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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 Translation from Vestn. Leningr. Univ., Ser. I 1986, No.3, 120-121 (Russian) (1986; Zbl 0606.18007). categories of fractions; presheaf of categories; affine scheme; commutative ring Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects), Schemes and morphisms, Localization of categories, calculus of fractions A ''presheaf'' of categories of fractions and 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 The authors consider spectral curves \((C, B, x, y)\) that locally resemble the curve \(x y^2 = 1\) near some zeros of the differential form \(d x\). The local behavior of the invariants \(\omega_n^g\) are in general not determined by intersection numbers, as is the case for the Airy curve [\textit{B. Eynard} and \textit{N. Orantin}, J. Phys. A, Math. Theor. 42, No. 29, Article ID 293001, 117 p. (2009; Zbl 1177.82049)]. Instead, the authors relate this local behavior to the enumeration of dessins d'enfant.
Given a tuple of integers, \(\mu = (\mu_1, \ldots, \mu_n)\), one can count the number \(B_{g, n} (\mu)\) of dessins of genus \(g\) with ramification behavior \(\mu\) at infinity, weighted by their automorphism group. The authors show that when \(g\) and \(n\) are fixed, the generating function \(F_{g, n} = \sum_{\mu} B_{g, n} (\mu) \prod_i x_i^{\mu_i}\) is a rational function in \(z_i\) when writing \(x_i = z_i + z_i^{-1} + 2\). Together with \(x\), the derivative \(y\) of \(F_{0,1}\) with respect to \(x\) defines the spectral curve \(x y^2 - x y + 1\). The functions \(F_{g, n}\) then constitute the analytic expansion of the invariants \(\omega_n^g\) of this spectral curve at its point at infinity. This yields structure theorems and explicit formulae for the numbers \(B_{g, n}\).
By studying the asymptotic behavior of the functions \(F_{g, n}\) near the pole \((z_1, \ldots, z_n) = (-1, \ldots, -1)\), the authors then determine the one-point invariants \(\omega_1^g\) of the spectral curve \(x y^2 = 1\) in terms of a three-term recursion for the numbers \(B_{g, 1} (n)\) of dessins d'enfant with one face. This new recursion has analogues in [\textit{J. Harer} and \textit{D. Zagier}, Invent. Math. 85, 457--485 (1986; Zbl 0616.14017)]. topological recursion; dessins d'enfant; spectral curves Enumerative problems (combinatorial problems) in algebraic geometry, Exact enumeration problems, generating functions, Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Dessins d'enfants theory Topological recursion for irregular spectral 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 aim of the paper is to reconcile Grothendick's construction of the arithmetic fundamental group of a \(k\)-scheme \(X\) [cf. \textit{A. Grothendieck}, Revetêments étales et groupe fondamental, SGA 1, Lectures Notes in Mathematics (1971; Zbl 0234.14002)] and Nori's Tannaka construction of the fundamental group scheme of a proper, reduced, strongly connected \(k\)-scheme \(X\) endowed wih a \(k\)-rational point \(x\in X(k)\) [cf. \textit{M. V. Nori}, Compos. Math. 33, 29--41 (1976; Zbl 0337.14016)] by using Deligne's Tannaka formalism [cf. \textit{P. Deligne}, Catégories tannakiennes. The Grothendieck Festschrift, Collect. Artic. in Honor of the 60th Birthday of A. Grothendieck. Vol. II, Prog. Math. 87, 111--195 (1990; Zbl 0727.14010)].
In order to do this they give a linear structure to Grothendieck's construction when \(X\) is a smooth scheme of finite type over a characteristic \(0\) field \(k\) and \(k=H^0_{DR}(X)\): first the authors extend Nori's construction to not necessarily proper schemes by definining over \(X\) the category \(FConn(X)\) of finite flat connections; then they define the fundamental groupoid scheme \(\prod(X,\overline{x})\) of \(X\) with base point a geometric point \(\overline{x}:Spec(\overline{k})\to X\) as the \(k\)-groupoid scheme acting on \(Spec(\overline{k})\) given by \(Aut^{\otimes}(\rho_{\overline{x}})\) where
\[
\rho_{\overline{x}}:FConn(X)\to Vect_{\overline{x}}
\]
is the fiber functor assigning to each connection the fiber of the underlining bundle at \(\overline{x}\); finally they compare their fundamental groupoid scheme to the Grothendieck's arithmetic fundamental group \(\pi_{1}(X,x)\). This allows the authors to link the existence of sections of the Galois group \(Gal(\bar k/k)\) to \(\pi_{1}(X,x)\) with the existence of a neutral fiber functor on the category which linearizes it. Finally they apply the construction to affine curves and neutral fiber functors coming from a tangent vector at a rational point at infinity, in order to follow this rational point in the universal covering of the affine curve. finite connection; tensor category; tangential fiber functor André, Y.: Différentielles non commutatives et Théorie de Galois différentielle ou aux différences. Ann. Scient. Ec. Norm. Sup., 4 serie, t. 34, pp. 685-739 (2001) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Affine algebraic groups, hyperalgebra constructions, Monoidal categories (= multiplicative categories) [See also 19D23] The fundamental groupoid scheme and applications | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(R\) be a complete discrete valuation ring of mixed characteristic, \(k\) its residue field of characteristic \(p\), \(\pi\) a uniformizing element, \(e\) the absolute ramification index. Let \(a\) be a positive integer, \(\overline R=R/\pi^ aR\). Let \(G\) be \(a\) a finite commutative \(p\)-group scheme over \(R\); here ``\(p\)-group'' means that \(G\) is killed by a certain power of \(p\). Let \(G\) be a lifting of \(\overline G\) to a commutative finite \(p\)-group over \(R\). When \(a=1\) (i.e., \(\overline R=k)\) and \(e=1\) (i.e., \(\overline R=W(k))\) the classification of the liftings was done by \textit{J. H. Fontaine} [``Groups \(p\)-divisibles sur les corps loceaux'', Astérisque 47-48 (1977; Zbl 0377.14009); see also \textit{A. Badra}, Thèse de 3ème cycle (Université de Rennes 1979)] not only for finite \(p\)-groups but also for \(p\)-divisible groups in terms of the so called Honda systems \((M,L)\). Here \(M\) is the Dieudonné module of \(\overline G\) and \(L\) is an \(R\)-submodule of \(M\) satisfying certain conditions imposed by the action of Frobenius on \(M\). The author develops a rather involved theory of Honda systems in the general case and uses it for the classification of all liftings \(G\) of \(\overline G\) up to an isomorphism. As a corollary, he obtains that, when \(p\geq 5\) and \(e\geq 2\) every commutative finite \(p\)-group scheme over \(k\) can be lifted to \(R\). deformations; \(p\)-group scheme; Honda systems Group schemes, Formal groups, \(p\)-divisible groups, Formal methods and deformations in algebraic geometry Finite group schemes over a discrete valuation ring and associated Honda systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(K\) be a field of characteristic \(0\), \(n\in N\) and \(\mathbb{P}^n = \mathbb{P}^n(K)\). In this paper the authors study the postulation of general fat point schemes of \(\mathbb{P}^3\) with multiplicity up to \(5\). A fat point \(mP\) is a zero dimensional subscheme of \(\mathbb{P}^3\) supported on a point \(P.\) A general fat point scheme \(Y = m_1P_1+\dots+m_kP_k\), with \(m_1,\dots, m_k\) is a general zero-dimensional scheme such that its support \(Y_{\mathrm{red}}\) is a union of \(k\) points and for each \(i\) the connected component of \(Y\) supported on \(P_i\) is the fat point \(m_iP_i\). Studying the postulation of \(Y\) means to compute the dimension of the space of hypersurfaces of any degree containing the scheme \(Y.\) In other words this problem is equivalent to computing the dimension \(\delta\) of the space of homogeneous polynomials of any degree vanishing at each point \(P_i\) and with all their derivatives, up to multiplicity \(m_i- 1\), vanishing at \(P_i\). \(Y\) has good postulation if \(\delta\) is the expected dimension, that is, either the difference between the dimension of the polynomial space and the number of imposed conditions or just the dimension of the polynomial space (when \(\delta\) would exceed it).
In this paper, they focus on the case of general fat point schemes \(Y\subset \mathbb{P}^3\). In this case a general conjecture which characterizes all the general fat point schemes not having good postulation was proposed by Laface and Ugaglia. The good postulation of general fat point schemes of multiplicity \(4\) was proved for degrees \(d\geq 41\) by the first two authors. Then Dumnicki showed how to check the cases with degree \(9\leq d \leq 40\). The authors, here, prove that if \(Y\subset \mathbb{P}^3\) is a general union of \(w\) \(5\)-points, \(x\) \(4\)-points, \(y\) \(3\)-points and \(z\) \(2\)-points with fixed non- negative integers \(d,w, x, y, z\) such that \(d\geq 11\), then \(Y\) has good postulation with respect to degree-\(d\) forms. They also classify the exceptions in degree \(9\) and \(10\). Polinomial interpolation; fat points; zero-dimensional scheme; projective space E. Ballico, M. C. Brambilla, F. Caruso, and M. Sala, Postulation of general quintuple fat point schemes in \Bbb P³, J. Algebra 363 (2012), 113 -- 139. Projective techniques in algebraic geometry, Vector and tensor algebra, theory of invariants, Numerical interpolation Postulation of general quintuple fat point schemes in \(\mathbb P^3\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We introduce a new approach to constructing derived deformation groupoids, by considering them as parameter spaces for strong homotopy bialgebras. This allows them to be constructed for all classical deformation problems, such as deformations of an arbitrary scheme, in any characteristic.We also give a general approach for studying deformations of diagrams. Pridham, J. P.: Derived deformations of schemes, (2009) Formal methods and deformations in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Noncommutative algebraic geometry, Deformations and infinitesimal methods in commutative ring theory, Groupoids, semigroupoids, semigroups, groups (viewed as categories) Derived deformations 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 In a previous paper in Ann. Math., II. Ser. 135, No. 3, 527-549 (1992; Zbl 0762.52003) the authors developed a theory of fiber polytopes for projections of convex \(n\)-polytopes into \(\mathbb{R}^ d\). The faces of the fiber polytope are related to subdivisions of the projected polytope. In the present paper this is extended to the case of iterated projections or, more precisely, flags of projections. This leads to the notion of a flag polytope. Now the faces are related to discrete homotopies between subdivisions. As one of the main results, for projections of the \((n + 1)\)-simplex the first two flag polytopes turn out to be combinatorially isomorphic to the \(n\)-cube and to the \((n - 1)\)-dimensional permutohedron, respectively. coherent homotopy; Minkowski sum; toric varieties; projections; flag polytope L. J. Billera and B. Sturmfels, ''Iterated fiber polytopes,'' Mathematika, vol. 41, iss. 2, pp. 348-363, 1994. Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.), \(n\)-dimensional polytopes, Toric varieties, Newton polyhedra, Okounkov bodies Iterated fiber polytopes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Recently there is a growing interest in the construction of an algebraic geometry for a class of noncommutative algebras said to be ``quantized'' algebras. In one approach, probably the most abstract one, those noncommutative algebras are viewed as a ring of functions on a kind ``variety'' that remains virtual throughout the whole theory. Then the methods used can only be of categorical nature (derived categories,\dots) and focus eventually on cohomological methods (quantum cohomology). There is another approach dealing with more ring theoretical objects, i.e. starting from a nice class of algebras, called schematic algebras it is possible to define a space, a Zariski topology, points, lines etc.\dots, then it becomes possible to develop a scheme theory, completely extending the theory of schemes in classical algebraic geometry.
In this paper attention is restricted to rings \(R\) with a Zariskian filtration \(FR\) such that the associated graded ring \(G(R)\) is a positive graded affine \(K\)-algebra,
\[
G(R)=K\oplus G(R)_1\oplus G(R)_2\oplus \dots,
\]
which is commutative and generated by \(G(R)_1\) as a \(K\)-algebra. We consider the scheme on \(\text{Proj} G(R)=Y\) given by the usual structure sheaf \({\mathcal O}_Y\) of \(G(R)\). Then we construct two deformations of \((Y,{\mathcal O}_Y)\) obtaining a first only by using classical localizations at Ore sets, and another one by using the rings of quantum sections which can be obtained from the first one by locally completing. In fact the sheaf of quantum sections of \(R\) appears as a degeneration of a formal scheme. The ``geometric'' properties of the deformed schemes are controlled by the algebraic theory relating ``filtered properties'' of filtered \(R\)-modules to ``graded properties'' of the associated graded \(G(R)\)-modules via the graded properties of the Rees modules over the Rees ring (blow-up ring). schematic algebras; quantum sections; Rees ring Noncommutative algebraic geometry, Relevant commutative algebra, Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics A deformation of projective schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The theme of this article is the connection between the pro-unipotent fundamental group \(\pi_1(X; o)\) of a pointed algebraic curve \(X\), algebraic cycles, iterated integrals, and special values of \(L\)-functions. The extension of mixed Hodge structures arising in the second stage in the lower central series of \(\pi_1(X; o)\) gives rise to a supply of complex points on the Jacobian \(\mathrm{Jac}(X)\) of \(X\) indexed by Hodge cycles on \(X\times X\). The main results of this note relate these points to the Abel-Jacobi image of Gross-Kudla-Schoen's modified diagonal in \(X^3\), and express this Abel-Jacobi image in terms of iterated integrals. The resulting formula is the basis for the practical complex-analytic calculations of these points when \(X\) is a modular (or Shimura) curve, a setting where the recent work \textit{Xinyi Yuan}, \textit{Shou-Wu Zhang} and \textit{Wei Zhang} [Triple product \(L\)-series and Gross-Schoen cycles. I: Split case. (preprint), \url{https://math.berkeley.edu/~yxy/preprints/triple.pdf}] relates their non-triviality to special values of certain \(L\)-series attached to modular forms. pro-unipotent fundamental group; algebraic cycle; mixed Hodge structure; \(L\)-function H. Darmon, V. Rotger, and I. Sols, ''Iterated integrals, diagonal cycles and rational points on elliptic curves'' in Publications mathématiques de Besançon: Algèbre et théorie des nombres, 2012/2, Publ. Math. Besançon Algèbre Théorie Nr. 2012, Presses Univ. Franche-Comté, Besançon, 2012, 19--46. Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols, Elliptic curves over global fields, \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture, (Equivariant) Chow groups and rings; motives, Algebraic cycles, Rational points Iterated integrals, diagonal cycles and rational points 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 an epoch-making paper from 1994, \textit{A. Beilinson} and \textit{P. Deligne} gave an interpretation of polylogarithmic functions in terms of variations of mixed Hodge structures, which led them to a motivic proof of the famous Zagier conjecture based upon Deligne's theory of mixed motives [in: Motives. Proceedings of the summer research conference on motives, Proc. Symp. Pure Math. 55, 97--121 (1994; Zbl 0799.19004)].
The aim of the present extensive and profound paper is to generalize various results of Beilinson and Deligne to the author's theory of iterated integrals within the framework of cosimplicial objects, mixed motives, and monodromy representations of fundamental groups. The motivic character of iterated integrals, which appear as horizontal sections of fiber bundles equipped with a Gauss-Manin connection and satisfy certain basic functional equations, has been pointed out before by the author [in: Algebraic \(K\)-theory and algebraic topology, NATO ASI Ser., Ser. C, Math. Phys. Sci. 407, 287--327 (1993; Zbl 0916.14006)].
These facts suggest that the Zagier conjecture generalizes to iterated integrals on algebraic varieties, and that there should be a motivic proof analoguous to the original one by Beilinson and Deligne. In this regard, the present paper is thought as being the first part of a huge and general program to work out these ideas in full detail, with further generalizations to follow. As the author points out, the present paper is a revised version of his foregoing preprint [Mixed Hodge structures and iterated integrals, Prépublication de l'Université de Nice-Sophia Antipolis, 1997]. In general, he has kept the exposition nearly self-contained and pleasantly detailed, with as few quotations as possible, full proofs, significant examples, and a list of related problems that are still open.
Finally, as the author also points out in the introduction to this paper, his generalized approach follows the original one by Beilinson and Deligne rather closely, mutatis mutandis, which certainly makes the underlying higher principles of the entire theory more apparent. mixed Hodge structures; polylogarithmic functions; iterated integrals; monodromy representations; Zagier conjecture; cosimplicial objects; mixed motives Wojtkowiak, Z.: Mixed Hodge structures and iterated integrals I. Motives, polylogarithms and Hodge theory (I), 121-208 (2002) Variation of Hodge structures (algebro-geometric aspects), Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects), Polylogarithms and relations with \(K\)-theory, Other functions defined by series and integrals, Transcendental methods, Hodge theory (algebro-geometric aspects), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Transcendental methods of algebraic geometry (complex-analytic aspects) Mixed Hodge structures and iterated integrals. I | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(Q\) be a smooth quadric surface and \(Z\subset Q\) a zero-dimensional scheme. We study the postulation of a general union of \(Z\) and prescribed numbers of fat points with multiplicity 2 and 3. postulation; Hilbert function; fat point; smooth quadric surface Projective techniques in algebraic geometry, Curves in algebraic geometry Postulation of zero-dimensional schemes on a smooth quadric surface | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A classical theorem says that a planar cubic curve passing through eight of the nine intersection points of two triples of lines passes also through the nineth point (see Theorem 1 for a more precise formulation). Yet another classical theorem has a similar structure (see Theorem 2).
One aim of this paper is to reveal the common roots of these theorems and to present a very far generalization for algebraic hypersurfaces of arbitrary degree \(n\) in \(\mathbb{R}^{d}\) with any dimension \(d\). The essential tool to achieve this is the notion of a ''skew grid'' (see Def. 1).
The second aim of the present investigations, is to derive constructions of sets \(\mathbf S \subset \mathbb{R}^{d}\) which are not contained in any algebraic hypersurface. Such sets are well suited as knot sets for multivariate interpolation by polynomial functions in \(d\) variables. Algebraic varieties; (generalized) grids; multivariate interpolation Hypersurfaces and algebraic geometry, Projective techniques in algebraic geometry, Questions of classical algebraic geometry, Numerical interpolation Algebraic varieties over skew grids with applications to multivariate 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 \(S\) be a smooth projective algebraic surface over an algebraically closed field \(\kappa\) and let \(H_d= \text{Hilb}_bS\) be the Hilbert scheme of 0-dimensional subschemes of length \(d\) in \(S\). Taking a point \(x\in S\), we shall study the set \(H_d[x]= \{\xi\in H_d |\text{Supp} \xi =x\}\). The set \(H_d[x]\) is obviously endowed with a variety structure, i.e., it is a reduced irreducible scheme of dimension \(d-1\) over \(\kappa\). Moreover, the set \(H_d[x]\) is a subscheme in \(H_d\). It is referred to as the punctual Hilbert scheme of the surface. For \(d=1\) and 2, its description is trivial: \(H_1[x]= \{\text{point}\}\) and \(H_2[x]= P(T_xS) \simeq \mathbb{P}^1\). Even for \(d=3\), the set \(H_d[x]\) acquires singularities: The set \(H_3[x]\) is a surface isomorphic to a cone over the space cubic curve in \(\mathbb{P}^3\). For higher dimensions \(d\), the singularities of the set \(H_d[x]\) are quite complicate. The description can, possibly, be made in terms of Iarrobino's stratification.
In this paper, we use a different approach based on the natural birational model \(X_d\) of the scheme \(H_d[x]\), which is obtained by ``lifting'' the scheme \(H_d[x]\) to the Hilbert scheme of complete flags \(\Gamma_{12 \dots d} =\{(\xi_1, \dots, \xi_d)\in \prod^d_{k=1} H_k |\xi_1 \subset\xi_2 \subset \cdots \subset \xi_d\}\). The model thus constructed is helpful in giving an algebraic-geometric description of the three-dimensional variety \(H_4[x]\) and its singularities. punctual Hilbert scheme of a surface; complete flags; singularities A. S. Tikhomirov, ''A smooth model of punctual Hilbert schemes of a surface,''Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.],208, 318--334 (1995). Parametrization (Chow and Hilbert schemes), Surfaces and higher-dimensional varieties A smooth model for the punctual Hilbert scheme of 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 [For the entire collection see Zbl 0681.00016.]
The aim of this paper is the study of singly-exponential techniques for stratification of real semi-algebraic sets, that means to find a decomposition of \({\mathfrak R}^ d\) into simple-shaped cells of dimensions ranging from 0 to d for n d-variate polynomials \(f_ 1,...,f_ n\) (d fixed) with rational coefficients, such that each \(f_ i\) has constant sign over each cell. Collins has given in 1975 a sign-invariant decomposition with \(O(n^{2^ d-1})\) cells. The authors are given a sign-invariant decomposition with \(O(n^{2d-1}*\beta (n))\) cells where \(\beta\) (n) is a slow-growing function. As application of this study is the generalized point location problem as by Chazelle and Sharir in 1989. The both problems stratification and generalized point location are restricted problems related to the theory of reals and they can be solved in singly-exponential time as by Canny 1987 or Caniglia et co., Grigor'ev, et co., Renager 1988. From the logical point of view are these problems quantifier elimination which can be solved into doubly- exponential time as by Davenport and Heintz 1988. stratification of real semi-algebraic sets; decomposition; generalized point location problem B. Chazelle, H. Edelsbrunner, L. J. Guibas, and M. Sharir, A Singly-Exponential Stratification Scheme for Real Semi-Algebraic Varieties and Its Applications,Proc. 16th ICALP, Lecture Notes in Computer Science, Vol. 372, Springer-Verlag, Berlin, 1989, 179--193. Analysis of algorithms and problem complexity, Semialgebraic sets and related spaces, Symbolic computation and algebraic computation, Computational aspects of higher-dimensional varieties A singly-exponential stratification scheme for real semi-algebraic varieties and its applications | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A famous theorem of \textit{A. Bondal} and \textit{D. Orlov} [Compos.\ Math.\ 125, No.\ 3, 327--344 (2001; Zbl 0994.18007)] states that if \(X\) is a smooth and projective variety over a field \(k\) with an ample or antiample canonical sheaf, then any \(k\)-linear and graded equivalence \(\Phi: D^b(X)\cong D^b(Y)\) between the bounded derived categories of coherent sheaves on \(X\) and \(Y\), where \(Y\) is another smooth projective variety, induces an isomorphism \(X\cong Y\). The purpose of the paper under review is to generalize this result.
Firstly, the authors prove that if \(X\) is a connected equidimensional Gorenstein projective \(k\)-scheme with ample or antiample canonical sheaf, then if \(D^b(X)\cong D^b(Y)\) or \(D_{\text{perf}}(X)\cong D_{\text{perf}}(Y)\) for some other proper \(k\)-scheme \(Y\), then \(X\cong Y\). Here, \(D_{\text{perf}}(X)\) denotes the full subcategory of \(D^b(X)\) consisting of perfect objects, that is, those \(P\) such that \(\oplus_n \text{Hom}^n(P,Q)\) is finite-dimensional for all \(Q\). Secondly, a relative version is also proved. Roughly speaking, if \(X\) is as above, \(f: X\rightarrow T\) is a proper morphism of finite Tor-dimension, \(T\) is Cohen-Macaulay, the relative dualizing complex is isomorphic to a shift of an invertible sheaf which is either \(T\)-ample or \(T\)-antiample and \(X'\) is another Gorenstein \(T\)-scheme, then the existence of a \(T\)-linear equivalence \(D_{\text{perf}}(X)\rightarrow D_{\text{perf}}(X')\) implies that \(X\cong X'\) as \(T\)-schemes. Here, \(T\)-linearity roughly means that the equivalence respects tensor products with perfect objects coming from \(T\). In addition to the above results, in both cases the group of autoequivalences can be computed and is as expected from the smooth case.
In Bondal and Orlov's proof one first identifies so-called point-like objects, which are categorically defined and turn out to be skyscraper sheaves up to shift. From this invertible sheaves and the topology of \(X\) can be recovered and the Serre functor allows to reconstruct the canonical ring. In the paper under review the authors find the correct substitute for point-like objects in their setting, which they call Gorenstein zero cycle objects. Any such object is proved to correspond, up to shift, to the structure sheaf of a Gorenstein zero cycle supported at some closed point. From here the proof proceeds along the same lines as in the smooth case. derived categories; equivalences; reconstruction; linear functors; point-like object; skyscraper sheaves; Serre functor; Gorenstein zero cycle Sancho de Salas, C.; Sancho de Salas, F., Reconstructing schemes from the derived category, Proc. edinb. math. soc. (2), 3, 55, 781-796, (2012) Derived categories, triangulated categories, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Reconstructing schemes from the derived category | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(R\) be a complete discrete valuation ring with fraction field \(K\) and residue field \(k\) of characteristic \(p>0\).
{ Theorem (Raynaud).} There is an equivalence of categories between {\parindent=6mm \begin{itemize} \item[{\(\cdot\)}] the category of quasi-paracompact admissible formal \(R\)-schemes, localized by the class of formal blow ups, and \item [{\(\cdot\)}] the category of quasi-separated quasi-paracompact \(K\)-rigid spaces.
\end{itemize}} The main purpose of the present paper is to prove an analogue of Raynaud's theorem: with formal \(R\)-schemes replaced by weak formal \(R\)-schemes, and with \(K\)-rigid spaces replaced by \(K\)-dagger spaces.
Weak formal \(R\)-schemes have been introduced by \textit{D. B. Meredith} [Nagoya Math. J. 45, 1--38 (1972; Zbl 0207.51502)]. They allow a sheafification and globalization of the constructions of Monsky and Washnitzer who used overconvergent function algebras to define a \(p\)-adic cohomology theory for smooth affine \(k\)-schemes. On the other hand, \(K\)-dagger spaces have been introduced by \textit{E. Grosse-Klönne} [J. Reine Angew. Math. 519, 73--95 (2000; Zbl 0945.14013)]. They can be thought of as \(K\)-rigid spaces, but with the usual Tate algebras (and their quotients) replaced by certain dense subalgebras (and their quotients) consisting of 'overconvergent' (rather than just 'convergent') power series.
Thus, the main theorem here reads:
{ Theorem.} There is an equivalence of categories between {\parindent=6mm \begin{itemize} \item[{\(\cdot\)}] the category of quasi-paracompact admissible weak formal \(R\)-schemes, localized by the class of weak formal blow ups, and \item [{\(\cdot\)}] the category of quasi-separated quasi-paracompact \(K\)-dagger spaces.
\end{itemize}} weak formal scheme; dagger space; generic fibre Rigid analytic geometry, \(p\)-adic cohomology, crystalline cohomology An analogue of Raynaud's theorem: weak formal schemes and dagger spaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We show that an \(n\)-geometric stack may be regarded as a special kind of simplicial scheme, namely a Duskin \(n\)-hypergroupoid in affine schemes, where surjectivity is defined in terms of covering maps, yielding Artin \(n\)-stacks, Deligne-Mumford \(n\)-stacks and \(n\)-schemes as the notion of covering varies. This formulation adapts to all HAG contexts, so in particular works for derived \(n\)-stacks (replacing rings with simplicial rings). We exploit this to describe quasi-coherent sheaves and complexes on these stacks, and to draw comparisons with Kontsevich's dg-schemes. As an application, we show how the cotangent complex controls infinitesimal deformations of higher and derived stacks. derived algebraic geometry; higher stacks; simplicial schemes J. P. Pridham, Presenting higher stacks as simplicial schemes, Adv. Math., 238 (2013), 184--245.Zbl 1328.14028 MR 3033634 Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Derived categories, triangulated categories, Generalizations (algebraic spaces, stacks), Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects), Simplicial sets, simplicial objects (in a category) [See also 55U10] Presenting higher stacks as simplicial schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let X be the spectrum of a quasi-unmixed local Nagata ring over a field of characteristic zero, and let Y be a reduced closed subscheme of X. The main result of the paper is the following: if x is the closed point of X and if X is equimultiple along Y, then the embedding dimension of Y at x is less than or equal to dim(X). Then the author applies this result to get a stability theorem for standard bases and a structure theorem for tangent cones. multiplicity; spectrum of a quasi-unmixed local Nagata ring; embedding dimension; standard bases; tangent cones Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Multiplicity theory and related topics, Relevant commutative algebra On equimultiple subschemes of a local scheme over a field of characteristic zero | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 complex, complete algebraic curve \(C\), one can form the Hilbert schemes \(C^{[m]}\) parametrizing subschemes \(Z \subset C\) of length \(m\) and nested Hilbert schemes \(C^{[m,m+1]}\) parametrizing pairs \(\{(Y,Z): Y \in C^{[m]}, Z \in C^{[m+1]}, Y \subset Z\}\). If \(\pi: \mathcal C \to B\) is a proper flat family of curves, there are relative versions \(\pi^{[m]}: {\mathcal C}^{[m]} \to B\) and \(\pi^{[m,m+1]}: {\mathcal C}^{[m,m+1]} \to B\). For families of curves that are reduced and locally planar over a smooth base \textit{V. Shende} showed that if \(\mathcal C \to B\) is a locally versal family of reduced, locally planar curves over a smooth base \(B\), then the total space \({\mathcal C}^{[m]}\) is smooth over sufficiently small analytic open sets in \(B\) provided \(m \leq \dim B\) [Compos. Math. 148, 531--547 (2012; Zbl 1312.14015)]. The author uses similar methods to prove an analogous smoothness result for \({\mathcal C}^{[m,m+1]}\).
For a smooth family \(\pi^{[m]}: {\mathcal C}^{[m]} \to B\), the decomposition theorem of \textit{Beilinson, Bernstein and Deligne} [Astérisque 100, (1982; Zbl 0536.14011)] says that the complex \(R \pi_*^{[m]} \mathbb C\) decomposes as a direct sum of shifted intersection complexes. In this setting, \textit{L. Migliorini} and \textit{V. Shende} determined the decomposition of \(R \pi^{[m]}_* \mathbb Q [m + \dim B]\) [J. Eur. Math. Soc. (JEMS) 15, 2353--2367 (2013; Zbl 1303.14019)], showing that none of the summands have proper support in \(B\). The author determines the analogous decomposition of \(R \pi_*^{[m,m+1]} \mathbb Q [m+1+\dim B]\) assuming that \({\mathcal C}^{[m,m+1]}\) is smooth, again showing that none of the summands has proper support in \(B\). The proof uses the theory of higher discriminants introduced by \textit{L. Migliorini} and \textit{V. Shende} [Algebr. Geom. 5, No. 1, 114--130 (2018; Zbl 1406.14005)]. locally planar curves; nested Hilbert schemes; versal deformations Parametrization (Chow and Hilbert schemes), Plane and space curves, Algebraic moduli problems, moduli of vector bundles, Transcendental methods, Hodge theory (algebro-geometric aspects) A support theorem for nested Hilbert schemes of planar 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 their article ``Infinitesimal Lifting and Jacobi Criterion for Smoothness on Formal Schemes'' [Commun. Algebra 35 (4), 1341--1367 (2007; Zbl 1124.14006)], the authors adapted the theory of smooth morphisms to the case of noetherian formal schemes based on properties of the completion of the module of Kähler differentials. In particular the property of formal smoothness in terms of the infinitesimal liftings of maps in the category of algebraic schemes was described. It was proved that smooth morphisms of noetherian formal schemes are flat and the associated module of differentials is locally free. In this paper a detailed study of the relationship between the infinitesial lifting properties of a morphism of formal schemes and those of the corresponding maps of usual schemes associated to the directed systems that define the corresponding formal schemes is made. A characterization of completion morphisms as pseudo--closed immersions that are flat is given. The local structure of smooth and étale morphisms between locally noetherian formal schemes is described. formal scheme; smooth maps; étale morphisms Tarrío, L. Alonso; López, A. Jeremías; Rodríguez, M. Pérez: Local structure theorems for smooth maps of formal schemes. J. pure appl. Algebra 213, 1373-1398 (2009) Infinitesimal methods in algebraic geometry, Formal neighborhoods in algebraic geometry Local structure theorems for smooth maps of formal schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper begins with a nice overview of the theory of Macaulay inverse systems. The authors then consider the situation of the inverse system of the ideal of a zero-dimensional scheme \(\mathfrak Z\) in \(\mathbb P^n\), beginning with one that is supported at a point and then passing to the general case. This represents the focus of the paper -- how to pass from the local inverse systems of the irreducible components of \(\mathfrak Z\) to the global inverse system. When \(\mathfrak Z\) is locally Gorenstein, the authors give conditions under which for a general element \(F\) of degree \(d\) that is apolar to \(\mathfrak Z\), one can recover \(\mathfrak Z\) from \(F\). As a consequence, they show that a natural upper bound for the Hilbert function of Gorenstein Artin quotients of the coordinate ring of \(\mathfrak Z\) is achieved for large socle degree. They give some consequences for linkage, and for the uniqueness (in some cases) of generalized additive decompositions of a homogeneous form into powers of linear forms. Many of the results and remarks are labelled with short descriptions for the convenience of the reader. Macaulay inverse system; regularity degree; globalization; zero-dimensional scheme; Gorenstein Artin ring; irreducible components; generalized additive decomposition Cho, Y. H.; Iarrobino, A., Inverse systems of zero dimensional schemes in \(\mathbb{P}^n\), J. Algebra, 366, 42-77, (2012) Projective techniques in algebraic geometry, Commutative rings of differential operators and their modules, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Parametrization (Chow and Hilbert schemes) Inverse systems of zero-dimensional schemes in \(\mathbb P^n\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(S\) be a Dedekind scheme and \(f: X\to S\) be a regular surface fibered onto \(S\) with fibers of genus \(\geq 2\). The relative canonical bundle \(\omega_{X/S}\) may not be ample, due to the presence of \(-2\)-curves in the fibers, but passing to the relative canonical model \(f': X'\to S\) one obtains a mildly singular surface such that \(\omega_{X'/S}\) is relatively ample. So it is natural to ask for what \(n\) the line bundle \(\omega_{X'/S}^{\otimes n}\) is relatively very ample. When \(X=X'\) is smooth, then it is easy to check that this is true for \(n\geq 3\). In this paper it is proven that, much more generally, \(\omega_{X'/S}^{\otimes n}\) is relatively very ample for \(n\geq 3\) if the residue field of every closed point of \(S\) is a perfect field. This last assumption is quite weak and it is not known whether it is necessary. family of curves; relative canonical bundle; very ampleness Families, moduli of curves (algebraic), Arithmetic ground fields (finite, local, global) and families or fibrations, Divisors, linear systems, invertible sheaves On pluri-canonical systems of arithmetic surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We present a new certified and complete algorithm to compute arrangements of real planar algebraic curves. Our algorithm provides a geometric-topological analysis of the decomposition of the plane induced by a finite number of algebraic curves in terms of a cylindrical algebraic decomposition of the plane. Compared to previous approaches, we improve in two main aspects: Firstly, we significantly limit the types of involved exact operations, that is, our algorithms only use resultant and gcd computations as purely symbolic operations. Secondly, we introduce a new hybrid method in the lifting step of our algorithm which combines the use of a certified numerical complex root solver and information derived from the resultant computation. Additionally, we never consider any coordinate transformation and the output is also given with respect to the initial coordinate system.{
}We implemented our algorithm as a prototypical package of the C++-library \textsc{Cgal}. Our implementation exploits graphics hardware to expedite the resultant and gcd computation. We also compared our implementation with the current reference implementation, that is, \textsc{Cgal}'s curve analysis and arrangement for algebraic curves. For various series of challenging instances, our experiments show that the new implementation outperforms the existing one. algebraic curves; arrangements; certified algorithms; topology Berberich, E.; Emeliyanenko, P.; Kobel, A.; Sagraloff, M.: Arrangement computation for planar algebraic curves, (2011) Computer graphics; computational geometry (digital and algorithmic aspects), Computational aspects of algebraic curves, Symbolic computation and algebraic computation Arrangement computation for planar algebraic curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(J\) be the generalized Jacobian of an integral projective curve \(C\) over an algebraically closed field, and let \(\overline J\) its compactification parametrizing torsion-free sheaves of rank 1 and degree 0 on \(C\).
Suppose that \(C\) has at worst double points as singularities and is of positive genus, for a fixed line bundle \({\mathcal L}\) of degree 1, the Abel map \(A_{{\mathcal L}}: C\to\overline J\) is a closed embedding.
In a previous paper jointly with \textit{M. Cagné} [J. Lond. Math. Soc., II. Ser. 65, No. 3, 591--610 (2002; Zbl 1060.14045)] the authors proved that the pullback map \(A^*_{{\mathcal L}}: \text{Pic}^0_{\overline J}\to J\) is an isomorphism which is independent of \({\mathcal L}\). The closure of \(\text{Pic}^0_{\overline J}\) in the moduli space of torsion-free sheaves of rank 1 on \(\overline J\) is a natural compactification.
The main result of the paper under review is
Theorem 4.1. Let \(C/S\) be a flat projective family of integral curves with at worst ordinary nodes and cusps, then the extended map \(A^*_{{\mathcal L}}\) is an isomorphism. compactified Jacobian; autoduality; curves with double points M. Melo, A. Rapagnetta and F. Viviani. \textit{Fourier-Mukai and autoduality for compactified Jacobians. I}. At http://arxiv.org/abs/1207.7233, with an appendix by A.C. López-Martín. Picard schemes, higher Jacobians, Jacobians, Prym varieties The compactified Picard scheme of the compactified Jacobian | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Given a complex projective variety \(X,\) its dual \(X^\vee\) is defined as the closure, in the dual projective space, of all hyperplanes tangent to \(X\) at a smooth point. It is known that generically \(X^\vee\) is a hypersurface. If \(\mathrm{codim}\, X^\vee >1,\) \(X\) is said to be defective and the quantity \(\mathrm{def}(X):=\mathrm{codim}\,X^\vee -1\) is called the dual defect of \(X\).
In the present paper the authors study the dual defect \(\mathrm{def}(X_A)\) of a projective toric variety \(X_A\) associated to a configuration of points \(A.\) Turns out that \(\mathrm{def}(X_A)\) can be computed from \(A\).
Are explored four approaches to the study of the dual defect of \(X_A\) from \(A:\) non-splitting flags, iterated circuits, \(\underline{\sigma}\)-matrices and Cayley configurations. projective toric varieties; dual defect; Cayley decompositions; Gale dual; tropical variety; iterated circuits; non-splitting flags Toric varieties, Newton polyhedra, Okounkov bodies, Combinatorial aspects of algebraic geometry, Combinatorial aspects of tropical varieties Non-splitting flags, iterated circuits, \(\underline{\sigma}\)-matrices and Cayley configurations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems If \(X\) is a smooth quasiprojective scheme and \(G\) is a finite group acting faithfully on \(X\), the quotient space of orbits \(X/G\) is in general a singular scheme. A kind of more refined variant of a quotient of \(X\) by \(G\) is the \(G\)-Hilbert scheme of \((G,X)\) introduced in \textit{Y. Ito} and \textit{I. Nakamura} [Proc. Japan Acad., Ser. A 72, 135--138 (1996; Zbl 0881.14002)] by replacing the \(G\)-orbits with \(0\)-dimensional \(G\)-invariant subsets of \(X\). Again in general the \(G\)-Hilbert scheme is a singular variety but expectedly ``less'' singular than \(X/G\). The known cases of smooth \(G\)-Hilbert schemes appear as minimal resolutions of Klein singularities and crepant resolutions of the quotients of \({\mathbb{C}}^3\) by a finite subgroup of \(\text{SL}_3({\mathbb{C}})\) [cf. \textit{T. Bridgeland, A. King} and \textit{M. Reid}, J. Am. Math. Soc. 14, 535--554 (2001; Zbl 0966.14028)]. But the question for which finite subgroups \(G\) of \(\text{GL}_n({\mathbb{C}}), n \geq 4\) the \(G\)-Hilberts scheme of \(({\mathbb{C}}^n,G)\) is a crepant resolution of the quotient \({\mathbb{C}}^n/G\) still remains open. The only attempt was given by \textit{D. Dais, C. Haase} and \textit{G. Ziegler} [Tôhoku Math. J., II. Ser. 53, 95--107 (2001; Zbl 1050.14044)], and it is restricted to the 2-dimensional case. In the present paper a special example of an action of a finite subgroup of \(\text{GL}_4({\mathbb{C}})\) on \({\mathbb{C}}^4\)
is described, where the answer to the above question is positive. The group is the cyclic group \(\mu_{15}\) of order \(15\) with a generator \(\varepsilon = \text{exp}(2{\pi}i/15)\), which acts on \({\mathbb{C}}^4\) by weights \((1,2,4,8)\). The quotient \({\mathbb{C}}^4/{\mu}_{15}\) has a Gorenstein canonical singularity at the origin. The main result of the paper (theorem 2.9) affirms that the \(\mu_{15}\)-Hilbert scheme of \({\mathbb{C}}^4\) is smooth and gives a crepant resolution of the singularity of \({\mathbb{C}}^4/\mu_{15}\). quotient singularities; crepant resolutions; toric varieties Sebestean, M.: A smooth four-dimensional G-Hilbert scheme. Serdica math. J. 30, No. 2 -- 3, 283-292 (2004) Parametrization (Chow and Hilbert schemes), Group actions on varieties or schemes (quotients), Global theory and resolution of singularities (algebro-geometric aspects), \(4\)-folds A smooth four-dimensional \(G\)-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 is largely a survey paper. Its focus is on various Grothendieck topologies on affine schemes and their connections to recent works in commutative algebra surrounding the direct summand theorem and the existence of big Cohen-Macaulay algebras. In the first part, the authors review basics of Grothendieck topology, with an emphasize on the canonical topology on affince schemes, including a proof that the canonical topology on affine schemes is equivalent to the (effective) descent topology (which was originally due to Olivier). In the second part, the authors give a geometric interpretation of the direct summand theorem (resp. the existence of big Cohen-Macaulay algebras). Namely, the direct summand theorem (resp. existence of big Cohen-Macaulay algebras) is saying that any finite cover of a regular affine scheme is a covering for the canonical topology (resp. the fpqc topology). In the third part, the authors survey what was known about splinters, i.e., rings that splits off from all their module-finite extensions, including the derived variants and connections to F-singularities. Some interesting questions about an fpqc analog of splinters are raised. Examples are given throughout the article, including a negative answer to a question of Ferrand (Example 10.6). Grothendieck topology; canonical cover; fpqc cover; direct summand theorem; big Cohen-Macaulay algebras; splinters Relevant commutative algebra, Schemes and morphisms, Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Grothendieck topologies and Grothendieck topoi, Extension theory of commutative rings, Morphisms of commutative rings, Research exposition (monographs, survey articles) pertaining to algebraic geometry On the canonical, fpqc, and finite topologies on affine schemes. The state of the art | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(R = \mathbb{Z}[x,y]\) denote the ring of polynomials in \(x\) and \(y\) with integer coefficients and fix a grading by some abelian group \(A\). The main result of this paper establishes the smoothness and irreducibility of the multigraded Hilbert scheme parametrizing those ideals of \(R\) with a given Hilbert function.
The paper's generality is remarkable on two fronts: one is that the polynomial ring only assumes integer coefficients, which implies that the theorem extends to schemes over any fixed scheme via change of base; the other is that the the group \(A\) is permitted to be any abelian group, even one with non-zero torsion. As is pointed out in the paper, the major restriction is the use of two variables, which is essential. Indeed it is known that in general multigraded Hilbert schemes may fail to even be connected and can be highly non-singular.
The proof is broken down into several steps. The fundamental idea is to connect any point on the Hilbert scheme to a distinguished point and show that the tangent space to any point along the connecting path has constant dimension. The paper is well written with thorough proofs. The authors do an excellent job of bringing the reader up to the current state of knowledge on relevant issues concerning the multigraded Hilbert scheme. Hilbert schemes; multigraded rings; combinatorial commutative algebra DOI: 10.1016/j.aim.2009.10.003 Parametrization (Chow and Hilbert schemes), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) Smooth and irreducible multigraded Hilbert schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We study zero-dimensional fat points schemes on a smooth quadric \(Q\cong\mathbb{P}^1\times\mathbb{P}^1\), and we characterize those schemes which are arithmetically Cohen-Macaulay (aCM for short) as subschemes of \(Q\) giving their Hilbert matrix and bigraded Betti numbers. In particular, we can compute the Hilbert matrix and the bigraded Betti numbers for fat points schemes with homogeneous multiplicities and whose support is a complete intersection. Moreover, we find a minimal set of generators for schemes of double points whose support is a CM. zero-dimensional fat points schemes; smooth quadric; arithmetically Cohen-Macaulay; complete intersection; minimal set of generators Guardo E.: Fat points schemes on a smooth quadric. J. Pure Appl. Algebra 162, 183--208 (2001) Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Schemes and morphisms, Complete intersections, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Syzygies, resolutions, complexes and commutative rings, Linkage, complete intersections and determinantal ideals Fat points schemes on a smooth quadric | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a smooth connected projective curve of genus \(q\) over an algebraically closed field \(\mathbb{K}\). A coherent system of type \((n,d,k)\) on \(X\) is a pair \((E, V)\) where \(E\) is a vector bundle of rank \(n\) and degree \(d\) over \(X\) and \(V \subseteq H^{0}(X, E)\) is a linear subspace of dimension \(k\). For \(\alpha \in \mathbb{R}\), the \(\alpha\)-slope of \((E,V)\) is the number \(d/n + \alpha k/n\). Moduli of \(\alpha\)-semistable coherent systems have been studied by \textit{S. Bradlow, O. García-Prada, V. Muñoz} and \textit{P. E. Newstead} [Int.\ J.\ Math.\ 14, No.\ 7, 683--733 (2003; Zbl 1057.14041)].
In the present article, the author gives a sufficient condition on \(q\), \(n\), \(d\) and \(k\) for the existence of a coherent system of type \((n,d,k)\) such that \(V\) spans \(E\) and \(E\) is a stable bundle, \((E,V)\) is \(\alpha\)-stable for all \(\alpha > 0\), and both the Petri map \(V \otimes H^{0}(X, E^{*} \otimes \omega_{X}) \to H^{0}(X, \mathrm{End}(E) \otimes \omega_{X})\) and the natural map \(\bigwedge^{n}V \to H^{0}(X, \det(E))\) are injective.
Now let \(\mathbb{K}\) be of characteristic zero, and consider a double cover \(f \colon X \to Y\). The author shows that for all \(\alpha > 0\), the pullback \((f^{*}E, f^{*}V)\) of an \(\alpha\)-semistable coherent system \((E,V)\) of type \((n,d,k)\) on \(Y\) is a \(2\alpha\)-semistable coherent system of type \((n, 2d, k)\) on \(X\). Furthermore, if \(E\) is a stable vector bundle and the covering is ramified then in fact stability (of both abstract bundle and coherent system) is preserved by the pullback. As a corollary and using \textit{H. Lange} and \textit{P. E. Newstead}'s results [Int.\ J.\ Math.\ 16, No.\ 7, 787--805 (2005; Zbl 1078.14045)], sufficient conditions are given for the existence of various \(\alpha\)-semistable coherent systems over bielliptic curves. Finally, using [\textit{H. Lange} and \textit{P. E. Newstead}, Int.\ J.\ Math.\ 15, No.\ 4, 409--424 (2004; Zbl 1072.14039)], the author shows that over a curve which is a double cover of \(\mathbb{P}^1\), there exist \(\alpha\)-semistable coherent systems where the ambient bundle may fail to be semistable. vector bundles on curves; stable vector bundles; Brill-Noether theory for vector bundles Ballico E.: Coherent systems with many sections on projective curves. Int. J. Math. 17, 263--267 (2006) Vector bundles on curves and their moduli Coherent systems with many sections on projective curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper is, according to the authors, another step in the direction of providing singly exponential algorithms for specific real algebraic geometry problems. The main result is a stratification algorithm that gives a stratification of the real affine \(d\)-dimensional space, sign invariant with respect to a given family of \(n\) polynomials in \(d\) variables, which has \(O(n^{2d-2})\) number of cells (therefore singly exponential with respect to the number of variables), having each cell a constant description size. This number of cells is a major improvement if compared to the number of cells (doubly exponential) produced by Collin's algorithm, of a more general purpose. Moreover, it is closer to Milnor's optimal bound \(O(n^ d)\). The whole algorithm presented in this paper is also singly exponential in time, but produces polynomials which are of doubly exponential degree (in the dimension of the space). On the other hand the cell decomposition produced here has not as good topological properties as in the case of Collin's algorithm, but suffices for the applications regarded by the authors: i.e. preprocessing a set of given real algebraic varieties (each one described by a polynomial of the above mentioned family) to support a fast point location. The resulting method allows to improve current solutions for a variety of optimization problems in computational geometry. singly exponential algorithms for real algebraic geometry problems 23.B.~Chazelle, H.~Edelsbrunner, L.J. Guibas, M.~Sharir, A singly-exponential stratification scheme for real semi-algebraic varieties and its applications. Theoret. Comput. Sci. 84, 77-105 (1991). Also in \(Proceedings of the 16th International Colloquium on Automata, Languages and Programming\), pp.~179-193 Semialgebraic sets and related spaces, Analysis of algorithms and problem complexity, Computer graphics; computational geometry (digital and algorithmic aspects) A singly exponential stratification scheme for real semi-algebraic varieties and its applications | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Nonstandard mathematics furnishes a remarkable connection between analytic and algebraic geometry. We describe this interplay for the most basic notions like complex spaces/algebraic schemes, generic points, differential forms etc. We obtain -- by this point of view -- in particular new results on the prime spectrum of a Stein algebra. complex spaces; nonstandard schemes; generic points; analytic Nullstellensatz; Stein algebras; internal polynomials Khalfallah, Adel; Kosarew, Siegmund, Complex spaces and nonstandard schemes, J. Log. Anal., 2, Paper 9, 60 pp., (2010) Complex spaces, Nonstandard models in mathematics, Schemes and morphisms, Nonstandard analysis, Stein spaces, Differential forms in global analysis Complex spaces and nonstandard 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 There are different ways to study the singularities of a plane curve branch \(C \subset \mathbb{K}^2\), for \(\mathbb{K}\) algebraically close field of characteristic 0. In one way (J. Nash), various information about the singularities, their resolutions and numerical invariants could be extracted from the jet schemes and the arc space of \(C\). Another way (B. Teissier), by re-embedding \(C\) in \(\mathbb{K}^{N}\) it is possible to resolve the singularity by one toric morphism. In the article under review both approaches are used to construct an embedded resolution of \((C, \mathbb{K}^{g+1})\), \(g\) being the number of Puiseux exponents of \(C\), such that the strict transform of the plane is the minimal embedded resolution.
After recalling briefly the definition of jet schemes, by a previous result of the second author, all irreducible components for the singular fiber of \(\pi_m : C_m \rightarrow C\) are grouped in three types. The restriction of the canonical morphisms \(\pi_{m+1, m}: C_{m+1} \rightarrow C_m\) then gives three types of projective systems of components in the jet schemes, and with each component is associated a divisorial valuation over \(\mathbb{A}^2\). Also, for the minimal embedded resolution of \(C\), three sets of exceptional divisors are defined. The main theorem then asserts that a valuation corresponding to a component from any type is determined by a divisor in one of these sets, which is the same for each type.
In the last section, by combinatorial arguments is established a relation between regular subdivisions of some kind of the cone \(\mathbb{R}_{\geq 0}^{g+1}\), and embedded resolutions of \(C\), and is proposed an algorithm to obtain such a subdivision. The equivariant morphism defined by it is an embedded resolution of \(C \subset \mathbb{K}^{g+1}\), with the property that the minimal embedded resolution of \((C, \mathbb{A}^2)\) is induced by the strict transform of the plane. This permits also to interpret the maximal contact in terms of jet schemes. plane branch; jet schemes; embedded resolution of singularities; toric variety Lejeune-Jalabert, M.; Mourtada, H.; Reguera, A., Jet schemes and minimal embedded desingularization of plane branches, \textit{Rev. Real Acad. Ciencias Exactas Fisicas Nat. Ser. A. Math.}, 107, 1, 145-157, (2013) Singularities of curves, local rings, Global theory and resolution of singularities (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies Jet schemes and minimal embedded desingularization of plane branches | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 the graduate polynomial algebra \(k[X,Y,Z,T]\) over a fixed algebraically closed field \(k\). Let \(\rho\) be a function \(\mathbb{Z}\to \mathbb{N}\) with finite support and let \(E_\rho\) be the scheme parametrizing graduate \(R\)-modules \(M=\bigoplus M_n\) of finite length such that \(\dim_k M_n= \rho(n)\). By \textit{M. Martin-Deschamps} and \textit{D. Perrin} [``Sur la classification des courbes gauches'', Astérisque 184-185 (1990; Zbl 0717.14017)] it is known that any result on the smoothness, the irreducibility or the dimension of \(E_\rho\) implies an analogous result on the (Hilbert sub-)schemes \(H_{\gamma,\rho}\) parametrizing locally Cohen-Macaulay space curves \(C\subset \mathbb{P}^3\) whose cohomology groups \(H^1(\mathbb{P}^3,{\mathcal I}_C(n))\) are of fixed dimension (\({\mathcal I}_C=\) ideal sheaf of \(C\) in \(\mathbb{P}^3)\).
In the present note the author examines the irreducibility and the dimension of the scheme \(E_\rho\) parametrizing \(R\)-modules of finite length and width three. Rao modules; locally Cohen-Macaulay space curves Linkage, Parametrization (Chow and Hilbert schemes), Linkage, complete intersections and determinantal ideals, Plane and space curves, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) On the schemes of Rao modules of length 3 | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a fixed projective scheme which is flat over a base scheme \(S\). The association taking a quasi-projective \(S\)-scheme \(Y\) to the scheme parametrizing \(S\)-morphisms from \(X\) to \(Y\) is functorial. We prove that this functor preserves limits, and both open and closed immersions. As an application, we determine a partition of schemes parametrizing rational curves on the blow-ups of projective spaces at finitely many points. We compute the dimensions of its components containing rational curves outside the exceptional divisor and the ones strictly contained in it. Furthermore, we provide an upper bound for the dimension of the irreducible components intersecting the exceptional divisors properly. rational curves; moduli spaces; blow-ups Fine and coarse moduli spaces, Families, moduli of curves (algebraic), Rational and birational maps, Rational and unirational varieties Properties of schemes of morphisms and applications to blow-ups | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems \textit{N. Schwartz} [in: Ordered algebraic structures, The 1991 Conrad Conf., 169-202 (1993; Zbl 0827.13008)] gave the following definitions: Let \(A\) be a ring or \(\mathbb{Q}\)-algebra contained in its real closure \(R(A)\), the so-called ring of abstract semialgebraic functions. For \(\alpha\) in the real spectrum \(\text{Sper}(A)\), we denote by \(\rho(\alpha)\) the real closure of the quotient field \(qf(A/\text{supp}(\alpha))\) with respect to the total order specified by \(\alpha\). This makes \(\prod \rho(\alpha)\) a lattice ordered ring with respect to the pointwise lattice order containing \(R(A)\). An \(a \in R(A)\) is piecewise polynomial if the real spectrum \(\text{Sper}(A)\) of \(A\) can be covered by finitely many closed constructible sets \(C_1, \ldots, C_r\) such that for certain \(a_1, \ldots, a_r\in A\) there holds \(a |C_i = a_i\) for all \(i\). Denoting the collection of piecewise polynomial functions by \(PW(A)\) and the collection of elements generated in the sense below by the elements of \(A\), the so-called sup-inf definable functions, by \(L(A)\), one has a chain \(L(A)\subseteq PW(A) \subseteq R(A) \subseteq \prod \rho (\alpha)\) of lattice ordered (sub)rings. A ring \(A\) is Pierce-Birkhoff if \(L(A) = PW(A)\). Defining inductively \(L_0(A) = A\), \(L_{n+1}(A) = L_n(A) [|a |: a \in L_n(A)]\), one has \(L(A) = \bigcup_n L_n (A)\).
The unresolved conjecture these authors made in 1956 [cf. \textit{G. Birkhoff} and \textit{R. S. Pierce}, Anais Acad. Bras. Cic. 28, 41-69 (1956; Zbl 0070.26602)] is that this equality holds for the case \(A=\mathbb{Q}[X_1,\dots,X_n]\), in which indeed \(L(A)\) and \(PW(A)\) assume the meaning the terminology suggests [see \textit{J. J. Madden}, Arch. Math. 53, No. 6, 565-570 (1989; Zbl 0691.14012) or \textit{C. N. Delzell}, Rocky Mt. J. Math. 53, No. 3, 651-668 (1989; Zbl 0715.14047)]. In the paper under review a construction corresponding to the inductive algebraic construction of \(L(A)\) is provided for \(PW(A)\). Interesting corollaries for understanding the Pierce-Birkhoff conjecture are derived. Assume (real) intermediate rings \(A \subset B\subset C\subset PW(A)\). Let \((\text{Spec}B)_{\text{re}}=\{\text{real primes of }B\) Pierce-Birkhoff conjecture; lattice ordered ring; semialgebraic function; piecewise polynomial functions; sup-inf definable; constructible sets Semialgebraic sets and related spaces, Real algebra An algebraic construction of the ring of piecewise polynomial functions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a genus \(g\) hyperelliptic projective curve, \(\alpha\in\mathbb{R}\), \(\alpha\geq 0\), and integers \(n\geq 2\), \(k\) and \(d\) such that \(0\leq d\neq n(2g-2)\). Here we prove when there is an \(\alpha\)-stable coherent system \((E,V)\) on \(X\) of type \((n,d,k)\) such that both \(E\) and \(\omega_X\otimes E^*\) are spanned if and only if \(d\) is even and there is an \((\alpha/2)\)-stable coherent system of type \((n,d/2,k)\) on \(\mathbb{P}^1\). Vector bundles on curves and their moduli, Special divisors on curves (gonality, Brill-Noether theory) Coherent systems with many ``spread'' sections on curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For a finite group scheme, the subadditive functions on finite-dimensional representations are studied. It is shown that the projective variety of the cohomology ring can be recovered from the equivalence classes of subadditive functions. Using Crawley-Boevey's correspondence between subadditive functions and endofinite modules, we obtain an equivalence relation on the set of point modules introduced in our joint work with \textit{S. B. Iyengar} and \textit{J. Pevtsova} [J. Am. Math. Soc. 31, No. 1, 265--302 (2018; Zbl 1486.16011)]. This corresponds to the equivalence relation on \(\pi \)-points introduced by \textit{E. M. Friedlander} and \textit{J. Pevtsova} [Duke Math. J. 139, No. 2, 317--368 (2007; Zbl 1128.20031)]. subadditive function; endofinite module; stable module category; finite group scheme Group schemes, Cohomology of groups, Representations of associative Artinian rings, Modular representations and characters, Cohomology theory for linear algebraic groups The variety of subadditive functions for finite 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 A blueprint is a pair \(B=(A,\mathcal{R})\) where \(A\) is a multiplicative monoid with zero object \(0\in A\) and \(\mathcal{R}\) is an equivalence relation on the semiring \(\mathbb{N}[A]\), which satisfies certain axioms. These axioms ensure that the quotient \(B^+=A/\mathcal{R}\) is a semiring. There is a natural notion of prime ideals \(\mathfrak{p}\subset B\), and the set \(\roman{Spec}(B)\) of all prime ideals is naturally endowed with a Zariski topology and a structure sheaf, taking values in the category \(\mathrm{Blpr}\) of blueprints. Blueprints thus lead to the notion of geometric blue schemes, in the same way as rings lead to schemes in algebraic geometry. In a similar way, the opposite category \(\mathrm{Aff}_{\mathbb F_1}^{\mathrm{can}}=\mathrm{Blpr}^{\circ}\) is endowed with a Grothendieck topology. Sheaves on this site that admit an ''affine covering'' are called subcanonical blue schemes. So one has the category \(\mathrm{Sch}_{\mathbb F_1}^{\mathrm{can}}\) of subcanonical blue schemes and the category \(\mathrm{Sch}_{\mathbb F_1}^{\mathrm{geo}}\) of geometric blue schemes, and the goal of the paper is to compare these two categories. In contrast to the situation in algebraic geometry, these two categories are not equivalent. The main result is the construction of adjoint functors between them. \(\mathbb F_1\)-geometry; generalized scheme theory and Grothendieck topologies; monoids Lorscheid, O., Blue schemes, semiring schemes, and relative schemes after toën and vaquié, J. Algebra, 482, 264-302, (2017) Generalizations (algebraic spaces, stacks), Monoidal categories (= multiplicative categories) [See also 19D23] Blue schemes, semiring schemes, and relative schemes after Toën and Vaquié | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 component of the Hilbert scheme of \(n\) points on a scheme \(X\) is the closure of the open subset parameterizing \(n\) distinct points. In this article we construct the principal component as a certain blow-up of the symmetric product of \(X\). Our construction is based on a local explicit analysis of étale families, from which the appropriate universal property, needed to identify the principal component with the blow-up, is derived. Hilbert scheme of points; blow-up; symmetric product; étale families David Rydh and Roy Skjelnes, An intrinsic construction of the principal component of the Hilbert scheme, J. Lond. Math. Soc. (2) 82 (2010), no. 2, 459 -- 481. Parametrization (Chow and Hilbert schemes), Actions of groups on commutative rings; invariant theory An intrinsic construction of the principal component of the Hilbert scheme | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For a general canonically embedded curve \(C\) of genus \(g\geq 5\), let \(d\leq g-1\) be an integer such that the Brill-Noether number \(\rho(g,d,1)=g-2(g-d+1)\geq 1\). We study the family of \(d\)-secant \(\mathbb P^{d-2}\)'s to \(C\) induced by the smooth locus of the Brill-Noether locus \(W^1_d(C)\). Using the theory of foci and a structure theorem for the rank one locus of special 1-generic matrices by \textit{D. Eisenbud} and \textit{J. Harris} [J. Algebr. Geom. 1, No. 1, 31--59 (1992; Zbl 0798.14029)], we prove a Torelli-type theorem for general curves by reconstructing the curve from its Brill-Noether loci \(W^1_d(C)\) of dimension at least 1. focal scheme; Brill-Noether locus; Torelli-type theorem Special divisors on curves (gonality, Brill-Noether theory), Determinantal varieties, Torelli problem, Projective techniques in algebraic geometry Focal schemes to families of secant spaces to canonical 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 1981, Mukai constructed the Fourier-Mukai transform for abelian varieties over an algebraically closed field, which gives an equivalence of categories between quasi-coherent sheaves over \(A\) and the one over \(A^\vee\), its dual variety. Laumon generalized these results for abelian varieties over a locally noetherian base. In this article, we define a Fourier-Mukai transform for an abelian formal scheme \(A/S=\mathrm{Spf}(V)\), where \(V\) is a discrete valuation ring, and we extend the classical results of Fourier-Mukai transform to this case. Finally, we discuss the case of the generic fiber \(A_K\) of \(A\) to obtain an equivalence of categories between coherent sheaves over \(A_K\) and the ones over \(A_K^\vee\). Sheaves in algebraic geometry, Picard schemes, higher Jacobians, Algebraic moduli problems, moduli of vector bundles Fourier-Mukai transform on formal schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The positive steady states of chemical reaction systems modeled by mass action kinetics are investigated. This sparse polynomial system is given by a weighted directed graph and a weighted bipartite graph. In this application the number of real positive solutions within certain affine subspaces of \(\mathbb{R}^m\) is of particular interest.
We show that the simplest cases are equivalent to binomial systems and are explained with the help of toric varieties. The argumentation is constructive and suggests algorithms. In general the solution structure is highly determined by the properties of the two graphs. We explain how the graphs determine the Newton polytopes of the system of sparse polynomials and thus determine the solution structure. Results on positive solutions from real algebraic geometry are applied to this particular situation. Examples illustrate the theoretical results. mass action kinetics; weighted bipartite graph; toric varieties; sparse polynomials Gatermann, K., \& Huber, B. (2002). A family of sparse polynomial systems arising in chemical reaction systems. J. Symb. Comput., 33(3), 275--305. Classical flows, reactions, etc. in chemistry, Directed graphs (digraphs), tournaments, Applications of graph theory, Toric varieties, Newton polyhedra, Okounkov bodies, Integro-ordinary differential equations, Thermodynamics and heat transfer A family of sparse polynomial systems arising in chemical reaction systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors present a flexible technique, based on polynomial continuation, for solving not-necessarily-square polynomial systems over \(\mathbb{C}\). The method proceeds incrementally, with the isolated (or nonsingular) solutions of a subset of the equations being used to generate the starting points of the homotopies to solve the system consisting of that subset plus one or more additional equations from the system. Hyperplanes are used to slice positive dimensional solution components down to isolated points. Deflation is used to handle singular points; the authors modify the deflation approach based on using the Dayton-Zeng multiplicity matrix so that it works for specific (rather than just generic) points on components. The authors also give ways to specify the necessary linear product decompositions, optimize the choice of single or subsets of equations to be introduced at each step, and ``regenerate'' isolated roots of a subsystem so that the results can be used as starting points for the system in the next iteration.
The underlying theory guarantees that the homotopy paths will lead to all isolated solutions. Thus one can generate witness supersets for solution components of any dimension, which is the first step in computing an irreducible decomposition of the solution set.
The authors compare their method to polyhedral homotopy and an improved diagonal homotopy on several sets of equations. They conclude that while polyhedral homotopy is often best for small systems, for large systems regeneration needs to track fewer paths and takes less time. Regeneration is also, on average, better than the diagonal homotopy. They conclude by saying that regeneration appears to reveal much of the same sparse structure as polyhedral homotopy without the mixed volume computation. algebraic set; component of solutions; diagonal homotopy; embedding; equation-by-equation solver; generic point; homotopy continuation; irreducible component; numerical irreducible decomposition; numerical algebraic geometry; path following; polynomial system; witness point; positive dimensional solution; product decomposition; incremental regeneration; polyhedral homotopy; diagonal homotopy J. D. Hauenstein, A. J. Sommese, and C. W. Wampler, \textit{Regeneration homotopies for solving systems of polynomials}, Math. Comp., 80 (2011), pp. 345--377. Numerical computation of solutions to systems of equations, Global methods, including homotopy approaches to the numerical solution of nonlinear equations, Polynomials, factorization in commutative rings, Symbolic computation and algebraic computation, Numerical computation of roots of polynomial equations, Computational aspects of higher-dimensional varieties, Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral) Regeneration homotopies for solving systems of 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 Let \(K\) be a commutative integral domain, \(K^*\) its group of units and \(F = (F_ 1, \dots, F_ n) \in K[X]^ n\), \(X = (X_ 1, \dots, X_ n)\). \(F\) is called locally invertible if for every point \(a \in K^ n\) the system of polynomials \(F_ a (X) : = F(X + a) - F(a)\) has a formal inverse, i.e. there exists \(G_ a \in K [[X]]^ n\) such that \(F_ a \circ G_ a = G_ a \circ F_ a = X\) (this is equivalent to the jacobian condition, i.e. \(\text{Jac} (F) (a) \in K^*\) for every \(a \in K^ n)\). The main theorem of the paper is the following upper bound for the cardinality of the fibers of \(F\): if \(K\) is of characteristic zero and \(F \in K [X]^ n\) is locally invertible, then the field \(K(X)\) is a finite algebraic extension of the field \(K(F)\) and for any point \(b \in K^ n\) holds: \(\# F^{-1} (b) \leq [K(X) : K(F(X))]\). polynomial map; formal series; integral domain; formal inverse DOI: 10.1006/jabr.1994.1218 Polynomial rings and ideals; rings of integer-valued polynomials, Automorphisms of curves, Integral domains On locally invertible systems of 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 prove the existence of \(\alpha\)-stable coherent systems on a singular genus 1 curve. This paper is a continuation of [ibid. 2, No. 17, 847--850 (2007)]. Ballico, Coherent systems on singular genus one curves, Int. J. Contemp. Math. Sci. 2 (31) pp 1527-- (2007) Vector bundles on curves and their moduli Coherent systems on singular genus one 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 If \(X\) is a smooth irreducible projective curve, a coherent system of type (n,d,k) on \(X\) is a pair \((E,V)\), where \(E\) is a vector bundle on \(X\), \(V\) is linear subspace of \(H^0(X,E)\) and \(\text{rank}(E)=n\), \(\deg(E)=d\), \(\dim(V)=k\). Using a notion of stability for coherent systems of data (cf. the references in the paper under review), depending on a real \(\alpha\), one constructs (cf. the references) a moduli space \(G(\alpha;n,d,k)\). The subject of this paper is the case of coherent systems with \(k=n+1\) on curves with the property that, for any line bundle \(L\), the map \(H^0(L)\otimes H^0 (L^\ast \otimes K) \to H^0(K)\) is injective (the so-called \textit{Petri curves}).
From the authors' abstract: ``We describe the geometry of the moduli space of such coherent systems for large values of the parameter \(\alpha \). We determine the top critical value of \(\alpha \) and show that the corresponding `flip' has positive codimension. We investigate also the non-emptiness of the moduli space for smaller values of \(\alpha\), proving in many cases that the condition for non-emptiness is the same as for large \(\alpha\). We give some detailed results for \(g\leq 5\) and applications to higher rank Brill-Noether theory and the stability of kernels of evaluation maps, thus proving Butler's conjecture in some cases in which it was not previously known.'' Petri curve; Brill-Noether theory; \(\alpha \)-stability; moduli space Bhosle, U. N.; Brambila-Paz, L.; Newstead, P. E., On coherent systems of type \((n, d, n + 1)\) on Petri curves, Manuscripta Math., 126, 409-441, (2008) Vector bundles on curves and their moduli On coherent systems of type \((n, d, n+1)\) on Petri 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 semi-classical approximation is a formula of mathematical physics for the sum of Feynman diagrams with a single circuit. This paper studies the same problem in the setting of modular operads, using the results and techniques which were developed by \textit{E. Getzler} and \textit{M. M. Kapranov} [in: ``Modular operads'', Compos. Math. 110, No. 1, 65-126 (1998; Zbl 0894.18005)]. Instead of being a number, the interaction at a vertex of valence \(n\) is an \(S_n\)-module.
The motivation for this theory is the computation of the \(S_n\)-equivariant Hodge polynomials of the Deligne-Mumford-Knudsen moduli spaces of stable curves of genus 1 with \(n\) marked smooth points [see also \textit{E. Getzler}, ``Intersection theory on \(\overline{\mathcal{M}}_{1,4}\) and elliptic Gromov-Witten invariants'', J. Am. Math. Soc. 10, No. 4, 973-998 (1997; Zbl 0909.14002)].
As an application, the author gives in the present article formulas for the Betti numbers of these spaces. modular operad; moduli space; semi-classical approximation; Feynman diagrams E. Getzler, The semiclassical approximation for modular operads,Comm. Math. Phys. 194 (1998), 481--492. Categorical structures, Families, moduli of curves (algebraic), Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory, Monoidal categories (= multiplicative categories) [See also 19D23] The semi-classical approximation for modular operads | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 propose an explicit formula for the Segre classes of monomial subschemes of nonsingular varieties, such as schemes defined by monomial ideals in projective space. The Segre class is expressed as a formal integral on a region bounded by the corresponding Newton polyhedron. We prove this formula for monomial ideals in two variables and verify it for some families of examples in any number of variables. Aluffi, P., Segre classes of monomial schemes, Electron. res. announc. math. sci., 20, 55-70, (2013) Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Segre classes of monomial schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The biregular geometry of punctual Hilbert schemes in dimensions 2 and 3, i.e. of schemes parametrizing fixed-length zero-dimensionaI subschemes supported at a given point on a smooth surface or a smooth three-dimensional variety, is studied. A precise biregular description of these schemes has only been known for the trivial cases of lengths 3 and 4 in dimension 2. The next case of length 5 in dimension 2 and the two first nontrivial cases of lengths 3 and 4 in dimension 3 are considered. A detailed description of the biregular properties of punctual Hilbert schemes and of their natural desingularizations by varieties of complete punctual flags is given. Stein expansion; Briançon classification; punctual Hilbert schemes Parametrization (Chow and Hilbert schemes) Punctual Hilbert schemes of small length in dimensions 2 and 3 | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a smooth projective variety. We show that the map that sends a codimension one distribution on \(X\) to its singular scheme is a morphism from the moduli space of distributions into a Hilbert scheme. We describe its fibers and, when \(X = \mathbb{P}^n\), compute them via syzygies. As an application, we describe the moduli spaces of degree 1 distributions on \({\mathbb{P}^3}\). We also give the minimal graded free resolution for the ideal of the singular scheme of a generic distribution on \(\mathbb{P}^3\). distributions; Hilbert scheme; singular scheme; syzygy Algebraic moduli problems, moduli of vector bundles, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Fine and coarse moduli spaces, Sheaves in algebraic geometry, Syzygies, resolutions, complexes and commutative rings Moduli of distributions via singular 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 \(J\subset S= K[x_0,\dots, x_n]\) be a monomial strongly stable ideal. The collection \({\mathcal M}f(J)\) of the homogeneous polynomial ideals \(I\), such that the monomials outside \(J\) form a \(K\)-vector basis of \(S/I\), is called a \(J\)-marked family. It can be endowed with a structure of affine scheme, called a \(J\)-marked scheme. For special ideals \(J\), \(J\)-marked schemes provide an open cover of the Hilbert scheme \({\mathcal H}\text{ilb}^n_{p(t)}\), where \(p(t)\) is the Hilbert polynomial of \(S/J\). Those ideals more suitable to this aim are the \(m\)-truncation ideals \(\underline J_{\geq m}\) generated by the monomials of degree \(\geq m\) in a saturated strongly stable monomial ideal \(\underline J\).
Exploiting a characterization of the ideals in \({\mathcal M}f(\underline J_{\geq m})\) in terms of a Buchberger-like criterion, we compute the equations defining the \(\underline J_{\geq m}\)-marked scheme by a new reduction relation, called superminimal reduction, and obtain an embedding of \({\mathcal M}f(\underline J_{\geq m})\) in an affine space of low dimension.
In this setting, explicit computations are achievable in many nontrivial cases. Moreover, for every \(m\), we give a closed embedding \(\phi_m:{\mathcal M}f(\underline J_{\geq m})\hookrightarrow{\mathcal M}f(\underline J_{\geq m+1})\), characterize those \(\phi_m\) that are isomorphisms in terms of the monomial basis of \(\underline J\), especially we characterize the minimum integer \(m_0\) such that \(\phi_m\) is an isomorphism for every \(m\geq m_0\). Hilbert scheme; strongly stable ideal; polynomial reduction relation Bertone, C.; Cioffi, F.; Lella, P.; Roggero, M., Upgraded methods for the effective computation of marked schemes on a strongly stable ideal, J. Symbolic Comput., 50, 263-290, (2013) Parametrization (Chow and Hilbert schemes), Software, source code, etc. for problems pertaining to algebraic geometry, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) Upgraded methods for the effective computation of marked schemes on a strongly stable ideal | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(B=\Gamma(U,\mathcal O_X)\) denote the ring of global sections of \(X =\text{Spec }A\) with respect to an open subset, where \(A\) denotes a local Noetherian two-dimensional ring. In the paper, there is given a criterion to decide the finiteness of \(B\) in case of \(U = D(I)\), \(I\) an ideal of height one. It says that in case that \(U\) is not affine the ring of global sections of \(U\) is not finitely generated if and only if there exists an irreducible formal component where \(U\) is affine and a second one where \(U\) is not affine such that their intersection is one-dimensional. This generalizes a result of \textit{P. M. Eakin jun., W. Heinzer, D. Katz} and \textit{L. J. Ratliff jun.} [J. Algebra 110, 407-419 (1987; Zbl 0631.13003)], where it is shown that \(D(I)\) is affine if and only if \(B\) is Noetherian if and only if \(B\) is of finite type over \(A,\) provided \(A\) is an excellent Cohen-Macaulay ring. Moreover it provides a counterexample to a statement of the reviewer [\textit{P. Schenzel} in: Commutative algebra, Proc. Workshop, Salvador/Brazil 1988, Lect. Notes Math. 1430, 88-97 (1990; Zbl 0719.13003)], where it was claimed that the result of the paper cited above is true without the Cohen-Macaulay assumption on \(A.\) global section; local Noetherian two-dimensional ring; irreducible formal component; excellent Cohen-Macaulay ring Ideals and multiplicative ideal theory in commutative rings, Relevant commutative algebra, Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics, Vanishing theorems in algebraic geometry Rings of global sections in two-dimensional schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(V\) be an \(n\)-dimensional real vector space, let \(\Gamma \subset V\) be a lattice of rank \(r\). Denote by \({\mathfrak U}\) the corresponding central hyperplane arrangement over \({\mathbb Z}\). It is assumed that the intersection of all hyperplanes \(\cap {\mathfrak U}\) coincides with the origin, and their dual lines span \(V^\ast.\) Set \(U = V\setminus \cup {\mathfrak U}\), \(\Gamma_{\mathfrak U} = U \cap \Gamma\). A function \(f\) on \(V\) is called \({\mathfrak U}\)-rational if it is rational and its poles are contained in \(\cup {\mathfrak U}.\) The author develops an approach in calculating sums \(B_{f,{\mathfrak U}} = \sum_{j\in \Gamma_{\mathfrak U}} f(j)\), or more generally, the Fourier series \(B_{f,{\mathfrak U}}(t) = \sum_{j \in \Gamma_{\mathfrak U}} f(j)\exp\langle t,j\rangle\), \(j \in V^\ast\), where the function \(f\) is \({\mathfrak U}\)-rational. It is proved that \(V^\ast\) has a chamber decomposition which comes from one of \(T = V^\ast/\Lambda\) and the series \(B_{f,{\mathfrak U}}(t)\) restricts to a polynomial inside each chamber. Following \textit{D. Zagier} [Prog. Math. 120, 497-512 (1994; Zbl 0822.11001)] such polynomials are called multiple Bernoulli polynomials.
For any integral arrangement \(\mathfrak U\) and a chamber \(\Delta\) the author constructs a local residue form \(\omega_\Delta({\mathfrak U})\) and a pairing \(\langle\;,\;\rangle\) such that for any \(t \in \Delta\) one has \(B^\Delta_f(t) = \langle \omega_\Delta({\mathfrak U}), f\exp(t)\rangle\). His constructions are essentially based on the notion of iterated residue of \({\mathfrak U}_{\mathbb C}\)-meromorphic functions where \({\mathfrak U}_{\mathbb C}\) is the complexification of the arrangement \({\mathfrak U}\). The author underlines that his study of these sums was inspired by the works of \textit{E. Witten} [Commun. Math. Phys. 141, 153-209 (1991; Zbl 0762.53063) and J. Geom. Phys. 9, 303-368 (1992; Zbl 0768.53042)]. hyperplane arrangements; multiple Bernoulli polynomials; Fourier series; iterated residues; weight lattice; broken circuit bases; logarithmic forms; Lie arrangements; Cartan subalgebra; Verlinde formulas A. Szenes, ''Iterated residues and multiple Bernoulli polynomials,'' Internat. Math. Res. Notices, vol. 1998, p. no. 18, 937-956. Bernoulli and Euler numbers and polynomials, Configurations and arrangements of linear subspaces, Vector bundles on curves and their moduli, Other special orthogonal polynomials and functions, Relationships between algebraic curves and physics Iterated residues and multiple Bernoulli 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 This paper announces a generalization of the classical description of germs of plane curve singularities in terms of Puiseux exponents and iterated torus knots to higher dimensional, irreducible quasi-ordinary singularities, making explicit earlier work of Gau and Lipman [\textit{J. Lipman}, Mem. Am. Math. Soc. 388, 1--106 (1988; Zbl 0658.14003); \textit{Y.-N. Gau}, ibid. 388, 109--129 (1988; Zbl 0658.14004)]. Only 2-dimensional hypersurfaces in \(\mathbb{C}^3\) are considered here; extensions to higher dimensions and reducible quasi-ordinary singularities are left to a later paper. hypersurface singularity; quasi-ordinary; 2-torus knot Popescu-Pampu, P.: Two-dimensional iterated torus knots and quasi-ordinary surface singularities. C. R. Math. Acad. Sci. Paris 336(8), 651--656 (2003) Complex surface and hypersurface singularities, Local complex singularities, Singularities of surfaces or higher-dimensional varieties Two-dimensional iterated torus knots and quasi-ordinary surface 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 study degenerations of rank 3 quadratic forms and of rank 4 Azumaya algebras, and extend what is known for good forms and Azumaya algebras. By considering line-bundle-valued forms, we extend the theorem of Max-Albert Knus that the Witt-invariant -- the even Clifford algebra of a form -- suffices for classification. An algebra Zariski-locally the even Clifford algebra of a ternary form is so globally up to twisting by square roots of line bundles. The general, usual and special orthogonal groups of a form are determined in terms of automorphism groups of its Witt-invariant. Martin Kneser's characteristic-free notion of semiregular form is used. T.E. Venkata Balaji, Line-Bundle-Valued Ternary Quadratic Bundles over Schemes, http://arxiv.org/abs/math.AG/0506146 General ternary and quaternary quadratic forms; forms of more than two variables, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.), Linear algebraic groups over adèles and other rings and schemes Line-bundle-valued ternary quadratic forms over 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 Classification of global musical objects deals with the determination of isomorphism classes in adequate categories of global compositions, the type of objects which play the role of musical manifolds. It relies on two constructions: coefficient systems of affine functions and resolutions of global compositions. These structures are introduced and discussed. As a main result, we derive classifying schemes, and shortly comment on their musical meaning. From the musicological point of view, this is one of the most difficult chapters of mathematical music theory since the relation between classification and musicology, in particular: esthetics is quite implicit. But classification is a deep concern since it reveals the a priori extent of a structural framework and therefore its power as an expressive tool of artistic activity will be in composition, performance, or understanding. mathematical music theory; classification; global composition; resolution General applied mathematics, Fibrations, degenerations in algebraic geometry, Categories of machines, automata, Permutation groups, Applied homological algebra and category theory in algebraic topology Classifying algebraic schemes for musical manifolds | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Classification of global musical objects deals with the determination of isomorphism classes in adequate categories of global compositions, the type of objects which play the role of musical manifolds. It relies on two constructions: coefficient systems of affine functions and resolutions of global compositions. These structures are introduced and discussed. As a main result, we derive classifying schemes and shortly comment on their musical meaning. From the musicological point of view, this is one of the most difficult chapters of mathematical music theory since the relation between classification and musicology, in particular: esthetics, is quite implicit. But classification is a deep concern since it reveals the a priori extent of a structural framework and therefore its power as an expressive tool of artistic activity, be it in composition, performance, or understanding. mathematical music theory; classification; global composition; resolution Mathematics and music, Algebraic combinatorics, Fine and coarse moduli spaces, Families, moduli, classification: algebraic theory Classifying algebraic schemes for musical manifolds | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Several recent papers look to the question of whether, for a given Hilbert function \(H\), there is a unique minimum set of graded Betti numbers or not [e.g. \textit{B. Richert}, J. Algebra 244, No.~1, 236--259 (2001; Zbl 1027.13008)]. Moreover by semicontinuity of the graded Betti numbers different minima correspond to different irreducible components of the postulation Hilbert scheme, Hilb\((H)\), of zero-schemes [see \textit{A. Ragusa} and \textit{G. Zappalá}, Rend. Circ. Matem. di Palermo Serie II, 53, 401--406 (2004; Zbl 1099.13029)]. In the present paper the author gives infinitely many families of reduced zero-schemes of height 3, having at least two irreducible components of Hilb\((H)\) which he separates by the incomparability of the set of graded Betti numbers. Moreover the general element of one of the components has the weak Lefschetz property (WLP) while every zero-scheme of the second component fails to have WLP. Linkage is important in constructing the general elements of the two components.
In [J. Algebra 262, No.~1, 99--126 (2003; Zbl 1018.13001)] \textit{T. Harima}, the author, \textit{U. Nagel} and \textit{J. Watanabe} characterized Artinian quotients having WLP in terms of their Hilbert function \(h\). Comparing \(h\) of this result with the first difference of \(H\) of the classes of reducible Hilbert schemes of the present paper, the existence range is very similar except for the number in the ``flat part'' of \(h\), which may be much more limited. Still, due to this and several other papers [e.g. \textit{A. Iarrobino} and \textit{H. Srinivasan}, J. Pure Appl. Algebra 201, No.~1--3, 62--96 (2005; Zbl 1107.13020) and the reviewer in Trans. Am. Math. Soc. 358, No.~7, 3133--3167 (2006; Zbl 1103.14005)], Hilb\((H)\) seems to be very rich in reducible Hilbert schemes, even in height 3. The stratum of Hilb\((H)\) of fixed graded Betti numbers should have a much better chance of being irreducible, but also this fails by infinitely many families of zero-schemes (of height 4) by the mentioned paper of the reviewer. postulation Hilbert scheme; graded Betti numbers; linkage Migliore, J.: Families of reduced zero-dimensional schemes. Collect. math. 57, No. 2, 173-192 (2006) Syzygies, resolutions, complexes and commutative rings, Parametrization (Chow and Hilbert schemes), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Families of reduced zero-dimensional schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author introduces a meromorphic function M(s,f,g) depending bilinearly on two cusp forms f, g of weight 2 for \(\Gamma_ 0(N)\). This is not a Dirichlet series (and so does not have an Euler product) but it does satisfy a functional equation. This expresses \(M(s,f,g)+M(2-s,f| W_ N,g| W_ N)\) in terms of the Mellin transforms of f and g. In some cases M(s,f,g) can be expressed as a Rankin type convolution. In general it is defined as an iterated integral so that M(1,f,g) is the generalized period investigated by the author in several previous papers. Its significance is that it determines the image of X-i(X) \((i(x)=-x\) in J(X)) in the intermediate Jacobian of J(X) where \(X=\Gamma_ 0(N)\setminus {\mathbb{H}}\) and J(X) is the Jacobian variety of X. meromorphic function; cusp forms; functional equation; Mellin transforms; Rankin type convolution; generalized period; Jacobian variety Holomorphic modular forms of integral weight, Jacobians, Prym varieties An analytic function and iterated integrals | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(C\) be a locally planar curve. Its versal deformation admits a stratification by the genera of the fibres. The strata are singular; we show that their multiplicities at the central point are determined by the Euler numbers of the Hilbert schemes of points on \(C\). These Euler numbers have made two prior appearances. First, in certain simple cases, they control the contribution of \(C\) to the Pandharipande-Thomas curve counting invariants of three-folds. In this context, our result identifies the strata multiplicities as the local contributions to the Gopakumar-Vafa BPS invariants. Second, when \(C\) is smooth away from a unique singular point, a conjecture of Oblomkov and the present author identifies the Euler numbers of the Hilbert schemes with the `\(U(\infty )\)' invariant of the link of the singularity. We make contact with combinatorial ideas of Jaeger, and suggest an approach to the conjecture. V. Shende, Hilbert schemes of points on a locally planar curve and the Severi strata of its versal deformation , to appear in Compos. Math., preprint, [math.AG] 1009.0914v2 Parametrization (Chow and Hilbert schemes), Singularities of curves, local rings Hilbert schemes of points on a locally planar curve and the Severi strata of its versal deformation | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(f:X\to X\) be a separated, essentially finite-type, flat map at map of Noetherian schemes and \(\delta:X\to X\times_Z X\) the diagonal map. The fundamental class \(C_f\) (globalizing residues) is a map from the relative Hochschild functor \(\delta^*\delta_* f^*\) to the relative dualizing functor \(f^!\). We show a compatibility between \(C_f\) and the derived tensor product. The main result is that, in a suitable sense, \(C_f\) generalizes Verdier's classical isomorphism for
smooth \(f\) with fibers of dimension \(d\), an isomorphism that binds \(f^!\) to relative \(d\)-forms. Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials On the fundamental class of an essentially smooth scheme-map | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 first two parts we recall the construction of generalized hypergeometric functions and of the cellular complex homotopy equivalent to the complement of a family of hyperplanes in \(\mathbb{C}^ N\). In the third part we find a generalization of some results of \textit{V. A. Vasil'ev}, \textit{I. M. Gel'fand} and \textit{A. V. Zelevinskij} [Funkts. Anal. Prilozh. 21, No. 1, 23-38 (1987; Zbl 0614.33008)] about the homology of local systems on an affine space minus some hyperplanes. Our method is based on a paper of the first author [Invent. Math. 88, 603-618 (1987; Zbl 0594.57009)] and it gives also information about the cellular complex constructed there. In the last part explicit bases for the only non-vanishing homology group are described in terms of the cells of the above mentioned complex. The configurations of hyperplanes which we examine are those giving fundamental strata in the Grassmannian [\textit{Vasil'ev} et al., loc. cit.; \textit{I. M. Gel'fand, R. M. Goresky, R. D. MacPherson} and \textit{V. V. Serganova}, Adv. Math. 63, 301-316 (1987; Zbl 0622.57014)] and strata in \(G_{3,n}\) allowing also triple points. cellular complex; complement of a family of hyperplanes in \(\mathbb{C}^ N\); homology of local systems on an affine space; homology group; configurations of hyperplanes; fundamental strata; Grassmannian Prati, M. C.; Salvetti, M.: On local system over complements to arrangements of hyperplanes associated to grassman strata. Ann. mat. Pura appl. 159, 341-355 (1991) Algebraic topology on manifolds and differential topology, Homology with local coefficients, equivariant cohomology, Special varieties On local systems over complements to arrangements of hyperplanes associated to Grassmann strata | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 projective surface over an algebraically closed field and \(p\in S\) a non-singular point. Denote by \(\text{H}(n)\) the reduced punctual Hilbert scheme parameterizing length \(n\) zero-dimensional subschemes of \(S\) supported at \(p\). By a result of \textit{J. Briançon} [Invent. Math. 41, 45--89 (1977; Zbl 0353.14004)], \(\text{H}(n)\) is irreducible of dimension \(n-1\). In fact, \(\text{H}(1)\) is a point, \(\text{H}(2)\cong {\mathbb P}_1\) but \(\text{H}(n)\) is singular for \(n\geq 3\) and contains as a dense open smooth subset the curvilinear schemes.
This paper is concerned with the description of the birational model for \(\text{H}(n)\) introduced by \textit{A. S. Tikhomirov} [Proc. Steklov Inst. Math. 208, 280--295 (1995; Zbl 0884.14001)]: the reduced moduli space \(\text{HF}(n)\) of length \(n\) complete flags \(\xi_1\subset \cdots\subset\xi_n\) of subschemes of \(S\) supported at \(p\) projects onto \(\text{H}(n)\), so that the closure \(\text{HF}'(n)\) in \(\text{HF}(n)\) of the inverse image of the curvilinear points is the unique component of \(\text{HF}(n)\) mapping birationally onto \(\text{H}(n)\).
In order to get a better understanding of the variety \(\text{HF}'(n)\), the author introduces the reduced moduli space \(\text{HMF}(n)\) of multiplicative complete flags \(\xi_1\subset\cdots\subset\xi_n\), that is with the property that \(I_i I_j\subset I_{i+j}\) where \(I_i\) denotes the ideal sheaf of \(\xi_i\) (\S 4) and identifies the model \(\text{HF}'(n)\) as a component of \(\text{HMF}(n)\). He shows for \(n\leq 7\) that \(\text{HMF}(n)\) is irreducible, so that \(\text{HF}'(n)=\text{HMF}(n)\) (Question 5.5). He proves further for \(n\leq 4\) that \(\text{HMF}(n)\) is smooth (Theorem 6.1), so that \(\text{HF}'(n)\) is a resolution of singularities of \(\text{H}(n)\), but that \(\text{HMF}(5)\) is singular along a curve (Theorem 6.2).
In the first sections, the author gives a detailed construction of the moduli space of (multiplicative) complete flags. punctual Hilbert scheme; complete flags; curvilinear schemes Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects) A partial resolution of the punctual Hilbert scheme of a nonsingular 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 A coherent system of type \((n,d,k)\) on an algebraic variety \(X\) is a pair \((E,V)\) where \(E\) is a torsionfree sheaf of rank \(n\) and degree \(d\) on \(X\) and \(V\subset H^0(E)\) a subspace of dimension \(k\). There is a notion of stability depending on a real parameter \(\alpha\). The corresponding moduli space is denoted by \(G(\alpha;n,d, k)\). The present paper studies these moduli spaces in the case of a nodal curve of arithmetic genus 1.
The main results are: Let \(G'(\alpha;n,d, k)\) the open subset with \(E\) locally free. Then \(G(\alpha;n,d, k)\) is non-empty if and only if \(G'(\alpha;n,d, k)\) is non-empty. If \(d> k\) or \(d= k\) and \(\text{gcd}(n, d)= 1\), the set of \(\alpha\) with \(G(\alpha;n,d, k)\neq\emptyset\) is an open interval \((0,a)\) with \(a={d\over n-k}\) for \(0< k< n\) and \(=\infty\) for \(k= n\). The spaces \(G(\alpha;n,d, k)\) are irreducible and unirational.
Unlike in the case of an elliptic curve they are not smooth. In fact, they are singular along \(G(\alpha;n,d, k)\setminus G'(\alpha;n,d, k)\). As in the smooth case, for a given type \((n,d,k)\) there are only finitely many different moduli spaces \(G(\alpha;n,d, k)\) and one gets from one to another by a sequence of flips. Examples show that the spaces are not normal in general, however in many cases they are seminormal and often rational. vector bundle; coherent system; moduli space 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 a nodal curve of genus 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 For a simply connected connected simple algebraic group \(G\), a cell \(B_{w_0}^- = B^- \cap U \overline{w_0} U\) is a geometric crystal with a positive structure \(\theta_{\mathbf{i}}^- : (\mathbb{C}^\times )^{l(w_0)} \to B_{w_0}^-\). Applying the tropicalization functor to a rational function \({\Phi}_{B K}^h = \sum_{i \in I} {\Delta}_{w_0 {\Lambda}_i , s_i {\Lambda}_i}\) called the half decoration on \(B_{w_0}^-\), one can realize the crystal \(B(\infty)\) in \(\mathbb{Z}^{l(w_0)} \). By computing \({\Phi}_{B K}^h\), we get an explicit form of \(B(\infty)\) in \(\mathbb{Z}^{l(w_0)} \). In this paper, we give an algorithm to compute \({\Delta}_{w_0 {\Lambda}_i, s_i {\Lambda}_i} \circ \, \theta_{\mathbf{i}}^-\) explicitly for \(i \in I\) such that \(V( {\Lambda}_i)\) is a minuscule representation of \(\mathfrak{g} = \operatorname{Lie}(G)\). In particular, the algorithm works for all \(i \in I\) if \(\mathfrak{g}\) is of type \({A}_n\). The algorithm computes a directed graph \textit{DG}, called a \textit{decoration graph}, whose vertices are labelled by all monomials in \({\Delta}_{w_0 {\Lambda}_i, s_i {\Lambda}_i} \circ \, \theta_{\mathbf{i}}^-( t_1, \cdots, t_{l (w_0)})\). The decoration graph has some properties similar to crystal graphs of minuscule representations. We also verify that the algorithm works in some other cases, for example, the case \(\mathfrak{g}\) is of type \({G}_2\) though \(V( {\Lambda}_i)\) is non-minuscule. geometric crystals; Berenstein-Kazhdan decoration; combinatorics; crystal bases Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Group actions on varieties or schemes (quotients) An algorithm for Berenstein-Kazhdan decoration functions and trails for minuscule representations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 the situation of a smooth complex projective variety over the complex numbers. For any divisor \(D\) on \(X\), the author studies the space of unitary local systems on the complement of \(D\) (from here on denoted by \(U\)). He considers this space in terms of \textit{parabolic line bundles} (also called \textit{realizations of boundaries of \(X\) on \(D\)}), see Theorem 1.2 in the paper and the list of references where it was proved in various generalities. He then stratifies this space by using multiplier ideals (see the Motivation section of the introduction). For a precise statement see Theorem 1.3 in the paper. This generalizes the results of a number of people including \textit{M. Green} and \textit{R. Lazarsfeld} [Invent. Math. 90, 389--407 (1987; Zbl 0659.14007) and J. Am. Math. Soc. 4, No. 1, 87--103 (1991; Zbl 0735.14004)], \textit{D. Arapura} [Bull. Am. Math. Soc., New Ser. 26, No. 2, 310--314 (1992; Zbl 0759.14016)], \textit{R. Pink} and \textit{D. Roessler} [Math. Ann. 330, No. 2, 293--308 (2004; Zbl 1064.14048)], and \textit{C. Simpson} [Ann. Sci. Éc. Norm. Supér. (4) 26, No. 3, 361--401 (1993; Zbl 0798.14005)]. Although, as the author points out, he relies on Simpson's result in his proof (see section 6).
As an application, the author uses these ideas to show the polynomial periodicity of Hodge numbers \(h^{q,0}\) of congruence covers. Further generalizations in this direction (for the Hodge rank function) are also obtained. See in particular Theorems 1.8, 1.9 and the surrounding discussion for a history of this problem.
Finally, the author uses some of these ideas to prove characterizations of abelian covers (and explains how some these results even follow immediately from the more classical results). See Corollaries 1.10 through 1.13. local system; multiplier ideal; parabolic line bundle; abelian cover; polynomial periodicity of line bundles Budur N. Unitary local systems, multiplier ideals, and polynomial periodicity of Hodge numbers. Adv Math, 2009, 221: 217--250 Multiplier ideals, Global theory of complex singularities; cohomological properties Unitary local systems, multiplier ideals, and polynomial periodicity of Hodge numbers | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(M\) be the complement of a line arrangement in the complex projective plane. A rank one local system \(L\) on \(M\) is admissible if the twisted cohomology groups \(H^*(M,L)\) can be computed using the Aomoto complex associated to the complex cohomology algebra \(H^*(M,C)\). The author gives a sufficient condition such that all the rank one local systems on \(M\) are admissible, of similar flavor as those given by \textit{T. A. T. Dinh} [Can. Math. Bull. 54, No. 1, 56--67 (2011; Zbl 1215.14011)]. This implies certain properties of the characteristic variety and resonance variety of \(M\). rank one local systems; line arrangement; characteristic variety; resonance variety (Co)homology theory in algebraic geometry, Relations with arrangements of hyperplanes, Enumerative problems (combinatorial problems) in algebraic geometry, Arrangements of points, flats, hyperplanes (aspects of discrete geometry) Admissibility of local systems for some classes of line arrangements | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author studies the scheme \(M_ d(X;p,n)\) parametrizing morphisms of degree \( d\) from an elliptic curve X to a Grassmann variety Gr(p,n). He proves that \(M_ d(X;p,n)\) is smooth if and only if \(p=1\), or \(p=n-1\), or \(d=2\); it is irreducible of dimension nd if \(d\geq n\); if \(d>n\), it is irreducible if and only if it is smooth. More precisely, if \(d>n\), the author shows that the irreducible components are the Zariski closures of certain locally closed subschemes \(M_ d^{a,b}(X;p,n)\), for \(0\leq a<p\), \(0\leq b<n-p\) and \(a+b=n-d\), of dimension \(nd+ab\). These subschemes are defined as follows: If \(f\in M_ d(X;p,n)\) is a closed point, let \(0\to V_ f\to E_ X\to Q_ f\to 0\) denote the pullback of the universal sequence on Gr(p,n). Then set \(M_ d^{a,b}(X;p,n)=\{f;\quad h^ 0(V_ f)=a,\quad h^ 0(Q_ f)=b\}\).
The last part of the paper gives sufficient conditions for three sheaves on X to fit into an exact sequence. If the sheaves are locally free and the sheaf of highest rank is trivial, this is related to the existence of maps from X into a corresponding Grassmann variety, with fixed pullbacks of the trivial bundles. unidecomposable sheaf; morphisms from an elliptic curve to a Grassmann variety Bruguières, A., The scheme of morphisms from an elliptic curve to a Grassmannian, Compositio Math., 63, 1, 15-40, (1987) Algebraic moduli problems, moduli of vector bundles, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Special algebraic curves and curves of low genus, Elliptic curves The scheme of morphisms from an elliptic curve to 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 Let \(X\) be a regular integral separated Noetherian scheme of finite Krull dimension, in which \(2\) is everywhere invertible. \textit{P. Balmer} [\(K\)-Theory 19, No. 4, 311-363 (2000; Zbl 0953.18003); Math. Z. 236, No. 2, 351-382 (2001; Zbl 1004.18010)] defined derived Witt groups \(W^n(X)\) of \(X\), which depend only on \(n\bmod 4\), and showed that \(W^0(X)=W(X)\) and \(W^2(X)=W^-(X)\) can be identified with the usual Witt groups of symmetric resp. skew-symmetric vector bundles on \(X\). The authors show that there exists a spectral sequence converging to \(E^n = W^n(X)\), such that \(E_1^{pq}=0\) unless \(0 \leq p \leq\dim(X)\), and such that the horizontal line
\[
0 \rightarrow E_1^{0q} \overset {d_1} \rightarrow E_1^{1q} \overset {d_1} \rightarrow E_1^{2q} \overset {d_1} \rightarrow \cdots \overset {d_1} \rightarrow E_1^{\dim(X),q} \rightarrow 0
\]
of the spectral sequence vanishes for \(q \not \equiv 0 \mod 4\), whereas for \(q \equiv 0 \mod 4\) this line equals the Gersten-Witt complex
\[
\mathcal W_X: 0 \rightarrow W(K) \rightarrow \bigoplus_{x_1 \in X^{(1)}} W(k(x_1)) \rightarrow \cdots \rightarrow \bigoplus_{x_e \in X^{(e)}} W(k(x_e)) \rightarrow 0.
\]
Here \(K\) is the field of rational functions on \(X\), \(X^{(p)}\) denotes the set of points of codimension \(p\) in \(X\), \(e = \dim(X)\) and \(k(x)\) is the residue field of \(X\) at the point \(x\). In fact, a more general version is proved for twisted Witt groups, where the duality is twisted by a line bundle. The periodicity of the Gersten-Witt spectral sequence and the vanishing of the horizontal lines in three out of 4 cases imply (and improve) most of the known Gersten-Witt related results in low dimension. For \(\dim X \leq 7\) the authors establish a long exact sequence relating the Witt groups \(W^n:=W^n(X)\) of \(X\) and the cohohomology groups \(H^n(\mathcal W):= H^n(\mathcal W_X)\) of the Gersten-Witt complex \(\mathcal W_X\):
\[
\begin{tikzcd} 0 \ar[r] & H^4(\mathcal W) \ar[r] & W^0 \ar[r] & H^0(\mathcal W) \ar[r] & H^5(\mathcal W) \ar[r] & W^1\ar[r] & H^1(\mathcal W) \ar[d]\\ 0 & H^3(\mathcal W) \ar[l] & W^3 \ar[l] & H^7(\mathcal W) \ar[l] & H^2(\mathcal W) \ar[l] & W^2\ar[l] & H^6(\mathcal W) \ar[l] \rlap{\,.} \end{tikzcd}
\]
Since \(H^0(\mathcal W_X)\) is shown to be equal to the unramified Witt group \(W_{\text{nr}}(X)\), immediate consequences are weak purity for \(\dim X \leq 4\), i.e., the surjectivity of \(W(X) \rightarrow W_{\text{nr}}(X)\), as well as purity for \(\dim X \leq 3\), i.e., \(W(X) \cong W_{\text{nr}}(X)\). The authors also show that for \(X = \operatorname {Spec}(R)\) with \(R\) a regular local ring of Krull dimension \(\leq 4\) containing \(\frac{1}{2}\) the Gersten conjecture holds, i.e., the Gersten-Witt complex augmented by the natural map \(W(X) \rightarrow W(K)\) is exact. Witt groups of regular schemes; Gersten-Witt complex; Gersten-Witt spectral sequence; purity Balmer, P.; Walter, C., A Gersten-Witt spectral sequence for regular schemes, Ann. Sci. Éc. Norm. Supér. (4), 35, 1, 127-152, (2002) Witt groups of rings, Algebraic theory of quadratic forms; Witt groups and rings, Applications of methods of algebraic \(K\)-theory in algebraic geometry, \(K\)-theory of schemes A Gersten-Witt spectral sequence for regular schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems ``If \(f : X^n \to Y^p\) is a morphism of smooth complex analytic varieties with \(n < p\), then the multiple points of order \(k\) of \(f\) in the target are those \(y \in Y^p\) with \(k\) preimages, each preimage counted with multiplicity. ... In this paper we give a new construction, using punctual Hilbert schemes, which we offer as an alternative to multi-jets in the study of multiple point singularities. As an illustration of its usefulness, we use it to find a resolution of the closure of the triple point set of any ``good'' map \(f\), and of the multiple point set of a ``good'' \(f\) of any order, provided the map \(f\) has kernel rank at most 2. ... This construction also provides a useful starting point for finding resolutions of multiple point sets for general \(f\), by reducing the problem of resolving the singularities of \(f\) to the problem of resolving the singularities of the corresponding Hilbert schemes''. punctual Hilbert schemes; multiple point singularities; triple point Gaffney, T.; Counting Double Point Singularities, preprint. Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry Punctual Hilbert schemes and resolutions of multiple point 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 The authors generalize the Alexander-Hirschowitz theorem to arbitrary zero-dimensional schemes contained in a general union of double points. They work with polynomials of degree \(\leq d\) in \(n\) variables. They study five exceptional cases if \(d>2\). If \(d=2\) the exceptional cases are studied. polynomial interpolation; zero dimensional schemes; differential Horace method; Alexander-Hirschowitz theorem Brambilla, MC; Ottaviani, G, On partial polynomial interpolation, Linear Algebra Appl., 435, 1415-1445, (2011) Numerical interpolation, Divisors, linear systems, invertible sheaves, Interpolation in approximation theory On partial polynomial 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 The purpose of this largely expository note is to give some examples of smooth projective varieties with zero first Betti number but nontrivial families of local systems of higher ranks. Homotopy theory and fundamental groups in algebraic geometry, Variation of Hodge structures (algebro-geometric aspects), Structure of families (Picard-Lefschetz, monodromy, etc.), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Simpson's construction of varieties with many 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 We give a point-free definition of a Grothendieck scheme whose underlying topological space is spectral. Affine schemes aside, the prime examples are the
projective spectrum of a graded ring and the space of valuations corresponding to an abstract nonsingular curve. With the appropriate notion of a morphism between
spectral schemes, elementary proofs of the universal properties become possible. point-free topology; Grothendieck scheme; graded ring; valuations; nonsingular curve; spectral schemes; universal properties; projective spectrum Coquand T., Lombardi H., Schuster P.\(Spectral Schemes as Ringed Lattices\). Annals of Mathematics and Artificial Intelligence. 56, (2009), 339-360. Other constructive mathematics, Distributive lattices, Schemes and morphisms Spectral schemes as ringed lattices | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In several previous papers [for example \textit{M. Artin} and \textit{J. J. Zhang}, Adv. Math. 109, 228-287 (1994; Zbl 0833.14002)], the authors have developed a theory of noncommutative algebraic geometry. Starting with a not necessarily commutative ring \(A\) (graded by the natural numbers \(\mathbb{N}\)) they do not define the scheme Proj(\(A\)) directly as a physical object, but instead define the category of ``coherent (respectively quasicoherent) sheaves on Proj(\(A\))'' as the quotient of the category of finitely generated (respectively arbitrary) graded \(A\)-modules by the subcategory of torsion modules (torsion means killed by some power of the graded maximal ideal of \(A\)). If \(A\) is commutative then under suitable hypotheses the resulting categories are equivalent to the categories of coherent or quasicoherent sheaves on the usual scheme Proj(\(A\)).
The following is a somewhat simplified description of what the authors do in the present paper. They replace the categories of \(A\) modules involved in the construction of Proj by a \(k\)-linear Grothendieck category \({\mathcal C}\), where \(k\) is a commutative ring. Among other things the category \({\mathcal C}\) is abelian, and for objects \(V\) and \(W\) in \({\mathcal C}\), Hom\(_{\mathcal C}(V,W)\) is a \(k\)-module in a natural way. Typical examples are Proj(\(A\)) (regarded as an abelian category as described above), the category Mod(\(A\)) of right \(A\)-modules, and the category Gr(\(A\)) of graded right \(A\)-modules. If \(R\) is a Noetherian \(k\)-algebra and \(P\) is a Noetherian object in \({\mathcal C}\) then the Hilbert functor Quot\(_P\) associated to \(P\) assigns to \(R\) the set isomorphism classes of quotients of \(P\otimes_k R\) which are flat over \(R\). The authors' goal, following Grothendieck in his construction of the classical Hilbert scheme, is to show that (under suitable hypotheses) the Hilbert functor is representable. The paper is very long, much of it taken up by carefully stating the hypotheses necessary to carry out the authors' program and developing the necessary categorical machinery (tensor products, Hom, Tor, Ext, base change, completion, deformations). The authors' conclusion in section E3 is that under suitable hypotheses the Hilbert functor Quot\(_P\) is represented by a separated algebraic space locally of finite type over \(k\). More specifically (section E4) if \({\mathcal C}=\) Gr then the Hilbert functor of quotients with specified Hilbert function is represented by a projective scheme over \(k\), and (section E5) if \({\mathcal C}=\) Proj(\(A\)) the Hilbert functor is represented by a scheme which is a countable union of projective closed subschemes. The authors conjecture that the Hilbert functor of quotients with specified Hilbert function is represented by a projective scheme. base change; Grothendieck category; Hilbert scheme; noncommutative projective scheme Artin, M., Zhang, J.: Abstract Hilbert schemes. Algebr. Represent. Theory 4, 305--394 (2001) Parametrization (Chow and Hilbert schemes), Category-theoretic methods and results in associative algebras (except as in 16D90), Noncommutative algebraic geometry, Grothendieck categories Abstract Hilbert schemes. I | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A polyhedral method to solve a system of polynomial equations exploits its sparse structure via the Newton polytopes of the polynomials. We propose a hybrid symbolic-numeric method to compute a Puiseux series expansion for every space curve that is a solution of a polynomial system. The focus of this paper concerns the difficult case when the leading powers of the Puiseux series of the space curve are contained in the relative interior of a higher dimensional cone of the tropical prevariety. We show that this difficult case does not occur for polynomials with generic coefficients. To resolve this case, we propose to apply polyhedral end games to recover tropisms hidden in the tropical prevariety. Newton polytope; polyhedral end game; polyhedral method; polynomial system; Puiseux series; space curve; tropical basis; tropical prevariety; tropism Solving polynomial systems; resultants, Toric varieties, Newton polyhedra, Okounkov bodies, Combinatorial aspects of tropical varieties, Numerical computation of solutions to systems of equations, Global methods, including homotopy approaches to the numerical solution of nonlinear equations, Symbolic computation and algebraic computation Computing all space curve solutions of polynomial systems by polyhedral methods | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper under review studies deformations of schemes with normal crossing singularities, a kind of singularities appearing naturally in many different problems in algebraic geometry.
The well-known Mumford's semi-stable reduction theorem states that, after a finite base change and a birational modification, any flat projective morphism \(f: \mathcal X \rightarrow C\) from a variety \(\mathcal X\) to a curve \(C\) can be brought to \(f': \mathcal X' \rightarrow C'\), where \(\mathcal X'\) is smooth and the special fibers are simple normal crossing varieties.
In the compactification of the moduli space of varieties of general type, stable varieties appear in the boundary. Recall that a stable variety is a proper reduced scheme \(X\) such that \(X\) has only semi-log-canonical singularities and \(\omega_X^{[k]}\) is locally free and ample for some \(k>0\). The simplest class of non-normal semi-log-canonical singularities are the normal crossing singularities.
In the Minimal Model Program, normal crossing singularities also play an important role. One of the outcomes of the Minimal Model Program starting with a smooth \(n\)-dimensional projective variety \(X\) is a Mori fiber space \(f: Y \rightarrow Z\), where \(f\) is a projective morphism, \(Y\) is a \(\mathbb Q\)-factorial terminal projective variety such that \(-K_Y\) is \(f\)-ample, and \(Z\) is normal with \(\dim Z \leq n-1\). If \(Z\) is a curve, then \(Y_z=f^{-1} (z)\) for \(z \in Z\) is a Fano variety of dimension \(n-1\) and \(Y\) is a \(\mathbb Q\)-Gorenstein smoothing of \(Y_z\). In general \(Y_z\) has non-isolated singularities and may not even be normal. It is difficult to describe the singularities of the special fibers but normal crossing singularities naturally occur and are the simplest possible non-normal singularities.
Motivated by the above different problems, the author investigates the deformation spaces of varieties with normal crossing singularities. Let \(X\) be a reduced scheme with normal crossing singularities defined over a field \(k\) and let \(T^1 (X)\) denote the sheaf of first order deformations of \(X\). Then \(T^1 (X)\) is an invertible sheaf supported on the singular locus \(D\) of \(X\). In general, \(D\) is not smooth, and may not even be Cohen-Macaulay. So the author works in a suitable log resolution of \((X, D)\) and obtains explicit formulas for \(T^1 (X)\) in this setting.
The paper under review is well written. In section \(1\) the author explains the motivation to study deformations of schemes with normal crossing singularities and states the main theorem. In section \(2\) the author recalls some technical results for future use. He proves the first part of the main theorem in section \(3\), where the case that \(X\) has only double point singularities is treated. The general case that \(X\) has higher multiplicity singularities is done in section \(4\). The log resolution \((X', D')\) of \((X, D)\) is obtained by successively blowing ups the singular locus of highest multiplicity of \(X\). This works in all dimensions but is not unique. In dimension at most three, the author obtains a unique log resolution \((\tilde{X}, \tilde{D})\) by running an explicit minimal model program on \((X', D')\) and proves the second part of the main theorem. At the end of the paper, the author constructs a Fano \(3\)-fold with normal crossing singularities and show that it is not smoothable. deformations of schemes; normal crossing singularities Formal methods and deformations in algebraic geometry, Fibrations, degenerations in algebraic geometry First order deformations of schemes with normal crossing 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 describe a new subdivision method to efficiently compute the topology and the arrangement of implicit planar curves. We emphasize that the output topology and arrangement are guaranteed to be correct. Although we focus on the implicit case, the algorithm can also treat parametric or piecewise linear curves without much additional work and no theoretical difficulties.The method isolates singular points from regular parts and deals with them independently. The topology near singular points is guaranteed through topological degree computation. In either case the topology inside regions is recovered from information on the boundary of a cell of the subdivision.Obtained regions are segmented to provide an efficient insertion operation while dynamically maintaining an arrangement structure.We use enveloping techniques of the polynomial represented in the Bernstein basis to achieve both efficiency and certification. It is finally shown on examples that this algorithm is able to handle curves defined by high degree polynomials with large coefficients, to identify regions of interest and use the resulting structure for either efficient rendering of implicit curves, point localization or boolean operation computation. topology; arrangement; implicit; parametric; curves Alberti L., Mourrain B., Wintz J.: Topology and arrangement computation of semi-algebraic planar curves. Comput. Aided Geom. Des. 25(8), 631--651 (2008) Topology of real algebraic varieties, Semialgebraic sets and related spaces, Computer-aided design (modeling of curves and surfaces), Computer graphics; computational geometry (digital and algorithmic aspects), Computational aspects of algebraic curves, Plane and space curves Topology and arrangement computation of semi-algebraic planar 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 Given a zero-dimensional scheme \(Z\), the higher-rank interpolation problem asks for the classification of slopes \(\mu\) such that there exists a vector bundle \(E\) of slope \(\mu\) satisfying \(H^i(E \otimes \mathcal{L}_Z) = 0\) for all \(i\). In this paper, we solve this problem for all zero-dimensional monomial schemes in \(\mathbb{P}^2\). As a corollary, we obtain detailed information on the stable base loci of Brill-Noether divisors on the Hilbert scheme of points on \(\mathbb{P}^2\). We prove the correspondence between walls in the Bridgeland stability manifold and walls in the Mori chamber decomposition of the effective cone conjectured in [\textit{D. Arcara} et al., Adv. Math. 235, 580--626 (2013; Zbl 1267.14023)] for monomial schemes. We determine the Harder-Narasimhan filtration of ideal sheaves of monomial schemes for suitable Bridgeland stability conditions and, as a consequence, obtain a new resolution better suited for cohomology computations than other standard resolutions such as the minimal free resolution. Hilbert scheme; monomial schemes; Bridgeland stability; stable base locus Parametrization (Chow and Hilbert schemes), Minimal model program (Mori theory, extremal rays), Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Syzygies, resolutions, complexes and commutative rings, Derived categories of sheaves, dg categories, and related constructions in algebraic geometry Interpolation, Bridgeland stability and monomial 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 The authors introduce the notion of lex-seq plus \(s\)-powers ideals, obtained from lex-seg ideals by adding \(s\) powers of distinct indeterminates; and that of lex-seg with holes ideals, derived from lex-seg ideals by removing suitable monomials (namely: those whose exponent on some of the indeterminates is higher than a bound previously fixed for that indeterminate). By taking a subcomplex of the one constructed by Eliahou and Kervaire, the authors exhibit a minimal resolution for lex-seg with holes ideals; then, by adding the powers one at a time and iterating mapping cones, they find a minimal resolution for lex-seg plus \(s\)-powers ideals. Lex-seg plus \(s\)-powers ideals were mentioned in a paper by \textit{D. Eisenbud, M. Green} and \textit{J. Harris} [in: Journées géométrie algébrique, Astérisque 218, 187-202 (1993; Zbl 0819.14001)] for making sharp \(a\) a conjecture of theirs (conjecture \(V_m)\) about a bound for the growth of the Hilbert function. mapping cones; lex-seg ideals; Hilbert function Charalambous, H.; Evans, G.: Resolutions obtained by iterated mapping cones. J. algebra 176, 750-754 (1995) Linkage, complete intersections and determinantal ideals, Determinantal varieties Resolutions obtained by iterated mapping cones | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 rank one local system \(L\) on a smooth complex algebraic variety \(M\) is \textit{admissible} if one has \(H^k(M,L)=H^k(H^*(M), \wedge \alpha)\) for all \(k\), where \(L=\exp(\alpha)\) in the usual sense.
A line arrangement \(A\) is said to be of type \(C_m\) for some \(m \geq 0\) if \(m\) is the minimal number of lines in \(A\) containing all the points of multiplicity at least \(3\).
It was known that any local system \(L\) on the complement \(M\) of a line arrangement of type \(C_m\) for \(m <2\) is admissible.
The authors show that the same result holds for line arrangements of type \(C_2\), and explain clearly why it fails for certain line arrangements of type \(C_3\), as for instance the deleted \(B_3\)-arrangement discovered by A. Suciu. admissible local system; line arrangement; characteristic variety. Nazir, S.; Raza, Z.: Admissible local systems for a class of line arrangements, Proc. am. Math. soc. 137, No. 4, 1307-1313 (2009) Relations with arrangements of hyperplanes, Pencils, nets, webs in algebraic geometry, (Co)homology theory in algebraic geometry, Plane and space curves, Rational and birational maps Admissible local systems for a class of line arrangements | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper provides an alternative method to the topological recursion for computing the one-point function \(W_1(x)\) in the case of Laguerre and generalized Laguerre ensembles, i.e. an Hermitian matrix model with potential \(V(x)=x\) on \(\mathbb{R}_+\). The main result of the paper is to provide an explicit three terms recursion, which is similar to the one of \textit{J. Harer} and \textit{D. Zagier} [Invent. Math. 85, 457--485 (1986; Zbl 0616.14017)] or \textit{N. Do} and \textit{P. Norbury} [Topological recursion for irregular spectral curves, \url{arXiv:1412.8334}] in the Gaussian case, for computing the coefficients of the one-point function. The main advantage of this method compared to the topological recursion, is that it does not require the knowledge of the other correlation functions and therefore computations are much easier and faster. Thus, as explained in the paper, it is particularly well-suited for the matching of the coefficients with integers arising in enumerative geometry. In the Laguerre case, the author presents the details of the connection with the number of unicellular two-colored maps. The proof of the three terms recursion relies first on a replica trick, then the use of the Harish-Chandra-Itzykson-Zuber integral, and finally some contour deformations and standard complex analysis techniques. Consequently, the paper is relatively short and pleasant to read. However, since the proof relies on the knowledge of some specific formulas, the method may not easily generalize to more complicated models. replica method; Laguerre ensemble; topological recursion Chekhov, L. O., The harer-Zagier recursion for an irregular spectral curve, J. Geom. Phys., 110, 30-43, (2016) Enumerative problems (combinatorial problems) in algebraic geometry, Random matrices (algebraic aspects), Exact enumeration problems, generating functions, Random matrices (probabilistic aspects) The Harer-Zagier recursion for an irregular spectral 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 Let \(p\) be a prime number, let \(k\) be a perfect field of characteristic \(p\) and let \(W(k)\) denote the ring of Witt vectors over \(k\). Let \(K_{00}=\text{Frac}(W(k))\) and let \(K_0\) be a totally ramified field extension of \(K_{00}\) of degree \(e\). Let \(t\) be an indeterminate, let \(S=k[[t]]\) and let \(\sigma: S\rightarrow S\) denote the map given as \(\sigma(s)=s^p,\forall s\in S\). Let \(\text{MF}^e_S\) denote the category of triples \((M^0,M^1,\varphi_1)\) where \(M^0\) is a free \(S\)-module of finite rank, \(M^1\) is an \(S\)-submodule of \(M^0\) which satisfies \(t^eM^0\subseteq M^1\), and \(\varphi_1: M^1\rightarrow M^0\) is an \(S\)-module morphism which satisfies the conditions \(\varphi_1\sigma=\sigma\varphi_1\) and \(S\cdot\varphi_1(M^1)=M^0\).
Next, let \(O_0\) denote the ring of integers in \(K_0\). Let \(\text{Gr}_{O_0}'\) denote the category of finite flat \(p\)-group schemes \(G\) over the base scheme \(\text{Spec}\;O_0\). In other words, \(G\) is a group scheme of the form \(G(-)=\text{Hom}_{O_0\text{-alg}}(H,-)\) where \(H\) is a flat commutative unitary \(O_0\)-Hopf algebra of rank a power of \(p\) over \(O_0\). Let \(\mu\), \(\Delta\) denote multiplication and comultiplication in \(H\), respectively. For an integer \(l\geq 2\) define \(\mu^{(l-1)}=\mu(I\otimes \mu)(I\otimes I\otimes \mu)\cdots (I\otimes I\otimes\cdots \otimes I\otimes \mu)\) (\(l-2\) copies of \(I\) in the last factor). Similarily, define \(\Delta^{(l-1)}=(I\otimes I\otimes\cdots \otimes I\otimes \Delta)\cdots (I\otimes I\otimes \Delta)(I\otimes \Delta)\Delta\). Then there exists an endomorphism of Hopf algebras \([p]: H\rightarrow H\) defined as \([p](h)=\mu^{(p-1)}(\Delta^{(p-1)}(h))\). By the Yoneda lemma, \([p]\) corresponds to an endomorphism of group schemes \(p: G\rightarrow G\) given as \(p_T(f)=f^p,\forall f\in \text{Hom}_{O_0\text{-alg}}(H,T)\), where \(T\) is a commutative unitary \(O_0\)-algebra.
Let \(\epsilon: H\rightarrow O_0\) denote the counit map of \(H\) and let \(\lambda: O_0\rightarrow H\) denote the \(O_0\)-algebra structure map of \(H\). Then the morphism \(\lambda\epsilon: H\rightarrow H\) corresponds to a morphism of group schemes \(0: G\rightarrow G\) (the trivial map). The group \(G\) is \textit{killed by \(p\)} if \(p=0\). Let \(\text{Gr}_{O_0}\) denote the full subcategory of \(\text{Gr}_{O_0}'\) consisting of the group schemes \(G\) which are killed by \(p\). The main result of the paper under review shows that there is an antiequivalence between the category \(\text{MF}^e_S\) and the category \(\text{Gr}_{O_0}\). Witt vectors; \(p\)-group scheme Victor Abrashkin, \emph{Group schemes of period \(p>2\)}, Proc. Lond. Math. Soc. (3), \textbf{101} (2010), 207--259. https://doi.org/10.1112/plms/pdp052; zbl 1200.14092; MR2661246 Group schemes, Drinfel'd modules; higher-dimensional motives, etc. Group schemes of period \(p>2\) | 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 The volume of a line bundle is defined in terms of a limsup. It is a fundamental question whether this limsup is a limit. It has been shown that this is always the case on generically reduced schemes. We show that volumes are limits in two classes of schemes that are not necessarily generically reduced: codimension one subschemes of projective varieties such that their components of maximal dimension contain normal points and projective schemes whose nilradical squared equals zero. volume of line bundle; projective scheme Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Riemann-Roch theorems Volumes of line bundles as limits on generically nonreduced schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper contains some fundamental results on coherent sheaves on algebraic stacks, which are also interesting in the special case the algebraic stack in question is a scheme. One says that an algebraic stack has the resolution property if any coherent sheaf on it is the quotient of a locally free coherent sheaf. The first main result is as follows: Suppose \(X\) is an algebraic stack that is normal, noetherian, and whose stabilizer groups at closed points are affine. Then \(X\) has the resolution property if and only if \(X\) is the quotient of a quasiaffine scheme by some GL\((n)\)-action. In the special case that \(X\) is of finite type over a field, this is also equivalent to the condition that \(X\) is the quotient of an affine scheme of finite type by some action of an affine group scheme of finite type.
The second main result deals with Deligne--Mumford stacks: Suppose that \(X\) is Deligne--Mumford stack over a field that is smooth, has finite and generically trivial stabilizer group, and whose coarse moduli space \(B\) is a scheme with affine diagonal. Then the stack \(X\) has the resolution property. The proofs make clever use of results on algebraic stacks from the book of \textit{G. Laumon} and \textit{L. Moret-Bailly} [``Algebraic spaces''. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 39 (Springer, Berlin) (2000; Zbl 0945.14005)], the work of \textit{D. Edidin, B. Hassett, A. Kresch}, and \textit{A. Vistoli} [Am. J. Math. 123, 761--777 (2001; Zbl 1036.14001)], of \textit{S. Keel} and \textit{S. Mori} [Ann. Math. (2) 145, 193--213 (1997; Zbl 0881.14018)], and of \textit{R. W. Thomason} [Adv. Math. 65, 16--34 (1987; Zbl 0624.14025)]. The paper also contains results and examples on the relation of naive \(K\)-groups of locally free sheaves and the true \(K\)-groups of perfect complexes. Totaro, B., \textit{the resolution property for schemes and stacks}, J. reine angew. Math., 577, 1-22, (2004) Generalizations (algebraic spaces, stacks) The resolution property for schemes and stacks | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems To any 0-dimensional, reduced, degree \(s\), projective scheme \(X\) we associate a set \(S_X\) of sequences of \(s\) natural numbers; these sequences turn out to be a suitable permutation of a nondecreasing sequence \((d_1,\dots,d_s)\) which is in \(1-1\) correspondence with the first difference \(\Gamma_X\) of the Hilbert function of \(X\). The structure of \(S_X\) allows us to read geometric properties of \(X\). In its turn, the set \(\mathcal S_X\) of all sequences allowed for at least one scheme \(X\) with \(\Gamma_X=\Gamma\) can be partitioned into equivalence classes, giving information on all the schemes \(X\) with \(\Gamma_X=\Gamma\). We start to afford the problem of producing all the sequences equivalent to a given one.
Cf. also the review [in: Zero-dimensional schemes and applications. Proceedings of the workshop, Naples, Italy, February 9-12, 2000. Kingston: Queen's University. Queen's Pap. Pure Appl. Math. 123, 221--230 (2002; Zbl 1008.14010)]. Beccari, G.; Massaza, C.: A new approach to the Hilbert function of a 0-dimensional projective scheme, J. pure appl. Algebra 165, No. 3, 235-253 (2001) Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Graded rings, Relevant commutative algebra, Computational aspects in algebraic geometry A new approach to the Hilbert function of a 0-dimensional projective scheme | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A classical way to study a finite set of points in a projective space \(\mathbb{P}^r\) over an algebraically closed field is to look at relations between its Castelnuovo function (i.e. the first difference function of the Hilbert function of its homogeneous coordinate ring) of geometric properties of the point set. In this extended abstract (containing no proofs), the authors refine that method as follows.
Given a sequence \(X= (P_1,\dots, P_s)\) of points in \(\mathbb{P}^r\), let \(S= (d_1,\dots, d_s)\in \mathbb{N}^s\) be defined by \(d_1=0\) and \(d_k=\) least degree of a hypersurface separating \(P_k\) from \(P_1,\dots, P_{k-1}\) for \(k>1\). Then the multiplicity sequence \(\gamma_S(n)= \#\{i\mid d_i=n\}\) equals the Castelnuovo function of \(X\) and does not depend on the order of the points. Hence it makes sense to study which sequences \(S\) are realizable and to try to classify all point sets \(X\) with given Castelnuovo function \(\Delta HF_X\) according to their sequences \(S\).
Here the authors announce some steps in this direction by examining the effect of neighbour transposition in the point sequence on the degree sequence. They discover that a set \(X\) gives rise to only one non-decreasing sequence \(S\) if and only if \(X\) is in uniform position. Moreover, maximal growth of the Castelnuovo function is shown to correspond to sequences of the form \(S= (0,1,\dots, n+h, S')\) with non-decreasing sequence \(S'\). All results are amply illustrated by examples. 0-dimensional scheme; Hilbert function; geometric properties of the point set; Castelnuovo function Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Cycles and subschemes A new approach to the Hilbert function of a 0-dimensional projective scheme | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The theme of polynomial interpolation has a long history in Mathematics, and its ``ubiquity'' has been remarked by many authors, especially when it regards studying vector spaced of forms of a given degree \(d\) passing through given fixed points \(P_q,\dots,P_s \in {\mathbb P}^n\) with assigned multiplicities \(m_1,\dots,m_s\) (i.e. considering the degree \(d\) part of the ideal of the scheme of ``fat points'' \(m_1P_1+\dots+m_sP_s\)).
In this paper the case of generic \(P_1,\dots,P_s \in {\mathbb P}^3\) is studied for \(m_1\leq m_2\leq \dots \leq m_s\leq 4\) is treated; the main result is that for \(d\geq 41\), such vector space have the expected dimension. The main tool for the proof is the ``differential Horace'' method, which allows an induction procedure on the multiplicities of the points. fat points; postulation; linear systems; 0-dimensional schemes Edoardo Ballico and Maria Chiara Brambilla, Postulation of general quartuple fat point schemes in \?³, J. Pure Appl. Algebra 213 (2009), no. 6, 1002 -- 1012. Projective techniques in algebraic geometry, Vector and tensor algebra, theory of invariants, Numerical interpolation Postulation of general quartuple fat point schemes in \(\mathbb{P}^3\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(A\) be an integer-valued \(d \times n\) matrix of rank \(d\) and \(\beta \in \mathbb C^d\). The \(A\)-hypergeometric system is a \(D\)-module defined by \(A\) and \(\beta\). Here \(D = \mathbb C[x_1, \dots, x_n] \langle \partial_1, \dots, \partial_n \rangle\) is the Weyl algebra. In the present paper, the authors studies the irregularity of \(M_A(\beta)\). To do this, they introduce a filtration \(L = \{L_k D\}_k\) of \(D\) and use the associated graded ring \(\text{gr}^L D\). Let \((l_1, \dots, l_{2n}) \in \mathbb Q^{2n}\) such that \(l_1 + l_{n+1}\), \dots, \(l_n + l_{2n} \geq 0\). Then let \(L_k D\) be the vector space spanned by monomials \(x_1^{e_1} \cdots x_n^{e_n} \partial_1^{f_1} \cdots \partial_n^{f_n}\) such that \(\sum e_i l_i + \sum f_i e_{i+n} \leq k\). For such a filtration, they define an \((A, L)\)-umbrella, which is a cell complex, and the characteristic variety \(\text{Ch}^L(M_A(\beta))\), which is a subvariety of \(\text{Spec} \text{gr}^L D\). Furthermore, they define slopes of \(M_A(\beta)\) by using a family of filtrations parameterized by rational numbers. For an irreducible subvariety \(C \subset \text{Spec} \text{gr}^L D\), let \(\mu^{L,C}_{A,0}(\beta)\) be the multiplicity of \(\text{gr}^L M_A(\beta)\). The main result of this paper is the computation of \(\mu^{L,C}_{A,0}(\beta)\). As its consequence, they prove the converse of Hotta's theorem. That is, any \(A\)-hypergeometric system is homogeneous with respect to some filtration.
The authors, moreover, define the Euler-Koszul \(L\)-characteristic along \(C\): \(\mu_{A,i}^{L,C}(-; \beta)\) (\(i = 0\), \(1\), \dots), which is a generalized notion of \(\mu_{A,0}^{L, C}(\beta)\). They also compute it. \((A, L)\)-umbrella; irregularity sheaf; Euler-Koszul characteristic M. Schulze and U. Walther, Irregularity of hypergeometric systems via slopes along coordinate subspaces , preprint,\arxivmath/0608668v2[math.AG] Commutative rings of differential operators and their modules, Toric varieties, Newton polyhedra, Okounkov bodies, Filtered associative rings; filtrational and graded techniques Irregularity of hypergeometric systems via slopes along coordinate subspaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For an arbitrary non-archimedean local field we classify reductive group schemes over the corresponding Fargues-Fontaine curve by group schemes over the category of isocrystals. We then classify torsors under such reductive group schemes by a generalization of Kottwitz' set \(B(G)\). In particular, we extend a theorem of Fargues on torsors under constant reductive groups to the case of equal characteristic. Fargues-Fontaine curve; isocrystals Class field theory; \(p\)-adic formal groups, Langlands-Weil conjectures, nonabelian class field theory, Vector bundles on curves and their moduli Reductive group schemes over the Fargues-Fontaine 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 Here we study the existence of \(\alpha\)-stable coherent systems of type \((r,d,r+1)\) on general nodal or cuspidal curves. Vector bundles on curves and their moduli, Singularities of curves, local rings, Special divisors on curves (gonality, Brill-Noether theory) Coherent systems of type \((r,d,r+1)\) on general nodal or cuspidal curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper studies tropicalizations of Hilbert schemes given as subschemes of the Grassmanian. The connection to the tropical Grassmanian enables the authors to draw nice consequences about degree bounds for tropical bases. The cases of hypersurfaces, and of pairs of points in the plane are considered in detail. tropicalization; Hilbert scheme; Grassmanian; tropical bases D. Alessandrini, M. Nesci, On the tropicalization of the Hilbert scheme, arxiv:0912.0082. , Fine and coarse moduli spaces On the tropicalization of 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 Investigations on separation of variables of integrable systems as a supplement of the previous paper [the author and \textit{T. Takebe}, J. Geom. Phys. 47, 1--20 (2003; Zbl 1073.37070)] are given. In a certain integrable system, its spectral curve becomes the graph \(C = \{(\lambda, z) = A(\lambda)\}\) of a function \(A(\lambda)\). The author deals with such integrable systems and shows that such integrable systems provide an interesting {\`\`}toy model{\'\'} of separation of variables. spectral curves; separation of variables; integrable system Takasaki, K.: Integrable systems whose spectral curve is the graph of a function, CRM proc. Lecture notes 37, 211-222 (2004) Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Momentum maps; symplectic reduction, Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Relationships between algebraic curves and integrable systems Integrable systems whose spectral curves are the graph of a function | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(K\) be an algebraically closed field of characteristic \(p\). A description is given of rational functions \(f\in K(X)\), some iterate of which is a polynomial. It has been known that if \(K\) is the complex number field, then either \(f(X)\) of \(f(f(X))\) must be a polynomial. The author shows that the same is true in the case of characteristic \(p\) provided \(f(X)\) is separable and proves that in the inseparable case such functions must be linearly conjugate to \((az^q+ b)/(cz_q+ d)\) with \(q\) being a power of \(p\) and \(ad- bc\neq 0\). rational functions with a polynomial iterate J. H. Silverman, ''Rational Functions with a Polynomial Iterate,'' J. Algebra 180, 102--110 (1996). Polynomials (irreducibility, etc.), Rational and birational maps, Polynomials in general fields (irreducibility, etc.) Rational functions with a polynomial iterate | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this very well-written article, the authors establish several results relating finite schemes and secant varieties over arbitrary fields. Some of these results were previously known only over the complex numbers. The paper is in part expository and contains background material on scheme theory, apolarity theory, Castelnuovo-Mumford regularity, Hilbert schemes, and secant varieties. Let \(\mathbb K\) be a field and \(R\) be a finite scheme over \(\mathbb K\). One of the main objectives is to study the \textit{smoothability} of \(R\) both as an abstract scheme and as an embedded scheme in some algebraic variety \(X\). The condition of smoothability can be easily seen over an algebraically closed field: a finite scheme \(R\) is smoothable if and only if it is a flat limit of distinct points. Theorem 1.1 gives the equivalence between the abstract smoothability and the embedded smoothability in some algebraic variety \(X\), whenever \(X\) is smooth. Moreover, smoothability over \(\mathbb K\) is equivalent to smoothability in the algebraic closure of \(\mathbb K\) (Proposition 1.2). Let \(\mathbb K\) be an algebraically closed field. Let \(X\) be an algebraic variety \(\mathbb K\) and let \(r\) be an integer. Condition \((\star)\) holds if every finite \textit{Gorenstein} subscheme over \(\mathbb K\) of \(X\) of degree at most \(r\) is smoothable in \(X\). One of the main results is Theorem 1.7. This relates the scheme theoretic condition above with the possibility of giving \textit{set-theoretic equations} for secants of sufficiently high Veronese embeddings of \(X\), by determinantal equations from vector bundles on \(X\). If condition \((\star)\) does not hold, then those equations are not enough to cut them. Interestingly, the locus of determinantal equations from vector bundles contain more general loci than secants: the \textit{cactus varieties}. This containment is the ultimate reason for the failure of present methods to give good enough lower bounds on tensor ranks. smoothable; secant varieties; finite Gorenstein scheme; cactus variety; Veronese reembedding; Hilbert scheme Buczyński, J.; Jelisiejew, J., Finite schemes and secant varieties over arbitrary characteristic, Differential Geom. Appl., 55, 13-67, (2017) Determinantal varieties, Local deformation theory, Artin approximation, etc., Parametrization (Chow and Hilbert schemes), Schemes and morphisms, Homogeneous spaces and generalizations Finite schemes and secant varieties over arbitrary characteristic | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a smooth projective variety of dimension \(n\) of general type defined over the field of complex numbers. Then for all sufficiently large \(m\), the pluricanonical system \(|mK_{X}|\) defines a birational map \(\varphi_{m}:X\to \mathbb{P}^{N}\) to a projective space \(\mathbb{P}^{N}\). Then it is natural to find an effective bound for such integers \(m\). When \(X\) is a curve, \(\varphi_{m}\) is an embedding for any \(m\geq 3\). For the case of \(\dim X=2\), the map \(\varphi_{m}\) is birational for any \(m\geq 5\). In this paper, the authors studied the case of irregular \(3\)-folds and they obtained the following result. Let \(X\) be an irregular smooth projective variety of general type with \(\dim X=3\) and \(\chi(\omega_{X})>0\). Then the rational map defined by \(|mK_{X}+P|\) is birational for any \(m\geq 5\) and any \(P\in \text{Pic}^{0}(X)\). irregular varieties; general type; pluricanonical systems Chen J.A. and Hacon C.D. (2007). Pluricanonical systems on irregular 3-folds of general type. Math. Z. 255: 343--355 Rational and birational maps, Divisors, linear systems, invertible sheaves, \(3\)-folds Pluricanonical systems on irregular 3-folds of general type | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Luna's slice theorem [\textit{D. Luna}, Bull. Soc. Math. Fr., Suppl., Mém. 33, 81--105 (1973; Zbl 0286.14014)] is a classical result to study the local structure of an affine variety equipped with a reductive group action. In this paper, the author obtains a Luna's slice type theorem for certain moduli spaces called \textit{invariant Hilbert schemes}. These moduli spaces were introduced by Haiman and Sturmfels for diagonalizable groups [\textit{M. Haiman} and \textit{B. Sturmfels}, J. Algebr. Geom. 13, No. 4, 725--769 (2004; Zbl 1072.14007)] and then by \textit{V. Alexeev} and \textit{M. Brion} for arbitrary reductive algebraic groups [J. Algebr. Geom. 14, No. 1, 83--117 (2005; Zbl 1081.14005); \textit{M. Brion}, in: Handbook of moduli. Volume I. Somerville, MA: International Press; Beijing: Higher Education Press. 64--117 (2015; Zbl 1322.14001)]. They appear in many situations: classification of spherical varieties, invariant deformation theory of affine schemes with reductive group actions, construction of canonical resolutions of quotient singularities, etc. However little is know about the geometric properties of the invariant Hilbert schemes (connectedness, smoothness, reducedness, etc) except in some particular cases.
The author obtains Luna's slice type results to study the local structure of invariant Hilbert schemes (Theorems 2.3 and 2.4). Then, he deduces numerical smoothness criteria for certain class of invariant Hilbert schemes (Corollaries 2.5 and 2.6). Also, a more involved example of invariant Hilbert scheme where these results apply to prove smoothness is provided in the last section of the paper. invariant Hilbert scheme; Luna's étale slice theorem; smoothness criterion Group actions on varieties or schemes (quotients), Parametrization (Chow and Hilbert schemes) On the invariant Hilbert schemes and Luna's étale slice theorem | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We extend the construction of DAHA-Jones polynomials for any reduced root systems and DAHA-superpolynomials in type \(A\) from iterated torus knots [the authors, Algebr. Geom. Topol. 16, No. 2, 843--898 (2016; Zbl 1375.14099)] to links, including arbitrary algebraic links. Such a passage essentially corresponds to the usage of the products of Macdonald polynomials and is directly connected to the splice diagrams [\textit{D. Eisenbud} and \textit{W. Neumann}, Three-dimensional link theory and invariants of plane curve singularities. Annals of Mathematics Studies, No. 110. Princeton, New Jersey: Princeton University Press. VII, 172 p. (1985; Zbl 0628.57002)]. The specialization \(t=q\) of our superpolynomials results in the HOMFLY-PT polynomials. The relation of our construction to the stable Khovanov-Rozansky polynomials and the so-called ORS-polynomials of the corresponding plane curve singularities is expected for algebraic links in the uncolored case. These 2 connections are less certain, since the Khovanov-Rozansky theory for links is not sufficiently developed and the ORS polynomials are quite involved. However we provide some confirmations. For Hopf links, our construction produces the DAHA-vertex, similar to the refined topological vertex, which is an important part of our work. double affine Hecke algebra; Jones polynomial; HOMFLY-PT polynomial; Khovanov-Rozansky homology; iterated torus link Cherednik, I., Danilenko, I.: DAHA approach to iterated torus links. arXiv:1509.08351 Knots and links in the 3-sphere, Hecke algebras and their representations, Plane and space curves, Root systems, Lie algebras of linear algebraic groups, Braid groups; Artin groups, Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.), Compact Riemann surfaces and uniformization, Singular homology and cohomology theory DAHA approach to iterated torus links | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems After the classical work of Hurwitz, the Hurwitz scheme in characteristic 0, parametrizing simply branched covers of the projective line, was first rigorously introduced by \textit{W. Fulton} [Ann. Math. (2) 90, 542--575 (1969; Zbl 0194.21901)]. Successively several variants of this scheme have been introduced and studied, e.g. the unparametrized Hurwitz scheme, which is the quotient of the Hurwitz scheme by the PGL(2) action on \(\mathbb P^1\). A natural compactification of this scheme, the space of admissible covers, was given by \textit{J. Harris} and \textit{D. Mumford} [Invent. Math. 67, 23--86 (1982; Zbl 0506.14016)]. Recently \textit{R. Pandharipande} [Math. Ann. 313, 715--729 (1999; Zbl 0933.14035)] identified it as a closed subscheme in the space of stable maps into the stack \(\overline{\mathcal M}_{0,n+1}\).
In the paper under review, following the idea of Pandharipande, the authors introduce compactified Hurwitz stacks in mixed characteristic. They study then in detail the example of the compactified stack of double covers of \(\mathbb P^1\) branched in 4 points, in characteristic 2, and compare it with the characteristic 0 scheme. branched cover; admissible cover; Hurwitz scheme; Hurwitz stacks D. Abramovich and F. Oort, Stable maps and Hurwitz schemes in mixed characteristics, Advances in algebraic geometry motivated by physics (Lowell 2000), Contemp. Math. 276, American Mathematical Society, Providence (2001), 89-100. Families, moduli of curves (algebraic), Generalizations (algebraic spaces, stacks), Coverings in algebraic geometry, Coverings of curves, fundamental group Stable maps and Hurwitz schemes in mixed characteristics | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 propose a variation of the classical Hilbert scheme of points, the \textit{double nested Hilbert scheme of points}, which parametrizes flags of zero-dimensional subschemes whose nesting is dictated by a Young diagram. Over a smooth quasi-projective curve, we compute the generating series of topological Euler characteristic of these spaces, by exploiting the combinatorics of reversed plane partitions. Moreover, we realize this moduli space as the zero locus of a section of a vector bundle over a smooth ambient space, which therefore admits a virtual fundamental class. We apply this construction to the stable pair theory of a local curve, that is the total space of the direct sum of two line bundles over a curve. We show that the invariants localize to virtual intersection numbers on double nested Hilbert scheme of points on the curve, and that the localized contributions to the invariants are controlled by three universal series for every Young diagram, which can be explicitly determined after the anti-diagonal restriction of the equivariant parameters. Under the anti-diagonal restriction, the invariants are matched with the Gromov-Witten invariants of local curves of Bryan-Pandharipande, as predicted by the Maulik-Nekrasov-Okounkov-Pandharipande (MNOP) correspondence. Finally, we discuss \(K\)-theoretic refinements à la Nekrasov-Okounkov. double nested Hilbert schemes; local curves; stable pair invariants Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Parametrization (Chow and Hilbert schemes), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Double nested Hilbert schemes and the local stable pairs theory of curves | 0 |
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