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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 Let \(B\subset {\mathbb{P}}^3\) a surface of even degree with \(\mu\) isolated ordinary double points (``nodes''). A subset \(I\) of the nodes of \(B\) is said to be ``even'' (respectively, ``weakly even'') if there exists a double cover of \(B\) branched precisely on \(I\) (respectively, the union of \(I\) and of a plane section of \(B\)). Another interpretation of even sets is the following: \(I\) is even (respectively, weakly even) iff there exists a surface \(S\) in \({\mathbb{P}}^3\) of even (respectively, odd) degree such that \(S\cdot B=2D\), where \(D\) is a curve, and the points of \(I\) are precisely the points of \(B\) where \(D\) is not Cartier. The surface \(S\) is called a ``contact surface''.
Now let \(f\colon V\to {\mathbb{P}}^3\) be the double cover branched on \(B\). The singularities of \(V\) are ordinary double points and the so-called ``big resolution'' \(\widetilde{V}\) of \(V\) can be obtained by blowing up \({\mathbb{P}}^3\) at the singular points of \(B\) and taking base change and normalization. The Picard group \(\text{ Pic}(\widetilde{V})=H^2(\widetilde{V},{\mathbb{Z}})\) is the direct sum of the pull-back of the Picard group of \({\mathbb{P}}^3\) blown up, which has rank \(\mu+1\), and of a free group \({\mathcal M}\). The author describes a set of generators of \({\mathcal M}\) in terms of some special contact surfaces of the (weakly) even sets of nodes of \(B\). double cover; Picard group; nodal surface; contact surface; big resolution Stephan Endraß, On the divisor class group of double solids, Manuscripta Math. 99 (1999), no. 3, 341 -- 358. Picard groups, Coverings in algebraic geometry, \(3\)-folds, Singularities of surfaces or higher-dimensional varieties On the divisor class group of double solids | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 0633.00007.]
Any unramified fibre of a finite and locally cyclic morphism of irreducible varieties consists of d points, where d is the degree of the field extension. Thus in this case, the definition of unramifiedness given in Shafarevich's book coincides with the usual one. unramified fibre; field extension Ramification problems in algebraic geometry, Coverings in algebraic geometry A cardinality theorem of the fibres | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 0694.00008.]
This paper contains, without proofs, some results of the author's book: ``The relative fundamental group. Galois theory and localization'', Alxebra 56 (1991; Zbl 0726.13007). relative fundamental group; Galois theory; localization (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.), Coverings in algebraic geometry, Galois theory and commutative ring extensions, Torsion theory for commutative rings, Homotopy theory and fundamental groups in algebraic geometry, Categories in geometry and topology The fundamental group of a generically closed subset of a scheme | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The goal of this paper is to present new example of Zariski pairs which are not distinguished by the Alexander polynomial. We develop some technique to compute the fundamental group of the complement of the curve in order to find different topological invariants of the members of the pair. We find examples of Zariski pairs where all the irreducible components are rational curves. Finally, we relate these examples with the embedded topological type of the associated superisolated germs of surfaces. complement of algebraic curves; fundamental group; Alexander polynomial; Zariski pairs; topological type; Zariski problem Artal-Bartolo E., Carmona-Ruber J.: Zariski pairs, fundamental groups and Alexander polynomials. J. Math. Soc. Jpn. 50(3), 521--543 (1998) Homotopy theory and fundamental groups in algebraic geometry, Coverings in algebraic geometry Zariski pairs, fundamental groups and Alexander 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 It is noted that one of the features of the \(p\)-adic variational theory is that it is closely related to a \(p\)-adic variation of \(L\)-functions and other arithmetic invariants of the varying modular eigenforms. To give a hint of all this the author first describes an analogous theory concerning finite-dimensional complex representations of ``abstract'' groups and the accompanying variational theory of their homology.
Sections of the article are: 1. The ``homological algebra'' of complex variations of group representations. 2. ``Classical'' \(p\)-motives. 3. Refinements. 4. Examples. 5. \(p\)-adic variations of refined \(p\)-motives. 6. Is there a ``homological algebra'' of \(p\)-adic variations of refined \(p\)-motives? Twelve unsolved problems (in a form of questions) related to the topics disscused are listed in the article. There are 47 references. \(L\)-function; \(p\)-adic variations; variations of group representations; \(p\)-motives Mazur, B., The theme of \(p\)-adic variation.Mathematics: frontiers and perspectives, 433-459, (2000), Amer. Math. Soc., Providence, RI Local ground fields in algebraic geometry, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Finite ground fields in algebraic geometry, Motivic cohomology; motivic homotopy theory, Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols, Modular representations and characters The theme of \(p\)-adic variation | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this article, we study the geometry of bisections of certain rational elliptic surfaces. As an application, we give examples of Zariski \(N\)-plets for conic arrangements, which generalize Namba-Tsuchihashi's result. rational elliptic surfaces; Mordell-Weil group; Zariski \(N\)-plet; dihedral covers Bannai, S.; Tokunaga, H.-o., Geometry of bisections of elliptic surfaces and Zariski \textit{N}-plets for conic arrangements, Geom. dedic., 178, 219-237, (2015) Elliptic surfaces, elliptic or Calabi-Yau fibrations, Coverings in algebraic geometry Geometry of bisections of elliptic surfaces and Zariski \(N\)-plets for conic 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 Let \(p\) be a prime number, and let \(k\) be an algebraically closed field of characteristic \(p\). We show that the tame fundamental group of a smooth affine curve over \(k\) is a projective profinite group. We prove that the fundamental group of a smooth projective variety over \(k\) is finitely presented; more generally, the tame fundamental group of a smooth quasi-projective variety over \(k\), which admits a good compactification, is finitely presented. finite presentation; tame fundamental group; numerical tameness; projective profinite group Homotopy theory and fundamental groups in algebraic geometry, Ramification and extension theory, Coverings in algebraic geometry Finite presentation of the tame fundamental group | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 reduce some key calculations of compositions of morphisms between Soergel bimodules (``Soergel calculus'') to calculations in the nil Hecke ring (``Schubert calculus''). This formula has several applications in modular representation theory. Soergel bimodules; intersection forms; nil Hecke ring; Schubert calculus Modular representations and characters, Hecke algebras and their representations, Classical problems, Schubert calculus, Representations of finite groups of Lie type, Intersection homology and cohomology in algebraic topology Soergel calculus and Schubert calculus | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 family \(\pi :\mathcal{X}\rightarrow S\) of complex projective varieties which is of relative dimension \(n\) and topologically a locally trivial fibration, one sets \(X=\pi ^{-1}(s)\) for a typical fiber and considers the \(\pi _1(S,s)\)-action on the middle cohomology group \(H^n(X,\mathcal{R})\).
The monodromy group of the family \(\pi\) is the image of \(\pi _1(S,s)\). For the universal family of cyclic covers of projective spaces branched along hyperplane arrangements in general position, the author considers its monodromy group acting on an eigenspace of the middle cohomology of the fiber and proves that the monodromy group is Zariski dense in the corresponding linear group.
As an application one can deduce that the fundamental group of the moduli space of hyperplane arrangements is large. This can be be viewed as a degenerate analogy of Carlson-Toledo's result about the monodromy groups of smooth hypersurfaces. In the proof the author uses a Picard-Lefschetz type formula for a suitable degeneration of this family. complex projective variety; monodromy group; hyperplane arrangement; Picard-Lefschetz type formula; cohomology; moduli space Structure of families (Picard-Lefschetz, monodromy, etc.), Global theory of complex singularities; cohomological properties, Coverings in algebraic geometry Zariski density of monodromy groups via a Picard-Lefschetz type formula | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\Gamma\) be an algebraic curve in \(\mathbb{P}^{2}\). A smooth conic \(\Delta\subset\mathbb{P}^{2}\) is called a contact conic of \(\Gamma\) if all of the intersection points of \(\Gamma\) and \(\Delta\) are smooth points of \(\Gamma\) and for them the intersection multiplicity \(I_{P}(\Gamma,\Delta)\geq2\).
\(\Gamma\) with even (means intersection multiplicities are even) contact conic \(\Delta\) is a splitting curve of type \((m,n),\) \(m\leq n,\) with respect to \(\Delta\) if for the double cover \(\pi_{\Delta}:X_{\Delta}\rightarrow \mathbb{P}^{2}\) branched along \(\Delta\) the pullback \(\pi_{\Delta}^{\ast}\Gamma=D^{+}+D^{-}\) for an \((m,n)\) divisor \(D^{+}\) on \(X_{\Delta}
\cong\mathbb{P}^{1}\times\mathbb{P}^{1}\), where \(D^{-}\) is \(D^{+} \) after the covering transformation of \(X_{\Delta}.\) The main theorem is a criterion for determining the splitting type of a nodal curve \(\Gamma\) (means singular points of \(\Gamma\) are nodes) with respect to an even contact conic in terms of the configuration of the nodes and tangent points of \(\Gamma\) and \(\Delta\).
The authors apply the results to Zariski pairs of curves. splitting curve; Zariski pair; double cover; nodal curve Singularities of curves, local rings, Coverings in algebraic geometry, Plane and space curves, Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants Nodal curves with a contact-conic and Zariski pairs | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) and \(Y\) be projective manifolds of the same dimension and let \(f\colon X\to Y\) be a cover, namely a finite morphism, of degree \(d\). To \(f\) one associates \({\mathcal E}:=(f_*\mathcal {O}_X/{\mathcal O}_Y)^*\), which is a vector bundle of rank \(d-1\) on \(Y\). The authors continue here the study of the positivity properties of \({\mathcal E}\) started in the first part [Commun. Algebra 28, No. 12, 5573--5599 (2000; Zbl 0982.14025)], with special attention to the case in which \(Y\) is a Fano manifold. Their first result is negative: they exhibit an example of a triple cover of a Fano threefold of degree 5 for which \({\mathcal E}\) is not ample (by results of the authors and Lazarsfeld [ibid. and appendix], \({\mathcal E}\) is known to be nef on the general curve of \(Y\) and to be spanned when \(Y\) is a del Pezzo manifold of degree \(\geq 5\).) In the second part of the paper they study topological properties of low degree covers. They find sufficient conditions for the surjectivity of the map \(H^i(Y)\to H^i(X)\), thus generalizing a well known result of \textit{R. Lazarsfeld} on covers of \({\mathbb P}^n\) of low degree [Math. Ann. 249, 153--162 (1980; Zbl 0434.32013)], and for the ampleness of the branch divisor \(R\subset X\). The divisor \(R\) was already known to be ample in the case in which \(Y\) is a projective space or a quadric [cf. \textit{A. Lanteri, M. Palleschi} and \textit{A. J. Sommese}, Osaka J. Math. 26, No.3, 647--664 (1989; Zbl 0715.14005) and \textit{R. Lazarsfeld}, loc. cit.] covering of low degree; Fano manifold; ramification divisor Thomas Peternell and Andrew J. Sommese, Ample vector bundles and branched coverings. II, The Fano Conference, Univ. Torino, Turin, 2004, pp. 625 -- 645. Coverings in algebraic geometry, Fano varieties, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Ample vector bundles and branched coverings, II | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A compact complex manifold \(M\) is rigid if it has no nontrivial (small) deformations, and it is infinitesimally rigid if \(H^1(M, \Theta_M)=0\). It is a classical result of Kuranishi theory that infinitesimal rigidity implies rigidity. \textit{J. Morrow} and \textit{K. Kodaira} in [Complex manifolds. Athena Series. Selected Topics in Mathematics. New York etc.: Holt, Rinehart and Winston (1971; Zbl 0325.32001)] asked for examples to disprove the converse, more precisely they posed the following:
Problem. ``Find an example of an \(M\) which is rigid, but \(H^1(M, \Theta_M)\neq 0\). (Not easy?)''.
In the paper under review the authors exhibit the first examples of compact complex manifolds which are rigid but not infinitesimally rigid, proving the following statement:
Theorem 4.1. For every even \(n\geq 8\) not divisible by 3, there exists a rigid, but not infinitesimally rigid minimal surface \(S_n\) of general type with
\[
K^2_{S_n}= 2(n-3)^2\,, \qquad p_g(S_n)=\left(\dfrac n2 -2\right)\left(\dfrac n2 -1\right)\,.
\]
The surface \(S_n\) is a \textit{product-quotient surface}, i.e.~the minimal resolution of the singularities of the quotient of a product \(C_1\times C_2\) of two algebraic curves by the action of a finite group \(G\) acting faithfully on each curve and diagonally on the product. Here \(C_1=C_2\) is the Fermat curve of degree \(n\), \(G=(\mathbb Z/n\mathbb Z)^2\) and the singular quotient \(X_n:=(C_1\times C_2)/G\) has 6 nodes. Each node gives a non trivial contribution to \(h^1(\Theta_{S_n})\) and so \(S_n\) is not infinitesimally rigid. To prove the rigidity of \(S_n\) the authors provide (Theorem 1.3) a more general criterion for the minimal resolution of the singularities \(S\) of a nodal surface to be rigid, which relies on the linear independence of certain elements of \(H^2(\Theta_{S})\) that have a simple explicit description in local coordinates due to \textit{A. Kas} [Topology 16, 51--64 (1977; Zbl 0346.32028)].
Taking the product of \(S_n\) with a compact complex rigid manifold \(X\) one can construct higher-dimensional rigid but not infinitesimally rigid examples (Lemma 5.2). Combining this with the results in [\textit{I. Bauer} and \textit{F. Catanese}, Adv. Math. 333, 620--669 (2018; Zbl 1407.14003)] and [\textit{I. Bauer} and \textit{C. Gleissner}, Doc. Math. 25, 1241--1262 (2020; Zbl 1452.14035)] the authors derive the second main result of the paper.
Theorem 5.1. There are rigid, but not infinitesimally rigid, manifolds of dimension \(d\) and Kodaira dimension \(\kappa\) for all possible pairs \((d,\kappa)\) with \(d\geq 5\) and \(\kappa\neq 0,1\) and for \((d,\kappa) =(3, -\infty), (4,-\infty),(4,4)\). branched or unramified coverings; deformation theory; rigid complex manifolds Local deformation theory, Artin approximation, etc., Coverings in algebraic geometry, Families, moduli, classification: algebraic theory, Surfaces of general type, Group actions on varieties or schemes (quotients), Deformations of complex structures Rigid but not infinitesimally rigid compact complex 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 In the Mathematical Subject Classification 2000 a new section ``Affine geometry'' was added and in it new subsections ``Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem)'' and ``Jacobian problem''. So far there has not appeared any monograph devoted to this subjects.
The book under review is the first one. It presents precisely the topics mentioned above. It summaries the actual knowledge of the main part of the theory of polynomial automorphisms. The topics selected reflect rather the own research interests of the author and the same can be said about the methods -- almost entirely algebraic. So, out of this book there are other methods used in the theory of polynomial automorphisms and the jacobian conjecture as for instance topological (Lê Dũng Tráng, W. Neumann -- study of fibers of polynomials), geometrical (Z. Jelonek -- identity and determining sets for polynomial automorphisms) and analytical (J.Chądzyński and the reviewer -- the Łojasiewicz exponent at infinity and components of polynomial automorphisms).
As it was said the presentation method is algebraic and the book is written from the perspective of the Jacobian conjecture. Because of the wide spectrum of results described, some of them are given without proofs. Nevertheless they are well presented and described (with good references). The main feature of results is that they are given in the form as general as possible (for instance ground fields are replaced by k-algebras). So, many proofs are new and some of them simpler than the known ones.
Let us now outline the contents of the book. It consists of three parts: 1. Methods, 2. Applications, 3. Appendices.
The first part, ``Methods'', presents algebraic tools and methods used in the attack of the Jacobian conjecture. In chapter 1 we have the formal inverse function theorem and the basic properties of derivations of rings (this last notion is a key notion for almost all results in the book). Chapter 2 is devoted to derivations connected with polynomial mappings satisfying the Jacobian condition and to conditions equivalent to the Jacobian conjecture expressed in terms of derivations. Chapter 3 is about formulas for the inverse of a polynomial mapping by using Gröbner basis and the elimination theory. Chapter 4 concerns the interesting topic of criteria of injectivity of regular mappings between affine varieties. In chapter 5 the author describes the difficult and interesting problem of the structure of the group of polynomial automorphisms of affine spaces (and topics connected with the problem of embeddings of affine spaces). Chapter 6 is connected with the previous one and studies the structure of polynomial automorphisms by extending them trivially to automorphisms of affine spaces of greater dimensions (so called stabilization problems).
The second part, ``Applications'', gives applications of methods and results of the first part to other branches of affine geometry which are also connected to the Jacobian conjecture. The chapter 8 gives applications to dynamical systems, precisely to conditions equivalent to Jacobian conjecture in terms of asymptotic stability of vector fields generated by a given polynomial mapping (in \({\mathbb R^n}\)). In chapter 9 we find applications to group actions on affine varieties. In particular the author describes the famous linearization and cancellation problem. Chapter 10 describes selected methods (and the results obtained) of attacks on the Jacobian conjecture (conceived by many authors). In particular the Newton polygon methods and Pinchuk's counterexample to the real Jacobian conjecture.
In the third part, ``Appendices'', the author collects some basic definitions and results (used in the text) from commutative algebra, algebraic geometry, and theory of the Gröbner bases. The last interesting ``Appendix F'' delivers many special examples and counterexamples connected with the problems considered in the main text, which the author (an expert in this topics) encountered in his mathematical life.
Summing up, the book is very useful to all interested in affine geometry and especially in the Jacobian conjecture. So far, it is the only book on this subject. Although it contains many results without proofs it may serve as a reference book. It can also be used in graduate courses (since the methods used are algebraic and rather elementary and almost each section ends with exercises). The text is well arranged and clear. One can feel from the text the personal connection and engagement of the author to the problems described. Evidences of this are personal reminiscences in the text (e.g. in ``Introduction on K. Adjamagbo'' and in chapter 8 on a solution of the Markus-Yamabe problem), although in the reviewer's opinion this is needless in monographs.
So, I may recommend this book to any interested in algebraic geometry who deals with affine geometry and especially with the Jacobian conjecture. polynomial automorphism; Jacobian conjecture; derivation; affine space; tame automorphism; Markus-Yamabe conjecture; Gröbner basis; injective endomorphism; affine variety; cancellation problem; locally nilpotent derivation; face polynomial; triangular automorphism van den Essen, Arno, Polynomial Automorphisms and the Jacobian Conjecture, Progress in Mathematics, vol. 190, (2000), Birkhäuser Verlag: Birkhäuser Verlag Basel Jacobian problem, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Rational and birational maps, Birational automorphisms, Cremona group and generalizations, Global stability of solutions to ordinary differential equations, Coverings in algebraic geometry, Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) Polynomial automorphisms and the Jacobian conjecture | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We define a subclass in the class of quasismooth complete intersections (QSCI) by imposing a general position condition relative to the singular locus \(P_{\sin g}\) of the ambient weighted projective space P. This condition is fulfilled iff the weights and the degrees of the QSCI satisfy some arithmetic conditions. The subclass obtained in this way contains strictly the weighted complete intersections introduced by \textit{S. Mori} [J. Math. Kyoto Univ. 15, 619-646 (1975; Zbl 0332.14019)]. - For a QSCI X belonging to this subclass and with dim \(X\leq 2\) we show that \(X_{\sin g}=X\cap P_{\sin g}\) and describe the coverings \(Y\to X\) (with Y a normal analytic space) ramified over \(B=X_{\sin g}\cup H\), H being a normal hypersurface in X. The similar problem for \(X=P\) was treated by the author and \textit{S. Dimiev} [Bull. Lond. Math. Soc. 17, 234-238 (1985; Zbl 0546.14006)]. quasismooth complete intersections; QSCI; singular locus; coverings Dimca, Alexandru, Singularities and coverings of weighted complete intersections, Journal für die Reine und Angewandte Mathematik. [Crelle's Journal], 366, 184-193, (1986) Complete intersections, Coverings in algebraic geometry, Singularities in algebraic geometry, Ramification problems in algebraic geometry Singularities and coverings of weighted complete intersections | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For a compact connected genus \(g\) complex curve \(C\), let \(\lambda: C\to \mathbb{C}\mathbb{P}^1\) be a meromorphic function. We treat this function as a ramified covering of the sphere. The critical values of topologically equivalent functions, i.e., the ramification points of the coverings, coincide, as do the genera of the covering curves. In a well-known paper Hurwitz initiated the topological classification of such coverings in the case when exactly one of the ramification points is degenerate, and the remaining points are non-degenerate. Below we refer to the degenerate ramification point as ``infinity'', and its preimages are called ``poles''. For a given set of orders \(k_1,\dots, k_n\) of \(n\) distinct poles, the number of the equivalence classes of topologically non-equivalent ramified coverings with these orders of poles and prescribed non-degenerate ramification points is finite. This number \(h_{g; k_1,\dots, k_n}\) (called the Hurwitz number) is independent of the exact location of the non-degenerate ramification points. The number of sheets in the covering is \(k= k_1+\cdots+ k_n\), and the space of all such coverings has dimension \(d= k+ n+ 2g- 2\) over \(\mathbb{C}\).
Hurwitz posed the problem of finding \(h_{g; k_1,\dots, k_n}\) explicitly. Our aim is to express Hurwitz numbers in terms of intersection numbers for the Chern classes of certain line bundles on the moduli space of complex curves with \(n\) marked points. Hurwitz numbers; Hodge integrals; number of coverings; moduli space of genus \(g\) curves Torsten Ekedahl, Sergei Lando, Michael Shapiro, and Alek Vainshtein, On Hurwitz numbers and Hodge integrals, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), no. 12, 1175 -- 1180 (English, with English and French summaries). Coverings in algebraic geometry, Families, moduli of curves (algebraic), Compact Riemann surfaces and uniformization On Hurwitz numbers and Hodge 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 This paper consists of an announcement of a criterion for a divisor in a complex manifold to be the branch locus of a Galois covering. The criterion is a condition on the set of all isomorphism classes of unitary flat vector bundles with prescribed singularities. It is a difficult condition to implement, but leads to a more amenable criterion in the case where M is projective and the divisor consists of irreducible hypersurfaces with certain intersection properties, a result which generalizes a theorem of \textit{M. Kato} [Mem. Fac. Sci., Kyushu Univ., Ser. A 38, 127-131 (1984; Zbl 0556.32007)]. divisor; branch locus of a Galois covering Ramification problems in algebraic geometry, Coverings in algebraic geometry, Divisors, linear systems, invertible sheaves, Group actions on varieties or schemes (quotients) On branched coverings of projective 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 A famous question by Shafarevitch asks whether the universal cover of a complex projective manifold is holomorphically convex. Here, this question is considered in the case of surfaces. Let \(S\) be a smooth projective surface, \(f: \widetilde{S}\to S\) the universal cover, \(\widetilde{C}\subset \widetilde{S}\) a compact (possibly non reduced) curve and write \(C:=f(\widetilde{C})\), \(\overline{C}=f^{-1}(C)\). The author starts by showing that if the answer to the question of Shafarevitch is yes, if \(\widetilde{S}\) is not compact and the images in \(\pi_1(S)\) of the fundamental groups of the normalizations of the irreducible components of \(C\) are finite subgroups, then the following properties hold:
(i) the image of \(\pi_1(C)\) in \(\pi_1(S)\) is a finite subgroup;
(ii) if \(C\) is not exceptional on \(S\), then there exists a finite étale Galois cover \(S_1\to S\) such that the Albanese image of \(S_1\) is a curve \(B\) and \(C\) is the image of a fibre of the Albanese pencil \(S_1\to B\).
The rest of the paper is devoted to proving results similar to the above without assuming that the Shafarevich question has positive answer. More precisely, let \(f: \widetilde{S}\to S\) be any non compact étale cover of a smooth projective surface \(S\), let \(\widetilde{C}\subset \widetilde{S}\) be a compact curve and define \(C\) and \(\overline{C}\) as above. Then either:
(i) \(\widetilde{C}\) is exceptional in \(\widetilde{S}\), or
(ii) for every curve \(D\) the support of which is contained in the support of \(C\) one has \(D^2\leq 0\) and there exists a curve \(D\) with the same support as \(\widetilde{C}\) such that \(D^2=0\).
In addition, every compact curve \(D\) of \(\widetilde{S}\) is either contained in \(\widetilde{C}\) or it is disjoint from \(\widetilde{C}\). Finally, consider a non compact Galois étale cover \(f: \widetilde{S}\to S\) with Galois group \(\Gamma\) and let \(C\subset S\) be a curve such that all the irreducible components of \(f^{-1}(C)\) are compact. Using the above results, the author obtains an explicit linear isoperimetric inequality \(\Gamma\), under the assumption there exists a curve \(D\subset S\) such that \(D^2>0\) and \(D\) has the same support as \(C\). covering; projective surface; Shafarevich conjecture; isoperimetric inequality Campana F.: Negativity of compact curves in infinite covers of projective surfaces. J. Algebraic Geom. 7(4), 673--693 (1998) Coverings in algebraic geometry, Special surfaces Negativity of compact curves in infinite covers of projective surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) and \(Y\) be smooth projective varieties defined over the complex number field. Then we say that a morphism \(\pi:X\to Y\) is a cover if \(\pi\) is a finite surjective morphism. Let \(d=\deg\pi\). For \(d\geq 2\) there exists a structure theorem for covers, which was proved by \textit{G. Casnati} and \textit{T. Ekedahl} [J. Algebr. Geom. 5, No. 3, 439-460 (1996; Zbl 0866.14009)]. The structure theorem is the following: There exists a locally free sheaf \({\mathcal E}\) on \(Y\) of rank \(d-1\), natural splittings \(\pi_* ({\mathcal O}_X) \cong{\mathcal O}_Y \oplus\check {\mathcal E}\) and an embedding \(i:X \hookrightarrow \mathbb{P}({\mathcal E})\) such that \(\pi=p\circ i\) and \({\mathcal O}_{\mathbb{P} ({\mathcal E})} (1)|_X\cong \omega_{X/Y}\), where \(p:\mathbb{P} ({\mathcal E})\to Y\) is the projection and \(\omega_{X/Y}=\omega_X\oplus\pi^*\omega^{-1}_Y\). If in addition \(d\geq 4\), there also exists a locally free sheaf \({\mathcal F}\) on \(Y\) fitting into a sequence of the form
\[
0\to{\mathcal F}\to S^2({\mathcal E})\to\pi_* \omega^2_{X/Y} \to 0.
\]
In this paper, the author treats the case where \(X\) is an abelian surface. Let \(D\) be a divisor of type \((1,t)\) on \(X\). Here we consider the rational map \(\pi:X \to\mathbb{P}^{t-1}\) which is defined by the complete linear system \(|D|\). If \(t=2\), then the base locus Bs\(|D|\) consists of 4 points and the map \(\pi\) has been studied by \textit{W. Barth} [in: Algebraic Geometry, Proc. Symp., Sendai 1985; Adv. Stud. Pure Math. 10, 41-84 (1987; Zbl 0639.14023)]. If \(t=4\) resp. \(t\geq 5\), the morphism \(\pi_{|D|}\) has been dealt with by \textit{C. Birkenhake}, \textit{H. Lange} and \textit{D. van Straten} [Math. Ann. 285, No. 4, 625-646 (1989; Zbl 0714.14028)] and by \textit{F. Tovena}, in: Abelian Varieties, Proc. Int. Conf., Egloffstein 1993, 303-321 (1995; Zbl 0842.14032)], respectively by \textit{S. Ramanan} [Proc. Lond. Math. Soc., III. Ser. 51, 231-245 (1985; Zbl 0603.14013)].
In this paper the author treats the case \(t=3\) and in this case we get that \(Bs|D|= \emptyset\) and \(\pi_{|D|}\) defines a cover of degree 6. By considering the above structure theorem, the author computes the sheaves \({\mathcal E}\) and \({\mathcal F}\) of the associated cover \(\pi_{|D|}: X\to\mathbb{P}^2\). Furthermore the author gives a complete description of the structure of the map for bielliptic surfaces. abelian surface; bielliptic surface; branch locus; cover; finite surjective morphism; divisor Casnati, The cover associated to a (1,3)-polarized bielliptic abelian surface and its branch locus, Proc. Edinb. Math. Soc. 42 pp 375-- (1999) Algebraic theory of abelian varieties, Ramification problems in algebraic geometry, Divisors, linear systems, invertible sheaves, Coverings in algebraic geometry The cover associated to a \((1,3)\)-polarized bielliptic abelian surface and its branch locus | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper, a method of using Hurwitz moduli spaces of covers of \({\mathbb P}^1\) for the regular inverse Galois problem is refined and applied to several examples. One of the new ingredients of this paper is to introduce a very general class of rational curves on Hurwitz spaces arising from plane curves, where tools on plane curve singularities, dual curves, Puiseux expansions of algebraic functions etc. can play effective roles to analyze the resulting subfamily of Hurwitz spaces through the associated group theoretical data. This enables the author to find rigid tuples for a wide class of finite groups \(G\) that make \(G\) realized as Galois groups over the rational function field \(\mathbb Q(t)\). Among several illustrative examples to present this method, rigid tuples for the Mathieu groups \(M_{23}\), \(M_{11}\) are detected from the concrete plane curves \(x(y-27/4\cdot(x^3+x^2))=0\). Also, it is shown that the method combined with the theory of middle convolutions (or braid companion functors) yields a rigid tuple for SL\(_5(9)\) from the plane curve \(x(y+x^2+1)(y-(x-1)^2)(y-(x+1)^2)=0\). Hurwitz moduli space; rigid tuples; middle convolution M. Dettweiler, Plane curve complements and curves on Hurwitz spaces, Journal für die Reine und Angewandte Mathematik 573 (2004), 19--43. Coverings of curves, fundamental group, Inverse Galois theory, Families, moduli, classification: algebraic theory, Coverings in algebraic geometry Plane curve complements and curves on Hurwitz spaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author replies the relationship between coverings and normal subgroups of the fundamental group of a topological space, and then formulates a Riemann-Roch type theorem for Riemann surfaces. fundamental group; Riemann-Roch; Riemann surfaces Coverings in algebraic geometry, Coverings of curves, fundamental group, Compact Riemann surfaces and uniformization, Riemann-Roch theorems On the decomposition of the fundamental group and the Riemann-Roch 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 This paper is a survey, which also includes new results, about the interaction between singularity theory and fundamental groups of complements of divisors in smooth projective varieties. The paper contains the history behind the subject (from Zariski's work about the fundamental group of the complement of plane curves to recent research), exposition of some basic tools and results (Braid monodromy presentation of fundamental groups, Alexander invariants, characteristic varieties, multiplier ideals, branched covers among others), many concrete examples, and 223 references. Braid monodromy; Alexander invariants; branched covers Singularities in algebraic geometry, Relations with arrangements of hyperplanes, Coverings in algebraic geometry, Low-dimensional topology of special (e.g., branched) coverings, Research exposition (monographs, survey articles) pertaining to algebraic geometry Complements to ample divisors and singularities | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper is a review of studies about the encoding of properties of an algebraic variety \(X\) in its algebraic fundamental group \(\pi _1(X\otimes _k \overline k)\), the profinite group which classifies finite étale coverings, defined by A. Grothendieck. The interesting case is when \(k\) is a proper subfield of \(\overline k \), since we can view the algebraic fundamental group as an extension
\[
0 \rightarrow \pi _1(X\otimes _k \overline k) \rightarrow \pi _1(X) \rightarrow \text{ Gal}(\overline k /k) \rightarrow 0.
\]
A result of \textit{S. Mochizuchi} [see Invent. Math. 138, No. 2, 319-423 (1999)] shows that for hyperbolic curves \(X_1\), \(X_2\) over \(k\), where \(k\) is a finitely generated extension of \({\mathbb Q}_p\), any open map \( \pi _1(X_1) \rightarrow \pi _1(X_2)\), as \(\pi _1(X_2\otimes _k \overline k)\)-conjugacy class of map of extensions, is induced by a unique and dominant map \(X_1 \rightarrow X_2\). The main tool used here is \(p\)-adic Hodge theory. hyperbolic curve; algebraic fundamental group; étale coverings; \(p\)-adic Hodge theory Faltings, G., Curves and their fundamental groups (following Grothendieck, Tamagawa and mochizuki), Astérisque, 252, pp., (1998) Coverings of curves, fundamental group, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Coverings in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry Curves and their fundamental groups [following Grothendieck, Tamagawa and Mochizuki] | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The main theorem of this paper is a very nice characterization of (quotients of) abelian varieties: let \(X\) be a complex projective manifold of dimension at least two. Then \(X\) is a finite étale quotient of an abelian variety if and only if \(\nu^* T_X\) is a semistable vector bundle for every holomorphic map \(\nu: C \rightarrow X\) from a smooth curve \(C\). This results gives a complete answer to a question of \textit{I. Biswas} [Geom. Dedicata 142, 37--46 (2009; Zbl 1174.14016)] who had shown the same result for surfaces and projective manifolds that are Kähler-Einstein.
The method of the proof is of independent interest: suppose that \(\nu^* T_X\) is semistable for every curve on \(X\). Then one shows easily that \(X\) does not contain any rational curve and \(X\) is not of general type. Thus we know that \(K_X\) is nef, but abundance being still unknown, the minimal model program does not provide any further information on the structure of \(X\). However it can be shown that the tangent bundle \(T_X\) is projectively flat, i.e., the projectivisation \(\mathbb P (T_X)\) is determined by a representation of the fundamental group
\[
\pi_1(X) \rightarrow \text{PGL}_{n-1}(\mathbb C).
\]
If the representation is finite we obtain that \(X\) is abelian after finite étale cover, so one is left to exclude the possibility of an infinite representation: using results of \textit{K. Zuo} [J. Reine Angew. Math. 472, 139--156 (1996; Zbl 0838.14017)] on the Shafarevich map the authors prove that \(K_X\) is semiample, so it defines a holomorphic Iitaka fibration. Classification theory now allows to derive a contradiction. minimal model program; abelian varieties; classification; stability Minimal model program (Mori theory, extremal rays), Families, moduli, classification: algebraic theory, Compact Kähler manifolds: generalizations, classification, Notions of stability for complex manifolds, Coverings in algebraic geometry, Sheaves and cohomology of sections of holomorphic vector bundles, general results Semistability of restricted tangent bundles and a question of I. Biswas | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 crystallographic Coxeter system \((W,S)\) (with an associated root datum), one has an associated Hecke algebra \({\mathcal H}\), which is a \({\mathbb Z}[v,v^{-1}]\)-algebra with basis indexed by \(W\). In the seminal paper of \textit{D. Kazhdan} and \textit{G. Lusztig} [Invent. Math. 53, No. 2, 165--184 (1979; Zbl 0499.20035)], a ``canonical'' basis for \({\mathcal H}\) was introduced. Canonical bases arise in various settings and have proven to be a key tool in representation theory, particularly in zero or large characteristic. Recent developments suggest that in order to address problems for small positive characteristic, the solution may be to develop the notion of a \(p\)-canonical basis which agrees with the canonical basis when the prime \(p\) is large, but may differ when \(p\) is small. The goal of the work under review is to present a survey of results and computations of the \(p\)-canonical basis for \({\mathcal H}\).
The authors review \(S\)-graphs, Soergel calculus, and the diagrammatic (monoidal) category \({\mathbf H}\) of Soergel bi-modules (which depends on the characteristic \(p\) of the underlying field). The category \({\mathbf H}\) is a Krull-Schmidt category and the split Grothendieck group \([{\mathbf H}]\) of \({\mathbf H}\) may be considered as a \({\mathbb Z}[v,v^{-1}]\)-algebra. There is an isomorphism of \({\mathbb Z}[v,v^{-1}]\)-algebras from \([{\mathbf H}]\) to \({\mathcal H}\). For \(w \in W\), the associated \(p\)-canonical basis element in \({\mathcal H}\) is defined to be the image under this isomorphism of an indecomposable object in \({\mathbf H}\) associated to \(w\). The definition was motivated by work of \textit{B. Elias} and the second author [Ann. of Math. (2) 180, No. 3, 1089--1136 (2014; Zbl 1326.20005)] in which they verified Soergel's conjecture that, in characteristic zero, such an image gave the canonical (Kazhdan-Lusztig) basis element.
The authors discuss how \(p\)-canonical basis elements may be computed (inductively on the length of a Weyl group element) using local intersection forms, sometimes aided by computations in the nil Hecke ring associated to \({\mathcal H}\) (whose definition is also reviewed). The authors also review some basic properties of the \(p\)-canonical basis, its relationship to the canonical basis, and its connection to parity sheaves on the affine Grassmannian and tilting modules for the associated semi-simple algebraic group.
Lastly, the authors provide explicit computations of the \(p\)-canonical basis elements in terms of canonical basis elements for root systems of type \(B_2\), \(G_2\), \(\tilde{A}_1\), \(B_3\), \(C_3\), \(D_4\), and \(A_7\) (the \(p\)-canonical basis agrees with the canonical basis in type \(A_n\) with \(n < 7\)). Hecke algebra; canonical basis; Coxeter system; Weyl group; root datum; semi-simple algebraic group; Langlands dual group; tilting module; Grassmannian Jensen, Lars Thorge; Williamson, Geordie, The \(p\)-canonical basis for Hecke algebras.Categorification and higher representation theory, Contemp. Math. 683, 333-361, (2017), Amer. Math. Soc., Providence, RI Hecke algebras and their representations, Modular representations and characters, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Grassmannians, Schubert varieties, flag manifolds, Reflection and Coxeter groups (group-theoretic aspects) The \(p\)-canonical basis for Hecke algebras | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems See the preview in Zbl 0697.14036. weighted complete intersections; weighted projective space; towers of coverings; infinitely extendable vector bundles Complete intersections, Coverings in algebraic geometry, Projective techniques in algebraic geometry, Low codimension problems in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Babylonian tower theorems for coverings. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 degree ten plane curve with five \([3,3]\) points and an ordinary 4-ple point, such that the six singularities do not lie on a conic, is called a Godeaux branch curve. A degree ten plane curve with six \([3,3]\) points, which again do not lie on a conic, is called a Campedelli branch curve.
A Godeaux or Campedelli branch curve \(C\) can be broken down into its irreducible components \(C_{m_1}\), \(C_{m_2}\), \(\dots\), where \(m_i\) denotes the degree of \(C_{m_i}\) and \(m_1+m_2+\cdots=10\). The sequence of integers \((m_1,m_2,\dots)\), \(m_i \leq m_j\) if \(i \leq j\), that we can call a \textit{configuration} of \(C\), is associated with the divisor \(C=C_{m_1}+C_{m_2}+\cdots\) of the plane.
In the paper under review, the author determines all the possible configurations of the above branch curves that are invariant under the plane involution \((x,y,z) \mapsto (x,-y,z)\), starting from an idea put forward by the reviewer [Ann. Univ. Ferrara, Nuova Ser., Sez. VII, 43, 1--26 (1997; Zbl 0927.14017)]. To be more precise, in the case of Godeaux branch curves, there are nine configurations and they are associated with the following nine divisors:
(1) \(C_4+C_6\)
(2) \(C_2+D_2+C_6\)
(3) \(C_1+C_4+C_5\)
(4) \(C_1+D_1+C_8\)
(5) \(C_1+D_1+C_4+D_4\)
(6) \(C_2+C_8\)
(7) \(C_2+C_4+D_4\)
(8) \(C_3+D_3+C_4\)
(9) \(C_2+D_2+C_3+D_3\).
In the case of Campedelli branch curves, the configurations are given by
(1) \(C_2+C_4+D_4\)
(2) \(C_1+D_1+C_4+D_4\)
(3) \(C_2+D_2+C_6\)
(4) \(C_2+D_2+C_3+D_3\)
(5) \(C_2+D_2+E_2+C_4\).
In the case of Godeaux branch curves, the configurations given in (1),(6) and (9) are known, while the others yield new configurations. In the case of Campedelli branch curves, there are no new configurations, only new constructions of divisors.
Moreover, the torsion group of the numerical Godeaux surfaces, which are defined by the nine Godeaux branch curves, is computed, whereas only the subgroup of torsion divisors of order \(2\) is computed in the case of the numerical Campedelli surfaces defined by the five Campedelli branch curves.
Reviewer's remark. Using an idea drawn from the reviewer, the author tacitly assumes that the \([3,3]\) points lying on the line of symmetry \(y=0\) (of the involution \((x,y,z) \mapsto (x,-y,z)\)) have the tangent singular line that is ``orthogonal'' (in the affine sense) to \(y=0\). \([3,3]\) points; double planes; branch curves of double planes; numerical Godeaux and Campedelli surfaces C. Werner, Branch curves for Campedelli double planes, Rocky Mountain J. Math. 36 (2006), 2057--2073. Surfaces of general type, Coverings in algebraic geometry, Singularities of curves, local rings, Special algebraic curves and curves of low genus Branch curves for Campedelli double planes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \((K,\mathcal{O},k)\) be a \(p\)-modular system with \(k\) algebraically closed and \(\mathcal{O}\) unramified, and let \(\Lambda\) be an \(\mathcal{O} \)-order in a separable \(K\)-algebra. We call a \(\Lambda \)-lattice \(L\) rigid if \(\operatorname{Ext}^1_{\Lambda}(L,L)=0\), in analogy with the definition of rigid modules over a finite-dimensional algebra. By partitioning the \(\Lambda \)-lattices of a given dimension into ``varieties of lattices'', we show that there are only finitely many rigid \(\Lambda \)-lattices \(L\) of any given dimension. As a consequence we show that if the first Hochschild cohomology of \(\Lambda\) vanishes, then the Picard group and the outer automorphism group of \(\Lambda\) are finite. In particular, the Picard groups of blocks of finite groups defined over \(\mathcal{O}\) are always finite. Modular representations and characters, Group rings of finite groups and their modules (group-theoretic aspects), Divisors, linear systems, invertible sheaves On the geometry of lattices and finiteness of Picard groups | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This article may be regarded as the continuation of the author's earlier work on torsion points of algebraic curves [cf. Ann. Math., II. Ser. 121, 111-168 (1985; Zbl 0578.14038); Compos. Math. 58, 191-208 (1986; Zbl 0604.14019) or Duke Math. 54, 615-640 (1987; Zbl 0626.14022)].
In the present paper, the author studies Abelian coverings of given degree \( m\) of the projective line \({\mathbb{P}}^ 1\) defined over the cyclotomic field \({\mathbb{Q}}(\mu_ m)\), which are unbranched outside the points \(0, 1\quad and\quad \infty\) in \({\mathbb{P}}^ 1\). If \(f:\quad X\to {\mathbb{R}}^ 1\) is such a covering, with X being of genus greater \(than\quad 1,\) then X can be embedded into its Jacobian such that one of the points \(0, 1\quad or\quad \infty\) is mapped to the origin. Under this assumption, the following question is of interest: What can be said about those torsion points in the Jacobian which are contained in X? - The main results obtained in this paper are as follows:
Theorem A:
(i) If the genus of X is greater than one, then the exponent of the finite Abelian group of divisor classes on X represented by divisors of degree zero, and supported on the torsion packet T that contains \(f^{- 1}(\{0,1,\infty \})\), divides a power of 2m.
(ii) If, in addition, X is non-hyperelliptic, then this exponent divides the double of a power of m.
Theorem B:
The set of torsion points of Jac(X)\(\cap X\) is defined over an extension of \({\mathbb{Q}}\) unramified outside 6m.
The method of proof is essentially based on the techniques developed by the author himself in his previous work (loc. cit.), i.e., on the theory of de Rham F-crystals and its application to cuspidal torsion packets. abelian coverings of the projective line; ramification; Galois extension; torsion points of algebraic curves; torsion points in the Jacobian; de Rham F-crystals; cuspidal torsion packets R. Coleman , Torsion Points on Abelian etale coverings of P1-{0,1, \infty } , Transactions of the AMS 311, No. 1 (1989), 185-208. Jacobians, Prym varieties, Coverings in algebraic geometry, Coverings of curves, fundamental group, Ramification problems in algebraic geometry, Complex multiplication and abelian varieties, Algebraic number theory: local fields, Classification theory of Riemann surfaces Torsion points on Abelian étale coverings of \(P^ 1-\{0,1,\infty \}\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 0717.00009.]
This article gives P. Deligne's long awaited version with complete proof of Saavedra Rivano's main theorem on Tannakian categories [\textit{N. Saavedra Rivano}, ``Catégories tannakiennes'', Lect. Notes Math. 265 (1972; Zbl 0241.14008)]. The inaccuracies in Saavedra's formulation are remedied, thereby justifying all results of the cited book. From the outset the notion of a groupoid acting transitively on a scheme replaces Saavedra's more intrinsic way to describe a Tannakian category as a category of representations of a gerb. The equivalence of both methods is demonstrated. The paper consists of nine sections, the first one giving the terminology, the statement of the theorem, the introduction of the concept of (coaction of) a coalgebroid and, grosso modo, the strategy for the proof of the theorem. In sections 2 to 6 (46 pages!) one is gradually led to this proof. The theory of Tannakian categories, groupoids, comonads, the theorem of Barr-Beck and tensor products of categories are explained in some detail. - Section 7 gives a characterization of a Tannakian category over a field of characteristic zero: A tensor category \({\mathcal T}\) over a field k of characteristic zero is a Tannakian category over k (cf. below) iff every object has non-negative integer dimension. The rudiments of algebraic geometry (affine \({\mathcal T}\)-schemes, \({\mathcal T}\)-group schemes, \({\mathcal T}\)-vector bundles,...) are carried over to tensor categories \({\mathcal T}\). In section 8 the fundamental group \(\pi\) (\({\mathcal T})\) of a Tannakian category \({\mathcal T}\) over a field k is introduced: \(\pi\) (\({\mathcal T})\) is a \({\mathcal T}\)-group, and for a fibre functor \(\omega\) of \({\mathcal T}\) on a k-scheme S (cf. below) one has \(\omega(\pi({\mathcal T}))\overset \sim \rightarrow \underline{Aut}_ S^{\otimes}(\omega)\). Furthermore, it is proved (actually something more general) that an object X of \({\mathcal T}\) is a sum of copies of the unit object 1 iff \(\pi({\mathcal T})\) acts trivially on X. The last section gives an application of the theory of Tannakian categories to the theory of Picard-Vessiot. For a differential field \((K,\partial)\) with algebraically closed field of constants \(K_ 0=\{x| \partial x=0\}\) of characteristic zero and an n-th-order ordinary differential equation with coefficients in K there exists an extension \((E,\partial)\) of \((K,\partial)\) with the same field of constants \(K_ 0\) such that the differential equation admits n solutions, linearly independent over \(K_ 0\), in \((E,\partial)\). The whole exposition is virtually self-contained.
Let k be a commutative field. By a \textit{tensor category} \({\mathcal T}\) over k is meant a rigid abelian k-linear \(\otimes\)-category ACU (i.e. subject to compatible associativity and commutativity constraints and with 1) satisfying \(X\otimes 1\overset \sim \rightarrow X\) and \(1\otimes X\overset \sim \rightarrow X)\) with \(k\overset \sim \rightarrow End(1)\). In particular, 1 is a simple object and \(\otimes: {\mathcal T}\times {\mathcal T}\to {\mathcal T}\) is a k-bilinear bifunctor, exact in each variable. `Rigid' means that \({\mathcal T}\) has internal \(\underline{Hom}\)'s or, more precisely, for every \(X\in {\mathcal O}b({\mathcal T})\) there exists a dual \(X^{\vee}\) and morphisms \(ev: X\otimes X^{\vee}\to 1\) and \(\delta: 1\to X^{\vee}\otimes X\) such that
\[
X\to^{X\otimes \delta}X\otimes X^{\vee}\otimes X\to^{ev\otimes X}X
\]
and
\[
X^{\vee}\to^{\delta \otimes X^{\vee}}X^{\vee}\otimes X\otimes X^{\vee}\to^{X^{\vee}\otimes ev}X^{\vee}
\]
are the identity. For a k-scheme S, an exact k-linear \(\otimes\)-functor \(\omega: {\mathcal T}\to {\mathcal Q}coh_ S\), where \({\mathcal Q}coh_ S\) is the category of quasi- coherent sheaves on S, is called a \textit{fibre functor} on S. In fact, \(\omega\) takes values in the category of locally free sheaves of finite rank on S, and it commutes with taking duals. If S is non-empty, \(\omega\) is also faithful. A fibre functor \(\omega\) on S extends to a \(\otimes\)- functor, also written \(\omega\), of the category of Ind-objects of \({\mathcal T}\), \(\omega: Ind{\mathcal T}\to {\mathcal Q}coh_ S\). For \(u: T\to S\), \(\omega_ T=u^*\omega\) is a fibre functor on T. For two fibre functors \(\omega_ 1\) and \(\omega_ 2\) on S one writes \(\underline{Isom}_ S^{\otimes}(\omega_ 1,\omega_ 2)\) for the functor that, with every \(u: T\to S\), associates the set of isomorphisms of fibre functors \(u^*\omega_ 1\overset \sim \rightarrow u^*\omega_ 2\). It is representable by an affine scheme over S, also denoted by \(\underline{Isom}_ S^{\otimes}(\omega_ 1,\omega_ 2)\). For fibre functors \(\omega_ i\) on schemes \(S_ i\), \(i=1,2\), one writes \(\underline{Isom}_ k^{\otimes}(\omega_ 2,\omega_ 1)\) for \(\underline{Isom}^{\otimes}_{S_ 1\times S_ 2}(pr^*_ 2\omega_ 2,pr^*_ 1\omega_ 1)\). For a fibre functor \(\omega\) on S, one writes \(\underline{Aut}_ S^{\otimes}(\omega)=\underline{Isom}_ S^{\otimes}(\omega,\omega)\) and \(\underline{Aut}_ k^{\otimes}(\omega)=\underline{Isom}_ k^{\otimes}(\omega,\omega)\). We use the same notation for the affine schemes representing these functors. If the tensor category \({\mathcal T}\) over k admits a fibre functor on a nonempty scheme S, \({\mathcal T}\) is called a \textit{Tannakian category} over k. A Tannakian category satisfies
(*): its objects have finite length and the Hom's are finite dimensional k-vector spaces.
A k-\textit{groupoid} acting on the k-scheme S is a k-scheme G equipped with two morphisms (`target' and `source') b,s: \(G\to S\) and a composition law \(\circ: G\times G\to G\) which is a morphism of schemes over \(S\times S\) and such that, for any k-scheme T, these data define a category (`groupoid') with objects \(S(T)=Hom(T,S)\) and arrows \(G(T)=Hom(T,G)\) such that every arrow is invertible. G is said to act transitively on S (for the fpqc-topology) if there exists T, faithfully flat and quasi-compact over \(S\times S\), with \(Hom_{S\times S}(T,G)\neq \emptyset\). A \textit{representation} of G is a quasi-coherent sheaf \({\mathcal V}\) on S with G-action, i.e. a morphism \(\rho\) (g): \({\mathcal V}_{s(g)}\to {\mathcal V}_{b(g)}\), \(g\in G(T)\), between the inverse images of \({\mathcal V}\) under s(g),b(g): \(T\to S\) for any k-scheme T. One assumes the \(\rho\) (g) to be compatible with base change and to satisfy some natural conditions to make them isomorphisms. For a non-empty k-scheme S with transitive G- action one writes Rep(S:G) for the tensor category of locally free sheaves of finite rank on S with G-action. For u: \(T\to S\) one has an induced groupoid \(G_ T\) on T and an equivalence of categories \(Rep(S:G)\overset \sim \rightarrow Rep(T:G_ T)\). For a tensor category \({\mathcal T}\) with fibre functor \(\omega\) on S, the scheme \(\underline{Aut}_ k^{\otimes}(\omega)\) is a k-groupoid acting on S. The main theorem of the article can now be stated:
Let \({\mathcal T}\) be a tensor category over k with fibre functor \(\omega\) on the nonempty k-scheme S. Then:
(i) the groupoid \(\underline{Aut}_ k^{\otimes}(\omega)\) is faithfully flat on \(S\times S;\)
(ii) \(\omega\) induces an equivalence \({\mathcal T}\overset \sim \rightarrow Rep(S:\underline{Aut}_ k^{\otimes}(\omega)).\)
Conversely, let G be a k-groupoid acting on \(S\neq \emptyset\), affine and faithfully flat on \(S\times S\), and let \(\omega\) be the `forgetful' fibre functor of Rep(S:G) on S, then:
(iii) \(G\overset \sim \rightarrow \underline{Aut}_ k^{\otimes}(\omega). (break?) \)
It suffices to consider the case of affine \(S=Spec(B)\) for a k-algebra B. Then \(\omega\) takes values in \(\Pr oj_ B\), the category of projective right B-modules of finite type. Now the proof can be divided into several steps:
A. Preliminaries. 1. Let R be a commutative ring, and let \(B_ 1\) and \(B_ 2\) be two R-algebras. For a (small) category \({\mathcal C}\) with functors \(\omega_ i: {\mathcal C}\to \Pr oj_{B_ i}\), \(i=1,2\), one has the coend \(L_ R(\omega_ 1,\omega_ 2)\). It is a \((B_ 1,B_ 2)\)- bimodule, whose induced R-module structures coincide, and which is equipped with a morphism of bimodules \(\omega_ 1(X)^{\vee}\otimes_ R\omega_ 2(X)\to L_ R(\omega_ 1,\omega_ 2)\), i.e. a morphism of \(B_ 2\)-modules \(\omega_ 2(X)\to \omega_ 1(X)\otimes_{B_ 1}L_ R(\omega_ 1,\omega_ 2)\), functorial in X, for all \(X\in {\mathcal O}b({\mathcal C})\). In particular, for \(B_ 1=B_ 2=B\) and \(\omega_ 1=\omega_ 2=\omega\), \(L_ R(\omega)=L_ R(\omega,\omega)\) becomes a R-coalgebroid acting on B. One has a coaction of \(L_ R(\omega)\) on the \(\omega\) (X), \(\omega (X)\mapsto \omega (X)\otimes_ BL_ R(\omega)\), functorial in X. \(L_ R(\omega)\) is called the \textit{coalgebroid of R- endomorphisms} of \(\omega\).
2. If \({\mathcal C}\) is a \(\otimes\)-category ACU and \(\omega_ 1,\omega_ 2: {\mathcal C}\to \Pr oj_ R\) are \(\otimes\)-functors, then one has a product \(L_ R(\omega_ 1,\omega_ 2)\otimes_ RL_ R(\omega_ 1,\omega_ 2)\to L_ R(\omega_ 1,\omega_ 2)\) and \(L_ R(\omega_ 1,\omega_ 2)\) becomes a commutative R-algebra. The functor \(\underline{Hom}_ S^{\otimes}(\omega_ 2,\omega_ 1)\), \(S=Spec(R)\), is represented by the affine scheme \(Spec(L_ R(\omega_ 1,\omega_ 2))\). For a \(\otimes\)-category ACU \({\mathcal C}\) over k, commutative k- algebras \(B_ 1\) and \(B_ 2\), and \(\otimes\)-functors \(\omega_ i: {\mathcal C}\to \Pr oj_{B_ i}\), \(i=1,2\), extension of scalars, written \(\omega_ 1\mapsto \omega_ 1\otimes 1 (\omega_ 2\mapsto 1\otimes \omega_ 2)\), turns the \(\omega_ i\) into \(\otimes\)-functors with values in \(\Pr oj_{B_ 1\otimes_ kB_ 2}\). One has: \(L_ k(\omega_ 1,\omega_ 2)=L_{B_ 1\otimes_ kB_ 2}(\omega_ 1\otimes 1,1\otimes \omega_ 2)\) and \(Spec(L_ k(\omega_ 1,\omega_ 2))\) represents the functor \(\underline{Hom}_ k^{\otimes}(\omega_ 2,\omega_ 1)=\underline{Isom}_ k^{\otimes}(\omega_ 2,\omega_ 1)\). Thus \(\underline{Aut}_ k^{\otimes}(\omega)\) is represented by \(Spec(L_ k(\omega))\). In particular, for \({\mathcal C}={\mathcal T}\) a tensor category over k, with fibre functor \(\omega\) on \(S=Spec(B)\), B a k- algebra, the action of \(\underline{Aut}_ k^{\otimes}(\omega)\) on the \(\omega(X)\) is given by the coaction of \(L_ k(\omega)\), \(\omega(X)\mapsto \omega(X) \otimes_ BL_ k(\omega)\), and the composition law of the groupoid \(\underline{Aut}_ k^{\otimes}(\omega)\) is defined by the comultiplication of the coalgebroid \(L_ k(\omega).\)
3. Let \({\mathcal A}\) be a k-linear abelian category satisfying (*) and let \(\omega: {\mathcal A}\to Proj_ B\), B a k-algebra, be a k-linear exact faithful functor. \({\mathcal A}\) is the filtered union of full abelian subcategories \(<X>\), \(X\in {\mathcal O}b({\mathcal A})\), where the objects of \(<X>\) are subquotients of the \(X^ n\), and \(L_ k(\omega)\) is the inductive limit of the \(L_ k(\omega | <X>)\). The following result is due to \textit{O. Gabber}: For \(X\in {\mathcal O}b({\mathcal A})\), \(<X>\) admits a projective generator, say P. The functor \(Y\mapsto Hom(P,Y)\) gives an equivalence of \(<X>\) with the category of right modules of finite type over the k-algebra \(A=End(P)\). Write \({}_ AM_ B=\omega(P)\). Then \({}_ AM_ B\) is an \((A,B)\)-bimodule, projective of finite type over B and faithfully flat over A. One proves \(L_ k(\omega | <X>)\overset \sim \rightarrow_ BM_ A^{\vee}\otimes_ A{}_ AM_ B\), and the theorem of Barr-Beck says that \(\omega\) induces an equivalence between \(<X>\) and the category of right B-modules of finite type with a coaction of \(L_ k(\omega | <X>)\). A limit argument leads to the result: \(\omega\) induces an equivalence between \({\mathcal A}\) and the category of right B-modules of finite type with coaction of \(L_ k(\omega).\)
4. One introduces the notion of a \textit{tensor product} \(\otimes {\mathcal A}_ i \) of a finite family of k-linear abelian categories \(\{{\mathcal A}_ i\}_{i\in I}\). By definition, a k-multilinear functor, right exact in each variable, \(\otimes: \prod_{i}{\mathcal A}_ i \to {\mathcal A}\), where \({\mathcal A}\) is a k-linear abelian category, makes \({\mathcal A}\) into a tensor product over k of the \({\mathcal A}_ i\) if, for any k-linear abelian category \({\mathcal C}\), the category of right exact functors from \({\mathcal A}\) to \({\mathcal C}\) is equivalent to the category of multilinear functors, right exact in each variable, from \(\prod_{i}{\mathcal A}_ i \) to \({\mathcal C}\). As a matter of fact, if such a product exists it is unique (up to isomorphism). If the \({\mathcal A}_ i\) satisfy (*), then the tensor product exists and has nice properties. For a finite family \(\{{\mathcal T}_ i\}_{i\in I}\) of Tannakian categories over k the tensor product \(\otimes {\mathcal T}_ i \) over k is a Tannakian category over k. It satisfies (*).
B. A.2. and A.3. above now imply part (ii) of the theorem.
C. For part (i) one proceeds as follows: for any (small) category \({\mathcal C}\), tensor category \({\mathcal T}\) over k and functors \(T_ 1,T_ 2: {\mathcal C}\to {\mathcal T}\), one constructs an Ind-object \(\Lambda_{{\mathcal T}}(T_ 1,T_ 2)\) of \({\mathcal T}\) with a morphism \(T_ 1(X)^{\vee}\otimes T_ 2(X)\to \Lambda_{{\mathcal T}}(T_ 1,T_ 2)\) for all \(X\in {\mathcal O}b({\mathcal C})\), in analogy with the coend construction of A.1. In case \({\mathcal T}={\mathcal T}_ 1\otimes {\mathcal T}_ 2\) is the tensor product (cf. A.4. above) over k of two tensor categories \({\mathcal T}_ 1\) and \({\mathcal T}_ 2\) over k, and \(T_ i: {\mathcal C}\to {\mathcal T}_ i(i=1,2)\) are functors, one writes \(\Lambda_ k(T_ 1,T_ 2)=\Lambda_{{\mathcal T}}(inj_ 1T_ 1,inj_ 2T_ 2)\), where the \(inj_ i\) are the injections of the \({\mathcal T}_ i\) in \({\mathcal T}\), i.e. \(inj_ 1: X\mapsto X\otimes 1\) and \(inj_ 2: X\mapsto 1\otimes X\). If the \({\mathcal T}_ i\) admit fibre functors \(\omega_ i\) on \(Spec(B_ i)\), \(B_ i\) commutative k-algebras, then the functor \((X,Y)\mapsto \omega_ 1(X)\otimes_ k\omega_ 2(Y)\) factors over a fibre functor \(\omega\) of \({\mathcal T}={\mathcal T}_ 1\otimes {\mathcal T}_ 2\) on \(Spec(B_ 1)\times Spec(B_ 2)\) and one has \(\omega \Lambda_ k(T_ 1,T_ 2)=L_ k(\omega_ 1T_ 1,\omega_ 2T_ 2)\). Apply this to the situation with \({\mathcal C}={\mathcal T}_ 1={\mathcal T}_ 2={\mathcal T}\) and \(T_ i\) the identity \(Id_{{\mathcal T}}\). This gives an Ind-object \(\Lambda =\Lambda_ k(Id_{{\mathcal T}},Id_{{\mathcal T}})\) of \({\mathcal T}\otimes {\mathcal T}\). One proves that \(\Lambda\neq 0\). A general result on Tannakian categories says that, if X is a nonzero Ind-object of a Tannakian category over k with fibre functor \(\omega\) on the k-scheme S, \(\omega\) (X) is faithfully flat on S. This implies (i) of the theorem.
D. Finally, for part (iii), let G be a groupoid acting transitively on S and affine on \(S\times S\), say \(G=Spec(L)\), where L is a \(B\otimes_ kB\)-module. The composition law of G makes L into a k-coalgebroid acting on B. It is sufficient to prove the isomorphism \(G\overset \sim \rightarrow \underline{Aut}_ k^{\otimes}(\omega)\) for one fibre: For \(T\to S\), \(T\neq 0\), \(Rep(S:G)\overset \sim \rightarrow Rep(T:G_ T)\) with forgetful functor \(\omega_ T\) on T and induced groupoid \(\underline{Aut}_ k^{\otimes}(\omega_ T)\) on T. Taking for T a point of S one may suppose T to be the spectrum of a field and the result follows from the fact that in this situation one can show that there is an isomorhism \(L_ k(\omega_ T)\overset \sim \rightarrow L\). This concludes the proof of the theorem. Tannakian category; gerb; theory of Picard-Vessiot; tensor category; groupoid; coalgebroid Deligne, P., Catégories tannakiennes, (The Grothendieck Festschrift, vol. II. The Grothendieck Festschrift, vol. II, Prog. Math., vol. 87, (1990), Birkhäuser Boston: Birkhäuser Boston Boston, MA, USA), 111-195 (Co)homology theory in algebraic geometry, Group schemes, Groupoids, semigroupoids, semigroups, groups (viewed as categories), Coverings in algebraic geometry, Groupoids (i.e. small categories in which all morphisms are isomorphisms), Commutative rings of differential operators and their modules Catégories tannakiennes. (Tannaka categories) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The book offers an extensive study on the convoluted history of the research of algebraic surfaces, focusing for the first time on one of its characterizing curves: the branch curve. Starting with separate beginnings during the 19th century with descriptive geometry as well as knot theory, the book focuses on the 20th century, covering the rise of the Italian school of algebraic geometry between the 1900s till the 1930s (with Federigo Enriques, Oscar Zariski and Beniamino Segre, among others), the decline of its classical approach during the 1940s and the 1950s (with Oscar Chisini and his students), and the emergence of new approaches with Boris Moishezon's program of braid monodromy factorization.
By focusing on how the research on one specific curve changed during the 20th century, the author provides insights concerning the dynamics of epistemic objects and configurations of mathematical research. It is in this sense that the book offers to take the branch curve as a cross-section through the history of algebraic geometry of the 20th century, considering this curve as an intersection of several research approaches and methods.
Researchers in the history of science and of mathematics as well as mathematicians will certainly find this book interesting and appealing, contributing to the growing research on the history of algebraic geometry and its changing images. ramified surfaces; branch curves; algebraic geometry in the 20th century Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, History of algebraic geometry, History of mathematics in the 20th century, Coverings in algebraic geometry Ramified surfaces. On branch curves and algebraic geometry in the 20th century | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 lecture of the proceedings the author gives a brief introduction to Galois covers of curves with formal and rigid patching. After reviewing the GAGA principle and Grothendieck's existence theorem, one proves a ``Van Kampen'' patching result in formal topology. These facts are then used to prove the existence of covers of a curve defined over a discrete valuation ring \(R\) and its fraction field \(K\). As an application, one of the main result proved, following \textit{D. Harbater} [Am. J. Math. 115, 487-508 (1993; Zbl 0790.14027)], is that every finite group \(G\) is the Galois group of a geometrically connected ramified cover of the projective line over \(K\). rigid geometry; Galois covers of curves; GAGA principle; formal geometry Rachel J. Pries, Construction of covers with formal and rigid geometry, Courbes semi-stables et groupe fondamental en géométrie algébrique (Luminy, 1998) Progr. Math., vol. 187, Birkhäuser, Basel, 2000, pp. 157 -- 167. Rigid analytic geometry, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Coverings in algebraic geometry, Coverings of curves, fundamental group, Inverse Galois theory Construction of covers with formal and rigid geometry | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems \textit{P. Lochak} and \textit{L. Schneps} [Invent. Math. 127, No. 3, 571--600 (1997; Zbl 0883.20016)] introduced the ``harmonic parameter \(g\)''
of the Grothendieck-Teichmüller group \(\widehat{\text{GT}}\). We closely study the behavior of \(g\) on the absolute Galois group \(G_\mathbb{Q}\) using a family of lemniscate elliptic curves. We obtain a relationship of the adelic beta function and the harmonic parameter specialized in
the matrix group \(\text{SL}_2(\hat\mathbb{Z})\).
[For part I, see \textit{H. Nakamura} and \textit{H. Tsunogai}, Forum Math. 15, 877--892 (2003; Zbl 1054.14038).] Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Coverings of curves, fundamental group, Coverings in algebraic geometry, Limits, profinite groups, Braid groups; Artin groups Harmonic and equianharmonic equations in the Grothendieck-Teichmüller group. II | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper the author studies some n-sheeted coverings of \({\mathbb{P}}^ 2\) branched along \(S_ 1\cup S_ 2\), where \(S_ 1\) and \(S_ 2\) are non singular curves with normal crossings, and computes their Chern classes. In particular, for \(n=3\), he proves that such a covering space X is a normal surface whose singularities are rational (double or triple) points and that the geometric genus of a non singular model \(\tilde X\) of X can be expressed in terms of the genus of \(S_ i\), \(i=1,2\). The proof is essentially based on torus embeddings. n-sheeted coverings of projective 2-space; Chern classes Yamamoto, S.: Covering spaces of P2 branched along two non-singular curves with normal crossings. Comment. math. Univ. st. Paul 33, 163-189 (1984) Coverings in algebraic geometry, Ramification problems in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Characteristic classes and numbers in differential topology Covering spaces of \(P^ 2\) branched along two non-singular curves with normal crossings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(X\) a compact Kähler manifold, the author studies relationships between \(\pi_ 1 (X)\), the positivity of \(\Omega^ 1_ X\) and generic compact subvarieties of the universal cover \(\widetilde X\) of \(X\), first considered by \textit{M. Gromov} [J. Differ. Geom. 3, No. 1, 263-292 (1991; Zbl 0719.53042)]. One gives in the first section a new proof for the fact that a K3 surface is simply connected and, in the second section, it is proved that \(\pi_ 1 (X)\) is finite if \(X\) is simple. In the third section it is shown the existence of a \(\widetilde \Gamma\)-reduction \(\widetilde \gamma : \widetilde X \to \widetilde Z\) for any connected compact Kähler manifold (analogous to the Stein reduction), where \(\widetilde Z\) parametrises essentially the maximal connected compact complex subvarieties of \(\widetilde X\). The \(\Gamma\)-reduction \(\gamma : X \to Z = \widetilde Z/ \Gamma\) \((\Gamma : = \pi_ 1 (X))\) obtained by taking the quotients, gives a new bimeromorphic invariant \(gd(X) : = \dim Z\), named \(\Gamma\)-dimension of \(X\). The main result of the fourth section is the following: If \(\chi ({\mathcal O}_ X) \neq 0\) and if \(gd(X) = n \leq 3\), then \(X\) is of general type (i.e. \(k(X) = n = gd(X))\). In the fifth section a geometric criterion for the finiteness of the fundamental group \(\pi_ 1 (X)\) is given. As a corollary, a simple proof is obtained for the fact that a rationally connected compact Kähler manifold is simply connected. positivity of \(\Omega^ 1_ X\); first fundamental group; compact Kähler manifold; universal cover Campana, F., Remarques sur le revêtement universel des variétés kählériennes compactes, Bull. Soc. Math. France, 122, 2, 255-284, (1994) Compact Kähler manifolds: generalizations, classification, Transcendental methods of algebraic geometry (complex-analytic aspects), Coverings in algebraic geometry, Global differential geometry of Hermitian and Kählerian manifolds Remarks on the universal covering of compact Kähler 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 A normal projective surface \(\bar V\) is a log (=logarithmic) del Pezzo surface if
(i) \(-K_{\bar V}\) is ample, and
(ii) every singularity of \(\bar V\) is a quotient singularity.
We consider three problems:
(1) Construct all these \(\bar V.\)
(2) Given \(\bar V,\) when \(\bar V\cong {\mathbb{P}}^ 2/G\) with a finite subgroup G of PGL(2,C)?
(3) The smooth part \(V^ 0=\bar V-Sing(\bar V)\supseteq {\mathbb{C}}^ 2\) if and only if the fundamental group \(\pi_ 1(V^ 0)=(1)?\)
The problems were completely solved by \textit{M. Miyanishi} and the author [J. Algebra 118, No.1, 63-84 (1988; Zbl 0664.14019)] when the Picard number \(\rho(\bar V)=1\) and \(\bar V\) is Gorenstein, i.e., every singularity of \(\bar V\) is a rational double singular point. Let \(U^ 0\) be the universal covering of \(V^ 0\) and \(\bar U\) the normalization of \(\bar V\) in \(C(U^ 0)\). Applying the author's classification theory [in Osaka J. Math. 25, 461-497 (1988)] we obtain the following theorem:
Let \(\bar V(\neq {\mathbb{P}}^ 2)\) be a log del Pezzo surface with \(\rho(\bar V)=1\) and with exactly one rational triple and several double singular points. Then we have the following:
(1) There are altogether 97 possible combinations of singularities of V (see appendix).
(2) Suppose \(\pi_ 1(V^ 0)\neq (1)\). Then \(\bar V\cong {\mathbb{P}}^ 2/G\) if and only if \(\rho(\bar U)=1.\)
(3) If \(\pi_ 1(V^ 0)=(1)\) then \(V^ 0\) contains \({\mathbb{C}}\times ({\mathbb{C}}-0)\) as a Zariski open set. Picard number; log del Pezzo surface; singularities D.-Q. Zhang, ''Logarithmic del Pezzo surfaces one with rational double and triple singular points,'' Tohoku Math. J. (2) 41(3), 399--452 (1989). Families, moduli, classification: algebraic theory, Special surfaces, Singularities in algebraic geometry, Coverings in algebraic geometry Logarithmic del Pezzo surfaces with rational double and triple singular points | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For part IV of this paper see \textit{G. Casnati}, Forum Math. 13, 21-36 (2001; Zbl 0971.14014)]. The author continues his analysis of coverings of low degree of algebraic varieties. He approaches the problem ``from the bottom'', namely, given a variety \(Y\), he tries to encode finite flat coverings \(f\colon X\to Y\) of fixed degree via a set of geometrical ``data'' (i.e., vector bundles, global sections,\dots) on \(Y\). This problem has a complete answer, at least for Gorenstein covers, if the degree of the cover is at most 5 [cf. \textit{G. Casnati} and \textit{T. Ekedahl}, J. Algebr. Geom. 5, 439-460 (1996; Zbl 0866.14009), \textit{G. Casnati}, J. Algebr. Geom. 5, 461-477 (1996; Zbl 0921.14006)]. In part IV of the paper (loc. cit.), the author has given a construction of a class of degree 6 covers. Here he describes constructions giving covers of degree 8 and 9.
The common feature of these examples and of those described in the previous papers is the following:
Let \(d\) be the degree of the cover; there are a \({\mathbb{P}}^{n+d-2}\)-bundle \(\tilde{\mathbb{P}}\) on \(Y\), a subbundle \({\mathbb{P}}\subset \tilde{\mathbb{P}}\) of rank \(d-2\) and a variety \(V\subset {\mathbb{P}}\) such that:
(a) \(V\) has relative dimension \(n\) and degree \(d\);
(b) for every \(y\in Y\) the fibre \(V_y\) of \(V\) over \(y\) is a Del Pezzo \(n\)-fold;
(c) \({\mathbb{P}}\cap V_y\) has dimension 0 for every \(y\in Y\).
Then one sets \(X:=V\cap {\mathbb{P}}\) and the natural map \(X\to Y\) is a cover of degree \(d\). In the case of degree 8, for fixed \(y\in Y\) one can identify \(\tilde{\mathbb{P}}_y\) with the projectivization of the space of symmetric matrices of order 4 and \(V_y\) with the set of matrices of rank 1. In the case of degree 9, \(\tilde{\mathbb{P}}_y\) is the projectivized space of trilinear symmetric forms on a \(3\)-dimensional vector space and \(V_y\) is a suitably defined degeneracy locus for these maps.
The paper contains also some Bertini type statements, showing that, given a curve or a surface \(Y\), the proposed constructions of covers of degree 8 and 9 actually give smooth covers of \(Y\). finite cover; coverings of low degree Casnati G. (2003). Cover of algebraic varieties V Examples of covers of degree 8 and 9 as catalecticant loci. J. Pure Appl. Algebra 182:17--32 Coverings in algebraic geometry, Ramification problems in algebraic geometry, Coverings of curves, fundamental group, Special surfaces, Algebraic theory of abelian varieties Covers of algebraic varieties. V: Examples of covers of degree 8 and 9 as catalecticant loci | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This is the first of the famous series of ``Séminaire de géométrie algébrique'' of Alexander Grothendieck and his collaborators, started in 1960-61, in which Grothendieck constructs (and studies the main properties of) the algebraic fundamental group \(\pi^{\text{alg}}_1(X)\) of a scheme \(X\). If \(X\) is a complex algebraic variety then the profinite completion of the topological fundamental group \(\pi_1(X)\) coincides with \(\pi^{\text{alg}}_1(X)\). This fundamental construction is extremely important when for example \(X\) is an algebraic scheme over a finite field extension \(K\) of \(\mathbb Q\) because the knowledge of \(\pi^{\text{alg}}_1(X)\) contains already a lot of information about \(X\) (in some cases, enough to reconstruct \(X\) itself). To quote Serre (from his talk in the Bourbaki Seminar of the fall of 1991), this seminar was the first great success of the theory of schemes.
Due to these facts it certainly deserved to be republished by the Société Mathématique de France as a volume of ``Documents Mathématiques''. The present volume is a (slightly) corrected and updated version of the previous edition [Lect. Notes Math. 224 (1971; Zbl 0234.14002)]. Some updated remarks have been added by M. Raynaud, which are bounded by brackets [ ] and indicated by the symbol (MR). étale covering; algebraic fundamental group; descent theory; smooth morphisms Raynaud, M., Catégories fibrée et descente, Revêtements étales et groupe fondamental (SGA 1), 3, (2003) Research exposition (monographs, survey articles) pertaining to algebraic geometry, Local structure of morphisms in algebraic geometry: étale, flat, etc., Coverings in algebraic geometry, Étale and other Grothendieck topologies and (co)homologies, Homotopy theory and fundamental groups in algebraic geometry Seminar on algebraic geometry at Bois Marie 1960-61. Étale coverings and fundamental group (SGA 1). A seminar directed by Alexander Grothendieck. Enlarged by two reports of M. Raynaud | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(\Lambda\) be a finite dimensional algebra over an algebraically closed field \(k\), given by a pair \((Q,I)\), where \(Q\) is a finite oriented graph without double arrows and \(I\) is an admissible ideal of the path algebra \(kQ\) of \(Q\), \(\pi\) be its fundamental group and \({\tilde \Lambda}\) its universal Galois covering. Then each group homomorphism \(\psi\colon \pi\to G\) produces a Galois covering \(p_{\psi}\colon \Lambda_{\psi}\to \Lambda\) with group \(G\), moreover \(\psi\) induces a certain action of the group \(X\) of characters of \(G\) on \(\Lambda\). The author shows that if \(G\) is a finite abelian group, whose order is prime with the characteristic of \(k\), then the Galois covering \(p_{\psi}\) is isomorphic to a certain covering \(p'\colon P\to \Lambda\), where \(P\) is a category of indecomposable projectives over a skew group algebra \(\Lambda X\) of \(X\) over \(\Lambda\), and consequently \(\Lambda_{\psi}\) and \(\Lambda X\) are Morita equivalent. Another proof of the last fact is given by \textit{I. Reiten} and \textit{C. Riedtmann} [J. Algebra 92, 224--282 (1985; Zbl 0549.16017)]. finite dimensional algebra; path algebra; fundamental group; universal Galois covering; characters; finite abelian group; category of indecomposable projectives; skew group algebra De La Peńa J.A., J. Algebra 102 (1) pp 129-- (1986) Representation theory of associative rings and algebras, Finite rings and finite-dimensional associative algebras, Group rings of finite groups and their modules (group-theoretic aspects), Coverings in algebraic geometry, Graded rings and modules (associative rings and algebras) On the Abelian Galois coverings of an algebra | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(G\), we continue our investigation of those finite-dimensional \(kG\)-modules that are of constant Jordan type. We introduce a Quillen exact category structure \(\mathcal C(kG)\) on these modules and investigate \(K_0(\mathcal C(kG))\). We study which Jordan types can be realized as the Jordan types of (virtual) modules of constant Jordan type. We also briefly consider thickenings of \(\mathcal C(kG)\) inside the triangulated category \(\text{stmod}(kG)\).
Together with \textit{J. Pevtsova}, the authors introduced [in J. Reine Angew. Math. 614, 191-234 (2008; Zbl 1144.20025)] an intriguing class of modules for a finite group \(G\) (or, more generally, for an arbitrary finite group scheme), the \(kG\)-modules of constant Jordan type. This class includes projective modules and endotrivial modules. It is closed under taking direct sums, direct summands, \(k\)-linear duals, and tensor products. We have several methods for constructing modules of constant Jordan type, typically using cohomological techniques. In the very special case that \(G=\mathbb Z/p\mathbb Z\times\mathbb Z/p\mathbb Z\), the authors and \textit{A. Suslin} have recently introduced several interesting constructions that associate modules of constant Jordan type to an arbitrary finite-dimensional \(kG\)-module and have identified cyclic \(kG\)-modules of constant Jordan type [Comment. Math. Helv. 86, No. 3, 609-657 (2011; Zbl 1229.20039)].
What strikes us as remarkable is how challenging the problem of classifying modules of constant Jordan type is even for relatively simple finite group schemes. In this paper we address two other aspects of the theory that also present formidable challenges. The first is the realization problem of determining which Jordan types can actually occur for modules of constant Jordan type. The second question concerns stratification of the entire module category by modules of constant Jordan type.
To consider realization, we give the class of \(kG\)-modules of constant Jordan type the structure of a Quillen exact category \(\mathcal C(kG)\) using ``locally split short exact sequences.'' This structure suggests itself naturally once \(kG\)-modules are treated from the point of view of \(\pi\)-points as in [\textit{E. M. Friedlander, J. Pevtsova}, Duke Math. J. 139, No. 2, 317-368 (2007; Zbl 1128.20031)], a point of view necessary to even define modules of constant Jordan type. With respect to this exact category structure, the Grothendieck group \(K_0(\mathcal C(kG))\) arises as a natural invariant. There are natural Jordan type functions JType, \(\overline{\text{JType}}\) defined on \(K_0(\mathcal C(kG))\) that are useful for formulating questions of realizability of (virtual) modules of constant Jordan type. The reader will find several results concerning the surjectivity of these functions.
A seemingly very difficult goal is the classification of \(kG\)-modules of constant Jordan type, or at least the determination of \(K_0(\mathcal C(kG))\). In this paper, we provide a calculation of \(K_0(\mathcal C(kG))\) for two very simple examples: the Klein four group and the first infinitesimal kernel of \(\mathrm{SL}_2\).
The category \(\mathcal C(kG)\) possesses many closure properties. However, the complexity of this category is reflected in the observation that an extension of modules of constant Jordan type need not be of constant Jordan type. We conclude this paper by a brief consideration of a stratification of the stable module category \(\text{stmod}(kG)\) by ``thickenings'' of \(\mathcal C(kG)\). finite group schemes; modules of constant Jordan type; exact categories; group algebras; Grothendieck groups; categories of finitely generated projective modules; thickenings; stable module categories; Frobenius kernels Jon F. Carlson and Eric M. Friedlander, Exact category of modules of constant Jordan type, Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. I, Progr. Math., vol. 269, Birkhäuser Boston, Inc., Boston, MA, 2009, pp. 267 -- 290. Representation theory for linear algebraic groups, Group schemes, Modular representations and characters, Cohomology theory for linear algebraic groups, Group rings of finite groups and their modules (group-theoretic aspects), Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Cohomology of groups, Representations of associative Artinian rings, Module categories in associative algebras, \(K_0\) of group rings and orders Exact category of modules of constant Jordan type. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(E\) denote an elementary Abelian \(p\)-group of rank \(n\), and let \(k\) denote an algebraically closed field of characteristic \(p>0\). The representation theory of elementary Abelian \(p\)-groups has played a key role in understanding the modular representations of more general finite groups, particularly through the use of cohomological support varieties. In the case of elementary Abelian groups, the Avrunin-Scott Theorem identifies cohomological support varieties with rank varieties which are defined via cyclic shifted subgroups of \(kE\). A cyclic shifted subgroup of \(kE\) is actually a subalgebra of \(kE\) and isomorphic to \(k[t]/(t^p)\). Subalgebras of the form \(k[t]/(t^p)\) have played a key role over the past thirty years in the development of the modular representation theory of various structures. This culminated in a sense with Friedlander and Pevtsova's introduction of the notion of \(\pi\)-points (appropriate flat maps from \(k[t]/(t^p)\) to \(kG\)) to develop a general theory for an arbitrary finite group scheme \(G\).
This work takes a renewed (and quite fruitful) look at the representation theory of elementary Abelian groups by considering the more general notion of a rank \(r\) shifted subgroup \(k[t_1,\dots,t_r]/(t_1^p,\dots,t_r^p)\) for \(1\leq r\leq n\). Let \(V\subset\text{Rad}(kE)\) be an \(n\)-dimensional subspace chosen so that the composite \(V\to\text{Rad}(kE)\to\text{Rad}(kE)/\text{Rad}^2(kE)\) is an isomorphism and consider the variety of Grassmannians \(\text{Grass}(r,V)\) of \(r\)-planes in \(V\). Associated to any \(U\in\text{Grass}(r,V)\) is an algebra \(C(U)\simeq k[t_1,\dots,t_r]/(t_1^p,\dots,t_r^p)\) and an associated flat map \(C(U)\to kE\). For a finite dimensional \(kE\)-module \(M\), the \(r\)-rank variety of \(M\) is defined to be the subset \(\text{Grass}(r,V)_M\) of \(\text{Grass}(r,V)\) consisting of those \(U\) for which the restriction of \(M\) to \(C(U)\) is not free. The variety \(\text{Grass}(r,V)_M\) is shown to be closed in \(\text{Grass}(r,V)\) and essentially dependent only on \(M\) not the choice of \(V\). Unlike the \(r=1\) case, when \(r=2\), every closed subvariety of \(\text{Grass}(2,V)\) cannot be realized as some \(\text{Grass}(2,V)_M\).
For a given \(U\in\text{Grass}(r,V)\) and \(kE\)-module \(M\), one can consider the radical (or socle) of \(M\) upon restriction to \(C(U)\). More generally, one can consider higher radicals (or socles). This can be used to extend Friedlander and Pevtsova's notion of generalized support varieties (for \(r=1\)) to higher \(r\). Specifically, the authors define non-maximal \(r\)-radical (and \(r\)-socle) support varieties. Some computations of such are given including an appendix by the first author that involves some computer calculations.
Further, the authors introduce an analogue of modules of constant Jordan type: modules of constant \(r\)-radical (or \(r\)-socle) type. Again, this is done in full generality for higher radicals (or socles). Having constant radical (or socle) type means that the dimension of the radical (or socle) of \(M\) restricted to \(C(U)\) is independent of the choice of \(U\in\text{Grass}(r,V)\). The authors present a number of examples of such modules and also investigate when the Carlson \(L_\zeta\) modules have such type.
In the second part of the paper, the authors use modules of constant \(r\)-radical or \(r\)-socle type to construct algebraic vector bundles on \(\text{Grass}(r,V)\). Again, numerous examples are given as well as examples of how various standard bundles can be realized by these constructions. elementary Abelian \(p\)-groups; Grassmannians; algebraic vector bundles; rank varieties; support varieties; modules of constant Jordan type; modules of constant radical type; modules of constant socle type; group algebras; cyclic shifted subgroups; finite group schemes Carlson, J. F.; Friedlander, E. M.; Pevtsova, J., Representations of elementary abelian \(p\)-groups and bundles on Grassmannians, Adv. Math., 229, 5, 2985-3051, (2012) Modular representations and characters, Grassmannians, Schubert varieties, flag manifolds, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Group rings of finite groups and their modules (group-theoretic aspects), Finite abelian groups, Group schemes, Torsion groups, primary groups and generalized primary groups Representations of elementary Abelian \(p\)-groups and bundles on Grassmannians. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 describe how the theory of ordinary and modular characters may be localized at the prime ideals of certain commutative rings acting on the representation ring of a finite group over a field. In addition, this localized character theory, and a Lefschetz-Riemann-Roch theorem, are applied to study the Galois module structure of the cohomology of the structure sheaves of semi-stable curves over rings of algebraic integers. modular characters; representation rings; finite groups; localized character theory; Lefschetz-Riemann-Roch theorem; Galois modules; sheaves; semi-stable curves Chinburg, Ted; Erez, Boas; Pappas, Georgios; Taylor, Martin: Localizations of Grothendieck groups and Galois structure, Contemp. math. 224, 47-63 (1999) Ordinary representations and characters, Frobenius induction, Burnside and representation rings, Modular representations and characters, Relations of \(K\)-theory with cohomology theories, Riemann-Roch theorems, Arithmetic ground fields for curves, de Rham cohomology and algebraic geometry, Integral representations related to algebraic numbers; Galois module structure of rings of integers Localizations of Grothendieck groups and Galois structure | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems See the preview in Zbl 0504.14021. compact bordered Klein surface; maximal symmetry; topological type; species of genus; \(M^*\)-group; full covering; fully wound covering; \(M^*\)-simple group Coverings of curves, fundamental group, Compact Riemann surfaces and uniformization, Coverings in algebraic geometry, Covering spaces and low-dimensional topology The species of bordered Klein surfaces with maximal symmetry of low genus | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be an algebraic variety and let \(S\) be a set of birational automorphisms of \(X\). One says that \(S\) is regularizable on a variety \(V\) if there is a birational map \(f: X\dashrightarrow V\) such that \(f\circ S\circ f^{-1}\) is a set of biregular automorphisms of \(V\). For instance, the Mori minimal model program implies that if \(X\) is a variety of general type of dimension \(2\) or \(3\) then the group \(\text{Bir}(X)\) of birational automorphisms is regularizable on a certain variety \(V\). The aim of the paper under review is to prove various results concerning regularization of birational automorphisms. For instance, any finite subgroup \(G\subset \text{Bir}(X)\) is regularizable. Another result asserts that if \(X\) is a birationally rigid Fano variety of dimension \(2\) or \(3\) and if \(G\subset \text{Bir}(X)\) is a finite subgroup of \(\text{Bir}(X)\) then there exists a birational map \(f\colon X\dashrightarrow V\), with \(V\) a Fano variety with terminal singularities, such that \(f\circ G\circ f^{-1}\) is a subgroup of biregular automorphisms of \(V\). As an application of his results the author answers a question raised by Manin concerning the birational automorphisms of a cubic surface \(X\) over a (non algebraically closed field) \(k\) with Pic\((X)=\mathbb Z\). terminal singularities; cubic surface; Fano varieties; birationally rigid; movable log pair Rational and birational maps, Coverings in algebraic geometry, Birational automorphisms, Cremona group and generalizations Regularization of birational automorphisms | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 following divisors in the space \(\text{Sym}^{12}\mathbb{P}^1\) of twelve points on \(\mathbb{P}^1\) are actually the same:
(A) the possible locus of the twelve nodal fibers in a rational elliptic fibration (i.e. a pencil of plane cubic curves);
(B) degree 12 binary forms that can be expressed as a cube plus a square;
(C) the locus of the twelve tangents to a smooth plane quartic from a general point of the plane;
(D) the branch locus of a degree 4 map from a hyperelliptic genus 3 curve to \(\mathbb{P}^1\);
(E) the branch locus of a degree 3 map from a genus 4 cuve to \(\mathbb{P}^1\) induced by a theta-characteristic; and several more.
The corresponding moduli spaces are smooth, but they are not all isomorphic; some are finite étale covers of others. We describe the web of interconnections among these spaces, and give monodromy, rationality, and Prym-related consequences. Enumerative consequences include:
(i) the degree of this locus is 3762 (e.g. there are 3762 rational elliptic fibralions with nodes above 11 given general points of the base);
(ii) if \(C\to\mathbb{P}^1\) is a cover as in (D), then there are 135 different such covers branched at the same points;
(iii) the general set of 12 tangent lines that arise in (C) turn up in 120 essentially different ways.
Some parts of this story are well known, and some other parts were known classically. The unified picture is surprisingly intricate and connects many beautiful constructions. branched covers; rational elliptic fibration; tangents; branch locus; monodromy Vakil R 2001 Twelve points on the projective line, branched covers, and rational elliptic fibrations \textit{Math. Ann.}320 33--54 Fibrations, degenerations in algebraic geometry, Coverings in algebraic geometry, Families, moduli of curves (algebraic), Ramification problems in algebraic geometry Twelve points on the projective line, branched covers, and rational elliptic fibrations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This mini-workshop focused on chromatic phenomena and duality as unifying themes in algebra, geometry, and topology. The overarching goal was to establish a fruitful exchange of ideas between experts from various areas, fostering the study of the local and global structure of the fundamental categories appearing in algebraic geometry, homotopy theory, and representation theory. The workshop started with introductory talks to bring researches from different backgrounds to the same page, and later highlighted recent progress in these areas with an emphasis on the interdisciplinary nature of the results and structures found. Moreover, new directions were explored in focused group work throughout the week, as well as in an evening discussion identifying promising long-term goals in the subject. Topics included support theories and their applications to the classification of localizing ideals in triangulated categories, equivariant and homotopical enhancements of important structural results, descent and Galois theory, numerous notions of duality, Picard and Brauer groups, as well as computational techniques. Collections of abstracts of lectures, Proceedings of conferences of miscellaneous specific interest, Proceedings, conferences, collections, etc. pertaining to algebraic topology, Proceedings, conferences, collections, etc. pertaining to category theory, Abstract and axiomatic homotopy theory in algebraic topology, Categorical algebra, Homotopy theory, Motivic cohomology; motivic homotopy theory, Module categories in associative algebras, Modular representations and characters Mini-workshop: Chromatic phenomena and duality in homotopy theory and representation theory. Abstracts from the mini-workshop held March 4--10, 2018 | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The main purpose of this paper is to study the structure of a nonsingular projective 3-fold \(X\) with a surjective morphism \(f:X\to X\) onto itself which is not an isomorphism, called a nontrivial surjective endomorphism of \(X\). Let \(f:X\to X\) be a surjective morphism from a nonsingular projective variety \(X\) onto itself. Then \(f\) is a finite morphism and if the Kodaira dimension \(\kappa(X)\) of \(X\) is non-negative, \(f\) is a finite étale covering. The structure of an algebraic surface \(S\) which admits a nontrivial surjective endomorphism is fairly simple. If \(\kappa(S)\geq 0\), \(S\) is minimal and a suitable finite étale covering of \(S\) is isomorphic to an abelian surface or the direct product of an elliptic curve and a smooth curve of genus \(\geq 2\).
Let \(X\) be a smooth projective 3-fold with \(\kappa(X)=0\) or 2 which admits a nontrivial surjective endomorphism \(f:X\to X\). The author shows that a suitable finite étale covering \(\widetilde X\) of \(X\) has the structure of a smooth abelian scheme over a nonsingular projective variety \(W\) with \(0\leq\dim (W)<\dim(X)\). Moreover, \(\widetilde X\) can be chosen to be isomorphic to an abelian 3-fold or the direct product \(E\times W\) of an elliptic curve \(E\) and a smooth projectice surface \(W\) with \(\kappa(W)=\kappa(X)\). nontrivial surjective endomorphism; finite étale covering [9] Fujimoto Y., Endomorphisms of smooth projective 3-folds with non-negative Kodaira dimension, Publ. Res. Inst. Math. Sci., 2002, 38(1), 33-92 \(3\)-folds, Moduli, classification: analytic theory; relations with modular forms, Coverings in algebraic geometry, Minimal model program (Mori theory, extremal rays) Endomorphisms of smooth projective 3-folds with non-negative Kodaira dimension. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The \(p\)-rank of an algebraic curve \(X\) over an algebraically closed field \(k\) of characteristic \(p>0\) is the dimension of the vector space \(H^1(X_{\text{ét}},\mathbb F_p)\). We study the representations of finite subgroups \(G \subset \text{Aut}(X)\) induced on \(H^1(X_{\text{ét}},\mathbb F_p)\otimes k\), and obtain two main results.
First, the sum of the nonprojective direct summands of the representation, i.e., its core, is determined explicitly by local data given by the fixed point structure of the group acting on the curve. As a corollary, we derive a congruence formula for the \(p\)-rank.
Secondly, the multiplicities of the projective direct summands of quotients curves, i.e., their Borne invariants, are calculated in terms of the Borne invariants of the original curve and ramification data. In particular, this is a generalization of both Nakajima's equivariant Deuring-Shafarevich formula and a previous result of Borne in the case of free actions. \(p\)-rank; Galois module Automorphisms of curves, Modular representations and characters On \(p\)-rank 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 The objective of the paper is to investigate tame fundamental groups of schemes of finite type over Spec\((\mathbb{Z})\). More precisely, let \(X\) be a connected scheme of finite type over Spec\((\mathbb{Z})\) and let \(\bar X\) be a compactification of \(X\). Then the tame fundamental group of \(X\) classifies finite étale coverings of \(X\) which are tamely ramified along the boundary \(\bar X - X\), in particular, the tame fundamental group \(\pi_1^t(\bar X,\bar X -X)\) is a quotient of the étale fundamental group \(\pi_1(X)\).
The first result (Theorem 1) of the paper says that the maximal pro-nilpotent quotient of the tame fundamental group \(\pi_1^t(\bar X,\bar X-X)^{\text{pro-nil}}\) is independent of the choice of the compactification \(\bar X\). Let \(\mathcal O\) be the ring of integers in a finite extension \(k\) of \(\mathbb{Q}\). Let \(\bar X\) be a normal flat \(\mathcal{O}\)-scheme of finite type whose geometric fibre \(\bar X\otimes_\mathcal{O}\bar k\) is connected and \(\bar X\to \text{Spec}(O)\) is surjective. The second result (Theorem 2) says that the abelianized tame fundamental group \(\pi_1^t(\bar X,\bar X -X)^{\text{ab}}\) is finite. In the special case \(\bar X=X\) it implies the finiteness of the étale fundamental group \(\pi_1(\bar X)^{\text{ab}}\) which sharpens a theorem of Katz and Lang by weaking the assumption `smooth' to normal. Schmidt A. (2002). Tame coverings of arithmetic schemes. Math. Ann. 322: 1--18 Arithmetic varieties and schemes; Arakelov theory; heights, Coverings in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry Tame coverings of arithmetic schemes. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper, we study numerical properties of Chern classes of certain covering manifolds. One of the main results is the following: Let \(\psi: X\to \mathbb{P}^n\) be a finite covering of the \(n\)-dimensional complex projective space branched along a hypersurface with only simple normal crossings and suppose \(X\) is nonsingular. Let \(c_i(X)\) be the \(i\)-th Chern class of \(X\). Then
(i) if the canonical divisor \(K_X\) is numerically effective, then \((-1)^k c_k(X)\) \((k\geq 2)\) is numerically positive, and
(ii) if \(X\) is of general type, then \((-1)^n c_{i_1}(X)\cdots c_{i_r}(X) > 0\), where \(i_1+\dots+i_r = n\). Furthermore we show that the same properties hold for certain Kummer coverings. Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Coverings in algebraic geometry Numerical properties of Chern classes of certain covering manifolds. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author's recent theorem on Galois rational coverings \(X\dashrightarrow V\) for primitive Fano varieties \(V\) is strengthened in two directions: first, the class of Galois groups \(G\) is enlarged to the maximal class for which the proof works, and second, the conditions on the variety \(V\) are relaxed to the divisorial canonicity requirement. Fano variety; divisorial canonicity; cyclic covering; branch divisor; rational map Coverings in algebraic geometry, Fano varieties, Divisors, linear systems, invertible sheaves Remarks on Galois rational coverings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 constructs six new pencils of \(K3\) surfaces with general element having Picard number 19, each pencil containing 4 surfaces with Picard number 20. Starting points are two pencils of symmetric surfaces in \({\mathbb P}^3\) constructed by the author in her previous paper [J. Algebra 246, No. 1, 429--452 (2001; Zbl 1064.14038)], surfaces admitting an action of a subgroup of \(\text{SO}(4,\mathbb R)\), \(G_n\), where \(n=6\) or \(8\) (depending on the pencil). In each of the two cases the author finds three normal subgroups of \(G_n\), all containing the Heisenberg group, such that the quotient of the pencil by the action of this subgroup is a pencil of \(K3\)s as described. The Picard lattice of many of these surfaces is described. K3 surfaces; Picard lattices Sarti, A, Group actions, cyclic coverings and families of \(K3\)-surfaces, Can. Math. Bull., 49, 592-608, (2006) \(K3\) surfaces and Enriques surfaces, Group actions on varieties or schemes (quotients), Coverings in algebraic geometry, Picard groups Group actions, cyclic coverings and families of \(K3\)-surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Numerical Campedelli surfaces are minimal, general type surfaces with \(p_g=0\) and \(K^2=2\). It is known that their algebraic fundamental groups are of order \(\leq 9\). Miles Reid pointed out that the dihedral group of order 8 cannot occur and constructed an example for the quaternionic group. In this paper, the non-existence set forth in the title is proved following an idea already present in Miles Reid's unpublished paper: To study the existence or the non-existence of a surface with given numerical invariants and finite algebraic fundamental group, one considers the Galois cover associated to the fundamental group and looks at the canonical image of this cover and to the interplay between the intrinsic geometry determined by the invariants and the extrinsic geometry determined by the group action. numerical Campedelli surfaces; algebraic fundamental group; Galois cover; invariants; group action Naie D.: Numerical Campedelli surfaces cannot have the symmetric group as the algebraic fundamental group. J. Lond. Math. Soc. 59, 813--827 (1999) Surfaces of general type, Coverings in algebraic geometry, Group actions on varieties or schemes (quotients), Geometric invariant theory Numerical Campedelli surfaces cannot have the symmetric group as the algebraic fundamental group | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 0744.00034.]
This note is a continuation of the author's paper in J. Algebr. Geom. 1, No. 2, 293-323 (1992; see the preceding review). The aim is again to prove various forms of the cone theorem using methods similar to the original geometric arguments of \textit{S. Mori} [cf. Ann. Math., II. Ser. 110, 593-606 (1979; Zbl 0423.14006) and 116, 133-176 (1982; Zbl 0557.14021)]. The basic idea is the same as in the author's paper cited above. Instead of deforming a curve \(C \subset X\) directly, we construct a covering \(Y \to X\) and deform another morphism \(D \to Y\) where \(D\) is a suitable covering of \(C\). In the author's mentioned paper, \(Y\) was a bug- eyed cover of \(X\), therefore a nonseparated algebraic space. The method of the present article is to replace \(X\) by its formal completion along \(C\) and then find a suitable cyclic covering \(Y\) of this formal scheme.
The new method has both disadvantages and advantages. It can handle only quotients by cyclic groups and not by arbitrary groups. The method however is still strong enough to recover the most general cone theorem for normal threefolds. The main advantage is that it can be used to investigate deformations of curves that are contained in the singular locus of \(X\). In fact this method shows that the deformation problem of \(C \to X\) where the canonical class of \(X\) is \(\mathbb{Q}\)-Cartier can be reduced to another deformation problem \(D\to Y\) where the canonical class of \(Y\) is Cartier.
Theorem. Let \(X\) be a projective threefold over a field of characteristic zero. Assume that \(K_ X\) is \(\mathbb{Q}\)-Cartier and that \(X\) has only log- terminal singularities with the exception of finitely many points. Then
(1) The set of extremal rays is locally finite in the open subset \(\{z \in N_ 1 (X) | z \cdot K_ X < 0\}\).
(2) For every extremal ray \(R\) the locus of \(R\) is covered by rational curves \(L_ R\) such that \([L_ R] \in R\) and \(-6 \leq L_ R \cdot K_ X < 0\).
The locus of a ray \(R\) is defined to be the set theoretic union of the closed points of all curves \(D \subset X\) such that \([D] \in R\). Presumably the 6 in the statement can be replaced by 4. cone theorem; formal completion; cyclic covering; normal threefolds; deformations of curves; log-terminal singularities J. Kollár. Cone theorems and cyclic covers. In Algebraic geometry and analytic geometry (Tokyo, 1990), ICM-90 Satell. Conf. Proc., pages 101--110. Springer, Tokyo, 1991. Coverings in algebraic geometry, Formal methods and deformations in algebraic geometry, \(3\)-folds Cone theorems and cyclic covers | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(G\) be a connected algebraic group over an algebraically closed field of characteristic \(p\). Then \(G\) is an extension of an abelian variety \(A\) and a linear algebraic group \(G_{\mathrm{aff}}\). Let \(g(G)\) be the dimension of \(A\) and let \(r(G)\) be the dimension of a maximal torus of \(G_{\mathrm{aff}}\). The authors prove that the maximal prime-to-\(p\) quotient \(\pi_1^{(p')}(G)\) of the étale fundamental group of \(G\) is commutative, and is a quotient of \(\mathbb Z_{(p')}^{2g(G)+r(G)}\), where \(\mathbb Z_{(p')}\) is the product of the rings \(\mathbb Z_\ell\) for all primes \(\ell\) distinct from \(p\). The authors also prove a generalization to homogeneous spaces. Let \(X\) be a variety on which \(G\) acts transitively with connected stabilizers, and let \(H\) be the stabilizer of a point in \(X\). Then \(\pi_1^{(p')}(X)\) is commutative, and is a quotient of \(\mathbb Z_{(p')}^{2(g(G)-g(H))+r(G)-r(H)}\). As an application, the authors show that for commutative \(G\), the \(\ell\)-primary torsion part of the Brauer group \(\mathrm{Br}(G)\) of \(G\) is isomorphic to \((\mathbb Q_\ell/\mathbb Z_\ell)^{(2g(G)+r(G))(2g(G)+r(G)-1)/2-\rho}\), where \(\rho\) is the rank of the Néron-Severi group of \(A\). etale fundamental groups; algebraic group; Brauer group Brion, M., Szamuely, T.: Prime-to-
\[
p
\]
étale covers of algebraic groups and homogeneous spaces. Bull. Lond. Math. Soc. 45(3), 602--612 (2013). doi: 10.1112/blms/bds110 Group varieties, Coverings in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry, Homogeneous spaces and generalizations Prime-to-\(p\) étale covers of algebraic groups and homogeneous spaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors study thoroughly the structure of \(n\)-dimensional normal complex-projective varieties \(X\) with non-isomorphic regular morphisms \(f:X\rightarrow X\) which are polarized in the sense that \(f^*(H)\) is linear equivalent to \(qH\), \(q>0\), for some ample divisor \(H\) on \(X\). This implies \(f\) finite and surjective, \(q\in\mathbb N\), deg \(f=q^n\). The first main result describes the general structure of \(X\). There exists a normal projective variety \(V\) with a polarized endomorphism \(f_V\), a finite morphism \(\tau:V\rightarrow X\), étale in codimension one, and a dominant rational map \(\pi\) from \(V\) to a product \(A\times S\) of an abelian variety \(A\) (with a polarized endomorphism \(f_A\)) and a weak Calabi-Yau variety \(S\) (with a polarized endomorphism \(f_S\)) such that \(\tau\circ f_V=f\circ\tau\), \(\pi\circ f_V=(f_A\times f_S)\circ\pi\). Moreover the Kodaira dimension \(\kappa(X)\) is non-positive. The authors conjecture that \(\dim S=0\). If \(X\) is not uniruled then \(\pi\) is an isomorphism and \(\kappa(X)=0\). In the uniruled case the projection of the graph of \(\pi\) to \(A\times S\) is birationally equivalent to the maximal rationally connected fibration of a smooth model of \(V\).
For a more detailed description of \(X\) the authors study the family \(\mathfrak{F}\) of all \((X',f',\tau)\) with \(X'\) a normal projective variety, \(f'\) an endomorphism of \(X'\), \(\tau: X'\rightarrow X\) a finite surjective morphism, étale of codimension one, with \(\tau\circ f'=f\circ\tau\). The structure of \(X\) is described in dependence of \(q^\natural(X,f)\) which is defined as \(\sup\{\dim H^1(X',{\mathcal{O}}_{X'})\,|\,(X',f',\tau)\in\mathfrak{F}\}\). There are specific results for \(q^\natural(X,f)\in\{0, n-1, n\}\) and for \(q^\natural(X,f)\geq n-3\). In particular, if \(q^\natural(X,f)=0\) and \(n\leq 3\) then \(X\) is rationally connected, and \(q^\natural(X,f)=n\) if and only if there is an abelian variety \(A\) and a finite surjective morphism \(A\rightarrow X\), étale in codimension one. \newline Some of the results in the paper are more generally shown for quasi-polarized endomorphisms (in the definition of ``polarized'' substitute ``ample'' by ``big and nef''). The authors use a broad spectrum of algebro-geometric methods and techniques for the proofs, in particular those from the minimal model program.
There is a series of results about the structure of complex-projective manifolds which admit a non-isomorphic surjective regular morphism \(f:X\rightarrow X\), see e.g. \textit{E. Amerik} [Manuscr. Math. 111, No. 1, 17--28 (2003; Zbl 1016.14008)], \textit{N. Fakhruddin} [J. Ramanujan Math. Soc. 18, No. 2, 109--122 (2003; Zbl 1053.14025)], \textit{Y. Fujimoto} and \textit{N. Nakayama} [J. Math. Kyoto Univ. 47, No. 1, 79--114 (2007; Zbl 1138.14023)], \textit{S. Cantat} [Enseign. Math., II. Sér. 49, No. 3--4, 237--262 (2003; Zbl 1059.32003)], \textit{J.-M. Hwang} and \textit{N. Mok} [J. Algebr. Geom. 12, No. 4, 627--651 (2003; Zbl 1038.14018)]. polarized endomorphism; finite surjective morphism; étale in codimension one; weak Calabi-Yau manifold Nakayama, Noboru and Zhang, De-Qi Polarized endomorphisms of complex normal varieties \textit{Math. Ann.}346 (2012) 991--1018 Families, moduli, classification: algebraic theory, Coverings in algebraic geometry, Normal analytic spaces, Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables Polarized endomorphisms of complex normal varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The article deals with the structure of fundamental groups of compact Kähler manifolds and can be considered as an extension of part I of this paper [\textit{F. Campana}, Contemp. Math. 241, 85-96 (1999; Zbl 0965.32021)] by adding results about solvable quotients of Kähler groups (without proofs), namely conditions that certain solvable quotients of the fundamental group \(\pi_{1}(X)\) of a compact Kähler manifold \(X\) are almost nilpotent. fundamental groups; Kähler manifolds; almost nilpotent groups Compact Kähler manifolds: generalizations, classification, Global differential geometry of Hermitian and Kählerian manifolds, Parametrization (Chow and Hilbert schemes), Normal analytic spaces, Coverings in algebraic geometry \(\mathcal G\)-connectedness of compact Kähler manifolds. II: Solvable quotients of Kähler groups | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let V be a smooth affine surface with coordinate ring \(\Gamma\) (V) a U.F.D. This happens for example if V is contractible. Then: \(V\cong {\mathbb{C}}^ 2\) if either:
(1) \(\pi_ 1^{\infty}(V)\), the fundamental group at infinity, is trivial, or:
(2) \({\bar \kappa}\)(V)\(=-\infty\), where \({\bar \kappa}\):\(=\log arithmic\) Kodaira dimension.
Ramanujam (essentially) proved (1) by considering the fundamental group of the real 3-dimensional boundary \(\partial T\) of a tubular neighborhood of the divisor at infinity of a compactification \(\bar V\) of V. Fujita, Miyanishi and Sugie proved (2) by purely geometrical arguments resting on the existence of Zariski decomposition of pseudo-effective divisors.
This expository article describes these approaches, as well as the direct proof of the relationship: (2)\(\Rightarrow (1)\) (the author). The existence of V with \({\bar \kappa}(V)=1\quad or\quad 2\) which are homology cells (the author and Miyanishi), as well as the rationality of these (the author and Shastri) are related questions. characterization of complex plane; fundamental group; logarithmic Kodaira dimension; rationality Special surfaces, Topological properties in algebraic geometry, Transcendental methods of algebraic geometry (complex-analytic aspects), Coverings in algebraic geometry, Families, moduli, classification: algebraic theory A tribute to a work of C. P. Ramanujam | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 normal complex surface and let \(\pi: S\to {\mathbb P}^2({\mathbb C})\) be a triple covering, namely a finite degree 3 map. Assume in addition that the branch locus of \(\pi\) (considered with the reduced structure) is the union of a quartic \(Q\) and a line \(l\), and that \(\pi\) is simply ramified over a general point of \(Q\) and totally ramified over any point of \(l\).
The main result of the paper is that, in the above situation, the surface \(S\) is a cubic surface of \({\mathbb P}^3\) and the morphism \(\pi\) is induced by projection from a point \(p\in {\mathbb P}^3\setminus S\). triple cover; plane quintic; cubic surface; projection from a point Coverings in algebraic geometry, Singularities of surfaces or higher-dimensional varieties Triple coverings of the projective plane branched along quintic curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We study residual finiteness for cyclic central extensions of cocompact arithmetic lattices \(\Gamma<\mathrm{PU}(n,1)\) of simple type. We prove that the preimage of \(\Gamma\) in any connected cover of \(\mathrm{PU}(n,1)\), in particular the universal cover, is residually finite. This follows from a more general theorem on residual finiteness of extensions whose characteristic class is contained in the span in \(H^2(\Gamma,\mathbb{Z})\) of the Poincaré duals to totally geodesic divisors on the ball quotient \(\Gamma\setminus\mathbb{B}^n\). For \(n\geq 4\), if \(\Gamma\) is a congruence lattice, we prove residual finiteness of the central extension associated with any element of \(H^2(\Gamma,\mathbb{Z})\).
Our main application is to existence of cyclic covers of ball quotients branched over totally geodesic divisors. This gives examples of smooth projective varieties admitting a metric of negative sectional curvature that are not homotopy equivalent to a locally symmetric manifold. The existence of such examples is new for all dimensions \(n\geq 4\). Coverings in algebraic geometry Residual finiteness for central extensions of lattices in \(\mathrm{PU}(n,1)\) and negatively curved projective varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems \textit{J.-P.~Serre} proved in [Comment. Math. Helv. 59, 651--676 (1984; Zbl 0565.12014)] a formula that related the Hasse-Witt invariant of the trace form of an étale algebra over a field of characteristic not \(2\) to the second Stiefel-Whitney class of the permutation representation of the Galois group which corresponds to that algebra. This formula has been generalized by \textit{H. Esnault, B. Kahn} and \textit{E. Viehweg} [J. Reine Angew. Math. 441, 145--188 (1993; Zbl 0772.57028)] who considered symmetric bundles obtained from certain tame finite flat coverings of Dedekind schemes with odd ramification everywhere. This latter result has been generalized further by the present authors in [J. Théor. Nombres Bordx. 12, 597--660 (2000; Zbl 1120.11300)] where a formula was obtained under certain regularity conditions but without further restrictions on the dimensions of the schemes. In [J. Reine Angew. Math. 360, 84--123 (1985; Zbl 0556.12005)], \textit{A. Fröhlich} retrieved Serre's result as a special case of a more general difference formula between the Hasse-Witt invariants of a quadratic form and the twist of this form by an orthogonal representation, involving first and second Stiefel-Whitney classes and spinor norms. The main purpose of the present paper is to obtain a theorem ``à la Fröhlich'' in the geometric setting considered in the authors' earlier paper.
From the authors' abstract: We establish comparison results between the Hasse-Witt invariants \(w_t(E)\) of a symmetric bundle \(E\) over a scheme and the invariants of one of its twists \(E_{\alpha}\). For general twists we describe the difference between \(w_t(E)\) and \(w_t(E_{\alpha})\) up to terms of degree \(3\). Next we consider a special kind of twist, which has been studied by A.~Fröhlich. This arises from twisting by a cocycle obtained from an orthogonal representation. A simple important example of this twisting procedure is the bilinear trace form of an étale algebra, which is obtained by twisting the standard/sum-of-squares form by the orthogonal representation attached to the algebra. We show how to explicitly describe the twist for representations arising from very general tame actions. This involves the `square root of the inverse different' which Serre, Esnault, Kahn, Viehweg and ourselves had studied before. For torsors we show that, in our geometric set-up, Jardine's generalisation of Fröhlich's formula holds. Namely let \((X, G)\) be a torsor with quotient \(Y\), let \(E\) be a symmetric bundle over \(Y\), let \(\rho: G \to\mathbf{O}(E)\) be an orthogonal representation and let \(E_{\rho,X}\) be the corresponding twist of \(E\). Then we verify up to degree 3 that the formula \(w_t(E_{\rho,X}) \text{sp}_t(\rho)=w_t(E)w_t(\rho)\) holds. Here \(\text{sp}_t(\rho)\) and \(w_t(\rho)\) are respectively the spinor invariant and the Stiefel-Whitney class of \(\rho\). The case of genuinely tamely ramified actions is geometrically more involved and leads us to introduce an invariant of ramification, which in a sense gives a decomposition in terms of representations of the inertia groups of the invariant introduced by Serre for curves. The comparison result in the tamely ramified case proceeds by reduction to the case of a torsor. The reduction is carried out by means of a partial normalisation procedure, which we had introduced in a previous paper. An important lemma of Esnault, Kahn and Viehweg allows us to express the differe nce between the invariants of bundles before and after the normalisation procedure in terms of Chern classes of certain sub-bundles. As noted elsewhere, this result can be best understood in the context of symmetric complexes and their invariants. Our results are new even for bundles over curves and they allow us to weaken the regularity assumptions that we had to impose in previous work of ours. Hasse-Witt invariant; Stiefel-Whitney class; étale cohomology; vector bundle; symmetric bundle; orthogonal representation \(K\)-theory of quadratic and Hermitian forms, Galois cohomology of linear algebraic groups, Coverings in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Étale and other Grothendieck topologies and (co)homologies Twists of symmetric bundles | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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, \(X\subset \mathbb{P}^r\) an integral nondegenerate proper curve over \(K\) and \(U\subset (\mathbb{P}^r)^*\) a dense open subvariety such that every hyperplane \(H\in U\) intersects \(X\) in \(d\) smooth points. The monodromy action \(\varphi :\pi _1^{\text{ét}}(U)\rightarrow S_d\) is defined when varying \(H\) in \(U\). Set \(G_X:=\)im\((\varphi )\). The permutation group \(G_X\) is called the sectional monodromy group (SMG) of \(X\). In characteristic \(0\), one has \(G_X=S_d\), but in characteristic \(p\), SMGs can be smaller than \(S_d\). For \(r\geq 3\) and for a large class of space curves, the author classifies all possibilities for the SMG \(G\) as well as the curves with \(G_X=G\). Applying similar methods he studies a family of rational curves in \(\mathbb{P}^2\) which gives the answer to an old question about Galois groups of generic trinomials. projective curve; sectional monodromy group; rational curve Plane and space curves, Coverings in algebraic geometry, Separable extensions, Galois theory Sectional monodromy groups of 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 critical value; miniversal deformation; parabolic singularities Deformations of complex singularities; vanishing cycles, Local complex singularities, Complex singularities, Coverings in algebraic geometry, Singularities in algebraic geometry Distribution of critical values of miniversal deformations of parabolic 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 One of the main remaining problems in the theory of algebraic surfaces is the problem of constructing algebraic surfaces with prescribed values of their Chern numbers \(c\) \(2_ 1\) and \(c_ 2\). This monograph describes in detail one of the approaches to this problem which consists of constructing surfaces as coverings of the projective plane ramified over an arrangement of lines. This method was first used by the second author to construct some examples of surfaces of general type which are uniformizable by the complex 2-ball [Contemp. Math. 9, 55-71 (1982; Zbl 0487.14009)]. According to a theorem of S. Yau, these surfaces can be characterized by the condition \(c\) \(2_ 1=3c_ 2\). In general, \(c\) \(2_ 1\leq 3c_ 2\). More systematically this construction was studied later in the dissertation of the third author, who also was first to consider weighted arrangements of lines. The present monograph is based on this dissertation and also on notes of the first author taken from a series of lectures given by the second author.
The following is a brief exposition of the contents of the book. In chapter 1, the authors compute the Chern numbers of surfaces obtained as nonsingular models of finite coverings of the projective plane uniformly ramified along an arrangement of lines. The corresponding extension of the field of rational functions is a Kummer extension with the Galois group \(({\mathbb{Z}}/n{\mathbb{Z}})^{k-1}\), where k is the number of lines and n is the ramification index over each line. Chapter 2 contains examples of some interesting arrangements of lines and their combinatorics. Among them are the arrangement of twelve lines which form the four degenerate members of a Hesse pencil of cubic curves (the Hesse configuration), the arrangement of reflection lines of the group of projective transformations arising from a complex 3-dimensional reflection group, and some other purely combinatorial arrangements.
In chapter 3 the authors use some of the arrangements of lines introduced in the previous paper to construct surfaces uniformizable by the complex 2-ball. They also discuss the Kummer coverings branched along an arrangement of lines from the point of view of the classification of algebraic surfaces. Finally, they show how the known inequalities between Chern numbers imply purely combinatorial results on arrangements of lines.
In chapters 4 and 5 the authors study coverings of the plane branched along an arrangement of lines with prescribed ramification index at each line (a weighted arrangement). The problem of classifying the corresponding weighted arrangements seems to be very difficult. Some of these arrangements arise naturally from the theory of hypergeometric differential equations in two variables. A special attention is given to the description of weighted arrangements formed by the sides and the diagonals of a complete quadrilateral.
The book ends with three appendices. In the first two some background material from the theory of algebraic surfaces and differential geometry is presented. In the third the techniques of finite ramified coverings is discussed. constructing algebraic surfaces with prescribed values of their Chern numbers; weighted arrangements of lines; Hesse configuration; surfaces uniformizable by the complex 2-ball; prescribed ramification index; hypergeometric differential equations Barthel, G.; Hirzebruch, F.; Höfer, T., Geradenkonfigurationen und algebraische flächen, Aspects of Mathematics, vol. D4, (1987), Friedr. Vieweg & Sohn Braunschweig, MR912097 Special surfaces, Projective techniques in algebraic geometry, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Designs and configurations, Characteristic classes and numbers in differential topology, Coverings in algebraic geometry, Ramification problems in algebraic geometry Geradenkonfigurationen und algebraische Flächen. (Configurations of lines and algebraic surfaces). Eine Veröffentlichung des Max-Planck-Instituts für Mathematik, Bonn | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper, the authors construct some ball-quotient surfaces as Galois coverings of the projective plane whose branch loci are some quadric-line arrangements, which are the first examples with a nonlinear locus. The construction uses Holzapfel's proportionality theorem, a generalization of Hirzebruch's construction of ball-quotient surfaces as coverings branched along a linear locus.
The simplest of these quadric-line arrangements is the Apollonius cycle, that is the arrangement of a smooth quadric with three tangent lines. The complement of the Apollonius cycle in the plane is in fact the second braid space of the thrice-punctured sphere. It is shown that this cycle supports several ball-quotient orbifolds and that these orbifolds are all commensurable with the ones obtained from the Barthel-Hirzebruch-Hofer list or equivalently the Deligne-Mostow list. There are also a few other arrangements, some of them are related to the Apollonius cycle via a covering. The authors remark that this construction gives another proof of the fact that the ``world of complex algebraic surfaces is Picard-Einstein with a universal degeneration lifted finitely'' from the Apollonius configuration. We refer the reader to the article for the precise meaning of this assertion. ball-quotient; Picard-Einstein metric; complex hyperbolic geometry Holzapfel, R.-P., Vladov, N.: Quadric-Line Configurations Degenerating Plane Picard Einstein Metrics I-II, Sitzungsber. der Berliner Math. Ges., Jg. 1997-2000, pp. 79-141. Berlin (2001) Complex multiplication and moduli of abelian varieties, Arithmetic aspects of modular and Shimura varieties, Automorphism groups of lattices, Quadratic extensions, Structure of families (Picard-Lefschetz, monodromy, etc.), Fine and coarse moduli spaces, Coverings in algebraic geometry Quadric-line configurations degenerating plane Picard Einstein metrics. I, II | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We interpret Galois covers in terms of particular monoidal functors, extending the correspondence between torsors and fiber functors. As applications we characterize tame \(G\)-covers between normal varieties for finite and étale group schemes and we prove that, if \(G\) is a finite, flat and finitely presented nonabelian and linearly reductive group scheme over a ring, then the moduli stack of \(G\)-covers is reducible. Tonini, Fabio: Ramified Galois covers via monoidal functors. Transform. groups (July 2015) Coverings in algebraic geometry, Stacks and moduli problems, Group schemes, Categories in geometry and topology Ramified Galois covers via monoidal functors | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper is devoted to the problem of classifying all (branched) Galois coverings of \(\mathbb{P}^2\) by smooth \(K3\) surfaces. The main results of the paper are the following:
a classification of all coverings as above with abelian Galois group;
a classification of all coverings as above with branch locus of degree at most \(5\);
a list of coverings as above with branch locus of degree \(6\).
The author conjectures that this last list is complete, but he does not prove it since he should apply his method to Yang's list of sextics with simple singularities [\textit{J.-G. Yang}, Tôhoku Math. J., II. Ser. 48, No. 2, 203--227 (1996; Zbl 0866.14014)], that is very long. Since there are no coverings as above with branch locus of degree higher than \(6\), the conjecture is equivalent to the fact that this paper describes all Galois coverings of \(\mathbb{P}^2\) by smooth \(K3\) surfaces. Uludağ, AM, Galois coverings of the plane by K3 surfaces, Kyushu J. Math., 59, 393-419, (2005) \(K3\) surfaces and Enriques surfaces, Coverings in algebraic geometry Galois coverings of the plane by \(K3\) surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems It is proved that \(X\) is a pseudo-sequence-covering \(\pi\)-image of a metric space if and only if \(X\) has a point-star network consisting of wcs-covers. (A cover \(\mathcal P\) is a wcs-cover if for every convergent sequence \(S\) converging to \(x\in X\) there exists a finite subfamily \(\mathcal P'\) of \((\mathcal P)_x\), such that \(S\) is eventually in \(\bigcup{\mathcal P'}\)). \(\pi\)-mapping; c\(\pi\)-image mapping Y. Ge, On pseudo-sequence-covering {\(\pi\)}-images of Metric Spaces, Matematicki Vesnik 57, 113 (2005). Approximation by positive operators, Coverings in algebraic geometry On pseudo-sequence coverings, \(\pi\)-images of metric 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 Suppose \(X\rightarrow Y\) is a double covering with \(Y\) smooth and \(X\) a normal complex surface. It is known that one can produce a birationally equivalent double covering \({\widetilde X} \rightarrow {\widetilde Y}\), with \({\widetilde X}\) smooth, by repeatedly blowing up singularities of the branch curve on \(Y\) and normalising the pull-back of \(X\). Here the authors prove that the same holds in the case of triple coverings of algebraic surfaces. The proof uses local computations which are based on results of \textit{R. Miranda} [Am. J. Math. 107, 1123--1158 (1985; Zbl 0611.14011)]. They also provide an example which shows that this procedure does not work in general for coverings of degree four. triple covering; normal surface; desingularisation; canonical resolution; blow-up Coverings in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties A canonical resolution of the singularities of a triple covering of algebraic surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This thesis is devoted to extending to higher dimension a construction of algebraic surfaces as covers of the plane ramified along an arrangement of lines which is due to Hirzebruch. Surprisingly this generalization goes rather far in spite of many technical difficulties like, for example, the analysis and resolution of singularities of the covers and the computation of the invariants of their nonsingular models. Arrangements of lines here are replaced by arrangements of hyperplanes. Some interesting examples of arrangements of planes in \(\mathbb P^3\) are provided by the reflection groups in dimension \(4.\) Those of them which are the Weyl groups of type \(A_4\), \(B_4\), \(D_4\), and \(F_4\) give arrangements with 10, 12, 16 and 24 planes respectively. The author shows that every ball quotient which is a Galois covering of \(\mathbb P^3\) of Kummer type branched over an arrangement of planes has the arrangement of one of the four types from above. Some of the examples of \textit{P. Deligne} and \textit{G. D. Mostow} [Publ. Math. Inst. Haut. Étud. Sci 63, 5--89 (1986; Zbl 0615.22008)] of ball quotients by non-arithmetic lattices are obtained in this way. 3-folds; resolution of singularities of the covers; arrangements of hyperplanes; ball quotients B. Hunt , Coverings and ball quotients , Bonn. Math. Schrif. 174 (1986). \(3\)-folds, Coverings in algebraic geometry, Projective and enumerative algebraic geometry, Homogeneous spaces and generalizations, Global theory and resolution of singularities (algebro-geometric aspects) Coverings and ball quotients with special emphasis on the 3-dimensional case | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A smooth scheme \(X\) over a field \(k\) of positive characteristic is said to be strongly liftable if \(X\) and all prime divisors on \(X\) can be lifted simultaneously to \(W_2(k)\), the ring of Witt vectors of length two of \(k\). The author continues the study of strongly liftable schemes introduced in [Math.\ Res.\ Lett.\ 17, No. 3, 563--572 (2010; Zbl 1223.14026)] in connection with the study of the Kawamata-Viehweg vanishing theorem in positive characteristic. First he shows that any smooth toric variety is strongly liftable. As a corollary he obtains the Kawamata-Viehweg vanishing theorem for smooth projective toric varieties. Second, he proves the Kawamata-Viehweg vanishing theorem for normal projective surfaces birational to a strongly liftable smooth projective surface, with no singularity assumption, extending the corresponding result obtained in the previous paper. Finally, the author deduces the cyclic cover trick over \(W_2(k)\) and uses it to study the behavior of cyclic covers over strongly liftable schemes. Furthermore, the author takes the opportunity to correct some statements in the previous paper. positive characteristic; strongly liftable scheme; Kawamata-Viehweg vanishing theorem Xie, Q. H., Strongly liftable schemes and the Kawamata-Viehweg vanishing in positive characteristic II, Math. Res. Lett, 18, 315-328, (2011) Positive characteristic ground fields in algebraic geometry, Vanishing theorems in algebraic geometry, Arithmetic ground fields for curves, Coverings in algebraic geometry Strongly liftable schemes and the Kawamata-Viehweg vanishing in positive characteristic. II | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(C= \bigcup^r_{i=1} C_i\) be a plane curve, set \(X= \mathbb{C}^2\setminus C\). The first homology \(M^K_C:= H_1(\widetilde X,\mathbb{C})\) of the maximal abelian cover \(\widetilde X\) of \(X\) has a \(\Lambda\)-module structure, where \(\Lambda= \mathbb{C}[t^{\pm 1}_1,\dots, t^{\pm 1}_r]\). The (reduced) support of the \(k\)th exterior power of \(M_c\) is the \(k\)th characteristic variety of \(C\), a subvariety \(\text{Char}_k(C)\) of the complex torus \((\mathbb{C}^*)^r\). An irreducible component \(V\) of \(\text{Char}_k(C)\) is called a coordinate component of \(V\) if \(V\) is in a coordinate torus \(\{t\in(\mathbb{C}^*)^r; t_i= 1\) for some \(i\}\); \(V\) is called non-essential if \(V\subset \text{Char}_k(b_{(j)})\) for some \(j\), where \(b_{(j)}= \bigcup_{i\neq j} C_i\), and it is called essential otherwise. Essential non-coordinate components of \(\text{Char}_k(C)\) may be determined as in \textit{A. Libgober} [in: Applications of algebraic geometry of coding theory, physics and computation. Kluwer Acad. Publ., NATO Sci. Ser. II, Math. Phys. Chem. 36, 215--254 (2001; Zbl 1045.14016)].
In the paper under review the authors characterize essential coordinate components of the first characteristic variety \(\text{Char}_1(C)\). Their results are illustrated by an example of a Zariski pair of curves. coverings; fundamental group; Zariski pair Artal, E., Carmona, J. and Cogolludo-Agustín, J. I.: Essential coordinate components of characteristic varieties. Math. Proc. Cambridge Philos. Soc. 136 (2004), no. 2, 287-299. Coverings of curves, fundamental group, Homotopy theory and fundamental groups in algebraic geometry, Coverings in algebraic geometry Essential coordinate components of characteristic varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Translation from Vestn. Mosk. Univ., Ser. I 1986, No.1, 5-8 (Russian) (1986; Zbl 0611.14038). characterization of the projective space; finite morphism; ampleness of the tangent bundle Projective techniques in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Coverings in algebraic geometry A corollary of Mori's tangent-bundle 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 The author establishes a formula to compute the irregularity of abelian coverings of smooth projective surfaces. The paper contains various examples and applications.
Such a formula was first developed by \textit{O. Zariski} [Ann. Math. (2) 32, 485--511 (1931; Zbl 0001.40301)] for cyclic coverings of the projective plane. Generalizations and applications of Zariski's formula were studied by many people (Budur, Esnault, Libgober, Naie, Vaquié).
The author generalizes the formula to the case of abelian coverings of smooth projective surfaces, assuming that the branch curve \(C=\sum_i C_i\) is endowed with an \(H\)-partition: there exists an ample divisor \(H\) s.t. for each \(i\), \(C_i\) is linearly equivalent to a multiple of \(H\). With this hypothesis, the irregularity is expressed as linear combination of superabundances of linear systems defined in terms of some mixed multiplier ideals associated to the partition \((C_1,\ldots, C_n)\). In \S 2 the author introduces the mixed multiplier ideals and characterize the jumping walls associated to them. In \S 3 there are summarized some properties of abelian covering of smooth projective surface. In \S the main theorem is proved, and finally \S 5 contains examples and applications. algebraic surface; abelian covering; mixed multiplier ideal Coverings in algebraic geometry, Singularities of curves, local rings, Surfaces and higher-dimensional varieties Mixed multiplier ideals and the irregularity of abelian coverings of smooth projective surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\to {\mathbb P}^2\) be a Galois cover with Galois group \(S_3\) whose branch locus \(B\) is a plane quintic. Then there is a component \(C\) of \(B\) along which the cover has ramification order \(3\) and \(C\) is either a cubic or a line. The latter case was classified in [\textit{H. Tokunaga}, J. Math. Kyoto Univ. 44, No. 2, 255--270 (2004; Zbl 1084.14019)], where 18 types are listed.
In the paper under review, the authors study how these covers can obtained by the following procedure:
1) pulling back by a suitable rational map the triple cover of \({\mathbb C}^2\) defined by \(z^3+xz+y=0\), where \(x,y\) are affine coordinates;
2) taking normalization and Galois closure of the cover constructed in 10.
For each of the 18 types, an explicit descriptions of the rational map in 1) is given. triple covers; pull back construction Coverings in algebraic geometry, Singularities of surfaces or higher-dimensional varieties Non-Galois triple covering of \(\mathbb{P}^2\) branched along quintic curves and their cubic equations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper one studies deformations as in the title, with a special interest in comparing Galois actions on the fundamental groups of the initial and the deformed curves.
The crucial construction is of an explicit parametrization of a universal deformation of such a degenerate curve and especially of a certain 1-parameter subfamily.
Let \(M_{g,n}\) be the moduli stack over \(\mathbb Q\) of \(n\)-point marked smooth curves of genus \(g\). The monodromy representation of \(\pi _1(M_{g,n})\) on the pro-\(l\) fundamental group of an \(n\)-point punctured smooth curve of genus \(g\) gives rise to a tower of coverings of \(M_{g,n}\). As an important application, one proves here the Oda's prediction about the independence of the constant field of this tower with respect to \((g,n)\), for the hyperbolic case (i.e. \(2-2g-n < 0\)). universal deformation; fundamental group; moduli of \(n\)-marked curves; Dedekind domain; normal crossing; moduli stack; tower of coverings Y. Ihara and H. Nakamura,On deformation of maximally degenerate stable marked curves and Oda's problem, Journal für die reine und angewandte Mathematik487 (1997), 125--151. Coverings of curves, fundamental group, Coverings in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry, Galois theory, Algebraic moduli problems, moduli of vector bundles On deformation of maximally degenerate stable marked curves and Oda's problem | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The class of branched coverings over \(\mathbb C^2\) traditionally called exotic arouses interest because of its connection with the Jacobian conjecture. In this paper, we construct a series of examples of such coverings; in particular, methods of construction of coverings with arbitrarily many sheets, as well as with unknotted branch curves, are described. In addition, some topological characteristics of these coverings are computed, which allows us to answer some questions about a possible counterexample to the Jacobian conjecture. exotic coverings; homology groups Coverings in algebraic geometry, Jacobian problem, Complex manifolds Branched coverings over \(\mathbb C^2\) and the Jacobian conjecture | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Applying an idea of \textit{C. Voisin} [Invent. Math. 201, No. 1, 207--237 (2015; Zbl 1327.14223)], we prove that a double cover of $\mathbb{P}^4$ or $\mathbb{P}^5$ branched along a very general quartic hypersurface is not stably rational. Coverings in algebraic geometry, Rationality questions in algebraic geometry, Rational and unirational varieties A very general quartic double fourfold or fivefold is not stably rational | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 prime characteristic \(p\). Consider an operator \(x\) on a finite dimensional vector space \(M\) with \(x^p=0\). The Jordan canonical form of \(x\) consists of blocks of size at most \(p\) and is determined by the number of blocks of each size. This information is encoded by saying that the Jordan type of the action is \([p]^{a_p}[p-1]^{a_{p-1}}\cdots[2]^{a_2}[1]^{a_1}\) where there are \(a_i\) blocks of size \(i\). Let \(E\) be an elementary Abelian group. Given a \(kE\)-module \(M\), \(M\) is said to be of \textit{constant Jordan type} if the Jordan types of certain nilpotent operators \(x\in KE\) over all field extensions \(K/k\) are in fact the same. This concept was introduced (in the more general setting of finite group schemes) by \textit{J. F. Carlson, E. M. Friedlander}, and the second author [J. Reine. Angew. Math. 614, 191-234 (2008; Zbl 1144.20025)]. These modules have seen extensive study and application in recent years. One of these applications has been to the construction of algebraic vector bundles on projective varieties, which is the goal of the paper under review.
For each \(1\leq i\leq p\), the authors construct a functor \(\mathcal F_i\) from the category of finitely generated \(kE\)-modules to the category of coherent sheaves on the projective space \(\mathbb P^{r-1}\), where \(r\) is the rank of \(E\). For a module \(M\) of constant Jordan type \([p]^{a_p}[p-1]^{a_{p-1}}\cdots[2]^{a_2}[1]^{a_1}\), \(\mathcal F_i(M)\) is a locally free sheaf (equivalently, algebraic vector bundle) of rank \(a_i\). The authors consider how the functors behave with respect to syzygies of modules and Serre shifts of sheaves. In particular, it is shown, for \(1\leq i\leq p-1\), that \(\mathcal F_i(M)(-p+i)\cong\mathcal F_{p-i}(\Omega M)\) where the sheaf on the left denotes the \((-p+i)\)-Serre shift of \(\mathcal F_i(M)\). It is also shown that \(\mathcal F_i\) takes the dual of a module to a shift of the dual sheaf.
The main result of the paper is that the functor \(\mathcal F_1\) can be used to almost realize vector bundles on \(\mathbb P^{r-1}\). In particular, for \(p=2\) and any vector bundle of \(\mathcal F\) of rank \(s\) on \(\mathbb P^{r-1}\), there exists a finitely generated \(kE\)-module \(M\) of Jordan type \([p]^t[1]^s\) for some \(t\) (or what is known as \textit{stable} Jordan type \([1]^s\)) such that \(\mathcal F_1(M)\cong\mathcal F\). For odd primes, it is shown that the pullback of \(\mathcal F\) along the Frobenius morphism \(F\colon\mathbb P^{r-1}\to\mathbb P^{r-1}\) can similarly be realized.
In the odd prime case, the authors explain in a sense why \(\mathcal F\) itself cannot be realized. If a module \(M\) has stable Jordan type \([1]^s\), it is shown that the Chern numbers \(c_m(\mathcal F_1(M))\) are divisible by \(p\) for \(1\leq m\leq p-2\). The authors note that the divisibility criterion does not necessarily hold for modules that have other constant Jordan types. Indeed, the authors provide examples throughout the paper to illustrate the ideas and results. elementary Abelian groups; modules of constant Jordan type; vector bundles; projective varieties; coherent sheaves; Chern numbers Benson, D; Pevtsova, J, A realization theorem for modules of constant Jordan type and vector bundles, Trans. Amer. Math. Soc., 364, 6459-6478, (2012) Modular representations and characters, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Group rings of finite groups and their modules (group-theoretic aspects), Finite abelian groups, Group schemes, Grassmannians, Schubert varieties, flag manifolds A realization theorem for modules of constant Jordan type and vector bundles. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 considers versal \(D_8\)-covers where \(D_8\) is the dihedral group of order 8. It is known that any versal \(D_8\)-cover has dimension at least 2 [cf. \textit{J. Buhler} and \textit{Z. Reichstein}, Compos. Math. 106, 159--179 (1997; Zbl 0905.12003)], and one of such models was given by the author [2-dimensional versal \(S_4\)-covers and rational elliptic surfaces, to appear in Séminaire et Congrés, Société Mathematique de France]. The purpose of this article is to give another new model, which is described as follows: Let \(\varphi_{141}: X_{141}\to\mathbb{P}^1\) be the rational elliptic surface obtained by blowing-up base points of the \((2,2)\)-pencil on \(\mathbb{P}^1 \times\mathbb{P}^1\) given by
\[
\bigl\{\lambda_0(s_0-s_1)^2(t_0-t_1)^2+\lambda_1 (s_0 s_1t_0t_1)=0\bigr\}_{[\lambda_0:\lambda_1]\in\mathbb{P}^1}.
\]
It is well-known that
(i) \(X_{141}\) is the elliptic modular surface attached to
\[
\Gamma_1(4):= \left\{\left(\begin{matrix} a& b\\ c& d\end{matrix}\right)\in \text{SL}(2,\mathbb{Z}) \left|\left (\begin{matrix} a & b\\ c &d\end{matrix}\right)\right. \equiv\left(\begin{matrix} 1 & *\\ 0 & 1 \end{matrix}\right)\text{mod}\,4\right \},
\]
and \(\varphi_{141}\) has three singular fibers and their types are of \(I^*_1\), \(I_4\) and \(I_1\)
(ii) The group of sections, \(MW(X_{141})\), is isomorphic to \(\mathbb{Z}/4\mathbb{Z}\). Let \(\sigma_{\varphi 141}\) be the involution on \(X_{141}\) induced by the inversion with respect to the group law and let \(\tau_s\) be the translation by a 4-torsion section \(s\). \(\sigma_{141}\) and \(\tau_s\) generate a finite fiber preserving automorphism group isomorphic to \(D_8\). Let \(\Sigma_{141}:= X_{141}/ \langle \sigma_{\varphi 141},\tau_s)\rangle\) be the quotient surface and we denote its quotient morphism by \(\pi_{141}:X_{141}\to \Sigma_{141}\). The author proves that this morphism is a versal \(D_8\)-cover. Moreover, this model has a nice description with respect to the action of the Galois group. rational elliptic surface; elliptic modular surface Tokunaga, H, Note on a \(2\)-dimensional versal \(D_8\)-cover, Osaka J. Math., 41, 831-838, (2004) Coverings in algebraic geometry, Elliptic surfaces, elliptic or Calabi-Yau fibrations Note on a 2-dimensional versal \(D_8\)-cover | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 connected curve defined over an algebraically closed field of characteristic \(p>0\) and let \(\pi_1(X)\) be its fundamental group. In [Pac. J. Math. 192, 143--158 (2000; Zbl 1068.14504)] the reviewer and \textit{K. Stevenson} considered a finite group \(G\) which is an extension of a group \(H\) of order prime to \(p\) by a \(p\)-group \(P\) and asked when such a group is a finite quotient of \(\pi_1(X)\), or equivalently whether the embedding problem \((\pi_1(X)\twoheadrightarrow H,G\twoheadrightarrow H)\) can be properly solved.
The condition for this to happen was expressed in representation theoretic terms by means of the generalized Hasse-Witt invariants [\textit{S. Nakajima}, in: Galois groups and their representations, Proc. Symp. Nagoya 1981, Adv. Stud. Pure Math. 2, 69--88 (1983; Zbl 0529.14016); \textit{H.-G. Rück}, J. Number Theory 22, 177--189 (1986; Zbl 0581.14025)]. In the paper under review the author extends this result excluding the hypothesis on \(H\) to be of order prime to \(p\). He does this by means of modular representation theory. fundamental groups Borne, N., A relative Shafarevich theorem, Math. Z., 248, 351-367, (2004) Coverings of curves, fundamental group, Modular representations and characters A relative Shafarevich 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 Contents: -- Introduction -- 1. Galois theory of fields 2. Fundamental groups in topology 3. Riemann surfaces 4. Fundamental groups of algebraic curves 5. Fundamental groups of schemes 6. Tannakian fundamental groups -- Bibliography and index.
Ever since the concepts of Galois groups in algebra and fundamental groups in topology emerged during the nineteenth century, mathematicians have known of the strong analogies between the two concepts. This book presents the connection starting at the graduate level, showing how the judicious use of algebraic geometry gives access to the powerful interplay between algebra and topology that underpins much modern research in geometry and number theory.
The first three chapters of the book cover basic algebraic, topological and analytical concepts, presented from the modern viewpoint advocated by Grothendieck. The first chapter consists of a concise introduction to classical Galois theory. The second chapter focusses on topology, discussing Galois covers, local systems and the monodromy action. The third chapter unites the algebraic and the topological picture by introducing Riemann surfaces and establishing the correspondence between finite Galois branched covers of a Riemann surface \(X\) and finite Galois extensions of the field of meromorphic functions on \(X\).
This background enables then a systematic and very accessible development of the theory of fundamental groups of algebraic curves in a first step, and fundamental groups of schemes in a second step. At the heart of the book under review lies Grothendieck's construction of the étale fundamental group \(\pi_1^{\mathrm{et}}(S,s)\) of a pointed connected scheme \(S\), and the correspondence between finite étale covers of \(S\) and finite continuous left \(\pi_1^{\mathrm{et}}(S,s)\)-sets. Further topics are the homotopy exact sequence splitting the étale fundamental group in its arithmetic and its geometric part, structure results for étale fundamental groups, comparison with topological fundamental groups and the abelianised fundamental group.
Input from algebraic geometry is kept at a minimum and is always explained in a honest manner, with appropriate references, and illustrated by examples.
The last chapter of the book is about Tannakian fundamental groups. Large parts of it are logically independent from the rest of the book, except for the last two section which discuss differential fundamental groups, Nori's fundamental group scheme and its relation with the étale fundamental group.
The book is consistently written in a clear and rigorous style, with a healthy balance between the abstract and the concrete. Much of the covered material appears for the first time in book form. The references given in the text satisfy those readers who seek further up-to-date information as well as those who are interested in the historical development of the theory. The typesetting is flawless.
The book suits well both, self study and class work. Each chapter is followed by an ample collection of instructive exercises. Key applications, for example on the inverse Galois problem, and remarks on recent results and open problems, for example on Grothendieck's section conjecture are given throughout. Thus, also the more advanced reader who already knows about étale fundamental groups will enjoy this reading matter. Galois groups; fundamental groups; Tannakian categories T. \textsc{Szamuely}, \textit{Galois Groups and Fundamental Groups}, Cambridge Studies in Advanced Mathematics, vol.~117, Cambridge Univ. Press, Cambridge, 2009. Research exposition (monographs, survey articles) pertaining to algebraic geometry, Coverings in algebraic geometry, Coverings of curves, fundamental group, Homotopy theory and fundamental groups in algebraic geometry, Separable extensions, Galois theory, Monoidal categories (= multiplicative categories) [See also 19D23] Galois groups and fundamental groups | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(S_ 0\) be a finite subset of the complex projective line \({\mathbb{P}}^ 1\) containing the points \(0, 1\quad and\quad \infty.\) Let \(\ell\) be a fixed prime number. In this situation, the authors study the category \(X(S_ 0)\) of all those finite branched coverings f: \(Y\to {\mathbb{P}}^ 1\), with Y being a complete, smooth, irreducible complex curve, such that the degree of the Galois closure of f is a \(power\quad of\quad \ell\) and f is unramified outside \(S_ 0\). More precisely, they show that there is a smallest common field \(\Omega =\Omega (S_ 0)\) of definition for all objects and morphisms in \(X(S_ 0)\), which implies that each object and morphism admits a canonical \(\Omega\)-model, and the basic aim of their studies is to obtain an explicit description of \(\Omega\) in terms of circular \(\ell\)-units. The main result in this direction is the proof of the fact that the common field of definition \(\Omega\) coincides with an explicitly constructible infinite, non-abelian \(pro-\ell \quad extension\) of the field \({\mathbb{Q}}(\mu_{\ell^{\infty}})\) unramified outside \(\ell\) and containing the group of circular \(\ell\)-units.
This very deep result is basically obtained by the following ingenious trick. It is shown that there are ``sufficiently many'' elementary objects of the form f: \({\mathbb{P}}^ 1\to {\mathbb{P}}^ 1\) in \(X(S_ 0)\), and those, together with their cusps, are studied first. This, in combination with investigating the Puiseux expansion at a cusp of an elementary object, is the basic idea for proving the main theorem. Since such interesting curves like the Fermat curve of level \(\ell^ n\), the Heisenberg curve of level \(\ell^ n\), and the modular curve of level \(2^ n\) are contained in the category \(X(S_ 0)\), the author's results imply that the Jacobian of any one of these curves has only \(\ell\)-power division points which are rational over the common field of definition of \(X(S_ 0)\). It is announced that the consequences and generalizations of the latter results will be discussed in a forthcoming paper under the same title [Part II: Int. J. Math. 1, No.2, 119-148 (1990)]. finite branched coverings of complex curve; ramification; algebraic curve; algebraic fundamental group; common field of definition Anderson, G.; Ihara, Y., Pro-\textit{l} branched coverings of \(\mathbb{P}^1\) and higher circular \textit{l}-units, Ann. of Math. (2), 128, 2, 271-293, (1988) Coverings of curves, fundamental group, Arithmetic ground fields for curves, Coverings in algebraic geometry, Arithmetic theory of algebraic function fields, Global ground fields in algebraic geometry Pro-\(\ell\) branched coverings of \(P^ 1\) and higher circular \(\ell\)- units | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 Shafarevich conjecture for holomorphic convexity states that the universal cover of a smooth complex projective variety is holomorphically convex.
The author gives an account of what is known concerning this conjecture in the case of surfaces and gives a proof of the following result that he presents as joint work with B.P. Purnaprajna:
\textbf{Theorem:} Let \(X\) be a smooth complex surface with \(K_X\) ample and let \(f: X\to C\) be a smooth genus 2 fibration without reducible fibers; denote by \(F\) a general fiber of \(f\). Then:
(1) If \(f\) is not smooth, then the map \(\pi_1(F)\to \pi_1(X)\) is trivial.
(2) The Shafarevich conjecture is true for \(X\).
Reviewer's remark: The same statement, without the assumption that \(f\) has no reducible fiber, is the main result of the later joint paper [``On the Shafarevich conjecture for genus-2 fibrations'', Math. Ann. 343, No. 4, 791--800 (2009; Zbl 1170.14027)] by the author and \textit{B. P. Purnaprajna}. Shafarevich conjecture; holomorphically convex; projective surface; genus \(2\) fibration Coverings in algebraic geometry, Fibrations, degenerations in algebraic geometry, Surfaces of general type Shafarevich conjecture for genus-2 fibrations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(f_1, \dots, f_p\) be irreducible analytic functions defined in a neighborhood of the origin in \({\mathbb{C}}^n.\) Then \(f = f_1^{\lambda_1} \cdots f_p^{\lambda_p}, \lambda_i \in {\mathbb{C}},\) is a multivalent function germ. Suppose that the unit is not contained in the additive group generated by the exponents \(\lambda_i.\) Then the range of \(f\) belongs to the quotient of \({\mathbb{C}}^\ast\) by the multiplicative group generated by \(\exp(2\pi\sqrt{-1}\lambda_i), i = 1, \dots, p.\) Thus, analogs of the notions of Milnor fibration and regular fibers of \(f\) are well-defined [\textit{E. Paul}, Ann. Inst. Fourier 45, No. 1, 31-63 (1995; Zbl 0819.32014)]. The author describes sufficient conditions on the exponents \(\lambda_i\) under which the fibres of \(f\) are connected. In fact, he generalizes the corresponding results of \textit{D. Cerveau} and \textit{J. F. Mattei}, `Formes intégrables holomorphes singulières', Astérisque 97 (1982; Zbl 0545.32006), and some others obtained in the holomorphic (univalent) case. Liouvillian function; Milnor fibration; closed logarithmic forms; residues; solvable holonomy; non-dicritical curves; desingularization without dicritical components of a divisor; divisors with normal crossings; Galois covering; Clemens structure Paul, E, Connectedness of the fibers of a Liouvillian function, Publ. Res. Inst. Math. Sci., 33, 465-481, (1997) Singularities of holomorphic vector fields and foliations, Meromorphic functions of several complex variables, Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\), Dynamics induced by flows and semiflows, Global theory and resolution of singularities (algebro-geometric aspects), Coverings in algebraic geometry, Hypersurfaces and algebraic geometry Connectedness of the fibers of a Liouvillian 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 Nach dem Vorbild von Hirzebruchs Konstruktion der Kummer-Überlagerungen zu Geradenkonfigurationen [vgl. \textit{G. Barthel}, \textit{F. Hirzebruch} and \textit{T. Höfer}, ``Geradenkonfigurationen und algebraische Flächen'' (Braunschweig 1987; Zbl 0645.14016)] betrachtet der Verfasser allgemeiner solche Überlagerungen \(Y_ n\) zu einer Konfiguration linear äquivalenter Kurven in einer komplexen Fläche und untersucht die Probleme: Wann gibt es hinreichend viele Faserungen von \(Y_ n\), so daß alle holomorphen Einsformen durch zurückgeholte Formen erzeugt werden? Wie hängt die Irregularität \(q(Y_ n)\) von n ab? Wann ist \(q(Y_ n)\) eine kombinatorische Invariante der Konfiguration?
Im 2. Kapitel werden entsprechende Probleme für nicht Kummersche Überlagerungen der Produktfläche \(T\times T\) eines Torus betrachtet.
Im 3. Kapitel wird ein Verschwindungssatz für infinitesimale Deformationen der Kummerschen Flächen \(Y_ n\) bewiesen. Daraus folgt die Starrheit von \(Y_ n\) für gewisse Geradenkonfigurationen. deformations; configuration of curves in a complex surface; irregularity; Kummer covering [Zu] Zuo, K.: Kummer-Überlagerungen algebraischer Flächen. Bonn. Math. Schr.193 (1989) Coverings in algebraic geometry, Special surfaces, Configuration theorems in linear incidence geometry, Enumerative problems (combinatorial problems) in algebraic geometry Kummer-Überlagerungen algebraischer Flächen. (Kummer coverings of algebraic surfaces) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper is motivated by the following question of Catanese: ``Does the absolute Galois group \(\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\) act faithfully on the connected components of the moduli space of surfaces of general type?'' The authors give a partial answer to this question, by showing that \(\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\) acts faithfully on the irreducible components of the moduli space.
The idea of the proof is the following: given \(z\in \bar{\mathbb{Q}}\), one associates to \(z\) a surface \(X_z\) defined over \(\bar{\mathbb{Q}}\) in such a way that for \(\sigma\in \text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\) we have \(\sigma(X_z)=X_{\sigma(z)}\). The surface \(X_z\) is an abelian cover of the plane and \(z\) can be recovered from the branch locus of the covering map as the cross ratio of certain special points. In addition, the construction is carried out in such a way that the covering map from \(X_z\to\mathbb{P}^2\) is intrinsically attached to \(X_z\) and extends to all surfaces in the irreducible component of the moduli space containing \(X_z\). As a result, given \(\sigma \in \text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\) and \(z\in \bar{\mathbb{Q}}\) not fixed by \(\sigma\), one is able to show that \(X_z\) and \(X_{\sigma(z)}\) lie in different irreducible components of the moduli space. surfaces of general type; moduli; Belyi theorem Easton, R.W., Vakil, R.: Absolute Galois acts faithfully on the components of the moduli space of surfaces: a Belyi-type theorem in higher dimension. Int. Math. Res. Not. IMRN, (20), 10, Art. ID rnm080 (2007) 10.1093/imrn/rnm080 Surfaces of general type, Coverings in algebraic geometry Absolute Galois acts faithfully on the components of the moduli space of surfaces: a Belyi-type theorem in higher dimension | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors study the affine curves with bounded genus and number of points at infinity on smooth affine surface \(X\), where \(\pi: X \to \mathbb{G}_{m}^{2}\) is a finite morphism, and obtain bounds for their degree in terms of Euler characteristic \(\chi_{\mathcal{C}} = 2g - 2 + \sharp S\). Here \(g\) is the genus of the smooth projective curve \(\tilde \mathcal{C}\), \(S\) is a finite nonempty subset of \(\tilde \mathcal{C} \) of the cardinality \(\sharp S\), \(\mathcal{C} = {\tilde \mathcal{C}} \backslash S\).
A typical example where these bounds hold is represented by the complement of a three-component curve in the projective plane of total degree at least 4. The corresponding results may be interpreted as bounding the height of integral points on \(X\) over a functional fields. In the language of Diophantine Equations, authors results may be rephrased in terms of bounding the height of the solutions of \(f(u,v,y) = 0\), with \(u, v, y\) over a functional field, \(u, v\) \(S\)-units. It turns out that all of this contain some cases of strong version of a conjecture by \textit{P. Vojta} [Diophantine approximations and value distribution theory. Berlin etc.: Springer-Verlag (1987; Zbl 0609.14011)] over function fields in the split case. Moreover, the authors' methods would apply also to the nonsplit case. The authors of the paper under review remark that special cases of their results in the holomorphic context were studied by M. Green already in the seventies, and recently in greater generality by Noguchi, Winkelmann and Yamanoi [\textit{J. Noguchi} et al., J. Math. Pures Appl. (9) 88, No. 3, 293--306 (2007; Zbl 1135.32018)]; however, the algebraic context was left open and seems not to fall in the existing techniques.
Let us now describe contents and results of this paper in more detail. Let \(\kappa\) be an algebraically closed field of characteristic zero, let \(\mathcal{O}_{S}= \kappa [\mathcal{C}]\) be the ring of regular functions on the affine curve \(\mathcal{C}\). The integral points \(X(\mathcal{O}_{S})\) on the surface \(X\) correspond to curves \(Y\) (not necessary smooth) lying on \(X\), parameterized by the smooth curve \(\mathcal{C}\), and authors put \(\chi_{Y} = \chi_{\mathcal{C}}\).
Theorem 1. (Voita's conjecture for \({\mathbb P}_{2}\) minus three divisors). Let \(D = D_{1} + \dots + D_{r} \subset {\mathbb P}_{2}\) be a curve whose components \(D_{i}\) meet transversally at each point of intersection. Suppose \(r \geq 3\) and \(\deg(D) \geq 4\). Then there exists a number \(C_{1} = C_{1}(D)\) such that for every curve \(Y \subset {\mathbb P}_{2} \backslash D \), we have the following bound: \(\deg(Y) \leq C_{1}\cdot \max\{1, \chi_{Y}\}\).
Let \(\tilde X\) be a smooth complete surface, \(K_{\tilde X }\) be a canonical divisor on \(\tilde X, X = \tilde X \backslash D.\) Recall follow to \textit{O. Debarre} [Higher-dimensional algebraic geometry. New York, NY: Springer (2001; Zbl 0978.14001)] that \(X\) is log-general type if \(h^{0}(n(D + K)) \gg n^2.\)
Theorem 2. Let \(Z \subset X\) be the ramification divisor of the finite map \(\pi: X \to \mathbb{G}_{m}^{2}.\) Assume that the closure of \(\pi(Z)\) in \({\mathbb P}_{2}\) does not intersect the set of singular points of the boundary of \(\mathbb{G}_{m}^{2}\) in \({\mathbb P}_{2}\) -- i.e., \((0:0:1), (0:1:0), (1:0:0).\) If \(X\) is of log-general type, then there exists a number \(C_2 = C_2(X, \pi)\) such that for every curve \(Y \subset X\) of Euler characteristic \(\chi_{Y}\) the following inequality holds: \(\deg(Y) \leq C_{2}\cdot \max\{1, \chi_{Y}\}\).
The surface \(X\) is defined in \(\mathbb{G}_{m}^{2} \times \mathbb{A}^1\) up to birationality, an equation of the form \(f(u, v, y) = 0.\)
Theorem 3. Let \(f(U, V, Y) \in \kappa[U, V, Y] \) be an irreducible polynomial, monic in \(Y.\) Suppose that the discriminant \(\Delta(U, V) \in \kappa[U, V] \) of \(f\) with respect to \(Y\) has no multiple nonmonomial factors. Then, for an affine curve \(\mathcal{C}\) as above, one of the following cases occurs:
(a) There exist numbers \(C_{3}, C_{4},\) effectively computable in terms of \(f\) and \(\chi = \chi_{\mathcal{C}} \), such that the solution \((u, v, y) \in \mathcal{O}_{S}^{*} \times \mathcal{O}_{S}^{*} \times \mathcal{O}_{S}\) of the equation \(f(u, v, y) = 0\) satisfy either
\[
\max \{\deg(u), \deg(v), \deg(y)\} \leq C_3
\]
or a multiplicative dependence relation \(u^a = \lambda \cdot v^b\) for a pair \((a, b) \in \mathbb{Z} \backslash \{(0, 0) \}\) with \(\max \{ |a|, |b|\} \leq C_4\) and \(\lambda \in \kappa^*. \)
(b) After an automorphism of \(\mathbb{G}_{m}^{2},\) there is a \(Q \in \kappa[U]\) such that \(\Delta(U, V) = Q(U)V^a\) and \(f(U, V, Y) = V^{l}P(U, V^{m}Y + A(U, V)),\) where \(P \in \kappa[U^{\pm 1}, W], A(U, V) \in \kappa[U^{\pm 1}, V^{\pm 1}, a, l, m \in \mathbb{Z}, a \geq 0.\) Also, either \(f(U, V, Y) = (Y + A(U, V))^d - bV^n U^p,\) where \(b \in \kappa^* , d, n, p\) are positive integers, or \(\deg(u) \leq C_5,\) and the number of possible \(u\) is finite, bounded only in terms of \(\deg f\) and \(\chi.\)
In Section 2 the authors of the paper under review are concerned with the connections between ``geometric'' and ``Diophantine'' languages, present preliminaries on heights and the proof of Theorem 2. The main result on the ramification divisor \(Z \subset X\) of \(\pi: X \to \mathbb{G}_{m}^{2}\) which is needed in the proof of Theorem 2 is as following:
Proposition 1. Let \(X, Z\) be as above; for every \(\epsilon > 0\) and every integer \(\chi \) there exists a number \(C = C(X, \pi, \epsilon, \chi) = C'(X, \pi, \epsilon) \cdot \max(1, \chi)\) such that for every morphism \(\varphi : \mathcal{C} \to X\) (with \( \chi_{\mathcal{C}} = \chi\)) of height \(H > C,\) with \(\varphi(\mathcal{C}) \not \subset Z,\) the degree of the divisor \(\varphi^{*}(Z) \) satisfies \(\deg (\varphi^{*}(Z)) \leq \epsilon H.\) In particular, the number of points \(p \in \mathcal{C}\) such that \(\varphi(p) \in Z\) is bounded by \(\epsilon H.\)
The proof of Theorem 2 rests also on the \(\mathrm{gcd}\) estimates of the authors [J. Algebr. Geom. 17, No. 2, 295--333 (2008); addendum Asian J. Math. 14, No. 4, 581--584 (2010; Zbl 1221.11146)].
In Sections 3 and 4 they present correspondingly proofs of Theorem 1 and Theorem 3.
Finally, Section 5 contains interesting examples as well as a counter-example, showing that the condition on the normal crossing of the of the divisor \( D_{1} + \dots + D_{r}\) in Theorem 1 cannot be omitted.
The paper should be read by everybody interested in algebraic hyperbolicity and Vojta's conjecture, not only because of the plenitude of information that it contains but also because of the examples and counter-example at the end. smooth affine surface; affine curve; finite morphism; degree of finite morphism; Euler characteristic; ramification divisor; height; log-Kodaira dimension; Vojta's conjecture; algebraically degenerate holomorphic curve Corvaja, Pietro; Zannier, Umberto, Algebraic hyperbolicity of ramified covers of \(\mathbb {G}^2_m\) (and integral points on affine subsets of \(\mathbb {P}_2\)), J. Differential Geom., 0022-040X, 93, 3, 355\textendash 377 pp., (2013) Arithmetic varieties and schemes; Arakelov theory; heights, Algebraic functions and function fields in algebraic geometry, Elliptic curves over global fields, Coverings in algebraic geometry, Height functions; Green functions; invariant measures in arithmetic and non-Archimedean dynamical systems Algebraic hyperbolicity of ramified covers of \(\mathbb{G}^2_m\) (and integral points on affine subsets of \(\mathbb{P}_2\)) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We study special algebraic curves being a covering over tori and associated with elliptic solitons. To describe the cover, we use an ansatz for the elliptic Baker-Akhiezer functions which accentuates the work of Hermite. We give a description of the elliptic solitons yielded by two-gap Lamé and Treibich-Verdier potentials; we also consider three and four-gap Lamé potentials and the Halphen equation with \(n=4\). Examples of elliptic solutions for integrable dynamical systems are constructed against the background of the developed approach. coverings of tori; Baker-Akhiezer functions; Lamé potentials; Halphen equation V. Z. Ènol\(^{\prime}\)skiĭ and N. A. Kostov, On the geometry of elliptic solitons, Acta Appl. Math. 36 (1994), no. 1-2, 57 -- 86. KdV equations (Korteweg-de Vries equations), Special algebraic curves and curves of low genus, Coverings in algebraic geometry On the geometry of elliptic solitons | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 Mat. Zametki 33, No.3, 459-462 (1983; Zbl 0516.14008). double plane; irregularity; ramification; stratawise separating covering S. I. Khashin, The irregularity of double surfaces. Mathematical notes of the Academy of Sciences of the USSR March 1983, Volume 33, Issue 3, 233-235. Ramification problems in algebraic geometry, Special surfaces, Coverings in algebraic geometry The irregularity of double surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be an irreducible projective variety, \(L\) a nef line bundle on \(X\) and \(x\) a point on \(X\). The Seshadri constant \(\varepsilon(L,x)\) of \(L\) at \(x\) is defined as the infimum of the set \(\{L \cdot C/\)mult\(_x(C)\}\) where \(C\) belongs to the set of reduced irreducible curves on \(X\) trough \(x\) and measures the local positivity of \(X\) at \(x\). For \(X\) a smooth surface it is known that \(\varepsilon(L,x)\leq \sqrt{L^2}\) and when the constant does not reach this bound there is a reduced irreducible curve \(C \subset X\) computing the constant. This curve \(C\) is called \textit{Seshadri exceptional curve}.
The paper under review provides bounds for the Seshadri constants of \(n\)-cyclic branched covering of smooth surfaces \(Y\) of Picard number one. The idea is to consider points \(x\) in the ramification divisor and divisors that are invariant for the order \(n\) automorphism of \(X\) induced by the covering to adapt a previous study of the Seshadri constants of surfaces of Picard number one to the current situation.
The precise statement is the following (see Thm. 3.3): Let \(\pi:X \to Y\) be a \(n\)-cyclic branched covering of a smooth surface \(Y\) with \(\rho(Y)=1\) and \(L\) an ample generator of \(\text{NS}(Y)\). If \(x \in X\) is very general then \([\sqrt{(\pi^*L)^2}]\leq \varepsilon(\pi^*L,x)\leq \sqrt{(\pi^*L)^2}\). As an application, a computation of explicit values of the Seshadri constant of some cyclic branched coverings of the projective planes are provided. Some relations with Nagata conjecture are also presented. Seshadri constants; smooth surfaces; branched cyclic coverings Coverings in algebraic geometry, Divisors, linear systems, invertible sheaves Cyclic coverings and Seshadri constants on smooth 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 A well known result of \textit{R. Lazarsfeld} [Math. Ann. 249, 153-162 (1980; Zbl 0547.32006)] states that if \(f: X\to Y=\mathbb{P}^n\) is a finite map and \(X\) is smooth then \((f_*{\mathcal O}_X/{\mathcal O}_Y)^*(-1)\) is generated by global sections, hence, in particular, \(E_f:=(f_*{\mathcal O}_X/{\mathcal O}_Y)^*\) is ample. In this article, the authors consider in general the question of determining the manifolds \(Y\) such that for every finite map \(f: X\to Y\) with \(X\) smooth the sheaf \(E_f\) is globally generated, nef or ample. They give a number of partial results and counterexamples. For instance they prove:
(i) if \(Y\) is an abelian variety then \(E_f\) is always nef;
(ii) if \(Y\) is a curve, then \(E_f\) is ample iff \(f\) does not factorize through an unramified cover of \(Y\);
(iii) if \(Y\) is a Del Pezzo surface with \(K^2_Y\geq 5\) then \(E_f\) is globally generated;
(iv) if \(Y\) is a Hirzebruch surface \({\mathbf F}_r\) with \(r\geq 2\) and the negative section of \(Y\) is not contained in the branch locus of \(f\) then \(E_f\) is globally generated;
(v) if \(Y\) is a Fano manifold with Picard number equal to 1, then \(E_f\) is ample;
(vi) if \(Y\) is a Del Pezzo manifold such that \(-K_Y=(n-1)H\) with \(H^n\geq 5\), then \(E_f\) is globally generated. branched cover; ampleness of vector bundle; finite map; nef vector bundle; abelian variety; Hirzebruch surface; Fano manifold; Del Pezzo manifold Peternell T, Sommese A J. Ample vector bundles and branched coverings. Comm Algebra, 28: 5573--5597 (2000) Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Coverings in algebraic geometry, Ramification problems in algebraic geometry Ample vector bundles and branched coverings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper we prove that a normal Gorenstein surface dominated by \(\mathbb{P}^2\) is isomorphic to a quotient \(\mathbb{P}^2/G\), where \(G\) is a finite group of automorphisms of \(\mathbb{P}^2\) (except possibly for one surface \(V_8')\). We can completely classify all such quotients. Some natural conjectures when the surface is not Gorenstein are also stated. fundamental groups; Gorenstein normal projective surface; Gorenstein quotients Rational and ruled surfaces, Coverings in algebraic geometry, Group actions on varieties or schemes (quotients) On Gorenstein surfaces dominated by \(\mathbb P^2\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \({\mathcal D}\) be an algebraic curve over \({\mathbb{C}}\) which is also defined over a number field F. Let G be a finite group and let \({\mathcal C}\to {\mathcal D}\) be a G-galois branched covering of \({\mathcal D}\) whose branch points are defined over a finite extension of F. The main result is that the set of finite primes of F which ramify in the field of moduli of the covering (the fixed field of those automorphisms of \(\bar F\) over F which preserve the covering and its G-action) is contained in the union of the set of primes of bad reduction of \({\mathcal D}\), the set of primes dividing the order of G and the set of primes p for which the branch locus becomes singular modulo p. The proof proceeds by first examining the primes of bad reduction of the branched covering. field of definition; galois branched covering; field of moduli of the covering; bad reduction S. BECKMANN, Ramified primes in the field of moduli of branched coverings of curves. J. Algebra, 125, no. 1 (1989), pp. 236-255. Zbl0698.14024 MR1012673 Coverings of curves, fundamental group, Coverings in algebraic geometry, Families, moduli of curves (algebraic), Algebraic moduli problems, moduli of vector bundles, Arithmetic ground fields for curves Ramified primes in the field of moduli of branched coverings of curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems As a result of his proof of the Calabi conjecture, Yau established an inequality known since as the Miyaoka-Yau inequality, concerning Chern classes of holomorphic vector bundles of compact Kähler manifolds, generalizing an inequality known as the Bogomolov-Gieseker inequality. In the paper under review, the authors establish the Miyaoka-Yau inequality in terms of orbifold Chern classes for the tangent sheaf of any complex projective variety of general type with Kawamata log-terminal singularities and nef canonical divisor. In the case of equality for a variety with at worst terminal singularities, they prove that the associated canonical model is the quotient of the unit ball by a discrete group action. The proof involves the introduction of an appropriate definition of Higgs sheaves on singular spaces and an associated notion of stability. classification theory; uniformization; ball quotient; minimal model of general type; Miyaoka-Yau inequality; Higgs sheaf; KLT singularity; canonical model; stability; hyperbolicity; flat vector bundle Uniformization of complex manifolds, Rational and birational maps, Notions of stability for complex manifolds, Coverings in algebraic geometry, Minimal model program (Mori theory, extremal rays), Projective connections, Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills), (Equivariant) Chow groups and rings; motives, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry The Miyaoka-Yau inequality and uniformisation of canonical models | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For the entire collection see Zbl 0642.00005.]
This is a sequel to Part I by the first and the third author [J. Algebra 99, 324-336 (1986; Zbl 0592.18005)]. In the present Part II, a notion of spectrum of a \(C^{\infty}\)-ring is studied, in analogy with the Grothendieck spectrum of a commutative ring. It is a generic way of constructing a sheaf of local \(C^{\infty}\)-rings. It should be compared with the spectrum construction of \textit{E. J. Dubuc} [Am. J. Math. 103, 683-690 (1981; Zbl 0483.58003)], which gives an Archimedian local \(C^{\infty}\)-ring; thus, the residue field in the Dubuc case is always \({\mathbb{R}}\), whereas the residue fields of the present authors usually will be a larger (non-Archimedian) real closed fields. They aim at a combination of the methods of synthetic differential geometry with non- standard analysis, cf. the earlier paper by the first and the third author in Adv. Math. 65, 229-253 (1987; Zbl 0648.18006). spectrum of a \(C^{\infty }\)-ring; sheaf of local \(C^{\infty }\)-rings Moerdijk, I.; Van Quê, Ngo; Reyes, G. E.: Rings of smooth functions and their localizations, II. (1984) Categories in geometry and topology, Schemes and morphisms, Rings and algebras of continuous, differentiable or analytic functions Rings of smooth functions and their localizations. II | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A double solid is a double cover of \(\mathbb{P}^3\) branched along a surface of even degree. This paper deals with \textit{ordinary} double solids, which are those where the ramification surface has at worst ordinary singularities. In local holomorphic coordinates \(u,v,w,t\), the singularities of such a double solid are thus of three types: \(t^2 = uv\) (type \(A\)), \(t^2 = uvw\) (type \(T\)), and \(t^2 = u^2 -vw^2\) (type \(D\)). The authors describe the mixed Hodge structures on the homology groups of the more general class of ADT threefolds \(X\): they are pure with the exception of \(H^3(X)\), which is an extension of a Hodge structure of weight \(3\) by a Hodge structure of type \((1,1)\), with extension data determined by the Abel-Jacobi mapping to the intermediate Jacobian of the natural resolution of singularities of \(X\). They then study in detail the cyclide double solid (ramified along an irreducible quartic surface whose singular locus is a smooth plane conic), and show that the Torelli mapping, sending \(X\) to the polarized mixed Hodge structure on \(H_3(X)\), is six-to-one. double solid; ordinary singularities; Torelli theorem; ADT threefold M. Grooten and J. H. M. Steenbrink, Quartic double solids with ordinary singularities , preprint, 2008; available at Coverings in algebraic geometry, \(3\)-folds, Transcendental methods, Hodge theory (algebro-geometric aspects) Quartic double solids with ordinary 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 In the paper under review, the author provides a classification result on singular fibers of finite cyclic covering fibrations of a ruled surfaces with use of singularity diagrams. Let \(f: S \rightarrow B\) be a surjective morphism from a smooth complex projective surface \(S\) to a smooth projective curve \(B\) with connected fibers. The datum \((S,f,B)\) or simply \(f\) is called a fibrations. Here the author focuses on primitive cyclic covering fibrations of type \((g,h,n)\), i.e., it is a fibration of genus \(g\) obtained as the relatively minimal model of an \(n\) sheeted cyclic branched covering of another fibration of genus \(h\). In example, hyperelliptic fibrations are fibrations of type \((g,0,2)\). The main result of the paper provides a complete classification of all fibers of \(n=3\) cyclic covering fibrations of genus \(g=4\) of a ruled surface (\(h=0\)). Moreover, the author shows that the signature of a complex surface with this fibration is non-positive by computing the local signature for any fiber. Another result classifies all fibers of hyperelliptic fibrations of genus \(g=3\) into \(12\) types according to the Horikawa index, and furthermore the author shows that finite cyclic covering fibrations of a ruled surface have no multiple fibers if the degree of the covering is greater than \(3\). singular fiber; cyclic covering; fibered surface Fibrations, degenerations in algebraic geometry, Coverings in algebraic geometry Fibers of cyclic covering fibrations of a ruled surface | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We present a method for deciding when a regular abelian cover of a finite CW-complex has finite Betti numbers. To start with, we describe a natural parameter space for all regular covers of a finite CW-complex \(X\), with group of deck transformations a fixed abelian group \(A\), which in the case of free abelian covers of rank \(r\) coincides with the Grassmanian of \(r\)-planes in \(H^1(X,\mathbb{Q})\). Inside this parameter space, there is a subset \(\Omega^i_A(X)\) consisting of all the covers with finite Betti numbers up to degree \(i\).{
} Building up on work of Dwyer and Fried, we show how to compute these sets in terms of the jump loci for homology with coefficients in rank-1 local systems on \(X\). For certain spaces, such as smooth, quasi-projective varieties, the generalized Dwyer-Fried invariants that we introduce here can be computed in terms of intersections of algebraic subtori in the character group. For many spaces of interest, the homological finiteness of abelian covers can be tested through the corresponding free abelian covers. Yet in general, abelian covers exhibit different homological finiteness properties than their free abelian counterparts. Suciu, A. I., Yang, Y. and Zhao, G., ' Homological finiteness of abelian covers', \textit{Ann. Sc. Norm. Super. Pisa Cl. Sci.} (5) 14 ( 2015) 101- 153. MR3379489. Homology with local coefficients, equivariant cohomology, Coverings in algebraic geometry, Homological methods in group theory, Topological methods in group theory Homological finiteness of abelian covers | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper provides new examples that make explicit the Grothendieck correspondence between dessins d'enfants and Belyi pairs in some special cases. All examples are curves of genus 0 defined over \(\mathbb{Q}\), more precisely plane quadrics \(C_{m,n,p,q}\) depending on integer parameters \(m,n,p\) and \(q\), together with Belyi functions \(\lambda_{m,n,p,q}: C_{m,n,p,q} \to\mathbb{P}^1\). The author presents the corresponding (rather involved!) dessins by interpreting his curves as certain Hurwitz spaces and calculating the monodromy of the associated covering. He also studies in some detail the reduction of a plane quadric with integer coefficients at various primes; in particular he proves criteria for good and potentially good reduction of such a curve, and applies them to his examples. ramified covering of \(\mathbb{P}^1-\{0,1,\infty\}\); monodromy group; dessins d'enfants; Belyi pairs Couveignes, J. -M.: Quelques revêtements définis surq. Manuscripta math. 94, 409-445 (1997) Coverings of curves, fundamental group, Global ground fields in algebraic geometry, Coverings in algebraic geometry, Arithmetic ground fields for curves Some coverings defined over \(\mathbb{Q}\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 0518.00005.]
Let \(p : V \to \mathbb{P}^2\) be a general projection of a complex algebraic surface V in a projective space \(\mathbb{P}^3\) and \(p' : \mathbb{C}^3 \to \mathbb{C}^2\) be the corresponding projection of the affine spaces which induces p over the complement of a general line in \(\mathbb{P}^2\). Choose a linear projection \(\pi : \mathbb{C}^2 \to \mathbb{C}\) and a point \(a\in \mathbb{C}\) which does not belong to the image S of the branch curve W of p. Let \(\ell =\pi^{-1}(a)\), \(b\in \ell \setminus (\ell \cap W)\) and \(\ell '=p'{}^{-1}(b)\). The author proposes to study the monodromy homomorphisms:
\[
\phi : \pi_ 1({\mathbb{C}}\setminus S,a)\to Diff(\ell;p^{-1}(a)),\quad \psi: \pi_ 1(\ell \setminus (\ell \cap W)b)\to Diff(\ell ';p^{'-1}(b))
\]
as certain invariants of the surface V. Let \(\{\delta_ 1,...,\delta_ k\}\) (resp. \(\{\gamma_ 1,..,\gamma_ n\})\) be a basis of \(\pi_ 1({\mathbb{C}}\setminus S,a)\) (resp. \(\pi_ 1(\ell \setminus (\ell \cap W),b))\) obtained by taking loops which turn around each point of the set S (resp. \(\ell \cap W)\). Identifying the diffeomorphism groups with the corresponding Artin braid groups one can show that the product \(\phi (\delta_ 1)...\phi (\delta_ k)\) (resp. \(\psi (\gamma_ 1)...\psi (\gamma_ n))\) is equal to the standard generator \(\Delta^ 2\) of the center of the corresponding braid group. This leads the author to the brilliant idea that the geometry of V is ''coded'' in terms of the arithmetic of the braid group. More precisely, V defines an equivalence class of factorization of \(\Delta^ 2\) into products of elements of the braid group. So different surfaces V can be distinguished by the corresponding equivalence classes. In particular, considering general projections of pluricanonical models of surfaces of general type, the author hopes to be able to separate connected components of the moduli spaces. The realization of this idea is obviously a very difficult task and requires a lot of cumbersome combinatorial computations with braid groups. In the present paper the author begins this program by studying the braid monodromies of nonsingular surfaces of degree n in \({\mathbb{P}}^ 3\). By degenerating such a surface to the union of n planes in general position he investigates how the normal forms of the factorization of \(\Delta^ 2\) are transformed under such degeneration. The precise statement of the main result of the paper is too technical to reproduce here. braid monodromies of nonsingular surfaces of degree; n in projective 3- space; monodromy homomorphisms; diffeomorphism groups; Artin braid groups; general projections of pluricanonical models of surfaces of; general type; general projections of pluricanonical models of surfaces of general type Moishezon, B.G.: Algebraic surfaces and the arithmetic of braids. I, Arithmetic and geometry, vol. II, Progress in Mathematics, vol.~36. Birkhäuser, Boston, pp.~199-269 (1983) Coverings in algebraic geometry, Braid groups; Artin groups, Differential topological aspects of diffeomorphisms, Moduli, classification: analytic theory; relations with modular forms, Generators, relations, and presentations of groups Algebraic surfaces and the arithmetic of braids. I | 0 |
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