, Sér. I 294, 447-450 (1982; Zbl 0495.14031) and 295, 629-633 (1982; Zbl 0542.14030), and Invariant theory, Proc. 1st 1982 Sess. C.I.M.E., Montecatini/Italy, Lect. Notes Math. 996, 118-144 (1982; Zbl 0542.14031)]. Chow ring; intersection theory; Schubert varieties; Giambelli; determinantal formula; Pieri formula; intersection theory of quotients of reductive algebraic groups by; Borel subgroups; basis theorem; Grassmann manifolds; intersection theory of quotients of reductive algebraic groups by Borel subgroups Grassmannians, Schubert varieties, flag manifolds, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Classical groups (algebro-geometric aspects), (Equivariant) Chow groups and rings; motives On the Schubert calculus interpolation | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let $k$ be a field and let $R\rightarrow k$ be a surjection from a commutative ring $R$. It is said that a $k$-scheme $X$ is liftable to $R\rightarrow k$ if there exists a flat $R$-scheme $\tilde{X}$ and a closed embedding $X\rightarrow \tilde{X}$ including an isomorphism $X\rightarrow \tilde{X}\times_R k$. The lifting of a $k$-algebra is nothing but the lifting of its corresponding affine scheme.\newline \par Throughout this review, characteristic zero rings are of mixed characteristic. A desirable property for an affine or a projective variety over a (perfect) prime characteristic field, is to have a lifting to characteristic zero (mixed characteristic), or to the ring, $W_2(k)$, of Witt vectors of length $2$. One reason, e.g. would be the ``Kodaira Vanishing property'' of those nonsingular projective varieties that lift to $W_2(k)$ (see, [\textit{P. Deligne} and \textit{L. Illusie}, Invent. Math. 89, 247--270 (1987; Zbl 0632.14017)]). Another remarkable result in this direction is the fact that any nonsingular projective curve over a perfect field of positive characteristic is liftable to characteristic $0$ (see, corollary 22.2 of [\textit{R. Hartshorne}, Grad. Texts Math. (2010; Zbl 1186.14004)]). \par The aim of the paper under review is to present new examples and theorems of certain positive characteristic zero-dimensional schemes, even zero dimensional Gorenstein rings, which do not lift to some specific rings of mixed characteristic; however they may still be liftable to a more ramified mixed characteristic ring. The main ingredient of the proofs seems to be an interesting relation between the existence of the lifting (to a mixed characteristic ring with a specific ramification index) and the existence of the divided power structure on the maximal ideal of those prime characteristic zero dimensional rings. It is proved that both of the aforementioned related facts, about the divided powers and liftings, under some conditions, are equivalent to the membership of certain element(s) to a specific primary ideal to the maximal ideal. The motivation of considering only the zero dimensional schemes, as described in the introduction of the paper, comes from the open question that, does a zero-dimensional prime characteristic non-liftable scheme exist (for a reference to this question, see, last line of page 148 of [\textit{R. Hartshorne}, Grad. Texts Math. (2010; Zbl 1186.14004)])? \par There are many different kind of examples and theorems in the paper in this regard, we shall list some of those. \par Corollary 4.4. Let $k$ be an algebraically closed field of characteristic $p>0$. Let $X\subseteq \mathbb{A}^n$ be a general hypersurface singular at $0$. Let us assume $n\ge 5$ if $p\ge 3$ or $n\ge 6$ if $p=2$. Then the first Frobenius neighborhood of $0\in X$ does not lift to $W_2(k)$. \par Corollary 4.9. Let $k$ be a field of characteristic $p>0$. There exists a direct system $\{X_n\}_{n\in \mathbb{N}}$ of zero-dimensional $k$-schemes such that for any Noetherian local ring $(R,\mathfrak{m}_R)$ with $pR\neq 0$ and residue field $k=R/\mathfrak{m}_R$ the schemes $X_n$ do not lift to $R\rightarrow k$ for all sufficiently large $n$. \par The following result is deduced by applying the linkage of Cohen-Macaulay almost complete intersections and Gorenstein rings. \par Corollary 5.2. Let $R$ be a Noetherian local ring with residue field of characteristic $p>0$. If $pR\neq 0$ then there exists a zero-dimensional Gorenstein $k$-scheme $Z$ that cannot be lifted to $R$. \par Example 4.3. The author applies his results, and as a special case, recovers Proposition 3.4 of [\textit{M. Zdanowicz}, Int. Math. Res. Not. 2018, 4513--4577 (2018; Zbl 1423.14168)] and shows that $A=k[x_1,\ldots,x_6]/(x_1^p,\ldots,x_6^p,x_1x_2+x_3x_4+x_5x_6)$ does not lift to $W_2(k)$. Then, despite of this non-liftability, the author shows that if $\text{Char}(k)=2$ then $A$ has a lifting to a ring of characteristic $0$. deformation theory; divided power structure; lifting of prime characteristic schemes Deformations and infinitesimal methods in commutative ring theory, Linkage, complete intersections and determinantal ideals, Deformations of singularities, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure Lifting zero-dimensional schemes and divided powers | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\mathcal L\) be a field and \({\mathcal L}^*:={\mathcal L}\setminus \{0\}\). If \(G:=(g_1,\dots,g_k)\), \(g_i\in{\mathcal L}[x_1,\dots,x_n]\setminus \{0\}\) and the number of monomial terms appearing in at least one \(g_i\) is exactly \(m\), then \(G\) is called a \(k\times n\) \(m\)-sparse polynomial system. The author proves:
Theorem. For any prime number \(p\) and positive integer \(d\), let \(L\) be any degree \(d\) algebraic extension of \({\mathbb Q}_p\). Let \(G\) be any \(k\times n\) \(m\)-sparse polynomial system over \(L\). Then there is an absolute constant \(\gamma(n,m)\) such that the number of isolated roots of \(G\) in \(L^n\) is no more than \(p^{dn}(1-\frac{1}{p^{f_L}})^n \gamma(n,m)\).
Corollary. Let \(K\) be any degree \(d\) algebraic extension of \(\mathbb Q\) and let \(G\) be any \(k\times n\) \(m\)-sparse polynomial system over \(K\). Then the number of isolated roots of \(G\) in \(K^n\) is no more than \(2^{dn}(1-\frac{1}{2^d})^n \gamma(n,m)\). isolated roots Varieties over finite and local fields, Varieties over global fields, Local ground fields in algebraic geometry Finiteness for arithmetic fewnomial systems. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The combinatorial model of the moduli space \(\mathcal{M}_{g,n}\) of smooth genus \(g\) algebraic curves with \(n\) marked points is given by metric ribbon graphs. The latter are thickened \(1\)-skeletons of the curve's cell decomposition with a length assigned to each edge. Denoting their moduli space \(RG_{g,n}\) one has an orbifold isomorphism \(RG_{g,n}\simeq\mathcal{M}_{g,n}\times\mathbb{R}^n_+\). It is known that the generating function of the Euler characteristics \(\chi(RG_{g,n})\) is given by the Penner's matrix model. As a generalization, the authors study suitably defined Poincaré polynomials \(F_{g,n}(t_1,\dots,t_n)\) of \(RG_{g,n}\) in this paper. In particular, they derive a recursion of Eynard-Orantin type for them and interpret their top degree terms via the intersection numbers on the Deligne-Mumford stack \(\overline{\mathcal{M}}_{g,n}\).
Let \(N_{g,n}(p)\) denote the number of ribbon graphs in \(RG_{g,n}\) of perimeter \(p\) with integral lengths, these graphs can be interpreted as Grothendieck's dessins d'enfants. It turns out that up to an explicit change of variables \(F_{g,n}\) are the Laplace transforms of \(N_{g,n}\). This is analogous to an earlier result for the Hurwitz numbers, the Laplace transforms plays the role of a mirror map.
Known recursions for \(N_{g,n}\) lead to a differential equation for \(F_{g,n}\), which is also recursive in \(2g-2+n\), the absolute value of the Euler characteristic of an \(n\)-punctured surface of genus \(g\). In the stable range \(2g-2+n>0\) this recursion recovers all \(F_{g,n}\) from \(F_{0,3}\) and \(F_{1,1}\). Moreover, \(F_{g,n}\) turns out to be a Laurent polynomial in \(t_1,\dots,t_n\) of degree \(3(2g-2+n)\) with the intersection numbers of \(\psi\)-classes on \(\overline{\mathcal{M}}_{g,n}\) as coefficients in the top degree terms. The above recursion restricts to these terms and recovers the DVV recursion (Virasoro constraint) for the intersection numbers. moduli of curves; metric ribbon graphs; Penner's matrix model; recursion of Eynard-Orantin type; Grothendieck's dessins d'enfants; Deligne-Mumford stack; psi-classes; Virasoro constraint M. Mulase and M. Penkava, \textit{Combinatorial Structure of the Moduli Space of Riemann Surfaces and the KP Equation}, unpublished [http://www.math.ucdavis.edu/~mulase/texfiles/1997moduli.pdf]. Families, moduli of curves (analytic), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Coverings of curves, fundamental group, Enumeration in graph theory, Lattice points in specified regions, String and superstring theories; other extended objects (e.g., branes) in quantum field theory Topological recursion for the Poincaré polynomial of the combinatorial moduli space 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 \(R\) be a noetherian ground ring. The author establishes that an \(R\)-scheme \(X\) of finite type that admits an ample family of line bundles allows a closed embedding into another \(R\)-scheme \(W\) of finite type that admits an ample family of line bundles and is furthermore smooth. This generalizes a result of \textit{H. Brenner} and \textit{S. Schröer} [Pac. J. Math. 208, No. 2, 209--230 (2003; Zbl 1095.14004)], who proved that \(X\) has an ample family if and only if it is isomorphic to a closed subscheme of some multihomogeneous homogeneous spectrum of some finitely generated polynomial ring, endowed with a \(\mathbb{Z}^n\)-grading. Note that such spectra are usually non-separated, unlike the classical case of \(\mathbb{N}\)-gradings.
Recall that \(\mathscr{L}_1,\ldots,\mathscr{L}_n\) is called an ample family is the non-zero loci of global sections in tensor combinations form a basis of the Zariski topology. For \(n=1\) this gives back the notion of an ample sheaf. The generalization
was introduced by \textit{M. Borelli} [Pac. J. Math. 13, 375--388 (1963; Zbl 0123.38102)]. Schemes admitting an ample family are also called divisorial. Examples are the regular noetherian schemes.
A crucial step in the paper is an essentially combinatorial lemma, which gives a sufficient conditions for homogeneous localizations of \(\mathbb{Z}^n\)-graded rings to be polynomial rings. As an application, Zanchetta shows that if \(X\) is divisorial, then each collection \(\mathscr{E}_1,\ldots,\mathscr{E}_n\) of locally free sheaves of finite rank arises as a pull-back of locally free sheaves under a morphism \(f:X\to Y\) to a smooth divisorial scheme. algebraic geometry; \(K\)-theory; toric geometry Schemes and morphisms, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies Embedding divisorial schemes into smooth ones | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 give a fast algorithm for the computation of the Arf closure of an algebroid curve with more than one branch, generalizing an algorithm presented by \textit{F. Arslan} and \textit{N. Şahin} [J. Algebra 417, 148--160 (2014; Zbl 1298.14031)] for the algebroid branch case. algebroid curves; Arf closure; Arf ring; good Arf semigroup; multiplicity sequence Commutative semigroups, Singularities of curves, local rings, Valuations and their generalizations for commutative rings, Software, source code, etc. for problems pertaining to group theory An algorithm for computing the Arf closure of an algebroid curve with more than one branch | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems It is shown that the computation of higher dimensional harmonic volume, defined by \textit{B. Harris} [Acta Math. 150, 91-123 (1983; Zbl 0527.30032)], can be reduced to Harris' computation in the one-dimensional case [see Q. J. Math., Oxf. II. Ser. 34, 67-75 (1983; Zbl 0528.10018)], so that higher dimensional harmonic volume may be computed essentially as an iterated integral. We then use this formula to produce a specific smooth curve \(\tilde C\), namely a specific double cover of the Fermat quartic, so that the image \(W_ 2(\tilde C)\) of the second symmetric product of \(\tilde C\) in its Jacobian via the Abel-Jacobi map is algebraically inequivalent to the image of \(W_ 2(\tilde C)\) under the group involution on the Jacobian. algebraic equivalence; computation of higher dimensional harmonic volume; Jacobian; Abel-Jacobi map Faucette W.M.: Higher dimensional harmonic volume can be computed as an iterated integral. Can. Math. Bull. 35(3), 328--340 (1992) Jacobians, Prym varieties, (Equivariant) Chow groups and rings; motives, Automorphic forms, one variable Higher dimensional harmonic volume can be computed as an iterated integral | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Inspired by Besser's work on Coleman integration, we use \(\nabla\)-modules to define iterated line integrals over Laurent series fields of characteristic \(p\) taking values in double cosets of unipotent \(n\times n\) matrices with coefficients in the Robba ring divided out by unipotent \(n\times n\) matrices with coefficients in the bounded Robba ring on the left and by unipotent \(n\times n\) matrices with coefficients in the constant field on the right.
We reach our definition by looking at the analogous theory for Laurent series fields of characteristic 0 first, and re-interpreting the classical formal logarithm in terms of \(\nabla\)-modules on formal schemes. To illustrate that the new \(p\)-adic theory is nontrivial, we show that it includes the \(p\)-adic formal logarithm as a special case. \(p\)-adic integration; Laurent series fields Arithmetic ground fields for abelian varieties, \(p\)-adic cohomology, crystalline cohomology, Homotopy theory and fundamental groups in algebraic geometry Iterated line integrals over Laurent series fields of characteristic \(p\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We generalize the main result of our previous paper [Adv. Math. 188, No. 1, 222--246 (2004; Zbl 1073.14002)] (to the effect that very general Noetherian log schemes may be reconstructed from naturally associated categories) to the case of log schemes locally of finite type over Zariski localizations of the ring of rational integers which are, moreover, equipped with certain ``Archimedean structures''. Arithmetic varieties and schemes; Arakelov theory; heights, Schemes and morphisms, Categories in geometry and topology Categories of log schemes with archimedean 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 Fix a complex projective elliptic curve \(E\). The choice of diffeomorphism \(\Phi\) from \(Y = \mathbb C^* \times \mathbb C^*\) to \(X = E \times \mathbb C\) (note that both are diffeomorphic to \(S^1 \times S^2 \times \mathbb R \times \mathbb R\)) induces an isomorphism \(\phi: H^*(X,\mathbb Q) \to H^*(Y,\mathbb Q)\) by pulling back. For \(n \geq 0\), the authors show that \(\phi\) extends to an isomorphism \(\phi^{[n]}: H^*(X^{[n]}, \mathbb Q) \to H^*(Y^{[n]}, \mathbb Q)\) on the \(\mathbb Q\)-cohomology of Hilbert schemes of \(n\) points. While work of \textit{C. Voisin} suggests that \(Y^{[n]} \cong X^{[n]}\) [Ann. Inst. Fourier 50, No. 2, 689--722 (2000; Zbl 0954.14002)], the authors do not use this but instead obtain \(\phi^{[n]}\) by the decomposition of the Hilbert scheme given by partitions \(\nu = (\nu_1, \nu_2, \dots, \nu_l)\) of \(n\); in particular \(\phi^{[n]}\) need not be an isomorphism of mixed Hodge structures.
The main result of the paper is the equality of two natural filtrations under \(\phi^{[n]}\). The weight filtration \(W_k \subset H^*(Y^{[n]}, \mathbb Q)\) has the property that \(W_{2k} = W_{2k+1}\) for each \(k\), so one may define the \textit{halved} filtration by \({}_{\frac{1}{2}}W_{Y^{[n]},k}=W_{Y^{[n]},2k}\). On the other hand the projection \(p:X \to \mathbb C\) induces a map \(h_n: X^{[n]} \to X^n/S_n \to \mathbb C^n/S_n\) which is projective and flat of relative dimension \(n\). Associated to the map \(h_n\) is the perverse Leray filtration \(P_{X^{[n]}}\) via the complex \({h_n}_* \mathbb Q_{X^{[n]}} [n]\). With this notation, the main theorem says that \(\phi^{[n]}(P_{X^{[n]}}) = {}_{\frac{1}{2}} W_{Y^{[n]}}\). This result gives more evidence of the general ``\(P=W\)'' conjecture. The authors have proved this for moduli space of rank 2 degree 1 stable Higgs bundles [\textit{M. A. A. De Cataldo} et al., Ann. Math. (2) 175, No. 3, 1329--1407 (2012; Zbl 1375.14047)]. A second result is an analog of the hard Lefschetz theorem for \(Y^{[n]}\). The closed 2-form \(\alpha_Y = \frac{1}{(2 \pi i)^2} \frac{dz \wedge dw}{zw}\) is an integral class in \(H^2(Y, \mathbb Q) \cap H^{2,2} (Y)\). Summing over the pull-backs of the projections \(p_i: Y^n \to Y\), quotienting by the action of \(S_n\) and pulling back by the Hilbert-Chow maps gives the cohomology class \(\alpha_{Y^{[n]}} \in H^2(Y^{[n]}, \mathbb Q) \cap H^{2,2} (Y^{[n]})\). Cup product with \(\alpha_{Y^{[n]}}\) gives isomorphisms
\[
\text{Gr}^{W_{Y^{[n]}}}_{2n-2k} H^*(Y^{[n]},\mathbb Q) \cong \text{Gr}^{W_{Y^{[n]}}}_{2n+2k} H^{*+2k} (Y^{[n]},\mathbb Q)
\]
which are analogs of the ``curious hard Lefschetz'' theorem of \textit{T. Hausel} and \textit{F. Rodriguez-Villegas} [Invent. Math. 174, No. 3, 555--624 (2008; Zbl 1213.14020)]. Under the identification given by the map \(\phi^{[n]}\), the authors show that these isomorphisms become the relative hard Lefschetz theorems on \(X^{[n]}\) found in work on perverse sheaves of \textit{A. A. Beilinson} et al. [Astérisque 100, 172 p. (1982; Zbl 0536.14011)]. perverse filtration; weight filtration; Hilbert scheme of points Cataldo, MA; Hausel, T; Migliorini, L, Exchange between perverse and weight filtration for the Hilbert schemes of points of two surfaces, J. Singul., 7, 23-38, (2013) Parametrization (Chow and Hilbert schemes), Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Exchange between perverse and weight filtration for the Hilbert schemes of points of two 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 Using the theory of jet schemes, we give a new approach to the description of a minimal generating sequence of a divisorial valuations on \(\mathbf{A}^2\). For this purpose, we show how to recover the approximate roots of an analytically irreducible plane curve from the equations of its jet schemes. As an application, for a given divisorial valuation \(v\) centered at the origin of \(\mathbf{A}^2\), we construct an algebraic embedding \(\mathbf{A}^2\hookrightarrow\mathbf{A}^N,\,N\geq2\) such that \(v\) is the trace of a monomial valuation on \(\mathbf{A}^N\). We explain how results in this direction give a constructive approach to a conjecture of Teissier on resolution of singularities by one toric morphism. Global theory and resolution of singularities (algebro-geometric aspects), Arcs and motivic integration, Toric varieties, Newton polyhedra, Okounkov bodies Jet schemes and generating sequences of divisorial valuations in dimension two | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In a recent paper, \textit{G. Prasad} and \textit{J. Yu} [J. Algebr. Geom. 15, No. 3, 507--549 (2006; Zbl 1112.14053)] introduced the notion of a quasireductive group scheme \(G\) over a discrete valuation ring \(R\), in the context of Langlands duality. They showed that such a group scheme \(G\) is necessarily of finite type over \(R\), with geometrically connected fibres, and its geometric generic fibre is a reductive algebraic group; however, they found examples where the special fibre is nonreduced, and the corresponding reduced subscheme is a reductive group of a different type. In this paper, the formalism of vanishing cycles in étale cohomology is used to show that the generic fibre of a quasireductive group scheme cannot be a restriction of scalars of a group scheme in a nontrivial way; this answers a question of Prasad, and implies that nonreductive quasireductive group schemes are essentially those found by Prasad and Yu. group scheme; quasireductive; nearby cycle Group schemes, Linear algebraic groups over adèles and other rings and schemes A topological property of quasireductive group schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For \(d\) divisible by \(3\) it is proved that there are real projective surfaces of degree \(d\) with \(\binom{d}{2}\lfloor \frac{d}{2}\rfloor+(1+\frac{d(d-3)}{3})\lfloor \frac{d-1}{2}\rfloor\) real nodes and no other singularities. The existing lower bounds are improved by \(\lfloor \frac{d-1}{2}\rfloor\). algebraic surfaces; real singularities Escudero, JG, A construction of algebraic surfaces with many real nodes, Ann. Mat. Pura Appl., 195, 571-583, (2016) Singularities of surfaces or higher-dimensional varieties, Hypersurfaces and algebraic geometry A construction of algebraic surfaces with many real nodes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Numerical Campedelli surfaces are smooth minimal surfaces of general type with \(p_g=0\) and \(K^2=2\). Although they have been studied by several authors, their complete classification is not known. In this paper the authors classify numerical Campedelli surfaces with an involution, i.e. an automorphism of order 2. Using results on involutions on surfaces of general type with \(p_g=0\) [cf. \textit{A. Calabri, C. Ciliberto} and \textit{M. M. Lopes}, Trans. Am. Math. Soc. 359, No. 4, 1605--1632 (2007; Zbl 1124.14036)], they show that an involution \(\sigma\) on a numerical Campedelli surface \(S\) has either four or six isolated fixed points, and the bicanonical map of \(S\) is composed with the involution if and only if the involution has six isolated fixed points. In the latter case they prove that the ramification divisor \(R\) on \(S\) is not 0, and the quotient surface \(S/\sigma\) is either birational to an Enriques surface or a rational surface; if \(S/\sigma\) is rational, then there are four possible cases and each of the four cases actually occurs. If the involution has four isolated fixed points, they show that the ramification divisor \(R\) is either \(0\) or constituted by one, two or three \(-2-\)curves. In this case there are more possibilities for the quotient surface \(S/\sigma\):
1)\(S/\sigma\) is of general type (a numerical Godeaux surface) if and only if the ramification divisor \(R\) is equal to 0;
2) if \(R\) is not empty and irreducible, then \(S/\sigma\) is properly elliptic;
3) if \(R\) has two or three components then \(S/\sigma\) may be rational or birational to an Enriques surface or properly elliptic. There are examples for all cases, except when the quotient surface is a rational surface for this case. The authors also study a family of numerical Campedelli surfaces with torsion \(\mathbb{Z}_3^2\), showing that every surface in this family has two involutions, one with four isolated fixed points and one with six isolated fixed points, whose quotients are respectively birational to a numerical Go deaux surface and a rational surface. Finally, they study the involutions of numerical Campedelli surfaces with torsion \(\mathbb{Z}_2^3\), the so-called ``classical Campedelli surfaces''. Using the description of these surfaces as a \(\mathbb{Z}_2^3\)-cover of \(\mathbb{P}^2\) branched on 7 lines, they show that these involutions are all composed with the bicanonical map. Campedelli surfaces; involutions on surfaces; double covers Calabri, A., Mendes Lopes, M., Pardini, R.: Involutions on numerical Campedelli surfaces. Tohoku Math. J. 60(1), 1--22 (2008) Surfaces of general type Involutions on numerical Campedelli surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The problem of integrable discretization (see [the second author, The problem of integrable discretization: Hamiltonian approach. Basel: Birkhäuser (2003; Zbl 1033.37030)]) consists in finding, for a given integrable system, a discretization which remains integrable. In this paper the authors find a novel one-parameter family of integrable quadratic Cremona maps of the plane preserving a pencil of curves of degree 6 and of genus 1. They turn out to serve as Kahan-type discretizations of a novel family of quadratic vector fields possessing a polynomial integral of degree 6 whose level curves are of genus 1, as well. These vector fields are non-homogeneous generalizations of reduced Nahm systems for magnetic monopoles with icosahedral symmetry, introduced by \textit{N. J. Hitchin} et al. [Nonlinearity 8, No. 5, 662--692 (1995; Zbl 0846.53016)]. The Kahan discretization of these nonhomogeneous systems is non-integrable. However, this drawback is repaired by introducing adjustments of order \(O(\epsilon^2)\) in the coefficients of the discretization, where \(\epsilon\) is the stepsize.
This paper is organized as follows. Section 1 is an introduction to the subject. Section 2 is devoted to a homogeneous \((1,2,3)\) system and its Kahan discretization. Section 3 deals with the reduction of the considered map into roots of special QRT maps. Section 4 deals with a generalization of the QRT root map. Here the authors try to generalize the map of the previous section. All obtained mappings will be one-parameter perturbations of the corresponding objects from the previous section. Section 5 and 6 are devoted to continuous limits and conclusions.
The results of the present paper confirm that the phenomenon discovered and described by \textit{M. Petrera} and \textit{R. Zander} [J. Phys. A, Math. Theor. 50, No. 20, Article ID 205203, 13 p. (2017; Zbl 1377.37086)] is not isolated, namely that in case of non-integrability of the standard Kahan discretization (when applied to an integrable system), its coefficients can be adjusted to restore integrability.
For Part I, see [\textit{M. Petrera} et al., ``How one can repair non-integrable Kahan discretizations'', J. Phys. A, Math. Theor. 53, No. 37 (2020; \url{doi:10.1088/1751-8121/aba308})]. birational maps; discrete integrable systems; elliptic pencil; rational elliptic surface; integrable discretization Completely integrable discrete dynamical systems, Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems, Integrable difference and lattice equations; integrability tests, Rational and birational maps, Relationships between algebraic curves and integrable systems How one can repair non-integrable Kahan discretizations. II: A planar system with invariant curves of degree 6 | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(Y\) be a Gorenstein trigonal curve with \(g:= p_a(Y)\geq 0\). Here we study the theory of special linear systems on \(Y\), extending the classical case of a smooth \(Y\) given by \textit{A. Maroni} [Ann. Mat. Pura Appl., IV. Ser. 25, 343-354 (1946; Zbl 0061.35407)]. As in the classical case, to study it we use the minimal degree surface scroll containing the canonical model of \(Y\). The answer is different if the degree 3 pencil on \(Y\) is associated to a line bundle or not. We also give the easier case of special linear series on hyperelliptic curves. The unique hyperelliptic curve of genus \(g\) which is not Gorenstein has no special spanned line bundle. Gorenstein trigonal curve; linear systems; scroll; canonical model Ballico E.,Trigonal Gorenstein curves and special linear systems, Israel J. Math.,119 (2000), 143--155. Special divisors on curves (gonality, Brill-Noether theory), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Divisors, linear systems, invertible sheaves Trigonal Gorenstein curves and special linear systems | 0 |
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