text stringlengths 571 40.6k | label int64 0 1 |
|---|---|
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The aim of this paper is to report on recent work on étale covers of the punctured disc. The paper surveys basic results on curves over \(p\)-adic fields and explains ideas of Riemann's existence problem for a \(p\)-adic field giving full details and as well new results. In [\textit{W. Lütkebohmert}, Invent. Math. 111, 309--330 (1993; Zbl 0780.32005)] the problem of the extension of an étale cover \(\phi^* : X^* \to \mathbb D^*\) of a punctured disc \(\mathbb D^*\), defined over a \(p\)-adic field \(K\), to a (ramified) cover \(\phi :X\to \mathbb D\) was proved in the case where the base field \(K\) has characteristic 0. The behavior of the discriminant of such an étale cover is now well understood. Moreover, new results of the second author [Étale Überlagerungen von \(p\)-adischen Kreisscheiben, Schriftenreihe d. Math. Inst. d. Uni. Münster; 3. Serie 29 (2001; Zbl 1002.14005)] in the case of positive characteristic \(\text{char}(K) > 0\) are presented as well. The paper ends with two interesting examples of covers in positive characteristic which are not extendable. Rigid analytic geometry, Coverings in algebraic geometry Étale covers of a punctured \(p\)-adic disc | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 is discussed the geometric case of the Van de Van- Bogomolov-Miyaoka-Yau inequality and it is given its arithmetic analogue. The author shows that the validity of the mentioned inequality yields a positive solution of the following problems of diophantine geometry of algebraic curves having
(I) genus \(=1:\)
(a) the boundedness of the torsion over any fixed base field,
(b) the lower bound of the Tate height of the infinite order points,
(c) an estimation of the number of integral points in terms of the rank of the curve,
(d) a new effective proof of Siegel theorem on integral points;
(II) genus \(>1:\)
(a) an effective proof of Mordell conjecture about rational points.
(b) the nonexistence of nontrivial points on Fermat curves having sufficiently large prime degree. Van de Van-Bogomolov-Miyaoka-Yau inequality; torsion; Tate height; number of integral points; Mordell conjecture [19] Parshin (A.~N.).-- On the application of ramified coverings in the theory of diophantine equations. Math. USSR Sbornik 60, p.~249-264 (1990). &MR~9 | &Zbl~0702. Arithmetic varieties and schemes; Arakelov theory; heights, Coverings in algebraic geometry, Arithmetic ground fields for curves, Rational points, Higher degree equations; Fermat's equation On the application of ramified coverings in the theory of diophantine 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 The existence of Ulrich bundles on projective varieties is relevant in the description of their geometric structure. Ulrich bundles are defined as vector bundles which are arithmetically Cohen-Macaulay, with maximal number of global sections in their first twist. It is conjectured that every smooth projective variety supports some Ulrich bundle \(E\), and indeed some large classes of projective varieties are known to satisfy the conjecture. On the other hand, it is hard in general to determine the minimal rank of Ulrich bundles, even for some specific natural classes of varieties \(X\). The athors consider the case where \(X\) is a double cover of \(\mathbb P^2\) branched along a curve \(B\) of even degree \(\geq 6\). A result of \textit{A. J. Parameswaran} and \textit{P. Narayanan} [J. Algebra 583, 187--208 (2021; Zbl 1473.14087)] proves that when \(B\) is general, then \(X\) has no Ulrich line bundles. The authors prove that for general \(B\) the double cover \(X\) supports Ulrich bundles of rank \(2\). The result is obtained by constructing on \(X\) special configurations of points, and then using the correspondence between finite sets with the Cayley-Bacharach property and rank \(2\) vector bundles on surfaces. Ulrich bundles; double planes Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Coverings in algebraic geometry Rank 2 Ulrich bundles on general double plane 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 Vermittelt man die conforme Abbildung zweier Ebenen durch die Gleichung \(Z= \log z\), so entspricht jedem unendlich langen Flächenstreifen der \(Z\)-Ebene, welcher parallel der reellen Axe und von der Breite \(2\pi\) ist, die ganze \(z\)-Ebene; dem Parallelen zur imaginären \((Y)\)-Axe entsprechen concentrische Kreise um den Nullpunkt, den Parallelen zur \(X\)-Axe, die durch den Nullpunkt gelegten Geraden der \(Z\)-Ebene; das Bild einer beliebigen Geraden der \(Z\)-Ebene ist eine logarithmische Spirale. Denkt man einen der oben genannten unendlich langen Flächenstreifen durch Parallelen zur \(X\)- und \(Y\)-Axe in congruente Rechtecke zerlegt, so ist das Bild dieses Netzes ein System ähnlicher und rechtwinkliger Flächenstücke, die von Radien und concentrischen Kreisen begrenzt werden; eine beliebige Gerade der \(Z\)-Ebene verbindet die Ecken der entsprechenden rechtwinkligen Flächenstücke in der \(Z\)-Ebene. Aus dieser Entstehungsart der Spirale lassen sich viele ihrer Grundeigenschaften schnell und übersichtlich ableiten, zu jeder Eigenschaft der Geraden in der \(Z\)-Ebene ergiebt sich eine analoge für die Spirale. Aus dem Satze z. B.: ``Die Curven, welche eine Schaar paralleler Geraden unter constantem Winkel schneiden, sind parallele Gerade'' folgt: ``Die Curven, welche eine Schaar gleichwinkliger logarithmischer Spiralen unter constantem Winkel schneiden, sind ebenfalls logarithmische Spiralen.'' Von praktischem Interesse erweist sich die logarithmische Abbildung, insofern sie den analytischen Zusammenhang zwischen der stereographischen und zwischen Mercators Projection giebt. Ist für erstere der Nordpol der abzubildenden Kugel Centrum, tangirt die Projectionsebene also den Südpol, so ist der Punkt \(Z\) der Mercator-Projection, welcher einem Punkte \(z\) des stereographischen Bildes entspricht, bestimmt durch die Gleichung \(Z=\log \left( \gamma + \frac{1}{z} \right)\), wo \(\gamma\) eine passend zu wählende Constante ist. Am Schluss der Arbeit befinden sich Bemerkungen über die durch doppelt periodische Functionen vermittelte Abbildung des Ellipsoids, resp. der Kugel auf das Innere eines Rechtecks; der Verfasser gelangt zu folgenden Sätzen: Bei der Abbildung des Ellipsoids auf die ganze Ebene entspricht dem System der Krümmungslinien eine Curvenschaar, welche durch stereographische Projection eines bestimmten Systems confocaler sphärischer Kegelschnitte entsteht. Bei der Abbildung der Ellipse auf das Innere oder das Aeussere eines Kreises entsprechen den confocalen Ellipsen und Hyperbeln stereographische Projectionen sphärischer Kegelschnitte; dieselben sind mit den von Siebeck (Borchardt J. LVII. u. LIX,) behandelten Curven \(4^{\text{ten}}\) Grades identisch, wenn der Mittelpunkt der Ellipse dem des Kreises entspricht. logarithmic covering Coverings in algebraic geometry On the logarithmic map and the orthogonal systems of curves to which it gives rise. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems If we want to study the constraints imposed on a complex manifold \(X\) by a large \(H_ 1 (M)\), we can use the Albanese map and so recover a variety (the Albanese image), which, in a sense, carries the information contained in that homology group. This suggests the following question. Does a formally similar set-up exist when the fundamental group replaces the first homology one? -- The starting point is the following conjecture by Shafarevich: Let \(X\) be a smooth complex projective variety and \(X^*\) its universal cover. There exists a proper surjective morphism \(s^* : X^* \to \text{Sh} (X^*)\) onto a normal Stein space \(\text{Sh} (X^*)\). If the conjecture is true, the fundamental group \(\pi_ 1 (X)\) acts on \(\text{Sh} (X^*)\), this giving rise to a morphism onto the quotient \(s : X \to \text{Sh} (X) = \text{Sh} (X^*)/ \pi_ 1 (X)\). Even when the action above mentioned has fixed points, technical devices allow to still consider \(\text{Sh} (X)\). So, if the conjecture is true, we recover a morphism \(s\) and a variety \(\text{Sh} (X)\), taking into account how large \(\pi_ 1 (X)\) is and thus helping to show the constraints imposed on the algebro-geometric properties of \(X\) by a large fundamental group. -- If \(s\) and \(\text{Sh} (X)\) exist, the fibres of \(s\) are characterized as those connected subvarieties \(Z\) -- assuming that \(s\) has connected fibres -- such that
\[
\text{the image of } \pi_ 1 (Z) \text{ in } \pi_ 1 (X) \text{ is finite}. \tag{*}
\]
All that suggests the following definition. Let \(X\) be a normal and proper variety. A normal variety \(\text{Sh} (X)\) and a rational map \(s : X \to \text{Sh} (X)\) are called the Shafarevich variety and the Shafarevich map of \(X\) if \(s\) has connected fibres and if a condition similar to \((*)\) characterizes the irreducible components of the fibres out of the union \(U\) of countably many closed proper subvarieties. -- As a first result the existence of the Shafarevich map is proved as well as that of variations of \(s\) defined by starting with the algebraic fundamental group or with normal subgroups. These maps are shown to be defined and proper on a large open set and their links with the Albanese morphism are pointed out according to the subgroups we start with. In view of condition \((*)\) the existence theorem decomposes the varieties into two classes according to whether \(\pi_ 1\) is finite or ``generically large'' \((s\) is birational, i.e. \(s\) contracts nothing out of \(U)\). In the former case this break-up applies to the fibres and so on. As to the smooth varieties \(X\) with generically large algebraic fundamental group, they should be built up by abelian varieties and varieties of general type. Actually such an \(X\) is conjectured to have a finite étale cover birational to a smooth family of abelian varieties over a projective variety of general type with generically large \(\pi_ 1\). The conjecture is proved if the Kodaira dimension of \(X\) is \(\geq \dim X - 2\).
Applications are given in many cases. Conditions on a normal analytic space are found ensuring that the surjection between the fundamental groups induced by a resolution of singularities is an isomorphism. A nonvanishing theorem is proved for varieties with generically large \(\pi_ 1\) and is applied to deduce information on the plurigenera of a smooth projective variety of general type. Further results on the plurigenera and on the pluricanonical map are given in the case of 3- folds of general type as well as numerical characterizations of varieties birational to Abelian ones. variety of general type; smooth complex projective variety; large fundamental group; Shafarevich variety; Shafarevich map; resolution of singularities; nonvanishing theorem; plurigenera; 3-folds of general type Kollár, J., Shafarevich maps and plurigenera of algebraic varieties, Invent. Math., 113, 176-215, (1993) Rational and birational maps, Coverings in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry, \(n\)-folds (\(n>4\)) Shafarevich maps and plurigenera of algebraic varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The question is posed whether an abstract group can be fundamental group of a compact Kähler or quasi-compact Kähler (obtained by deleting a normal crossing divisor) manifold? The answer is ``no'' for any cocompact lattice in \(\mathrm{SO}(n,1)\), \(n\geq 3\), [\textit{J. A. Carlson} and \textit{D. Toledo}, Publ. Math., Inst. Hautes Étud. Sci. 69, 173--201 (1989; Zbl 0695.58010)]. Another negative result is obtained in this paper, namely:
Let \(\Gamma\) be a nonuniform lattice of \(\mathrm{SO}(n,1)\), \(n\geq 4\). Let \(M\) be any compact Kähler manifold and let \(D\) be any normal crossing divisor of \(M\). Then \(\Gamma\), as an abstract group, is not isomorphic to \(\pi _1(M \setminus D)\).
The theorem is true in the case if \(M \setminus D)\) is quasi-projective variety according to the well known Hironaka's theorem.
The proposed proof uses the infinite energy harmonic maps theory of \textit{J. Jost} and \textit{K. Zuo} [J. Differ. Geom. 47, No. 3, 469--503 (1997; Zbl 0911.58012)]. non-uniform lattice; quasi-compact Kähler manifolds; fundamental group Yang Y. H., On non-Kählerianity of nonuniform lattices in SO(n, 1)(n 4), Manuscripta Mathematica, 2000, 103:401--407 Group structures and generalizations on infinite-dimensional manifolds, Harmonic maps, etc., Calabi-Yau theory (complex-analytic aspects), Coverings in algebraic geometry On non-Kählerianity of nonuniform lattices in \(\mathrm{SO}(n,1)\) \((n\geq 4)\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 review of the entire collection in [Zbl 1303.01004]. History of algebraic geometry, History of mathematics in the 20th century, Coverings in algebraic geometry On Grothendieck's work on the 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 Let \(X\) be a complex projective manifold, or more generally, let \(\Delta\) be an effective divisor on a normal projective variety \(X\) such that the pair \((X, \Delta)\) is log-canonical. The abundance conjecture claims that if the divisor \(K_X+\Delta\) is pseudoeffective, then some positive multiple \(m(K_X+\Delta)\) is even effective. This conjecture is known to hold for varieties of dimension at most three \textit{S. Keel, K. Matsuki} and \textit{J. McKernan} [Duke Math. J. 75, No.1, 99--119 (1994; Zbl 0818.14007)], but is wide open in higher dimension. In the paper under review the author proves this conjecture under the additional assumption that the numerical dimension (in the sense of [\textit{N. Nakayama}, Zariski-decomposition and abundance. MSJ Memoirs 14. Tokyo: Mathematical Society of Japan. (2004; Zbl 1061.14018)]) is equal to zero. More precisely the author proves that if \(K_X+\Delta\) is pseudoeffective and for every ample divisor \(A\) on \(X\) the sequence
\[
\dim H^0(X, m(K_X+\Delta)+A)
\]
is bounded, then we have \(H^0(X, m(K_X+\Delta)) \neq 0\) for some \(m \in \mathbb N\). This theorem generalises the work of N. Nakayama (ibid) and \textit{F. Campana, M. Toma} and \textit{T. Peternell} [Bull. Soc. Math. Fr. 139, No. 1, 41--74 (2011; Zbl 1218.14030)] for pairs \((X, \Delta)\) that are klt. Using Nakayama's divisorial Zariski decomposition the theorem can be reduced to the following statement: if we have \(H^0(X, m(K_X+\Delta+L)) \neq 0\) for some numerically trivial line bundle \(L\) on \(X\) and some \(m \in \mathbb N\), then there exists a \(m' \in \mathbb N\) such that \(H^0(X, m'(K_X+\Delta)) \neq 0\). This numerical character of the effectivity for adjoint line bundles has been proven independently by \textit{F. Campana, V. Koziarz} and \textit{M. Păun} [Ann. Inst. Fourier 62, No. 1, 107--119 (2012; Zbl 1250.14009)]. Both proofs are based on a fundamental result by \textit{C. Simpson} [Ann. Sci. Éc. Norm. Supér. (4) 26, No. 3, 361--401 (1993; Zbl 0798.14005)]. minimal model program; abundance conjecture; nonvanishing conjecture Y. Kawamata, On the abundance theorem in the case of numerical Kodaira dimension zero, Amer. J. Math. 135 (2013), no. 1, 115--124. Coverings in algebraic geometry, Fano varieties On the abundance theorem in the case of numerical Kodaira dimension zero | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The following theorem is proved: Let \(K/k\) be an extension of algebraically closed fields of infinite transcendence degree, and let \(Y\) be a \(\text{Spec}(k)\)-scheme. If \(X\) is an integral \(Y_K\)-scheme of finite type such that \(X^\sigma \cong_{Y_K} X\) for all \(\sigma\in \Aut_k(K)\), then \(X\) admits a unique \(Y\)-model. Field of moduli; Field of definition; Fibered product; Going-up theorem; Hilbert's Nullstellensatz Transcendental field extensions, Coverings in algebraic geometry Algebraically closed descent | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems String theory leads to a flavor scheme where modular symmetries play a crucial role. Together with the traditional flavor symmetries they combine to an eclectic flavor group, which we determine via outer automorphisms of the Narain space group. Unbroken flavor symmetries are subgroups of this eclectic group and their size depends on the location in moduli space. This mechanism of local flavor unification allows a different flavor structure for different sectors of the theory (such as quarks and leptons) and also explains the spontaneous breakdown of flavor- and \(\mathcal{CP}\)-symmetries (via a motion in moduli space). We derive the modular groups, including \(\mathcal{CP}\) and \(R\)-symmetries, for different sub-sectors of six-dimensional string compactifications and determine the general properties of the allowed flavor groups from this top-down perspective. It leads to a very predictive flavor scheme that should be confronted with the variety of existing bottom-up constructions of flavor symmetry in order to clarify which of them could have a consistent top-down completion.
For Part I, see [the authors, Phys. Lett., B 808, Article ID 135615, 7 p. (2020; Zbl 1473.81141)]. Kaluza-Klein and other higher-dimensional theories, Topology and geometry of orbifolds, Dimensional compactification in quantum field theory, Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations, Finite-dimensional groups and algebras motivated by physics and their representations, Symmetry breaking in quantum theory, Complex-analytic moduli problems, Strong interaction, including quantum chromodynamics, Modular representations and characters, Fine and coarse moduli spaces, Weak interaction in quantum theory Eclectic flavor scheme from ten-dimensional string theory. II: Detailed technical analysis | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 fundamental discoveries of \textit{S. Mori} [cf. Ann. Math., II. Ser. 110, 593-606 (1979; Zbl 0423.14006) and 116, 133-176 (1982; Zbl 0557.14021)] is the cone theorem. In its simplest form it can be stated as follows:
Theorem. Let \(X\) be a smooth projective variety and \(NE(X) \subset H_ 2 (X, \mathbb{R})\) the convex cone generated by the homology classes of algebraic curves on \(X\). Then \(NE(X)\) is ``locally polyhedral'' in the open half-space \(\{z \in H_ 2 (X, \mathbb{R}) | z \cdot K_ X < 0\}\), where \(K_ X\) is the canonical class.
Subsequently a cohomological approach to the cone theorem was developed in a series of articles (Kawamata, Reid, Shokurov and the author). This approach relies heavily on vanishing results of Kodaira type, and thus is applicable only in characteristic zero.
The aim of this article is to develop Mori's geometric approach further. There are at least two reasons to attempt this. The first is the hope of doing Mori's program in positive characteristic. Second, the geometric approach yields information about the cone of curves that is not yet available via the cohomological approach even in characteristic zero. I was unable to prove the formulas about deformations of curves in singular varieties that would make Mori's original proof work. Therefore the current proofs proceeds along slightly different lines. The first step is the observation that, in order to establish the cone theorem, one needs the desired formulas only for certain curves. More precisely, instead of deforming \(C \subset X\) it is sufficient to deform a morphism \(D \to C \subset X\) where \(D\) is any finite cover of \(C\) (possibly ramified or even inseparable). To study deformations of \(D \to X\) the following construction is used:
Lemma. Let \(X\) be a projective variety over \(\mathbb{C}\) with at worst isolated quotient singularities. Then there is a unique algebraic space \(Y\) and a morphism \(p : Y \to X\) with the following properties:
(1) \(Y\) is of finite type and \(Y \to \text{Spec} \mathbb{C}\) is universally closed;
(2) \(Y\) is smooth;
(3) \(p\) is an isomorphism over \(X - \text{Sing} X\);
(4) \(p\) is one-to-one on closed points.
I call \(Y\) the bug-eyed cover of \(X\) [compare the picture in \textit{M. Artin}'s paper in J. Algebra 29, 330-348 (1974; Zbl 0292.14013), p. 331]. -- For suitable choices of \(D\) there is a factorisation \(D \to Y \to X\). Since \(Y\) is smooth, the deformation theory of \(D\to Y\) is as expected. This way one obtains deformations of \(D \to X\) as well. These questions are discussed in \S2.
The main theorem (5.1) proves the cone theorem for varieties whose singularities are quotients of complete intersection singularities by an equivalence relation that is étale in codimension one. In characteristic zero this is a reasonable restriction; however in positive characteristic one would like to allow quotients by nonreduced group schemes as well (and possibly certain other inseparable covers). As an application of this approach one obtains the following result:
Theorem. Let \(X\) be a normal projective threefold over a field of characteristic zero. Assume that \(c_ 1(K_ X) \in N(S(X) \otimes \mathbb{Q}\) exists (e.g. \(K_ X\) is \(\mathbb{Q}\)-Cartier). Then \(NE(X)\) is ``locally polyhedral'' in the open half-space \(\{z \in N_ 1 (X) | z\cdot c_ 1 (K_ X) < 0\}\).
The aim of the last section is to investigate curves on a threefold \(X\) that have no deformations at all. A curve \(C \subset X\) is called very rigid if for every subscheme \(\overline C \subset X\) whose support is \(C\), every flat deformation of \(\overline C\) is supported on \(C\) (except possibly at finitely many points). Very rigid curves arise in connection with flips and flops. cone theorem; deformations of curves; bug-eyed cover; threefold János Kollár, Cone theorems and bug-eyed covers, J. Algebraic Geom. 1 (1992), no. 2, 293 -- 323. Coverings in algebraic geometry, \(3\)-folds, Formal methods and deformations in algebraic geometry, Families, moduli of curves (algebraic) Cone theorems and bug-eyed 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 X be an Enriques surface over an algebraically closed field k, char \(k\neq 2\). The canonical divisor \(K_ X\) which is the only non-zero element of order 2 in Pic X gives rise to an étale double covering \(u: Y\to X\), where Y is a K 3 surface. It is clear that the Enriques surfaces are in a one-to-one correspondence with the K 3 surfaces with a fixed points free involution. A standard example of K 3 surface with a fixed points free involution is given by an intersection of three quadrics in \({\mathbb{P}}^ 5\), where in a suitable system of coordinates the involution is given by the formula \(i(x_ 0:x_ 1:x_ 2:x_ 3:x_ 4:x_ 5)=(- x_ 0:-x_ 1:-x_ 2:-x_ 3:-x_ 4:-x_ 5).\) Roughly speaking, the goal of the paper under review is to show that each K 3 surface which can be represented in the form of étale double covering of an Enriques surface has a projective model of the form described in this example. Enriques surface; double covering; K 3 surface Verra, Alessandro: The étale double covering of an Enriques surface, Rend. semin. Mat. univ. Politec. Torino 41, No. 3, 131-167 (1983) Special surfaces, Coverings in algebraic geometry The étale double covering of an Enriques 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 The main result of this paper is the following:
Theorem: Let \(\widetilde S\to S\) be a non compact topological covering of a smooth algebraic surface and let \(\widetilde C\subset\widetilde S\) be a compact connected curve, then the intersection form on \(\widetilde C\) is semi-negative.
In his book ``Basic algebraic geometry'' (2nd edition 1994; Zbl 0797.14001), \textit{I. R. Shafarevich} conjectures that the universal covering of a smooth algebraic variety is holomorphically convex. Here the author remarks that this conjecture implies the theorem above in the case of the universal covering \(\widetilde S\to S\), provided that \(\widetilde S\) is not compact. So the results in this paper can be regarded as some kind of evidence for the Shafarevich conjecture. non compact covering; projective surface; semi-negative intersection form; holomorphically convex covering; Shafarevich conjecture Coverings in algebraic geometry, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Surfaces and higher-dimensional varieties Semi-negativity of compact curves in non compact covers of projective complex surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In J. Reine Angew. Math. 417, 191-213 (1991; Zbl 0721.14009), \textit{R. Pardini} has given a complete description of abelian covers of algebraic varieties in terms of the so-called building data, namely of certain line bundles and divisors on the base of the covering, satisfying suitable compatibility relations. Natural deformations of an abelian cover \(f:X \to Y\) are also introduced there and it is shown that they are complete, if \(Y\) is rigid, regular and of dimension \(\geq 2\), and if the building data are sufficiently ample.
In this paper we study natural deformations of an abelian cover \(f:X\to Y\) and prove that they are complete for varieties of dimension at least two if the branch divisors are sufficiently ample. The result requires no assumption on \(Y\), and in particular also holds when the cover has obstructed deformations; this is a key technical step towards the moduli space constructions described below. -- We then turn to the study of the automorphism group of the cover. Since the automorphism group of a variety of general type is finite, one would expect that in the case of a Galois cover it coincides with the Galois group, at least if the cover is generic. Our main theorem 4.6 shows that this is indeed the case for an abelian cover, if the branch divisors are generic and sufficiently ample. We construct explicitly coarse moduli spaces of abelian covers and complete families of natural deformations for a fixed base of the cover \(Y\).
The main application of the results described so far is the study of moduli of varieties with ample canonical class. Recently Viehweg proved the existence of a coarse moduli space for varieties with ample canonical class of arbitrary dimension, generalizing Gieseker's result for surfaces. Given an irreducible component \(M\) of the moduli space of varieties with ample canonical class, the automorphism group \(G_M\) of a generic variety in \(M\) is well-defined. In contrast with the case of curves (where this group is trivial for \(g\geq 3)\), it was already known in the case of surfaces that there exist infinitely many components \(M\) of the moduli with nontrivial automorphism group \(G_M\).
As a first application of theorem 4.6 we prove that for any finite abelian group \(G\) there are infinitely many irreducible components \(M\) of the moduli of varieties with ample canonical class such that \(G_M=G\); notice that we precisely determine \(G_M\) instead of just bounding it from below. -- We also prove that there are varieties with ample canonical class lying on arbitrarily many irreducible components of the moduli. We distinguish these components by means of their generic automorphism group; there are examples both in the equidimensional and in the non-equidimensional case. In the surface case, this answers a question raised by \textit{F. Catanese} [J. Differ. Geom. 19, 483-515 (1984; Zbl 0549.14012)].
Let \(S\) be a surface of general type; Xiao has given explicit upper bounds both for the cardinality of \(\Aut(S)\) and of an abelian subgroup of \(\Aut(S)\), in terms of the invariants of \(S\) [\textit{G. Xiao}, Invent. Math. 102, No. 3, 619-631 (1990; Zbl 0739.14024) and Ann. Math., II. Ser. 139, No. 1, 51-77 (1994; Zbl 0811.14011)]. Some upper bounds are also known for a higher-dimensional variety \(X\) with ample canonical class, although sharp bounds are still lacking. natural deformations of an abelian cover; automorphism group; coarse moduli spaces of abelian covers; moduli space of varieties with ample canonical class; irreducible components of the moduli Fantechi B, Pardini R. Automorphism and moduli spaces of varieties with ample canonical class via deformations of abelian covers. Comm Algebra, 1997, 25: 1413--1441 Automorphisms of curves, Coverings in algebraic geometry, Algebraic moduli problems, moduli of vector bundles Automorphisms and moduli spaces of varieties with ample canonical class via deformations 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 It is well-known that the set of n-dimensional algebras over an algebraically closed field K of characteristic prime to n can be parameterized via structure constants by an affine algebraic scheme. The author ``functorizes'' this by defining the corresponding functor on the category of affine K-algebras and proving its representability. Then she applies Schlessinger's theory to study formal deformations of K-algebras and proves the existence of the versal deformation space.
A computer implemented method for constructing the tangen space is given. The theory is illustrated in many interesting explicit examples. formal deformations; versal deformation space; computer implemented method for constructing the tangen space M. Schaps, Moduli of commutative and non-commutative covers , Israel J. Math. 58 (1987), 67-102. Formal methods and deformations in algebraic geometry, Software, source code, etc. for problems pertaining to algebraic geometry, Grothendieck groups, \(K\)-theory and commutative rings, Deformations of associative rings, Coverings in algebraic geometry, Finite rings and finite-dimensional associative algebras, Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.) Moduli of commutative and non-commutative finite 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 aims to provide further examples of higher dimensional varieties that are rationally connected, but not rational, and not even ruled. The original methods of \textit{V. A. Iskovskikh} and \textit{Yu. I. Manin} [Math. USSR, Sb. 15 (1971), 141-166 (1972); translation from Mat. Sb., Nov. Ser. 86(128), 140-166 (1971; Zbl 0222.14009)], and of \textit{C. H. Clemens} and \textit{Ph. A. Griffiths} [Ann. Math., II. Ser. 95, 281-356 (1972; Zbl 0214.48302)] have been further developed by many authors [see for example \textit{A. Béauville}, Ann. Sci. Éc. Norm. Supér., IV. Sér. 10, 309-391 (1977; Zbl 0368.14018), \textit{V. Iskovskikh}, J. Sov. Math. 13, 815-868 (1980); translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 12, 159-236 (1979; Zbl 0415.14025), \textit{F. Bardelli}, Ann. Mat. Pura Appl., IV. Ser. 137, 287-369 (1984; Zbl 0579.14033)], and give a fairly complete picture in dimension three. On the other hand, in higher dimension, only special examples were known until recently [see \textit{M. Artin} and \textit{D. Mumford}, Proc. Lond. Math. Soc., III. Ser. 25, 75-95 (1972; Zbl 0244.14017), \textit{V. G. Sarkisov}, Math. USSR, Izv. 17, 177-202 (1981); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 44, 918-945 (1980; Zbl 0453.14017) and Math. USSR, Izv. 20, 355-390 (1983); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 46, No. 2, 371-408 (1982; Zbl 0593.14034), \textit{A. V. Pukhlikov}, Invent. Math. 87, 303-329 (1987; Zbl 0613.14011), \textit{J.-L. Colliot-Thélène} and \textit{M. Ojanguren}, Invent. Math. 97, No. 1, 141-158 (1989; Zbl 0686.14050)].
In his previous paper [\textit{J. Kollár}, J. Am. Math. Soc. 8, No. 1, 241-249 (1995; Zbl 0839.14031)], the author proposes a new approach to the problem of non-rationality, proving that a ``very general'' hypersurface \(X_d \subset \mathbb{P}^n\) of degree \(d\) is not rational for \(2n+9 \leq 3d \leq 3n+3\); very general means that \(X_d\) is a point in the complement of countably many closed proper subsets in the space of all hypersurfaces. This technique involves reduction to characteristic \(p\), and a rather clever and surprising analysis of the stability of the tangent bundle in characteristic \(p\).
The same method applies to hypersurfaces \(X_{c,d} \subset \mathbb{C}\mathbb{P}^m \times \mathbb{C}\mathbb{P}^{n+1}\) of bidegree \((c,d)\). The more interesting is the study of some cases when the fibers of \(X_{c,d} \rightarrow \mathbb{P}^m\) are rational -- for example: (1) conic bundles, or (2) families of cubic surfaces. Especially for these two cases the result is (see theorem 1.4):
1. The very general conic bundle \(X_{c,2} \subset \mathbb{C}\mathbb{P}^m \times \mathbb{C}\mathbb{P}^2\), \(m \geq 2\), is not rational for \(c \geq m+3\);
2. The very general family of cubic surfaces \(X_{c,3} \subset \mathbb{C}\mathbb{P}^m \times \mathbb{C}\mathbb{P}^3\), \(m \geq 1\), is not rational for \(c \geq m+4\).
Especially the above result for conic bundles makes a progress after the works by \textit{V. G. Sarkisov} (see above). A similar result is obtained for cyclic covers \(X_{ap,bp} \rightarrow \mathbb{C}\mathbb{P}^m \times \mathbb{C}\mathbb{P}^n\) (of prime degree \(p\), ramified over a hypersurface \(B\) of bidegree \((ap,bp)\)). The author, in his 1995 paper cited above, showed that if \(ap > m+1\) and \(bp > n+1\) then \(X_{pa,pb}\) is not rational, and not even ruled, for a very general \(B\).
This paper studies the cases when \(bp = n+1\) and \(n = 1,2\) (which are the analogs of cases (1) and (2) avove), and gets (see theorems 1.2 and 1.3):
1. The very general double cover \(X_{2a,2} \rightarrow \mathbb{C}\mathbb{P}^m \times \mathbb{C}\mathbb{P}^1\), \(m \geq 2\), is non-rational for \(2a > m+1\);
2. The very general cyclic triple cover \(X_{3a,3} \rightarrow \mathbb{C}\mathbb{P}^m \times \mathbb{C}\mathbb{P}^2\), \(m \geq 1\), is non-rational, and not even ruled, for \(3a > m+1\). non-rational hypersurface; non-rational cyclic cover; rationally connected varieties János Kollár, Nonrational covers of \?\?^{\?}\times \?\?\(^{n}\), Explicit birational geometry of 3-folds, London Math. Soc. Lecture Note Ser., vol. 281, Cambridge Univ. Press, Cambridge, 2000, pp. 51 -- 71. Rational and unirational varieties, Coverings in algebraic geometry Nonrational covers of \(\mathbb{C}\mathbb{P}^m\times \mathbb{C}\mathbb{P}^n\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We give a description of a graded cyclic cover of a normal graded ring in terms of the Pinkham-Demazure description of normal graded rings \(R=R(X,D)\). With the geometric description of \(\text{Cl}(R)\), it is shown that our cyclic cover \(S\) possesses the Pinkham-Demazure description \(S\cong R(Y,\widetilde D)\) [Theorem 1.3], by which we obtain a description of an index one cover [Corollary 1.7] of \(R\). In \S2, as an application of this description, we give criteria for the normal graded singularities to be Kawamata log terminal or to be log canonical. Further, in \S3 we study the relations between cyclic covers of the Kummer type and cyclic covers obtained by using veronese subrings. Our results extend S. Mori's structure theorem regarding graded factorial domains. Tomari, M., Watanabe, K-i.: On cyclic cover of normal graded rings. (Preprint) Coverings in algebraic geometry, Integral closure of commutative rings and ideals, Singularities in algebraic geometry Cyclic covers of normal graded rings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 2-dimensional analogue of of the Riemann-Hurwitz formula for curves over an algebraically closed field of any characteristic is proved. In characteristic zero, this is due to \textit{B. Iversen} [Am. J. Math. 92, 968--996 (1970; Zbl 0232.14013)]. Euler characteristic; Chern class; ferocious ramification; wild different Surfaces of general type, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Coverings in algebraic geometry, Coverings of curves, fundamental group Iversen's formula for the second Chern classes of regular surfaces in any characteristic | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper, the author considers all degree \(n\in\mathbb{N}^*\) flat generically étale covers \(p:\Gamma\longrightarrow X\), marked at a subset \(D\) of cardinality \(d\in\mathbb{N}\), satisfying a natural tangency condition inside \(\mathrm{Jac}(\Gamma)\), where \((X,q)\) is a curve of arithmetic genus \(g>0\), marked at a smooth point and defined over an algebraic closed field. He focuses mainly in the case \(D\subset p^{-1}(q)\), with \(q\) a non-Weierstrass point and \(g\leq d<n\). All such covers are zero divisors of so-called \(d\)-tangential polynomials and factor through the same ruled surface over \(X\). Conversely, any generic \(d\)-tangential polynomial gives back such a cover. When \(X\) is a smooth complex curve, he generates all \(d\)-tangential polynomials at once, in terms of the Baker-Akhiezer function of \((X,q)\). At last, he considers new phenomena in positive characteristic, namely, infinite towers of Artin-Schreier \(1\)-tangential covers wildly ramified at a unique point. This paper is organized as follows : the first section is an introduction to the subject. In Section 2, the author defines (minimal/indecomposable) \(D\)-tangential covers and present a \(D\)-tangential criterion characterizing them by the existence of a meromorphic, so-called \(D\)-tangential function. He restricts to \(d\)-tangential ones and give examples of decomposable ones. In Section 3, the author studies the characteristic polynomials of the tangential functions of flat \(d\)-tangential covers. He shows that their coefficients are holomorphic outside \(q\in X\) and satisfy affine conditions defining the subvariety of \(d\)-tangential polynomials of degree \(n\). He also find conditions under which this subvariety is not empty, calculates its dimension and proves its generic element to be irreducible. Sections 4 and 5 deal with tangential covers as divisors of a ruled surface and with tangential polynomials via the Baker-Akhiezer function, respectively. Section 6 deals with towers of Artin-Schreier \(1\)-tangential covers (\(p>0\)). The author restricts at last to the positive characteristic case and obtains new phenomena. The paper is supported by an appendix where applications to \(d\times d\)-matrix elliptic (as well as rational and trigonometric) KP solitons are given. The author considers a complex rational curve \(X\) with a node or a cusp and obtain polynomial equations for the spectral curves associated to \(d\times d\) matrix KP trigonometric and rational solitons. coverings; curves; singularities, integrable systems; KP solitons Coverings of curves, fundamental group, Coverings in algebraic geometry, Singularities of curves, local rings, Relationships between algebraic curves and integrable systems Tangential covers and polynomials over higher genus curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(f:X\to \mathbb{P}^2_\mathbb{Q}\) be a finite cover of degree at least three, where \(X\) is integral. The author shows that the number of points of \(f(X(\mathbb{Q}))\) of multiplicative height at most \(B\) is \(O_{f,\varepsilon} (B^{2+ \varepsilon})\) for every \(\varepsilon >0\). height function; thin set; finite cover of degree at least three; multiplicative height N. Broberg, Rational points on finite covers of \(\mathbb P^1\) and \(\mathbb P^2\) , J. Number Theor. 101 (2003), 195-207. Rational points, Coverings of curves, fundamental group, Coverings in algebraic geometry, Varieties over global fields, Heights Rational points on finite covers of \({\mathbb P}^{1}\) and \({\mathbb P}^{2}\) | 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 curves defined over fields of formal power series; reciprocity law; bad reduction Douai, Jean-Claude; Touibi, Chedly: Courbes définies sur LES corps de séries formelles et loi de réciprocité. Acta arith. 42, No. 1, 101-106 (1982/1983) Arithmetic theory of algebraic function fields, Galois cohomology, Galois cohomology, Coverings in algebraic geometry, Étale and other Grothendieck topologies and (co)homologies, Global ground fields in algebraic geometry, Arithmetic ground fields for curves Curves defined over fields of formal power series, and a reciprocity law | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author investigates Galois covers of arithmetic surfaces, focussing on \(\mathbb{P}^1_{\mathbb{Z}[t]}\) on the basis of the arithmetic convergent power series ring \({\mathbb{Z}}\{t\}.\)
In section 1, the author develops the arithmetic of \(\mathbb{Z}\{t\}.\)
In section 2, he gives a criterion for power series, algebraic over the polynomial ring, to be rational and an analogous of Artin's approximation theorem, which gives henselization of structures over the convergent power series ring.
Under the results of sections 1 and 2, the author, in section 3, studies the Galois covers of arithmetic surfaces and shows, in particular, that every finite group is a Galois group over \(\text{Spec}(\mathbb{Z}\{t\})\), after proving an arithmetic version of Grothendieck existence theorem. Hilbert irreducibility theorem; arithmetic unit disc; inverse problem of Galois theory; Galois covers of arithmetic surfaces; arithmetic convergent power series; Artin's approximation; henselization Harbater, D.: Galois covers of an arithmetic surface. Amer. J. Math. 110, 849-885 (1988) Coverings in algebraic geometry, Special surfaces, Galois theory and commutative ring extensions, Representations of groups as automorphism groups of algebraic systems, Finite automorphism groups of algebraic, geometric, or combinatorial structures Galois covers of an arithmetic 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 Let \(C\) be a smooth projective curve of genus 2 over a discrete valuation field \(K\) of \(\text{char} (K) \neq 2\), then \(C\) is defined by a hyperelliptic equation \(y^ 2 = a_ 0 x^ 6 + a_ 1x^ 5 + \cdots + a_ 0\), \(a_ i \in K\). Let \({\mathcal X}\) be the minimal regular model of \(C\) over the ring of integers of \(K\). Our purpose is to determine explicitly the special fiber \({\mathcal X}_ s\) in terms of the coefficients \(a_ i\). This is done completely when the residual characteristic of \(K\) is \(\neq 2\). minimal regular model of smooth curve over a discrete valuation field Liu, Qing, Modèles minimaux des courbes de genre deux, J. Reine Angew. Math., 453, 137-164, (1994) Arithmetic ground fields for curves, Local ground fields in algebraic geometry, Coverings in algebraic geometry Minimal models of genus 2 curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a projective bundle. We prove that \(X\) admits an endomorphism of degree \(>1\) and commuting with the projection to the base, if and only if \(X\) trivializes after a finite covering. When \(X\) is the projectivization of a vector bundle \(E\) of rank 2, we prove that it has an endomorphism of degree \(>1\) on a general fiber only if \(E\) splits after a finite base change. projective bundle; finite covering; endomorphism Amerik E. On endomorphisms of projective bundles. Manuscripta Math, 2003, 111: 17--28 Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Coverings in algebraic geometry, Birational geometry On endomorphisms of projective 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 \(Z\) be a threefold: there exist two Koszul type complexes \(K^ 2\), \(K^ 3\) of cohomology groups of \(Z\) such that if \(K^ 2\), \(K^ 3\) are exact, then the Abel Jacobi map for 2-dimensional cycles of \(Z\) has torsion image. If \(Z\) is a cyclic cover of a threefold \(X\), then the Galois group of the covering map \(Z\to X\) acts naturally on these complexes, which therefore split as sums of complexes of cohomology groups on \(X\), indexed by the characters of the Galois group. So exactness can be proven by showing exactness of certain complexes on \(X\). This way, the author proves:
(1) If \(X = \mathbb{P}^ 3\) and \(Z\) is a cyclic cover of degree \(N\) of \(X\), branched on a general surface of degree \(Nd\), with \(d \geq 5\), then the Abel-Jacobi map of \(Z\) has torsion image.
(2) If \(X \neq \mathbb{P}^ 3\) and \(Z \to X\) is a cyclic covering branched on a sufficiently ample divisor (the notion of sufficiently ample is made precise in the paper), generic in its linear system, then the image of the Abel-Jacobi map is contained in the pull-back of the intermediate Jacobian of \(X\). Abel Jacobi map; threefold; cyclic cover; intermediate Jacobian Algebraic cycles, Coverings in algebraic geometry, \(3\)-folds Syzygies and the Abel-Jacobi map for cyclic 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 gives a method for producing examples of Calabi-Yau threefolds (i. e. smooth threefolds \(X\) with trivial canonical classes and \(h^1(X, \mathcal O_X)=0\)) as covers of degree \(3\leq d \leq 8\) of almost Fano threefolds. (An almost-Fano threefold is a smooth projective threefold with nef and big canonical class.) The almost-Fano threefolds used for this method should have ``even'' canonical class (i. e. canonical sheaf should be isomorphic to square of sheaf that is dual to the globally generated one). Let \(X\to Y\) be such cover and \(\omega_X\cong \mathcal L^{-2}\). The author gives the formula for the Euler-Poincaré characteristic:
\[
e(X)=de(Y)-24d-(d-5)(d-12)c_1(\mathcal L)^3.
\]
For the cases \(d=3,4,5,6,8\), the author uses \textit{G. Casnati} and \textit{T. Ekedahl} [J. Algebr. Geom. 5, 439--460 (1996; Zbl 0866.14009)], \textit{G. Casnati} and \textit{P. Supino} [Glasg. Math. J. 44, 65--79 (2002, Zbl 1046.14019)], and \textit{P. Pragacz} [Ann. Sci. Éc. Norm. Supér., IV Sér. 21. No. 3, 413--454 (1988, Zbl 0687.14043)] (for Euler-Poincaré characteristic computation). Section 2 is aimed to this. The degree 7 case is similar but more complicated; the author constructs such covers in the section 3. Finally, in section 4 author gives examples for all of these cases. Gorenstein cover; Calabi-Yau threefold; almost-Fano threefold G. Casnati, Examples of Calabi-Yau threefolds as covers of almost-Fano threefolds, Geom. Dedicata 119 (2006), 169 -- 179. Calabi-Yau manifolds (algebro-geometric aspects), Calabi-Yau theory (complex-analytic aspects), Fano varieties, \(3\)-folds, Coverings in algebraic geometry Examples of Calabi-Yau threefolds as covers of almost-Fano threefolds | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author computes explicitly the integral closure of a ring of the form \(A[x]/(x^3+ax+b)\), where \(A\) is a Noetherian unique factorization domain and \(p(x)=x^3+ax+b\) is an irreducible polynomial over \(A\). The main motivation and application of this result comes from the theory of triple covers, established by \textit{R. Miranda} [Am. J. Math. 107, 1123-1158 (1985; Zbl 0611.14011)], where it is shown that a finite flat map of degree 3, \(f: X\to Y\), is determined by a rank 2 vector bundle \(E\) and by a map \(S^3E\to \wedge^2 E\) [cf. also \textit{R. Pardini}, Ark. Mat 27, 319-341 (1989; Zbl 0707.14010) for the positive characteristic case]. Indeed, if \(f: X\to Y\) is a finite degree 3 map of normal varieties over an algebraically closed field of characteristic \(\neq 3\) with \(Y\) factorial, then \(f\) factorizes as \(X\to Z\to Y\) where \(X\to Z\) is a birational morphism and there exists a line bundle \({\mathcal L}\) on \(Y\) such that \(Z\subset V({\mathcal L})\) is defined by \(x^3+ax+b=0\), \(x\) being the tautological section and \(a\) and \(b\) sections of \({\mathcal L}^2\), \({\mathcal L}^3\), respectively. The map \(Z\to Y\) is of course the restriction of the bundle projection \(V({\mathcal L})\to Y\). The computation of the integral closure globalizes, so that it is possible to compute \(f_*{\mathcal O}_X\) and other invariants of \(f: X\to Y\) in terms of the triple \((a,b,{\mathcal L})\). In particular, when \(Y\) is a surface the map \(f\) is automatically flat and one recovers Miranda's description of \(f\). cubic extension; integral closure; normalization; triple cover; vector bundle Tan, S L, Integral closure of a cubic extension and applications, Proc Amer Math Soc, 129, 2553-2562, (2001) Integral closure of commutative rings and ideals, Coverings in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Coverings of curves, fundamental group Integral closure of a cubic extension and applications | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be an irreducible projective variety defined over an algebraically closed field \(k\). The first problem considered in the paper is:
Given a vector bundle over \(X\), when is there an étale cover \(\beta: Y \to X\) (of degree greater than \(1\)) and a vector bundle \(F\) on \(Y\) such that \(E = \beta_*F\)?
The authors show that the realisations \((\beta, F)\) of \(E\) as a direct image under an étale cover of \(X\) are in bijective correspondence with the torus subgroup-schemes of the adjoint bundle \(\mathrm{Ad}(E) = \mathrm{Aut}(E)\) of \(E\).
Let \((E, \theta)\) be a Higgs bundle, \(\theta \in H^0(\mathrm{End}(E)\otimes \Omega_X)\) being the Higgs field. Since \(\beta^*\Omega_X = \Omega_Y\), a Higgs field on \(F\) induces a Higgs field on \(E = \beta_* F\). The authors prove that a Higgs field \(\theta\) on \(E\) is induced by a Higgs field on \(F\) if and only if the action of \(\mathcal{T}\) on \(H^0(\mathrm{End}(E)\otimes \Omega_X)\) fixes \(\theta\), where \(\mathcal{T}\) is the subgroup-scheme corresponding to \((\beta,F)\).
Assume that \(X\) is smooth and \(E\) has a connection \(D\). Since \(\beta\) is étale, a connection on \(F\) induces a connection of \(\beta_*F = E\). On the other hand, the connection \(D\) on \(E\) induces one on \(\mathrm{Ad}(E)\). The authors prove that there is a connection on \(F\) inducing the connection \(D\) on \(E\) if and only if the subgroup-scheme \(\mathcal{T}\) is preserved by the connection on \(\mathrm{Ad}(E)\) induced by \(D\). connection; Higgs bundle; étale cover; direct image; torus Coverings in algebraic geometry, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Other connections Direct images of vector bundles and connections | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 0678.00025.]
This paper contains a discursive account, supported by historical and motivational digressions, of the author's work on a problem perhaps best described in his own abstract:
``Consider a rational function field \(C(x)\) in one variable. There have been quite a number of attempts to use Riemann's existence theorem to organize both the lattice of subfields and the algebraic extensions of it. This exposition describes a further attempt that includes exposition on ground... covered sporadically by Zariski... A rough phrasing of the particular problem: For each nonnegative integer g describe explicitly all of the ways that the function field of the ``generic curve'' of genus \(g\) contains \(C(x)...\)
Although a general program has been envisioned by John Thompson..., we narrow to the case where \(g\geq 2\) and the containment of fields gives a solvable Galois closure. This alone illustrates that Zariski's most definitive conjecture on this is wrong...
Theorem 3.5 gives a presentation of the fundamental group \(\pi_ 1(x)\) and first homology \(H_ 1(X,{\mathbb{Z}})\) of a Riemann surface appearing in a (not necessarily Galois) cover \(X\to {\mathbb{P}}^ 1\) of the sphere in terms of branch cycles for the cover. In particular this offers an action of the Hurwitz monodromy group \(H(r)\) on \(H_ 1(X,{\mathbb{Z}})\) where r is the number of branch point of the cover. The remainder of {\S} 3 interprets the dimension of the image of the deformations of the cover in the moduli space of curves of genus \( g=g(X)\) in terms of this group action.'' Artin braid group; Jacobian varieties; Hurwitz monodromy; moduli space of curves M. Fried, Combinatorial computation of moduli dimension of Nielsen classes of covers, Contemporary Mathematics 89 (1989), 61--79. Coverings of curves, fundamental group, Computational aspects of algebraic curves, Algebraic functions and function fields in algebraic geometry, Algebraic moduli problems, moduli of vector bundles, Separable extensions, Galois theory, Families, moduli of curves (algebraic), Coverings in algebraic geometry Combinatorial computation of moduli dimension of Nielsen classes of 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 [For the entire collection see Zbl 0583.00011.]
Let \(\Gamma\) be a discrete, irreducible subgroup without fixed point of the automorphism group of the polydisc \(\Delta^ n\). Observe next that the subgroup \(\Gamma_ 0=\Gamma \cap (Aut(\Delta))^ n\) is of finite index. So \(X_ 0=\Delta^ n/\Gamma_ 0\) is a finite covering of \(X=\Delta^ n/\Gamma\). If \((z_ 1,...,z_ n)\) are usual Euclidean coordinates on \(\Delta^ n\) than the line bundle \(L_ k\) is generated by \(\partial /\partial z_ k\). Equip \(L_ K\) with the hermitian metric induced by the Poincaré metric on \(\Delta^ n\). Since \(\Gamma_ 0\subset (Aut(\Delta))^ n\) the hermitian bundle \(L_ K\) is invariant under \(\Gamma_ 0\), thus inducing a hermitian line bundle \(L_ K(X_ 0)\) on \(X_ 0\). The main result of the paper is the next vanishing theorem:
Theorem 1. Let \(X=\Delta^ n/\Gamma\) be an irreducible quotient of the polydisc \(\Delta^ n(n\geq 2)\) of finite volume in the invariant Poincaré metric, E a hermitian vector bundle on X such that under the covering map \(\pi\) : \(X_ 0\to X\), \(\pi^*E\) is isomorphic to \(L^{S_ 1}_{p(1)}(X_ 0)\oplus...\oplus L^{S_ m}_{p(m)}(X_ 0)\) with \(S_ k>0\) for \(1\leq k\leq m\). Then, there does not exist a non- trivial \(L^ 2E\)-valued harmonic (0,p)-form with \(0\leq p\leq n.\)
This theorem is a generalization of Matsushima-Shimura vanishing theorem to noncompact (but finite volume) X. The proof makes use of lifting harmonic form \(\phi\) to \(\phi_ 0\) on \(X_ 0\), splitting \(\phi_ 0=\sum \phi_ A\) into a sum of harmonic components and Bochner-Kodaira formula for (1,0)-gradients. quotients of polydisc; vanishing theorem; Poincaré metric; hermitian vector bundle; Bochner-Kodaira formula Vanishing theorems, Sheaves and cohomology of sections of holomorphic vector bundles, general results, Local differential geometry of Hermitian and Kählerian structures, Coverings in algebraic geometry On a vanishing theorem on irreducible quotients of finite volume of polydiscs | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Assume one is given a solvable branched covering \(X\to P^ 1\) with Galois group G, let K be a field of definition of this covering and let \(H=gal(K/{\mathbb{Q}})\). The problem adressed to is: give information on H in terms of G for a suitable ``small'' K. The main theorem of the present paper shows that given a topological description of this covering (i.e. a family of elements in G corresponding to a standard homotopy basis of \(P^ 1\setminus discri\min ant)\) and ``knowing'' a chief series of G (i.e. a series \(G_ m\subset G_{m-1}\subset...\subset G_ 0=G\) with \(G_{i+1}\) normal in \(G_ i\) and \(G_{i+1}\) maximal among normal subgroups of G contained in \(G_ i)\) one can describe the factor groups \(H_ j/H_{j+1}\) of a subinvariant series \(H_ n\triangleleft H_{n- 1}\triangleleft...\triangleleft H_ 0=H\) for a suitable K. In particular it follows that if L is the field of definition for the branch points of the covering and M is the Galois closure over L of the field of moduli of the covering then gal(M/L) is an extension of abelian groups and subquotients of symplectic groups. small field of definition; solvable branched covering Beckmann, S.: Galois groups of fields of definition of solvable branched coverings. Compositio math. 66, 121-144 (1988) Relevant commutative algebra, Coverings in algebraic geometry, Families, moduli of curves (algebraic), Coverings of curves, fundamental group Galois groups of fields of definition of solvable 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 The authors prove that a resolution of singularities of any finite covering of the projective complex plane branched along a Hurwitz curve \(\overline H\), and possibly along the line ``at infinity'', can be embedded as a symplectic submanifold in some projective algebraic manifold equipped with an integer Kähler symplectic form. For cyclic coverings they realize these embeddings in a rational complex 3-fold. Properties of the Alexander polynomial of \(\overline H\) are investigated and applied to the calculation of the first Betti number \(b_1(\overline X_n)\), where \(\overline X_n\) is a resolution of singularities of an \(n\)-sheeted cyclic covering of \({\mathbb C}{\mathbb P}^2\) branched along \(\overline H\), and possibly along the line ``at infinity''. They prove that \(b_1(\overline X_n)\) is even if \(\overline H\) is an irreducible Hurwitz curve but, in contrast to the algebraic case, \(b_1(\overline X_n)\) may take any non-negative value in the case when \(\overline H\) consists of several components. Coverings in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Symplectic and contact topology in high or arbitrary dimension, Singularities of curves, local rings, Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants) On symplectic coverings of a projective plane. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(S\) be a surface of general type over \(\mathbb{C}\) and let \(\chi\), \(c^ 2_ 1\) be the Chern numbers of \(S\). If we assume that the canonical map of \(S\) is birational, then we have the following inequalities: \(3\chi- 10\leq c^ 2_ 1\leq 9\chi\), \(\chi>0\), \(c^ 2_ 1>0\). For every pair \(x,y\) of natural numbers \(3x-10\leq y\leq 8x-78\), the author constructs a surface of general type \(S\) with \(x=\chi\), \(y=c^ 2_ 1\) and such that the canonical map of \(S\) is birational.
Examples of surfaces of general type with invariants in the same region had already been constructed by Persson, but the canonical map of those was 2 to 1. It seems that birationality should be ``the rule'', so that the author's examples are interesting from this point of view. --- The examples are constructed as resolution of some singular genus 3 fibrations with elliptic singularities of type \(\tilde E_ 7\). An explicit method of resolution is described. birational canonical map; surface of general type; Chern numbers Ashikaga, T.: A remark on the geography of surfaces with birational canonical morphisms. Math. Ann. 290, 63--76 (1991) Surfaces of general type, Families, moduli, classification: algebraic theory, Characteristic classes and numbers in differential topology, Rational and birational maps, Coverings in algebraic geometry A remark on the geography of surfaces with birational canonical morphisms | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 every algebraic Kummer surface can be an unramified double cover of some Enriques surface. The main tools used in the proof are Nikulin's results on lattices, the Torelli theorem for K3-surfaces, and the surjectivity of the period map for marked K3-surfaces and Enriques surfaces. cover of Enriques surface; algebraic Kummer surface; Torelli theorem; period map \(K3\) surfaces and Enriques surfaces, Coverings in algebraic geometry, Families, moduli, classification: algebraic theory Every algebraic Kummer surface is the K3-cover of an Enriques 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 The author studies the following two types of transforms of a complex projective plane curve \(C=\{ F( X,Y,Z) =0\} \subset \mathbb{P}^{2}\).
The \(n-\)fold cyclic covering \(\varphi _{n}:\mathbb{C} ^{2}\to \mathbb{C}^{2}\), \(\varphi _{n}( x,y) =( x,( y-\beta) ^{n}+\beta) \) determines the cyclic transform \(\mathcal{C}_{n}(C;D)\) (\(D\) is the line \(\{ y=\beta \}\)) as the projective closure in \(\mathbb{P}^{2}\) of \(\varphi_{n}^{-1}( C\cap \{ Z\neq 0\}) \), and
the Jung automorphism of degree \( n\), \(J_{n}:\mathbb{C}^{2}\to \mathbb{C}^{2}\), \(J_{n}( x,y) =( x+y^{n},y) \) determines the Jung transform \(\mathcal{J} _{n}( C;L_{\infty }) \) (\(L_{\infty }\) is the line at infinity \( \{ Z=0\} \)) as the projective closure in \(\mathbb{P}^{2}\) of \(J_{n}^{-1}( C\cap \{ Z\neq 0\}) \).
The transforms have similar effect on the fundamental group of the complement of the curve: For sufficiently generic \(\varphi _{n}\), \(J_{n}\) and for \(C'\mathcal{C} _{n}(C)\), \(\mathcal{J}_{n}( C) \) there exist epimorphisms \(\pi _{1}( \mathbb{P}^{2}-C') \to \pi _{1}( \mathbb{P} ^{2}-C) \) such that their kernels are cyclic subgroups of order \(n\) in the centers of \(\pi _{1}( \mathbb{P}^{2}-C') \). The author investigates systematically the two transforms and the fundamental groups of their complements, and uses them to construct new examples of interesting plane curves. These include new plane curves of degree \(4n\) whose fundamental group of the complement is noncommutative finite of order \(12n\), a new Zariski pair of curves of degree \(12\), a rational curve of degree \(pq\) (\(p\) and \(q\) coprime) whose fundamental group of the complement is the free product \( \mathbb{Z}_{p}*\mathbb{Z}_{q}\). plane algebraic curve; fundamental group of the complement of the curve; Jung transform; cyclic covering Oka, M.: Two transforms of plane curves and their fundamental groups. J. math. Sci. univ. Tokyo 3, No. 2, 399-443 (1996) Homotopy theory and fundamental groups in algebraic geometry, Plane and space curves, Global theory of complex singularities; cohomological properties, Coverings in algebraic geometry Two transforms of plane curves and their 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 \(V_3\) be the Veronese surface of order 3, and \(S_3\) be the branch curve of a generic projection of \(V_3\) on \(\mathbb{C} P^2\). The authors compute the fundamental group \(\pi (\mathbb{C}^2\smallsetminus S_3)\). They use the technique developed in the preceding parts [Part: I: Contemp. Math. 78, 425-555 (1988; Zbl 0674.14019); Part II: Lect. Notes Math. 1479, 131-180 (1991; Zbl 0764.14014)]; Part III: Contemp. Math. 162, 313-332 (1994; Zbl 0815.14023); Part IV: ibid. 162, 333-358 (1994; Zbl 0815.14024)]. The authors also announce a similar result concerning \(\pi (\mathbb{C} P^2\smallsetminus S_3)\). Veronese surface; generic projection; fundamental group Moishezon, B., Teicher, M.: Braid group technique in complex geometry V: The fundamental group of a complement of a branch curve of a Veronese generic projection. Communications in Analysis and Geometry, 4(1), 1--120 (1996) Fundamental group, presentations, free differential calculus, Coverings in algebraic geometry, Special surfaces, Coverings of curves, fundamental group Braid group technique in complex geometry. V: The fundamental group of a complement of a branch curve of a Veronese generic projection | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Abelian varieties are very important objects that lie at the crossroads of several areas in mathematics: algebraic geometry, complex analysis, and number theory. So they can be studied from different points of view. The main idea we want to convey in this article is that one can construct and study abelian varieties using curves and its surrounding geometry. Jacobians, Prym varieties, Coverings in algebraic geometry, Abelian varieties and schemes The Prym map: from coverings to abelian varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a smooth real surface and let Br\((X) = H^ 2_{et} (X; \mathbb{G}_ m)\) denote its (cohomological) Brauer group. Let \(_ 2\text{Br}(X)\) be the group of elements of order 2 in Br\((X)\). It is known that there is the canonical map \((*)\) \(_ 2\text{Br}(X) \to (\mathbb{Z}/2)^ s\) where \(s\) denotes the number of connected components of \(X(\mathbb{R})\) (= the space of \(\mathbb{R}\)-rational points of \(X)\). In general, it is a difficult problem to determine the dimension of \(_ 2\text{Br}(X)\) explicitly.
In this paper, the dimension of the 2-part of Br\((X)\) is determined in terms of computable invariants for real Enriques surfaces. Let \(Y\) denote a real Enriques surface, and let \(Y_ \mathbb{C}\) be the corresponding surface over \(\mathbb{C}\). Let \({\mathcal G} = \text{Gal} (\mathbb{C}/ \mathbb{R}) = \langle \theta \rangle\). Define two invariants \(b(Y)\) and \(\varepsilon(Y)\) as follows:
\[
b(Y) = \dim H^ 2 \bigl( Y(\mathbb{C}), \mathbb{Z}/2 \bigr)^ \theta - \dim (\text{Pic} Y_ \mathbb{C})^ \theta/2 (\text{Pic} Y_ \mathbb{C})^ \theta + 1
\]
and
\[
\varepsilon (Y) = 1 \text{ if the differential } d_ 2^{0,2} : E_ 2^{0, 2} \to E_ 2^{2,1} \text{ vanishes, and \(\varepsilon(Y) = 0\) otherwise}.
\]
Theorem 1. Suppose that \(Y (\mathbb{R}) \neq \emptyset\). Then \(\dim_ 2\text{Br}(Y) = b(Y) + \varepsilon(Y)\). -- An estimate for the lower bound of \(b(Y)\) is given in terms of the number \(s\) of connected components of \(Y(\mathbb{R})\).
Theorem 2. \(b(Y) \geq 2s - 2\). Consequently, \(\dim {_ 2\text{Br}}(Y) \geq 2s - 2\).
Noting that the element \((1,\dots,1)\) \((s\) times) belongs to the image of the map \((*)\), it follows:
Corollary. \(\dim{_ 2\text{Br}} (Y) \geq s\).
A precise formula for the invariant \(b(Y)\) is presented using the universal covering K3 surfaces, which is rather intricate to be formulated here. Let \(H^ 2 : = H^ 2 (Y(\mathbb{C}); \mathbb{Z})/ \text{Tor}\). Then \(H^ 2\) is a unimodular lattice with respect to the intersection pairing. The involution \(\theta\) acts on \(H^ 2\), and one defines the invariant \(r(\theta) : = \text{rank} (H^ 2)^ \theta\). One also defines a group \(A_{(H^ 2)^ \theta} = ((H^ 2)^ \theta)^*/(H^ 2)^ \theta\), which is 2-elementary, i.e., \(\simeq (\mathbb{Z}/2)^{a (\theta)}\). Then \(\dim (H^ 2/2H^ 2)^ \theta = \text{rank} H^ 2 - a (\theta)\).
Theorem 3. \(b(Y)=0\) if and only if the surface \(Y(\mathbb{R})\) is connected non-orientable and for the invariants \(r(\theta)\) and \(a (\theta)\) we have the equality \(r(\theta) = a (\theta)\). -- Consequently, for such a surface \(Y\), \(\varepsilon (Y) = 1\) and \(\dim{_ 2\text{Br}}(Y) = 1\).
The idea of the proof of theorem 1 is to use the Kummer sequence and estimate the dimension of \(_ 2\text{Br}(Y)\) using the Hochschild-Serre spectral sequence. Theorem 2 is proved using the {Lefschetz} fixed point formula and Smith exact sequence. Theorem 3 is proved using the theory of involutions of lattices (integral quadratic forms) developed by the first named author in Math. USSR, Izv. 22, 99-172 (1984); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 47, No. 1, 109-188 (1983; Zbl 0547.10021). The theory is applied to the action of antiholomorphic involutions on the 2- cohomology lattice of \(Y(\mathbb{C})\) and its universal covering K3-surface \(X(\mathbb{C})\). Brauer group; real Enriques surfaces; universal covering K3 surfaces; 2- cohomology lattice V. V. Nikulin,J. Reine Angew. Math.,444, 115--154 (1993). \(K3\) surfaces and Enriques surfaces, Real algebraic and real-analytic geometry, Brauer groups of schemes, Coverings in algebraic geometry On Brauer groups of real Enriques surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For part I of this paper see \textit{G. Casnati} and \textit{T. E. Ekedahl}, J. Algebr. Geom. 5, No. 3, 439-460 (1996; Zbl 0866.14009); part II: \textit{G. Casnati}, ibid., 461-477 (1996; Zbl 0921.14006).
The aim of this article is to study the discriminant of a Gorenstein cover of degree 4. By part I of this paper we have a general structure theorem for Gorenstein covers: If \(\rho: X\to Y\) is a Gorenstein cover of schemes, \(Y\) integral, of degree \(d=4\) (resp. \(d=3\)) \(X\) is obtained inside a projective bundle \(\pi:{\mathbb{P}}={\mathbb{P}}( {\mathcal E})\to Y\) of rank 2 (resp. 1), where \({\mathcal E}\) is the dual of the Tschirnhausen module of \(\rho\), i.e. the cokernel of \(\rho^{\sharp}:{\mathcal O}_Y\to\rho_*{\mathcal O}_X\). More precisely, if \(d=3\), there is an exact sequence \(0\to\pi^*\text{det }{\mathcal E}(-3)\to{\mathcal O}_{\mathbb{P}}\to {\mathcal O}_X \to 0\) and \(\rho\) determines a section \(\eta\in H^0(Y,{\mathcal S}^3{\mathcal E}\otimes\text{det }{\mathcal E}^{-1})\).
If \(d=4\), there is a locally free \({\mathcal O}_Y\)-sheaf \({\mathcal F}\) of rank 2 such that \(\text{det }{\mathcal F}\simeq\text{det }{\mathcal E}\) and an exact sequence \(0\to\pi^*\text{det }{\mathcal E}(-4)\to\pi^* {\mathcal F}(-2)\to {\mathcal O}_{\mathbb{P}}\to{\mathcal O}_X\to 0\) and \(\rho\) determines a section \(\eta\in H^0(Y,\breve{\mathcal F}\otimes{\mathcal S}^2 {\mathcal E})\). Locally if \(Y=\text{Spec}A\) then \(X=\text{Spec}R\), with \(R=A[u,v]/(u^2 + 2\alpha v +\beta, v^2+2\gamma u+\delta)\) for suitable \(\alpha, \beta, \gamma, \delta \in A\).
The branch locus \(B_{\rho}\) of the cover \(\rho: X\to Y\) is defined by \(b=\text{det } Q\), where \(Q: R\times R\to A\) is the bilinear form defined by \(Q(x,y)=\text{Tr}(xy)\). To define the discriminant of the cover \(\rho\), we identify \(\eta\in H^0(Y,\breve {\mathcal F}\otimes {\mathcal S}^2 {\mathcal E})\) with a symmetric map \(\breve {\mathcal G}(-1)\to {\mathcal G}\otimes \text{det }{\mathcal G}^{-1}\), where \({\mathcal G} = \overline \pi^*{\mathcal E}\) and \(\overline\pi: {\mathbb{P}} ({\mathcal F}) \to Y\), and we call \(\Delta(\eta)\) the section of \(H^0 ({\mathbb{P}}({\mathcal F}),\overline \pi^*(\text{det }{\mathcal F} ^{-1})(3))\) obtained by taking the determinant of \(\eta\) and the projection formulas. The discriminant scheme \(\Delta (X)\) of \(X\) is the subscheme of \({\mathbb{P}}({\mathcal F})\) defined by \(\Delta(\eta)\), and the discriminant map \(\Delta(\rho):\Delta(X)\to Y\) is the restriction of \(\overline\pi\) to \(\Delta (X)\). We can give the theorem about the discriminant:
If \(\rho : X \to Y\) is a Gorenstein cover of degree 4, the discriminant \(\Delta(\rho):\Delta(X)\to Y\) is generically finite of degree 3.
The points \(y\in Y\) over which \(\Delta(\rho)\) does not have finite fibers are exactly the points such that \(\rho^{-1}(y)=\text{Spec} (k(y) [u,v]/ (u^2,v^2))\) and in this case \(\Delta(\rho)^{-1}(y)\simeq {\mathbb{P}}^1 _{k(y)}\). If there are no such \(y\)'s, then \(\Delta(\rho)\) is a Gorenstein cover of degree 3 with Tschirnhausen module \(\breve{\mathcal F}\). When the schemes \(X\) and \(Y\) are smooth, the authors study the reducedness, the connectedness and the smoothness of the discriminant cover.
Proposition: Let \(\rho: Y\to X\) be a Gorenstein cover of degree 4 with \(X\) smooth and \(Y\) smooth and integral. Then \(\Delta (X)\) is reduced and if \(\Delta (X)\) is reducible there exists an epimorphism \({\mathcal F}\to{\mathcal M}\) with \({\mathcal M}\) an invertible \({\mathcal O}_Y\)-sheaf. Moreover, if the branch locus \(B_{\rho}\) is reduced and \(\Delta(X)\) is integral, then it is also normal.
If we assume moreover that the discriminant \(\Delta(\rho)\) is a cover and that \(\rho\) does not factor as \(\beta \circ\alpha\) with \(\beta\) a non-trivial étale double cover of \(Y\), then we have:
If \(\rho\), \(X\) and \(Y\) are as above, then \(\Delta (X)\) is connected, and if \(B_{\rho}\) has at most ordinary cuspidal double points as singularities, then \(\Delta(X)\) is smooth.
To study the singularities of the discriminant \(\Delta(X)\), the author considers \(\overline\pi:\overline{\mathbb{P}} = {\mathbb{P}}({\mathcal F})\to Y\) and the sheaves \({\mathcal H}={\mathcal O}_{\overline{\mathbb{P}}}(1)\otimes \overline\pi^*({\mathcal E}\otimes \text{det}{\mathcal E}^{-1} )\) and \({\mathcal M}={\mathcal O}_{\overline{\mathbb{P}}}(-1)\otimes \overline \pi^*\text{det} {\mathcal F}\). Then he can associate to the section \(\eta\in H^0 (Y,\breve {\mathcal F} \otimes {\mathcal S}^2 {\mathcal E} )\) a symmetric map \(\varphi : \breve {\mathcal H} \to {\mathcal H}\otimes{\mathcal M}\) and an exact sequence \(0 \to \breve {\mathcal H} \to {\mathcal H}\otimes {\mathcal M} \to {\mathcal P} \to 0\), where \({\mathcal P}\) is supported on \(\Delta(X)\). We have \(\Delta (X) =D_2(\varphi)\) and \(\text{Sing} ( \Delta (X))=D_1 (\varphi)\).
Then, with a hypothesis of genericity, we have: Let \({\mathcal E}\) and \({\mathcal F}\) be locally free sheaves of ranks 3 and 2 respectively over an integral and smooth scheme \(Y\) of dimension 2. If \({\mathcal S}^2 {\mathcal E} \otimes \breve {\mathcal F}\) is globally generated, then the discriminant \(\Delta (X)\) of each general cover \(\rho : X\to Y\) of degree 4 with invariants \({\mathcal E}\) and \({\mathcal F}\) is smooth in codimension 1 and has at most nodes as singularities.
In the next part, the author generalizes the trigonal construction due to \textit{S. Recillas}. Let \({\mathcal T} ^4 _{Y, {\mathcal E} , {\mathcal F}}\) denote the set of Gorenstein covers \(\rho : X \to Y\) of degree 4 with invariants \({\mathcal E}\) and \({\mathcal F}\) and such that \(X\) is smooth and \(\Delta(\rho)\) is a cover, ``with an extra condition''. Let \({\mathcal R} ^3 _{Y, {\mathcal E},{\mathcal F}}\) denote the set of pairs \( ( {\mathcal P} , \tau : \Delta\to Y)\) such that \(\tau\) is a Gorenstein cover of degree 3 with invariant \({\mathcal F}\), \(\Delta\) is normal with at most pseudo-nodes as singularities, and \({\mathcal P}\) is a sheaf with appropriate properties such that \(\tau _* {\mathcal P}={\mathcal E}\). From what we have seen above, we have a map \(\text{Trig} : {\mathcal T} ^4_{Y,{\mathcal E},{\mathcal F}}\to {\mathcal R}^3 _{Y,{\mathcal E} , {\mathcal F}}\) that associates to each cover \(\rho\) the discriminant cover \(\Delta(\rho)\), where \({\mathcal P}\) is the sheaf defined above. This map is called the ``trigonal construction''. The authors show how to recover a Gorenstein cover of degree 4 from a fixed pair \(({\mathcal P},\tau : \Delta \to Y )\), and then they get the following result:
The map \(\text{Trig}: {\mathcal T} ^4 _{Y, {\mathcal E},{\mathcal F}}\to {\mathcal R}^3_{Y, {\mathcal E} , {\mathcal F}}\) is bijective.
In the last section of the article the author studies covers of the projective space \({\mathbb{P}} ^n_{\mathbb{C}}\), and proves the following theorem:
Let \(\rho : X \to {\mathbb{P}} ^n _{\mathbb{C}}\) be a cover of degree 4 with \(X\) smooth and integral and with \(n \geq 5\). If \(\rho _*{\mathcal O}_X /{\mathcal O}_{\mathbb{P} ^n _{\mathbb{C}}}\) is uniform, then \(\rho^*_i : H^i ( {\mathbb{P}} ^n _{\mathbb{C}},{\mathbb{C}}) \to H^i(X,{\mathbb{C}})\) is an isomorphism for \(0 \leq i \leq n-1\) and a monomorphism for \(i=n\).
By \textit{Fujita}'s result on covers of degree 3, the author gives a description of the locus of points of total ramification. Finally, he obtains the following theorem:
Let \(\rho : X \to {\mathbb{P}} ^n _{\mathbb{C}}\) be a cover of degree 4 with \(X\) smooth and \(n \geq 5\), then \(X\) is a quadrisecant of an ample line bundle if and only if for each \(y \in {\mathbb{P}} ^n _{\mathbb{C}}\) there is an embedding \(\rho ^{-1} (y) \hookrightarrow {\mathbb{A}} ^1 _{\mathbb{C}}\). More precisely either \(X\) is a quadrisecant of an ample line bundle, or there is a point \(y \in {\mathbb{P}} ^n _{\mathbb{C}}\) such that the fiber of \(\rho\) over \(y\) is isomorphic to Spec\( ( {\mathbb{C}} [u,v] / (u^2 , v^2))\). Tschirnhausen module; Gorenstein cover; discriminant scheme; discriminant map DOI: 10.1090/S0002-9947-98-02136-9 Coverings in algebraic geometry, Ramification problems in algebraic geometry, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Covers of algebraic varieties. III: The discriminant of a cover of degree 4 and the trigonal construction | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(E\cong (\mathbb Z/p)^r\) (\(r\geq 2\)) be an elementary Abelian \(p\)-group and let \(k\) be an algebraically closed field of characteristic \(p\). A finite dimensional \(kE\)-module \(M\) is said to have constant Jordan type if the restriction of \(M\) to every cyclic shifted subgroup of \(kE\) has the same Jordan canonical form. I begin by discussing theorems and conjectures which restrict the possible Jordan canonical form. Then I indicate methods of producing algebraic vector bundles on projective space from modules of constant Jordan type. I describe realisability and non-realisability theorems for such vector bundles, in terms of Chern classes and Frobenius twists. Finally, I discuss the closely related question: can a module of small dimension have interesting rank variety? The case \(p\) odd behaves throughout these discussions somewhat differently to the case \(p=2\). modular representations; elementary Abelian groups; modules of constant Jordan type; vector bundles; rank varieties; Chern classes; Frobenius twists; endotrivial modules Benson, D.: Modules for elementary abelian \(p\)-groups. In: Proceedings of the International Congress of Mathematicians (ICM 2010), pp. 113-124 (2010) Modular representations and characters, Group rings of finite groups and their modules (group-theoretic aspects), Vector bundles on curves and their moduli, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Modules for elementary Abelian \(p\)-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 The papers of this third volume of Zariski's work were originally published between 1925 and 1966, but the bulk of them are from the ten year period 1928--1937. Thus they were written during the last year of his stay in Rome and his early years at John Hopkins University. The introduction by M. Artin and B. Mazur contains an illuminating discussion of the impact of these papers on the later work of other mathematicians.
The papers themselves may be broadly divided into three sections: (1) solvability by radicals of the equations of plane-algebraic curves, (2) the fundamental group of the residual space of a plane algebraic curve and (3) the topology of the singularities of plane algebraic curves.
In addition three survey articles are reproduced. The first is Zariski's lecture to the International Congress of Mathematicians at Harvard in 1950 surveying his overall view of algebraic geometry [Zbl 0049.22701]. The second is an introduction to the application of valuation theory to algebraic geometry from lectures given in Rome in 1953. The third on Serre's coherent sheaves is a report of a seminar at an AMS summer institute in 1954 and is still a good place to find an introduction to algebraic sheaf theory. collected papers; solvability by radicals; valuation theory; algebraic sheaf theory; uniformization of algebraic functions; purity of branch locus; fundamental group of a curve O. Zariski,Collected Papers, Vol. III, MIT Press, 1978, pp. 43--49. History of algebraic geometry, Collected or selected works; reprintings or translations of classics, Coverings of curves, fundamental group, Coverings in algebraic geometry, General valuation theory for fields, Singularities of curves, local rings Collected papers. Vol. III: Topology of curves and surfaces, and special topics in the theory of algebraic varieties. Edited and with an introduction by M. Artin and B. Mazur | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 line bundle \(L\) on a smooth projective variety \(X\), the Gauss map
\[
\bigwedge^ 2 H^ 0(X,L) \to H^ 0 (X,L^{\otimes 2})
\]
is defined by sending \(\sum s_ i \otimes t_ i\) to \(\sum(ds_ i\otimes t_ i- s_ i\otimes dt_ i)\). If \(X \to Y\) is a cyclic covering, with \(Y\) and \(X\) smooth and projective, and \(L\) is a pull-back from \(Y\), then, using projection formulas, it is possible to express the Gauss map for \(L\) in terms of similar maps on \(Y\).
After having described this general setting, the author focuses on the case of double covers of \(\mathbb{P}^ m\) \((m>1)\) and computes the cokernel of the Gauss map for the pull-back of any effective line bundle on \(\mathbb{P}^ m\). The paper also contains several references on the general theory and applications of the Gauss maps. projection; Gauss map; double covers Duflot, J.: Gaussian maps for double coverings. Manuscript Math. 82, 71--87 (1994) Coverings in algebraic geometry, Projective techniques in algebraic geometry Gaussian maps for double 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 Let \(X\) be a normal projective threefold over \(\mathbb{C}\) with, at worst, isolated log terminal singularities, such that \(c(K_X+H) \sim 0\) for a minimal positive integer \(c\) and a Cartier ample divisor \(H\) on \(X\). In this note the author, improving previous results of Reid, Alexeev and Prokorov, shows that a general member \(S\) of \(| H|\) is a log Enriques surface and he gives a bound for \(c\). The proof is based on the construction of a suitable Galois \(\mathbb{Z}/c \mathbb{Z}\)-cover \(\pi: T\to S\) from a K3 surface \(T\) with, at worst, rational double point singularities.
In the second part of the paper, the author studies and classifies the quotients of such a K3 surface. quotients of K3 surfaces; log Enriques surface \(3\)-folds, \(K3\) surfaces and Enriques surfaces, Fano varieties, Coverings in algebraic geometry, Homogeneous spaces and generalizations Log Fano threefolds and quotients 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 A collection \(A=\{D_1,\dots,D_n\}\) of divisors on a smooth variety \(X\) is an \textit{arrangement} if the intersection of every subset of \(A\) is smooth. We show that, if \(X\) is defined over a field of characteristic not equal to 2, a double cover of \(X\) ramified on an arrangement has a crepant resolution under additional hypotheses. Namely, we assume that all intersection components that change the canonical divisor when blown up are \textit{splayed}, a property of the tangent spaces of the components first studied by Faber. This strengthens a result of Cynk and Hulek, which requires a stronger hypothesis on the intersection components. Further, we study the singular subscheme of the union of the divisors in \(A\) and prove that it has a primary decomposition where the primary components are supported on exactly the subvarieties which are blown up in the course of constructing the crepant resolution of the double cover. double covers; crepant resolution; primary decomposition; Calabi-Yau varieties Calabi-Yau manifolds (algebro-geometric aspects), Singularities in algebraic geometry, Coverings in algebraic geometry, Polynomial rings and ideals; rings of integer-valued polynomials On the Cynk-Hulek criterion for crepant resolutions of double 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 The purpose of this work is to study the classification of the ramified coverings of the projective line determined by the type of the direct image of the sheaf of regular functions. To be more precise, to each ramified covering of degree \(r\) we associate \(r-1\) positive integer invariants determined by the direct image of the sheaf of regular functions; we try first to determine the sets of \(r-1\) positive integers which are the set of invariants of a covering and then to study the family of coverings with a fixed set of invariants. ramified coverings of the projective line; sheaf of regular functions; family of coverings with a fixed set of invariants Gabriela Chaves, Revêtements ramifiés de la droite projective complexe, Math. Z. 226 (1997), no. 1, 67 -- 84 (French). Coverings in algebraic geometry, Ramification problems in algebraic geometry Revêtements ramifiés de la droite projective complexe. (Ramified coverings of the complex projective line) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Given an irreducible normal Noetherian scheme and a finite Galois extension of the field of rational functions, we discuss the comparison of the categories of vector bundles on the scheme and equivariant vector bundles on the integral closure in the extension. This is well understood in the tame case (geometric stabilizer groups of order invertible in the local rings), so we focus on the wild (non-tame) case, which may be reduced to the case of cyclic extensions of prime order. In this case, under an additional flatness hypothesis, we give a characterization of the equivariant vector bundles that arise by base change from vector bundles on the scheme. descent; vector bundles; wild ramification Ramification problems in algebraic geometry, Ramification and extension theory, Coverings in algebraic geometry Descent of vector bundles under wildly ramified extensions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 smooth surfaces that are coverings of the complex projective plane with ramification curve a union of conics, and computes their Chern classes. arrangements of conics; ramification curve; Chern classes Li Zhong Tang, ``Algebraic surfaces associated to arrangements of conics'', Soochow J. Math.21 (1995) no. 4, p. 427-440{
}{\copyright} Annales de L'Institut Fourier - ISSN (électronique) : 1777-5310 Special surfaces, Homology of classifying spaces and characteristic classes in algebraic topology, Coverings in algebraic geometry Algebraic surfaces associated to arrangements of conics | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 simple algebraic group over a field \(k\), defined and split over the finite field \(\mathbb F_p\) for a prime \(p\). The Lie algebra \(\mathfrak g\) of \(G\) admits the structure of a \(p\)-restricted Lie algebra with restricted enveloping algebra \(u(\mathfrak g)\). Let \(G(\mathbb F_q)\) denote the finite Chevalley group of \(\mathbb F_q\)-rational points of \(G\) (for \(q=p^d\)). Given a rational \(G\)-module \(M\), by restriction, \(M\) can be considered as either a \(kG(\mathbb F_q)\)-module or a \(u(\mathfrak g)\)-module.
The first main result, which extends work of \textit{J. F. Carlson, Z. Lin} and \textit{D. K. Nakano} [Trans. Am. Math. Soc. 360, No. 4, 1879-1906 (2008; Zbl 1182.20039)] for \(q=p\) to an arbitrary \(q=p^d\), is a relationship between the (projectivized) cohomological support variety of \(G(\mathbb F_q)\) and that of \(u(\mathfrak g_{\mathbb F_q}^{\oplus d})\) as long as \(p\) is at least the Coxeter number of \(G\). A crucial tool in proving this relationship is use of the Weil restriction functor. For a Galois extension of fields \(E/F\), the Weil restriction functor is a functor from affine \(E\)-schemes to affine \(F\)-schemes that is right adjoint to base change from \(F\) to \(E\). The Weil restriction functor is applied for example to \(G_{\mathbb F_q}\) relative to the extension \(\mathbb F_q/\mathbb F_p\).
A second main result is that the complexity of a rational \(G\)-module \(M\) when considered as a \(kG(\mathbb F_q)\)-module is bounded by one-half its complexity when considered as a \(u(\mathfrak g_{\mathbb F_q}\otimes_{\mathbb F_p}k)\)-module. From this it follows that if \(M\) is projective upon restriction to \(u(\mathfrak g_{\mathbb F_q}\otimes_{\mathbb F_p}k)\), then it is necessarily projective upon restriction to \(kG(\mathbb F_q)\). These results extend to arbitrary \(q\) results for \(q=p\) of \textit{Z. Lin} and \textit{D. K. Nakano} [Invent. Math. 138, No. 1, 85-101 (1999; Zbl 0937.17006)].
Lastly, the author obtains a comparison theorem on non-maximal support varieties for a rational \(G\)-module with ``small'' high weights; comparing the support over \(kG(\mathbb F_p)\) with that over \(u(\mathfrak g)\). It follows that if such a module has constant Jordan type as a \(u(\mathfrak g)\)-module, then it has constant Jordan type as a \(kG(\mathbb F_p)\)-module. group schemes; cohomological support varieties; restricted Lie algebras; complexity; \(\pi\)-points; Weil restriction; projectivity; finite Chevalley groups; non-maximal support varieties; constant Jordan type Eric M. Friedlander, Weil restriction and support varieties, J. Reine Angew. Math. 648 (2010), 183 -- 200. Representations of finite groups of Lie type, Modular Lie (super)algebras, Cohomology of Lie (super)algebras, Modular representations and characters, Cohomology of groups, Representation theory for linear algebraic groups, Cohomology theory for linear algebraic groups, Group schemes Weil restriction and support varieties. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In the present paper we study examples of double coverings of the projective space \(\mathbb{P}^3\) branched over an octic surface. A double covering of \(\mathbb{P}^3\) branched over a smooth octic is a Calabi-Yau threefold. If the octic is singular then so is the double covering and we study its resolution of singularities. In this paper we restrict our considerations to the case of octics with only non-isolated singularities of a special type, namely looking locally like plane arrangements. - The main results of this note are theorem 2.1 and theorem 3.5 which can be formulated together as follows.
Let \(S\subset \mathbb{P}^3\) be an octic arrangement with no \(q\)-fold curve for \(q\geq 4\) and no \(p\)-fold point for \(p\geq 6\). Then the double covering of \(\mathbb{P}^3\) branched along \(S\) has a non-singular model \(Y\) which is a Calabi-Yau threefold. - Moreover if \(S\) contains no triple elliptic curves and \(l_3\) triple lines then the Euler characteristic \(e(Y)\) of \(Y\) is given as follows
\[
\begin{multlined} e(Y)=8-\sum_i (d^3_i-4d^2_i +6d_i)+2\sum_{i\neq j} (4-d_i-d_j) d_id_j\\ -\sum_{i\neq j\neq k\neq i} d_id_j d_k+4p^0_4+3p^1_4+ 16p^0_5+18p^1_5 +20p^2_5+l_3, \end{multlined}
\]
where \(d_i\) denotes the degree of the arrangement surfaces and \(p^j_i\) the number of \(i\)-fold points contained in \(j\) triple curves.
For the arrangements we allow exactly six types of singularities. For each case we describe precisely the resolution of singularities in the double cover. Then we study the effect on the Euler number of every blowing-up. This leads to the formula on the Euler number of \(Y\) as stated in the above theorem. Using this formula we obtain a table of examples of Calabi-Yau threefolds with 63 different Euler numbers. In the view of the mirror symmetry it is important to have examples of Calabi-Yau threefolds with as many Euler numbers as possible (it is conjectured that there are only finitely many possible numbers to appear as the Euler number of a Calabi-Yau threefold). Calabi-Yau threefold; octic arrangement; Euler number Cynk, S.; Szemberg, T., Double covers of \(\mathbb{P}^3\) and Calabi-Yau varieties, Banach center publ., 44, 93-101, (1998) \(3\)-folds, Coverings in algebraic geometry, Topological properties in algebraic geometry, Calabi-Yau manifolds (algebro-geometric aspects) Double covers and Calabi-Yau 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 We show that the Braden-MacPherson algorithm computes the stalks of parity sheaves. As a consequence we deduce that the Braden-MacPherson algorithm may be used to calculate the characters of tilting modules for algebraic groups and show that the \(p\)-smooth locus of a (Kac-Moody) Schubert variety coincides with the rationally smooth locus, if the underlying Bruhat graph satisfies a GKM-condition. modular representation theory; equivariant cohomology; moment graphs; constructible sheaves; tilting modules; Schubert varieties; \(p\)-smooth locus Fiebig, P. & Williamson, G., Parity sheaves, moment graphs and the p-smooth locus of Schubert varieties. \textit{Ann. Inst. Fourier (Grenoble)}, 64 (2014), 489-536. Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Classical groups (algebro-geometric aspects), Modular representations and characters, Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.), Intersection homology and cohomology in algebraic topology, Equivariant homology and cohomology in algebraic topology, Grassmannians, Schubert varieties, flag manifolds Parity sheaves, moment graphs and the \(p\)-smooth locus of Schubert 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 This paper is a continuation of the authors' previous paper, [Geom. Dedicata 178, 219--237 (2015; Zbl 1327.14177)], regarding the existence of low degree curves in the projective plane with certain contact properties with respect to a fixed divisor \(D\). This has implications for the splitting behaviour of the preimage of a curve in finite covers branched on \(D\). This, in turn, has implications for the topology of \(\mathbb{P}^2-D\).
The authors construct contact conics (namely, conics that intersect \(D\) only at smooth points with even intersection multiplicity) to a variety of low degree curves, and use them to construct Zariski \(N\)-plets in low degree. rational elliptic surfaces; Mordell-Weil group; Zariski \(N\)-plets Elliptic surfaces, elliptic or Calabi-Yau fibrations, Coverings in algebraic geometry Geometry of bisections of elliptic surfaces and Zariski \(N\)-plets. 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 disprove a strong form of the Regular Inverse Galois Problem: there exist finite groups \(G\) which do not have a \(\mathbb{Q}(U)\)-parametric extension, i.e., a regular realization \(F/\mathbb{Q}(T)\) that induces all Galois extensions \(L/\mathbb{Q}(U)\) of group \(G\) by specializing \(T\) to \(f(U)\in\mathbb{Q}(U)\). A much weaker Lifting Property is even disproved for these groups: two realizations \(L/\mathbb{Q}(U)\) exist that cannot be induced by realizations with the same ramification type. Our examples of such groups \(G\) include symmetric groups \(S_n,n\geq6\), infinitely many \(\mathrm{PSL}_2(\mathbb{F}_p)\), the Monster.{
}Two variants of the question with \(\mathbb{Q}(U)\) replaced by \(\mathbb{C}(U)\) and \(\mathbb{Q}\) are answered similarly, the second one under a diophantine ``working hypothesis'' going back to a problem of Fried-Schinzel.{
}We introduce two new tools: a comparison theorem between the invariants of an extension \(F/\mathbb{C}(T)\) and those obtained by specializing \(T\) to \(f(U)\in\mathbb{C}(U)\); and, given two regular Galois extensions of \(k(T)\), a finite set of \(k(U)\)-curves that say whether these extensions have a common specialization \(E/k\). Galois extensions; inverse Galois theory; specialization; parametric extensions; twisting Dèbes, Pierre, Groups with no parametric Galois realizations, Ann. sci. éc. norm. supér., (2016), in press Inverse Galois theory, Arithmetic theory of algebraic function fields, Coverings in algebraic geometry, Ramification problems in algebraic geometry, Field arithmetic Groups with no parametric Galois realizations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Bei der Untersuchung von algebraischen Varietäten gehört die Picardgruppe, d.h. die Gruppe der Geradenbündel, der Varietät zu den wichtigen Charakteristika. Hierbei interessiert insbesondere die Frage, in welchem Maße die Picardgruppe einer nichtsingulären Varietät auch die Picardgruppe einer (allgemeinen) genügend amplen Untervarietät bestimmt. Der Lefschetz-Hyperebenensatz für die Untervarietäten der Dimension größer oder gleich drei beantwortet diese Frage -- mit Hilfe der Exponentialsequenz -- vollständig; allgemein ist hier gleichbedeutend mit nicht-singulär. Für den Fall von zweidimensionalen Untervarietäten (d.h. Flächen) in einer dreidimensionalen Varietät ist die Situation vollkommen anders. Die klassische Formulierung dieses Problems durch Max Noether wurde erst von S. Lefschetz vollständig bewiesen [siehe \textit{S. Lefschetz}, Trans. Am. Math. Soc. 22, 327-482 (1921; JFM 48.0428.03) und \textit{M. Noether}, ``Zur Grundlegung der Theorie der algebraischen Raumkurven'', Math.-Phys. Abhdl., Kgl. Preuss. Akad. Wiss. Berlin (1882; JFM 15.0684.01)]. Das Noether-Lefschetz-Theorem für Hyperflächen vom Grad größer gleich vier im dreidimensionalen projektivem Raum besagt, daß für eine ``allgemeine'' Fläche die Picardgruppe der Fläche isomorph zu der Picardgruppe des projektiven Raumes ist, erzeugt durch die Einschränkung des Hyperebenenschnitts (d.h. des Geradenbündels \({\mathcal O}(1)\)) des projektiven Raumes.
Wir betrachten diese Noether-Lefschetz-Probleme für zyklische Überlagerungen der projektiven Ebene \(\mathbb{P}^2\). Mit Hilfe der infinitesimalen Methode lassen sich auch hier die Noether-Lefschetz-Theoreme beweisen: Aber die Tatsache, daß die Hodgetheorie einer zyklischen Überlagerung durch die Hodgetheorie mit vertwisteten Koeffizienten der Basisvarietät (hier die projektive Ebene \(\mathbb{P}^2)\) beschrieben werden kann und sich die Operation der Galoisgruppe auf der Kohomologie genau darstellen läßt, bewirkt, daß sich die allgemeinen und expliziten Noether-Lefschetz-Theoreme sogar in den Fällen beweisen lassen, wo die infinitesimalen Noether-Lefschetz-Theoreme nicht gelten. Verwendet werden die Methoden, die von \textit{M. L. Green} [J. Differ. Geom. 27, No. 1, 155-159 (1988; Zbl 0674.14005) and 29, No. 2, 295-302 (1989; Zbl 0674.14003)] bei der Untersuchung des Noether-Lefschetz-Ortes eingeführt wurden. Man findet hierbei auch in den Fällen, wo sich die zyklische Überlagerung der projektiven Ebene \(\mathbb{P}^2\) als Fläche vom Grad \(N\) in den projektiven Raum \(\mathbb{P}^3\) einbetten läßt, die klassischen Resultate wieder. Hodge theory; cyclic covering; Picard group; Noether-Lefschetz theorem Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Coverings in algebraic geometry, Transcendental methods, Hodge theory (algebro-geometric aspects) Noether-Lefschetz-theorems for cyclic 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 We study cyclic coverings of homogeneous-rational manifolds and prove an infinitesimal Torelli theorem. The main ingrediences are the modules of meromorphic differentials with logarithmic poles along the branch divisor and the Bott vanishing theorem for homogeneous vector bundles. The results have appeared in Math. Ann. 274, 443-472 (1986; Zbl 0593.32017). cyclic coverings of homogeneous-rational manifolds; infinitesimal Torelli theorem Deformations of special (e.g., CR) structures, Sheaves and cohomology of sections of holomorphic vector bundles, general results, Coverings in algebraic geometry, Vanishing theorems, Divisors, linear systems, invertible sheaves Infinitesimal Torelli 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 Authors' abstract: Rigid algebraic varieties form an important class of complex varieties that exhibit interesting geometric phenomena. In this paper we propose a natural extension of rigidity to complex projective varieties with a finite group action (\(G\)-varieties) and focus on the first nontrivial case, namely, on \(G\)-rigid surfaces that can be represented as desingularizations of Galois coverings of the projective plane with Galois group \(G\). We obtain local and global \(G\)-rigidity criteria for these \(G\)-surfaces and present several series of such surfaces that are rigid with respect to the action of the deck transformation group. The paper contains many very useful explicit descriptions of the geometry of brach curves and singularities. A \(G\)-variety is said to be rigid if every smooth deformation in the category of \(G\)-varieties is trivial. For them a smooth deformation must be a submersion with a smooth base and a smooth total space on which \(G\) acts compatibly. \(G\)-variety; group action; deformation; Galois covering of the plane Kulikov, Vik. S.; Shustin, E. I., On, Proc. Steklov Inst. Math., 298, 144-164, (2017) Group actions on varieties or schemes (quotients), Automorphisms of surfaces and higher-dimensional varieties, Formal methods and deformations in algebraic geometry, Coverings in algebraic geometry On \(G\)-rigid 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 Two proper polynomial maps \(f_1,f_2:\mathbb C^2\rightarrow\mathbb C^2\) are called \textit{equivalent} if there are automorphisms \(\Phi_1,\Phi_2\) of \(\mathbb C^2\) such that \(f_2=\Phi_2\circ f_1\circ\Phi_1\). It is shown in [\textit{S. Lamy}, Publ. Mat., Barc. 49, No. 1, 3--20 (2005; Zbl 1094.14052)] that if the proper polynomial map \(f:\mathbb C^2\rightarrow\mathbb C^2\) has topological degree 2, then it is equivalent to the map \(\tilde f(x,y)=(x,y^2)\). Thus natural questions arise:
(1) Is every proper polynomial mapping \(f:\mathbb C^2\rightarrow\mathbb C^2\) equivalent to some map of type \((x,y)\mapsto(x,P(y))\) or, at least, \((x,y)\mapsto(x,Q(x,y))\), where \(P,Q\) are polynomials?
(2) How many equivalence classes of proper polynomial maps \(\mathbb C^2\rightarrow\mathbb C^2\) of fixed topological degree \(d\geq3\) are there?
The authors answer that questions and try to generalize the above result for arbitrary topological degree \(d\geq3\). Namely, they show that:
(1) For every \(d\geq3\) there exists at least one proper polynomial map \(f:\mathbb C^2\rightarrow\mathbb C^2\) of topological degree \(d\) such that \(f\) is not equivalent to a map of type \((x,y)\mapsto(x,Q(x,y))\).
(2) For any \(d,n\in\mathbb N\), \(d\geq3\), \(n\geq2\), consider the polynomial map \(f_{d,n}:\mathbb C^2\rightarrow\mathbb C^2\) given by
\[
f_{d,n}(x,y):=(x,y^d-dx^ny).
\]
Then \(f_{d,n}\) and \(f_{d,m}\) are equivalent if and only if \(m=n\). In particular, there are infinitely many classes of equivalence of proper polynomial maps \(f:\mathbb C^2\rightarrow\mathbb C^2\) of fixed topological degree \(d\geq3\).
Although a satisfactory description of all equivalence classes in the case of \(d\geq3\) is at the moment out of reach, the authors classify those proper polynomial maps enjoying some additional properties. Namely, they give full classification of the proper polynomial maps \(\mathbb C^2\rightarrow\mathbb C^2\) which are Galois coverings with finite Galois group. proper polynomial maps; Galois coverings; complex reflection groups Bisi, C.; Polizzi, F.: On proper polynomial maps of C2. J. geom. Anal. 20, No. 1, 72-89 (2010) Proper holomorphic mappings, finiteness theorems, Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem), Coverings in algebraic geometry, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Reflection groups, reflection geometries, Other geometric groups, including crystallographic groups On proper polynomial maps of \(\mathbb C^{2}\) | 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 This remarkable paper deals with the monodromy of solutions of generic systems of \(n\) algebraic equations in \(n\) non-zero complex variables. The Bernstein-Koushnirenko theorem states that the number of solutions of such a system does not depend on the (generic) coefficients and equals the mixed lattice volume of the Newton polytopes of the given equations. Fixing the set of monomials for each equation, one obtains the monodromy group or the Galois group of the system induced by the fundamental group of the complement to the discriminant on the space of all systems spanned by fixed monomials. Recently, the first author [Compos. Math. 155, No. 2, 229--245 (2019; Zbl 1451.14152)] showed that the Galois group of a reduced, irreducible system of generic equations is isomorphic to the symmetric group on the number of solutions. In the case of one univariate equation, one obtains the full symmetric group whenever the gcd of the exponents of the monomials equals \(1\). However, if the gcd is non-trivial, the Galois group appears to be the wreath product of the reduced symmetric group and a cyclic group.
The current paper is mainly devoted to the study of the latter phenomenon in the multivariate case. The expected multivariate answer is as follows. Let \(A_1,\dots,A_n\subset{\mathbb Z}^n\) be the sets of monomials for the considered system of algebraic equations, \(\widetilde d\) the mixed volume of the Newton polytopes \(P(A_1),\dots,P(A_n)\), and let \(\Lambda\subset{\mathbb Z}^n\) be a full rank sublattice of index \(k\) generated by the differences of the points of \(A_1,\dots,A_n\). Then the expected Galois group is the wreath product of the symmetric group \(S_d\), \(d=\widetilde d/k\), and the cyclic group \({\mathbb Z}_k\). The authors show that the actual answer is not always the expected one, and it is widely open though, in principle, depends only on the combinatorial data. The main theorem, proved in the paper, provides almost coinciding necessary and sufficient conditions for the Galois group to be the expected one. A very particular instance of it sounds like that: If the Newton polytopes \(P(A_1),\dots,P(A_n)\) share the same dual fan, then the Galois group is expected if the tuple \((A_1,\dots,A_n)\) is combinatorially ample, and the Galois group is strictly smaller than the expected one, otherwise.
To prove the main theorem and to provide applications to enumerative geometry and to the problem of connectivity of covering spaces, the authors develop new combinatorial and topological techniques. In particular, the connectivity of covering spaces in certain important cases can be reduced to the study of the first homology and its behavior under covering, instead of consideration of fundamental groups. An application to enumerative problems consists in computaion of Galois (or monodromy) groups whenever the counted objects appear in a family, and the enumeration is maid for a generic choice of the parameters. A typical example is the Galois group of the enumeration of roots of generic systems of algebraic equations. Another important example is the enumeration of lines in a projective space that meet and appropriate collection of projective subspaces or points. The novelty of the authors' approach is that they separate a class of wreath enumerative problems, for which they provide criteria to have an expected Galois group.
Numerous illustrating examples and a very accurate exposition help much to follow the argument. enumerative geometry; topological Galois theory; Galois covering; monodromy; Newton polytope Coverings in algebraic geometry, Algebraic functions and function fields in algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies, Enumerative problems (combinatorial problems) in algebraic geometry, Multiply transitive finite groups, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Monodromy on manifolds Sparse polynomial equations and other enumerative problems whose Galois groups are wreath products | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Galois stratification for formulas in the language of rings, introduced by \textit{M.~Fried} and \textit{G.~Sacerdote} [``Solving diophantine problems over all residue class fields of a number field and all finite fields'', Ann. Math. (2) 104, 203--233 (1976; Zbl 0376.02042)], provides an explicit description of definable sets over finite fields by giving a quantifier elimination procedure for so-called Galois formulas associated with Galois covers of varieties. This strengthened results by \textit{J.~Ax} [``The elementary theory of finite fields'', Ann. Math. (2) 88, 239--271 (1968; Zbl 0195.05701)] on the theory of finite fields. \textit{E.~Hrushovski} [``The elementary theory of the Frobenius automorphisms'', \url{arXiv:math/0406514}] generalized several aspects of Ax' work by developing a theory of difference schemes and studying the elementary theory of Frobenius difference fields, i.e.~algebraically closed fields of positive characteristic together with a power of the Frobenius endomorphism.
The present paper develops a theory of Galois stratification for formulas in the language of difference rings. For this, the notions of twisted Galois covers of difference schemes and twisted Galois stratifications are introduced, and a quantifier elimination is proven. This leads to an algebraic description of definable sets over Frobenius difference fields.
After publication, a significantly extended version under the same title appeared as [\url{arXiv:1112.0802}]. difference ring; difference scheme; Galois stratification; Frobenius automorphism Tomašić, Ivan: Twisted Galois stratification, (2012) Difference algebra, Model-theoretic algebra, Automorphisms and endomorphisms of algebraic structures, Coverings in algebraic geometry, Finite ground fields in algebraic geometry Twisted Galois stratification | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 concerned with the computation of Betti and Chern numbers of a minimal smooth model \(\hat X\) of an abelian covering of \(\mathbb{P}_ 2(\mathbb{C})\) branched along \({\mathcal L}\), a union of real lines. The basic object is \(b_ 1(\hat X)\); an algorithm for its computation is given:
1. Find a presentation for \(\pi_ 1(\mathbb{P}_ 2-{\mathcal L})\) (using Moishezon's and Libgober's technique of projection to a line and analysing the monodromy from braids). An algorithm for this is given, too.
2. Using Fox calculus, compute \(b^ u_ 1(\hat X)\), the first Betti number of the unbranched part of the covering.
3. Find lifting data for curves lying above the branch locus: this consists of an enumeration of these curves and the way they intersect. What is needed is the intersection matrix \(I\) of this set of curves, because \(b_ 1(\hat X)= b^ u_ 1(\hat X)-\text{Null}(I)\), where \(\text{Null}(I)\) is the nullity of \(I\).
This algorithm is described in \S4 in such a way that it can be implemented on a computer, what the author has done to compute examples presented in \S5 (coverings of degree 2 and 3, sometimes 5 of 10 configurations of 6 lines, 23 configurations of 7 lines and two others with 9 and 10 lines). Configurations with \(\hat X\) not of general type are also considered. abelian branched coverings; Hirzebruch surfaces; computation Chern numbers; computation of Betti numbers; algorithm implemented on a computer; intersection matrix Hironaka, E.: Abelian coverings of the complex projective plane branched along configurations of real lines. Mem. Am. Math. Soc. \textbf{105}(502), i-vi, 1-85 (1993) Computational aspects of algebraic surfaces, Special surfaces, Coverings in algebraic geometry Abelian coverings of the complex projective plane branched along configurations of real lines | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper is concerned with the study of the ramification theory of Artin-Schreier extensions of algebraic surfaces, with the aim of answering, in certain cases, a question posed by \textit{K. Kato} in [Am. J. Math. 116, No. 4, 757--784 (1994; Zbl 0864.11057)].\newline The question is about the behaviour of his higher codimension ramification invariants: specifically the invariant \(r_x\), for \(x\) a closed point on a proper surface, is conjectured to be bounded from above by an expression depending on the ramification theory in codimension one only (the Swan conductor of the character defining the Artin-Schreier extension, computed along divisors).\newline The main idea is to look for a combinatorial interpretation of Kato's invariant \(r_x\), by finding suitable rational functions defining the Artin-Schreier extension of function fields, and then studying the local behaviour, of such functions, around the ramification divisor inside the surface. The paper is organized as follows:\newline In chapter 1, the author sets up the notation. Let \(X\) be a smooth proper surface over an algebraically closed field of characteristic \(p\), \(D\) a snc divisor and \(U\) its complement. A \(p\)-torsion \(l\)-adic character \(\chi: \pi_1(U)\to \bar{Q_l}^*\) defines an Artin-Schreier extension of \(K(X)\), given by \(\alpha^p-\alpha=f,f\in K(X)\), which is étale over \(U\).\newline In chapter 2, the author recalls the relevant definitions. Given an Artin-Schreier extension as above, and given a closed point \(x\in X\setminus U\), there exists a tower of blow-ups, centered at points over \(x\), \(p:X_s\to\dots\to X_0=X\), such that the resulting Artin-Schreier extension of \(X_s\) is clean along the complement of \(p^{-1}(x)\). Being clean is a local condition of transversality between the divisors \(D\) and \(\mathrm{div}(f)\). The invariant \(r_x\) is defined as a sum of expressions \(\sum_i \mu_i\), where each \(\mu_i\) is itself a simple formula depending uniquely on the Swan conductor of the \(i\)-th exceptional divisor, and that of the proper transform of the divisor \(D\). This is reminiscent of the notion of multiplicity of a divisor at a closed point \(x\) in a proper surface, defined as the intersection between the divisor and the exceptional locus of the blow-up at \(x\).\newline The author proceeds to construct, via a series of lemmas, a so-called good representative for the Artin-Schreier extension. This is an element \(g\in K(X)\) such that the extension is defined by the equation \(\alpha^p-\alpha=g\), and satisfying certain constraints on its behaviour around the ramification divisor \(D\), see Definition 2.\newline In chapter 3 the author defines a map \(Y(X,D,x)\) associating, to every Artin-Schreier extension as above, a Young diagram. The map is defined in terms of the good representative, but it turns out to be independent of any choice.\newline In chapter 4 the author introduces, in Definition 4, the crucial notion of good Artin-Schreier extension. It is verified, in Theorem 4.2, that for a good Artin-Schreier extension \(\chi\), Kato's invariant \(r_x\) can be read off the structure of the Young diagram \(Y(X,D,x)(\chi)\) solely, in fact \(r_x=|Y(X,D,x)(\chi)|\).\newline In chapter 5, it is pointed out that the identity \(r_x=|Y(X,D,x)(\chi)|\) implies Kato's conjecture for good Artin-Schreier extensions.\newline In chapter 6 and 7, some applications are discussed, including a lower bound for the Euler characteristic with compact support of the étale sheaf corresponding to the Artin-Schreier extension on \(U\).
For part I,, cf. [the author, JSIAM Lett. 6, 33--36 (2014)]. ramification of surface; Artin-Schreier extension; Young diagram M. Oi, On ramifications of Artin-Schreier extensions of surfaces over algebraically closed fields of positive characteristic II, in preparation. Positive characteristic ground fields in algebraic geometry, Ramification problems in algebraic geometry, Ramification and extension theory, Coverings in algebraic geometry, Galois theory and commutative ring extensions On ramifications of Artin-Schreier extensions of surfaces over algebraically closed fields of 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 The Grothendieck-Teichmüller group \(\widehat{GT}\) acts naturally on the profinite completion \(\widehat K(0,n)\) of the mapping class group of a sphere with \(n\) marked points. The elements of \(\widehat{GT}\) are, by definition, parameterized by the pairs \((\lambda,f)\in\widehat\mathbb{Z}^\times\times\widehat F_2\) subject to certain conditions. In this paper, it is shown that the principal parameter \(f\) evaluated in \(\widehat K(0,4)\) (resp. \(\widehat K(0,5)\)) admits certain decompositions like non-commutative 1-coboundaries under certain standard actions of finite cyclic groups of order 2, 3 (resp. 5). As a by-product of their method, it is also proven that the ``complex conjugation'' in \(\widehat{GT}\) is self-centralizing. profinite groups; non-commutative cohomology; Grothendieck-Teichmüller group; profinite completions; mapping class groups; actions of finite cyclic groups P. Lochak and L. Schneps, ''A cohomological interpretation of the Grothendieck-Teichmüller group,'' Invent. Math., vol. 127, iss. 3, pp. 571-600, 1997. Limits, profinite groups, Cohomology of groups, Coverings in algebraic geometry, Other groups related to topology or analysis A cohomological interpretation of the Grothendieck-Teichmüller group. Appendix by C. Scheiderer | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(Y\) be a smooth projective variety and \({\mathcal Z}\) be an ample invertible sheaf on \(Y\). For \(N\) a sufficiently large positive number, the sheaf \({\mathcal L}^ N\) has (a lot of) smooth reduced sections. We choose one of them: \(X\), and we construct the \(N\)-cyclic covering \(f:Z \to Y\) branched over \(X\). The variational Torelli problem is to know if the pair \((Z,f^*{\mathcal Z})\) is uniquely determined by the algebraic part of its infinitesimal variation of Hodge structure. This theory was explained in a series of papers of J. Carlson, M. Green, P. Griffiths, J. Harris and mainly [\textit{R. Donagi} in Compos. Math. 50, 325-353 (1983; Zbl 0598.14007)].
The author's main theorem is that, in this context and for \(N\) sufficiently large, the variational Torelli theorem holds. The proof uses the Green's theory of symmetrizers and the computation of the Hodge theory of the cyclic covering due to \textit{H. Esnault} and \textit{E. Viehweg} [in Algebraic threefolds, Proc. 2nd Sess. C.I.M.E, Varenna 1981, Lect. Notes Math. 947, 241-250 (1982; Zbl 0493.14012)]. variational Torelli problem; infinitesimal variation of Hodge structure KÈ\copyright stutis Ivinskis, A variational Torelli theorem for cyclic coverings of high degree, Compositio Math. 85 (1993), no. 2, 201 -- 228. Torelli problem, Variation of Hodge structures (algebro-geometric aspects), Coverings in algebraic geometry A variational Torelli theorem for cyclic coverings of high degree | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 complex projective normal surface with anti-canonical divisor \(-K_S\) nef and big. It is known that the fundamental group of the smooth part \(S^0=S-\text{Sing }S\) is finite under the assumption that \(S\) has at worst log-terminal singularities [see \textit{D.-Q. Zhang}, Math. Proc. Camb. Philos. Soc. 117, No. 1, 161-163 (1995; Zbl 0838.14009) and \textit{R. V. Gurjar} and \textit{D.-Q. Zhang}, J. Math. Sci., Tokyo 1, No. 1, 137-180 (1994; Zbl 0841.14017) and 2, No. 1, 165-196 (1995; Zbl 0847.14021)]. The case in which \(S\) has at worst log-canonical singularities is considered in the present paper, where it is shown that the fundamental group \(\pi_1(S^0)\) may no longer be finite. The author describes \(\pi_1(S^0)\) and, in a sense, the surface \(S\) under the assumption that \(S\) has only rational singularities. The main theorem proven says that when \(\pi_1(S^0)\) is infinite \(S\) has a finite Galois covering unramified over \(S^0\) and the covering surface is described. log canonical singularities; nef divisor; big anti-canonical divisor; non-finite fundamental group Zhang, D. -Q.: Algebraic surfaces with log canonical singularities and the fundamental groups of their smooth parts. Trans. amer. Math. soc. 348, 4175-4184 (1996) Homotopy theory and fundamental groups in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Coverings in algebraic geometry Algebraic surfaces with log canonical singularities and the fundamental groups of their smooth parts | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 spite of several papers on triple covers, the basic problem of characterizing branch divisors of normal triple covers of the projective plane \(\mathbb P^2\) is still open. In the paper under review, the author deals with the case of a branch divisor of degree six, lower degree branch divisors being already studied in the literature. Let \(S\) and \(T\) be two reduced divisors on \(\mathbb P^2\), with no common components. The author aims at describing normal triple covers \(\pi:X \to \mathbb P^2\), with branch divisor \(\Delta = S+2T\) (i.e., with ramification index \(2\) over \(S\) and \(3\) over \(T\)). Let \(F \subset \mathbb P^2 \times \mathbb P^{2 \vee}\) be the flag variety of pairs of points and lines in \(\mathbb P^2\), and let \(p:F \to \mathbb P^2\) and \(q:F \to \mathbb P^{2 \vee}\) be the projections. Relying on the work on triple covers done by \textit{R. Miranda} [Am. J. Math. 107, 1123--1158 (1985; Zbl 0611.14011)], the author proves the following result.
Let \(\pi:X \to \mathbb P^2\) be a normal triple cover whose branch locus \(\Delta\) has degree six: then either (i) \(\Delta =S\) is an irreducible sextic curve with nine cusps, \(X=q^{-1}(S^{\vee})\) (a \(\mathbb P^1\) bundle over the smooth plane cubic \(S^{\vee}\), dual to \(S\)), and \(\pi\) is the restriction of \(p\) to \(X\), or (ii) \(X\) is a cubic surface in \(\mathbb P^3\) and \(\pi\) is a projection from a point of \(\mathbb P^3 \setminus X\). Furthermore (i) occurs if and only if the nine cusps are total branching points of \(\pi\). According to a previous result of \textit{H. Ishida} and \textit{H.-o Tokunaga} [Adv. Stud. Pure Math. 56, 169--185 (2009; Zbl 1198.14016)], if the branch divisor is a sextic curve with simple singularities, then either \(X\) is a quotient of an abelian surface by an involution, or \(X\) is as in (ii). triple cover; cubic surface; Tschirnhaus module; torus curve Coverings in algebraic geometry, Ramification problems in algebraic geometry, Divisors, linear systems, invertible sheaves A note on normal triple covers over \(\mathbb P^{2}\) with branch divisors of degree 6 | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a smooth projective variety of dimension \(n\), let \(L\) be an ample line bundle on \(X\), and let \(x_1,\ldots,x_r\) be distinct points in \(X\). Then the Seshadri constant of \(L\) at \(x_1,\ldots,x_r\) is:
\[
\epsilon(L;x_1,\ldots,x_r)=\sup\{\epsilon| \, f^*L-\epsilon \sum_{i=1}^rE_i\text{ is nef }\},
\]
where \(f\) is the blowing up of \(X\) at \(x_1,\ldots,x_r\) and \(E_1,\ldots,E_r\) are the exceptional divisors. The upper bound
\[
\epsilon(L; r)\leq \sqrt[n]{\frac{L^{n}}{r}}
\]
holds. However, explicit values are difficult to obtain even when \(r=1\).
Now suppose, in addition, that \((X,L)\) is a polarized abelian surface of type \((1,d)\) of Picard rank one. Let \(x\) be a point of \(X\). Let \(G\) be a finite subgroup of \(X\) of order \(g\). Consider the étale quotient: \(q\colon X\to X/G\). Let \(n\) be the smallest natural number such that \(nL=q^*M\) for some line bundle \(M\) on \(X/G\).
The paper under review proves that \(\epsilon(L; x+G)=\sqrt{\frac{2d}{g}}\) if \(\sqrt{2d/g}\) is rational, and
\[
\epsilon(L; x+G)= \frac{k_0}{l_0}\frac{2dn}{g}=\sqrt{1-\frac{1}{l_0^2}}\sqrt{\frac{L^2}{g}}
\]
if \(\sqrt{2d/g}\) is irrational, where \((l_0,k_0)\) is the primitive solution of Pell's equation \(l^2-\frac{2n^2d}{g}k^2=1\). García, L. F., Seshadri constants in finite subgroups of abelian surfaces, Geom. Dedicata, 127, 1, 43-48, (2007) Divisors, linear systems, invertible sheaves, Coverings in algebraic geometry Seshadri constants in finite subgroups of abelian surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The starting point of this work has been the use of bidouble covers of \(\mathbb{P}^1 \times \mathbb{P}^1\) made by \textit{F. Catanese} [in J. Differ. Geom. 19, 483-515 (1984; Zbl 0549.14012) and 24, 395-399 (1986; Zbl 0621.14014)] in order to show some properties of the moduli space of surfaces of general type. Since these covers appear to be very widely spread in the geography of surfaces of general type, it is a natural question to ask whether they satisfy the infinitesimal Torelli property or not. Actually it turned out that it is possible to answer, at least partially, the much more general question of when infinitesimal Torelli holds for an abelian cover of a surface. In theorem 3.1 we give sufficient conditions for this to be true and we obtain as a corollary that infinitesimal Torelli holds for abelian covers of the projective plane and of the Segre-Hirzebruch surfaces \(F_n\), under very mild restrictions. abelian covers of Segre-Hirzebruch surfaces; infinitesimal Torelli; abelian cover of surface; abelian covers of the projective plane Rita Pardini, Infinitesimal Torelli and abelian covers of algebraic surfaces, Problems in the theory of surfaces and their classification (Cortona, 1988) Sympos. Math., XXXII, Academic Press, London, 1991, pp. 247 -- 257. Torelli problem, Coverings in algebraic geometry, Special surfaces Infinitesimal Torelli and abelian covers 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 A configuration of six lines in \(\mathbb{P}^2\) determines a \(K3\) surface as the minimal model of a double covering of \(\mathbb{P}^2\) branched along these lines. Regarding an elliptic curve as a double covering of \(\mathbb{P}^1\) branched at four points, a classical theory relates the period integrals of an elliptic curve, Gauss hypergeometric function, theta functions, and limit formula of arithmetic-geometric means. Being an analogue of elliptic curves, peiod integrals of \(K3\) surfaces obtained as above are understood with certain hypergeometric functions and some mean iteration.
The paper under review focuses on \(K3\) surfaces correponding to six lines in \(\mathbb{P}^2\) as above and studies relations among their period integrals, hypergeometric functions, theta functions and limit of mean iterations. The main result is to give a relation between the period integrals of a \(K3\) surface and theta functions by writing down a Thomae-type formula that is induced from a \(2\tau\)-formula of the theta functions. As a corollary, defining hypergeometric functions \(F_S, F_T\), and expressing the period integrals by them, it is shown that the theta function is also written down with the functions \(F_S, F_T\). This result explains how the Thomae-type formula for Kummer locus relates to the limit formula for Borchardt's mean iteration. Finally as the last application, a functional equation for the hypergeometric function \(F_S\) is given. \(K3\) surfaces and Enriques surfaces, Projective techniques in algebraic geometry, Coverings in algebraic geometry Thomae type formula for \(K3\) surfaces given by double covers of the projective plane branching along six lines | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A complex-projective manifold \(X\) with \(K_X\) ample famously admits a Kähler-Einstein metric. As a consequence, we have the Miyaoka-Yau inequality
\[
\big( 2(n+1)\mathrm{c}_2(X) - n\mathrm{c}_1^2(X) \big) \cdot K_X^{n-2} \ge 0,
\]
where \(n = \dim X\).
Theorem A of the present paper extends this inequality to minimal models, i.e.~normal complex-projective varieties with terminal singularities such that \(K_X\) is \(\mathbb{Q}\)-Cartier and nef. Of course, the statement is empty if e.g. \(K_X\) is numerically trivial; Theorem B proves the following more general version:
\[
\big( 2(n+1)\mathrm{c}_2(X) - n\mathrm{c}_1^2(X) \big) \cdot K_X^i \cdot H^j \ge 0,
\]
where \(H\) is ample and \(i = \mathrm{min}(\nu, n-2)\), where \(\nu\) is the numerical dimension of \(K_X\).
Actually, Theorem B deals even more generally with dlt pairs \((X, D)\) with standard coefficients. (This generality is also needed in the proof.) The development of the necessary orbifold techniques is one of the main technical contributions of the paper. The other is Theorem C, which proves the semistability of the (orbifold) tangent sheaf in the above setting. The proof of the latter is analytic, relying on the theory of conical/cuspidal metrics. Miayoka-Yau inequality; minimal models; orbifold pairs; singular Kähler-Einstein metrics Coverings in algebraic geometry, Minimal model program (Mori theory, extremal rays), Kähler-Einstein manifolds, (Equivariant) Chow groups and rings; motives, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Notions of stability for complex manifolds, Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) Orbifold stability and Miyaoka-Yau inequality for minimal 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 \(K3\) surfaces, the class of surfaces which are simply connected and that have trivial canonical bundle, were intensively studies over the last 40 years. One of the earliest results is due to Piatetski-Shapiro (and Shafarevich), who proved that their period map is injective. However, there still open questions regarding the geometry of algebraic, projective \(K3\) surfaces.
In this note we present a new restriction on the position of the singularities of branch curves of projective \(K3\) surfaces. Using this restriction, we are able to find a new Zariski pair, when one of the curves is a branch curve of a projective \(K3\) of genus 4 surface, embedded in \(\mathbb{P}^4\), and the other is not.
Reformulating former results on branch curves of projective \(K3\) surfaces, we pose a conjecture about the reducibility of certain varieties of nodal-cuspidal curves, when one of its components is the component of branch curve of a projective \(K3\) of genus \(g\). \(K3\) surfaces and Enriques surfaces, Plane and space curves, Coverings in algebraic geometry, Families, moduli of curves (algebraic) On the singularities of branch curves of \(K3\) surfaces and applications | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Sei k ein endlich algebraischer Zahlkörper und X ein geometrisch zusammenhängendes glattes projektives k-Schema. Sei \(\pi_ 1^{ab}(X)\) die abelsch gemachte Fundamentalgruppe von X; sie beschreibt die abelschen étalen Überlagerungen von X. - In Fortführung der Arbeiten von \textit{S. Bloch} [Ann. Math., II. Ser. 114, 229-265 (1981; Zbl 0512.14009)] und \textit{K. Kato} und Verf. [ibid. 118, 241-275 (1983; Zbl 0562.14011)] behandelt die vorliegende Arbeit das Reziprozitätsgesetz für X, welches die Gruppe \(\pi_ 1^{ab}(X)\) mit der K-theoretisch definierten Idelklassengruppe C(X) von X in Verbindung setzt. - Die Idelklassengruppe C(X) ist der Kokern der kanonischen Abbildung \(SK_ 1(X)\to I(X)\) von \(SK_ 1(X)\) in die Idelgruppe I(X) von X. Die Idelgruppe I(X) ist das eingeschränkte direkte Produkt der lokalen Gruppen \(SK_ 1(X_ v)\) über alle Primstellen v von K bezüglich der Untergruppen \(T_ v=Ker(SK_ 1(X_ v)\to SK_ 0({\mathcal X}_ v))\) für Primstellen v aus einer passenden offenen Teilmenge U des Ganzheitsrings \({\mathfrak O}\) von k, wobei \({\mathcal X}\) eine glatte projektive Ausbreitung von X über U und \({\mathcal X}_ v\) die Faser von \({\mathcal X}/U\) in v ist. I(X) und alsdann auch C(X) ist in natürlicher Weise mit einer Topologie versehen. Durch Zusammensetzung gewisser lokaler Reziprozitätsabbildungen auf den \(SK_ 1(K_ v)\) erhält man eine Reziprozitätsabbildung \(\tau: C(X)\to \pi_ 1^{ab}(X).\)
Das Reziprozitätsgesetz betrifft das Prontrjagindual dieser Abbildung und lautet unter Verwendung von \(Hom(\pi_ 1^{ab}(X),{\mathbb{Q}}/{\mathbb{Z}})=H^ 1(X,{\mathbb{Q}}/{\mathbb{Z}}):\) Theorem (5.10): \(H^ 1(X,{\mathbb{Q}}/{\mathbb{Z}})\to^{\cong}Hom_{fin}(C(X),{\mathbb{Q}}/{\mathbb{Z}}),\) wobei rechts die Gruppe aller stetigen Homomorphismen C(X)\(\to {\mathbb{Q}}/{\mathbb{Z}}\) von endlicher Ordnung betrachtet wird. - Dieses Theorem ist in Wahrheit in einem allgemeineren Reziprozitätsgesetz (5.6) enthalten, welches die entsprechende Aussage für die Reziprozitätsabbildung \(\tau: C({\mathcal X})\to \pi_ 1^{ab}({\mathcal X})\) macht, wobei jetzt \({\mathcal X}\) ein reguläres flaches eigentliches Schema über einer offenen Teilmenge U von Spec(\({\mathfrak O})\) ist, dessen generische Faser geometrisch zusammenhängend, glatt und projektiv über k ist. Grundlegend für den Beweis des allgemeinen Reziprozitätsgesetzes ist die Verallgemeinerung (7.7) der Blochschen exakten Sequenz auf höhere Dimensionen. class field theory of arithmetical schemes; Bloch's exact sequence; reciprocity theorem; abelian fundamental group; idele class groups S. Saito, Unramified class field theory of arithmetic schemes, Ann. Math., 121 (1985), 251--281. Schemes and morphisms, Class field theory, Global ground fields in algebraic geometry, Coverings in algebraic geometry Unramified class field theory of arithmetical schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let k be an algebraically closed field of characteristic \(p>0\) and let X be a nonsingular projective surface over k. An \textit{Artin-Schreier covering} is a finite morphism \(\pi: Y\to X\) from a normal surface Y onto X such that the field extension \(k(Y)/k(X)\) is an Artin-Schreier extension, i.e., \(k(Y)/k(X)\) is a Galois extension whose Galois group is isomorphic to \({\mathbb{Z}}/p{\mathbb{Z}}.\)
A good class of Artin-Schreier coverings which is called \textit{of simple type} is defined and, in the case of simple type, some formulas are given to compute invariants of coverings. Moreover, smooth Artin-Schreier coverings of simple type with ample branch loci are determined. positive characteristic; invariants of coverings; Artin-Schreier coverings of simple type; ample branch loci Takeda, Y, \textit{Artin-Schreier coverings of algebraic surfaces}, J. Math. Soc. Japan, 41, 415-435, (1989) Coverings in algebraic geometry, Arithmetic ground fields for surfaces or higher-dimensional varieties, Finite ground fields in algebraic geometry Artin-Schreier 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 Let \(k\) be a field and let \(X\) be a smooth geometrically irreducible curve over \(k\). Let \(f: X\to\mathbb{P}_k^1\) be a \(k\)-regular cover. A rational pullback of \(f\) is a \(k\)-regular cover \(f_{T_0}: X_{T_0}\to\mathbb{P}_k^1\) obtained by pulling back \(f\) along some non-constant rational map \(T_0 :\mathbb{P}_k^1\to\mathbb{P}_k^1\). The Regular Inverse Galois Problem over \(k\) is the problem of realizing each finite group as the Galois group of a \(k\)-regular Galois cover of \(\mathbb{P}_k^1\). Rational pullbacks crete such covers.
The main result of this paper is to obtain all Galois covers \(X\to\mathbb{P}_k^1\) of a given group \(G\) from a proper subset of them by rational pullbacks. In this case, the subset is called a regular parametric cover.
Theorem. Let \(k=\mathbb{C}\). The finite subgroups of \(PGL_2(\mathbb{C})\) (i.e, cyclic and dihedral groups, \(A_4, S_4\) and \(A_5\)) are exactly those finite groups which have a regular parametric cover \(X\to\mathbb{P}_k^1\). More precisely, given a finite group \(G\), the following statements hold.
(a) If \(G\subset PGL_2(\mathbb{C})\), then \(G\) has a regular parametric cover.
(b) If \(G\) is not contained in \(PGL_2(\mathbb{C})\), then even the set of all Galois covers \(X\to\mathbb{P}_k^1\) of group \(G\) and with at most \(r_0\) branch points is not regularly parametric, for any \(r_0\geq 0\).
As an application it is shown that if \(k\) is algebraically closed of characteristic \(0\), then the finite groups of \(PGL_2(\mathbb{C})\) are exactly those finite groups for which the Beckman-Black regular lifting property over \(k\) holds. The Beckman-Black property is that any two Galois covers of \(\mathbb{P}_{\mathbb{C}}^1\) with the same group \(G\) are always pullbacks of another Galois cover of group \(G\)..
Methods for establishing the results are based on Riemann's Existence Theorem, in particular, the geometric structure of the Hurwitz moduli spaces. Galois covers; rational pullback; inverse Galois theory Ramification problems in algebraic geometry, Inverse Galois theory, Arithmetic theory of algebraic function fields, Coverings in algebraic geometry, Field arithmetic, Arithmetic algebraic geometry (Diophantine geometry) Rational pullbacks of Galois 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 We relate a certain category of sheaves of \(k\)-vector spaces on a complex affine Schubert variety to modules over the \(k\)-Lie algebra (for \(\text{char\,}k>0\)) or to modules over the small quantum group (for \(k=0\)) associated to the Langlands dual root datum. As an application we give a new proof of Lusztig's conjecture on quantum characters and on modular characters for almost all characteristics. Moreover, we relate the geometric and representation-theoretic sides to sheaves on the underlying moment graph, which allows us to extend the known instances of Lusztig's modular conjecture in two directions: We give an upper bound on the exceptional characteristics and verify its multiplicity-one case for all relevant primes.
One of the fundamental problems in representation theory is the calculation of the simple characters of a given group. This problem often turns out to be difficult and there is an abundance of situations in which a solution is out of reach. In the case of algebraic groups over fields of positive characteristic we have a partial, but not yet a full answer.
In 1979, George Lusztig conjectured a formula for the simple characters of a reductive algebraic group defined over a field of characteristic greater than the associated Coxeter number; [cf. \textit{G. Lusztig}, Proc. Symp. Pure Math. 37, 313-317 (1980; Zbl 0453.20005)]. Lusztig outlined in 1990 a program that led, in a combined effort of several authors, to a proof of the conjecture for almost all characteristics. This means that for a given root system \(R\) there exists a number \(N=N(R)\) such that the conjecture holds for all algebraic groups associated to the root system \(R\) if the underlying field is of characteristic greater than \(N\). This number, however, is unknown in all but low rank cases.
One of the essential steps in Lusztig's program was the construction of a functor between the category of intersection cohomology sheaves with complex coefficients on an affine flag manifold and the category of representations of a quantum group (this combines results of \textit{M. Kashiwara} and \textit{T. Tanisaki} [Duke Math. J. 77, No. 1, 21-62 (1995; Zbl 0829.17020)], and \textit{D. Kazhdan} and \textit{G. Lusztig} [J. Am. Math. Soc. 6, No. 4, 905-947, 949-1011 (1993; Zbl 0786.17017); ibid. 7, No. 2, 335-381, 383-453 (1994; Zbl 0802.17007, Zbl 0802.17008)]). This led to a proof of the quantum (i.e. characteristic 0) analog of the conjecture. \textit{H. H. Andersen, J. C. Jantzen} and \textit{W. Soergel} then showed that the characteristic zero case implies the characteristic \(p\) case for almost all \(p\) [cf. Representations of quantum groups at a \(p\)-th root of unity and of semisimple groups in characteristic \(p\): independence of \(p\). Astérisque 220 (1994; Zbl 0802.17009)].
One of the principal functors utilized in Lusztig's program was the affine version of the Beilinson-Bernstein localization functor. It amounts to realizing an affine Kac-Moody algebra inside the space of global differential operators on an affine flag manifold. A characteristic \(p\) version of this functor is a fundamental ingredient in Bezrukavnikov's program for modular representation theory [cf. \textit{R. Bezrukavnikov, I. Mirković} and \textit{D. Rumynin}, Ann. Math. (2) 167, No. 3, 945-991 (2008; Zbl 1220.17009)], and recently Frenkel and Gaitsgory used the Beilinson-Bernstein localization idea in order to study the critical level representations of an affine Kac-Moody algebra [cf. \textit{P. Fiebig}, Duke Math. J. 153, No. 3, 551-571 (2010; Zbl 1207.20040)].
There is, however, an alternative approach that links the geometry of an algebraic variety to representation theory. It was originally developed in the case of finite-dimensional complex simple Lie algebras by \textit{W. Soergel} [J. Am. Math. Soc. 3, No. 2, 421-445 (1990; Zbl 0747.17008)]. The idea was to give a ``combinatorial description'' of both the topological and the representation-theoretic categories in terms of the underlying root system using Jantzen's translation functors. This approach gives a new proof of the Kazhdan-Lusztig conjecture, but it is also important in its own right: when taken together with the Beilinson-Bernstein localization it establishes the celebrated Koszul duality for simple finite-dimensional complex Lie algebras [cf. \textit{W. Soergel}, loc. cit., and \textit{A. Beilinson, V. Ginzburg, W. Soergel}, J. Am. Math. Soc. 9, No. 2, 473-527 (1996; Zbl 0864.17006)].
In this paper we develop the combinatorial approach for quantum and modular representations. We relate a certain category of sheaves of \(k\)-vector spaces on an affine flag manifold to representations of the \(k\)-Lie algebra or the quantum group associated to Langlands' dual root datum (the occurrence of Langlands' duality is typical for this type of approach). As a corollary we obtain Lusztig's conjecture for quantum groups and for modular representations for large enough characteristics.
The main tool that we use is the theory of sheaves on moment graphs, which originally appeared in the work on the localization theorem for equivariant sheaves on topological spaces by \textit{M. Goresky, R. Kottwitz} and \textit{R. MacPherson} [Invent. Math. 131, No. 1, 25-83 (1998; Zbl 0897.22009)] and \textit{T. Braden} and \textit{R. MacPherson} [Math. Ann. 321, No. 3, 533-551 (2001; Zbl 1077.14522)]. In particular, we state a conjecture in terms of moment graphs that implies Lusztig's quantum and modular conjectures for all relevant characteristics.
Although there is no general proof of this moment graph conjecture yet, some important instances are known: The smooth locus of a moment graph is determined by \textit{P. Fiebig} [loc. cit.], which yields the multiplicity-one case of Lusztig's conjecture in full generality. Moreover, by developing a Lefschetz theory on a moment graph we obtain in [\textit{P. Fiebig}, J. Reine Angew. Math. 673, 1-31 (2012; Zbl 1266.20059)] an upper bound on the exceptional primes, i.e. an upper bound for the number \(N\) referred to above. Although this bound is huge (in particular, much greater than the Coxeter number), it can be calculated by an explicit formula in terms of the underlying root system. Kazhdan-Lusztig polynomials; irreducible characters; highest weight modules; simple Lie algebras; quantized enveloping algebras; reductive algebraic groups; positive characteristic; root systems; intersection cohomology sheaves; Schubert varieties; character formulae; Coxeter numbers; Lusztig conjecture; affine flag manifolds; affine Kac-Moody algebras; moment graphs Fiebig, Peter, Sheaves on affine Schubert varieties, modular representations, and Lusztig's conjecture, J. Amer. Math. Soc., 0894-0347, 24, 1, 133\textendash 181 pp., (2011) Representation theory for linear algebraic groups, Quantum groups (quantized enveloping algebras) and related deformations, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Cohomology theory for linear algebraic groups, Grassmannians, Schubert varieties, flag manifolds, Hecke algebras and their representations, Modular representations and characters, Sheaf cohomology in algebraic topology Sheaves on affine Schubert varieties, modular representations, and Lusztig's 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 An abelian cover is a finite morphism \(X \to Y\) of algebraic varieties which is the quotient map for a generically faithful action of a finite abelian group \(G\). The authors of the paper under review prove that the now standard theory of abelian covers in which \(X\) is normal and \(Y\) is smooth, developed by the second author in [J. Reine Angew. Math. 417, 191--213 (1991; Zbl 0721.14009)], can be extended to the case when \(X\) and \(Y\) are \(S_{2}\) varieties having double crossing singularities in codimension 1. This extension is made gradually: first they prove that the theory of standard covers of Pardini has a very natural extension to the case when \(Y\) is smooth and \(X\) is a \(S_{2}\) variety that has at worst normal crossings outside a subset of codimension greater or equal to two, then they extend this construction to the case when \(Y\) is normal, and finally with a gluing construction they are able to give a comprehensive treatment of the case when both \(X\) and \(Y\) are \(S_{2}\) varieties with at worst normal crossings outside a subset of codimension greater or equal to two. They determine the conditions for \(X\) to have semi log singularities, to be Cohen-Macaulay, and then they determine the index of the canonical divisor of \(X\). An interesting special case is when \(G=\mathbb{Z}^{r}_{2}\), \(\dim X=\dim Y=2\) and \(Y\) is smooth or has two smooth transversal branches, and the branch divisors and the double locus look locally like a collection of lines in the plane, since this case is used in the paper [\textit{V. Alexeev} and \textit{R. Pardini}, ``Explicit compactifications of moduli spaces of Campedelli and Burniat surfaces'', Preprint (2009), \url{arXiv:0901.4431}] to construct explicitly compactifications of some components of the moduli space of surfaces of general type. In this situation, the authors are able to give a complete classification of the covers and the singularities of \(X\), encompassing all cases in nine table where some of these covers, but not all, appear on the boundary of moduli of Campedelli and Burniat surfaces. non-normal abelian covers; \(S_2\) varieties; surfaces of general type; moduli spaces Alexeev, Valery; Pardini, Rita, Non-normal abelian covers, Compos. math., 148, 1051-1084, (2012), also available at Coverings in algebraic geometry, Singularities of surfaces or higher-dimensional varieties Non-normal 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 The paper under review studies étale endomorphisms of non-singular projective complex varieties. The approach is interesting and motivated by a standard subdivision of varieties according to the sign of the Kodaira dimension. The statements of theorems are too technical to appear in a review. I will try to summarize the ideas and give a hint at the flavour of the paper.
Fix a nonsingular projective variety \(X\) and an étale endomorphism \(f\).
If \(X\) is uniruled, the authors associate an endomorphism \(h\) of a non-uniruled normal variety \(Y\). This can even be improved, assuming standard minimal model program conjectures, to an étale endomorphism of a smooth variety of non negative Kodaira dimension.
If \(\kappa(X)>0\) then one can attach to \(f\) an étale endomorphism of the general fiber of a Iitaka fibration of \(X\) [see also \textit{K. Ueno}, Classification theory of algebraic varieties and compact complex spaces. Notes written in collaboration with P. Cherenack. Lecture Notes in Mathematics 439. Berlin-Heidelberg-New York: Springer-Verlag (1975; Zbl 0299.14007)].
If \(\kappa(X)=0\) then one can attach endomorphisms to weak CY varieties and abelian varieties.
Summing up all one can see the endomorphisms of weak CY and abelian varieties as the building blocks of all endomorphisms of projective varieties. It should be said that going backward (that is, recovering \(f\) from the attached endomorphisms) is quite complicated and not always possible [see \textit{Y. Fujimoto} and \textit{N. Nakayama}, J. Math. Kyoto Univ. 47, No. 1, 79--114 (2007; Zbl 1138.14023) and \textit{Y. Fujimoto}, Publ. Res. Inst. Math. Sci. 38, No. 1, 33--92 (2002; Zbl 1053.14049)]. endomorphisms; etale; reduction N. Nakayama and D.-Q. Zhang: Building blockcs of étale endomorphisms of complex projective manifolds, Proc. Lond. Math. Soc. 99 (2009), 725--756. Coverings in algebraic geometry, Birational automorphisms, Cremona group and generalizations, Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables Building blocks of étale endomorphisms of complex 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 We investigate representations of an algebraic surface \(X\) with \(A\)-\(D\)-\(E\)-singularities as a generic covering \(f:X\to\mathbb{P}^2\), that is, a finite morphism which has at most folds and pleats apart from singular points and is isomorphic to the projection of the surface \(z^2 = h(x, y)\) onto the plane \(x, y\) near each singular point, and whose branch curve \(B\subset\mathbb{P}^2\) has only nodes and ordinary cusps except for singularities originating from the singularities of \(X\). It is regarded as folklore that a generic projection of a non-singular surface \(X\subset\mathbb{P}^r\) is of this form.
In this paper we prove this result in the case when the embedding of a surface \(X\) with \(A\)-\(D\)-\(E\)-singularities is the composite of the original one and a Veronese embedding. We generalize the results of \textit{Vik. S. Kulikov} [Izv. Math. 63, No. 6, 1139-1170 (1999); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 63, No. 6, 83-116 (1999; Zbl 0962.14005)], which considers Chisini's conjecture on the unique reconstruction of \(f\) from the curve \(B\). To do this, we study fibre products of generic coverings. We get the main inequality bounding the degree of the covering in the case when there are two inequivalent coverings with branch curve \(B\). This inequality is used to prove Chisini's conjecture for \(m\)-canonical coverings of surfaces of general type for \(m\geqslant 5\). \(A\)-\(D\)-\(E\)-singularities; Chisini conjecture; degree of the covering V. S. Kulikov and Vik. S. Kulikov, Generic coverings of the plane with \?-\?-\?-singularities, Izv. Ross. Akad. Nauk Ser. Mat. 64 (2000), no. 6, 65 -- 106 (Russian, with Russian summary); English transl., Izv. Math. 64 (2000), no. 6, 1153 -- 1195. Ramification problems in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Coverings in algebraic geometry Generic coverings of the plane with A-D-E-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 Starting from \textit{O. Zariski}'s famous example [Amer. J. 51, 305--328 (1929; JFM 55.0806.01)] of two cuspidal plane sextics whose complements have different fundamental groups, the author investigates whether the topology of \(\mathbb{P}^2(\mathbb{C})\setminus B\), where \(B\) is a reduced plane curve, can be determined from
the local type of the singularities of \(B\). In particular, it is discussed when the fundamental group \(\pi_1(\mathbb{P}^2(\mathbb{C})\setminus B)\) is abelian.
A sharp criterion, in the case of curves having only nodes and cusps, for an abelian fundamental group of the complement was given by \textit{M. V. Nori} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 16, 305-344 (1983; Zbl 0527.14016)].
In the paper under review, the author gives the following criterion for a non-abelian fundamental group.
Theorem. Let \(B\) be a reduced plane curve with at most simple singularities of even degree \(d\). Denote by \(\mu_x\) the Milnor number at \(x\), and by \(l_p\) the number of singularities of type \(A_{kp-1}\) for a prime \(p\geq 5\) (respectively, if \(p=3\), let \(l_3\) be the number of singularities of type \(A_{3p-1}\) and \(E_6\)). If there exists an odd prime \(p\) such that
\[
l_p+\sum_{x\in\text{Sing}(B)}\mu_x>d^2-3d+3,
\]
then \(\pi_1(\mathbb{P}^2(\mathbb{C})\setminus B)\) is non-abelian.
Furthermore, the author gives examples to show that this bound is sharp for \(d=6\), but not in the general case for large \(d\). Singularities of curves, local rings, Homotopy theory and fundamental groups in algebraic geometry, Coverings in algebraic geometry, Singularities in algebraic geometry Local types of singularities of plane curves and the topology of their complements | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper outlines a theory of normal abelian covers of smooth algebraic varieties, generalizing the well known analysis of double covers and of some special instances of cyclic covers. - Given an abelian cover \(\pi\) : \(X\to Y\) with group G, as above, one defines the ``building data'' of such a cover to be the eigensheaves of \(\pi_*{\mathcal O}_ X\) corresponding to the different characters of G and certain subsets of the branch divisor of \(\pi\). The main result is a necessary and sufficient condition for the existence of a cover with prescribed building data.
Moreover the paper contains an analysis of the geometric properties of X (singularities and so on...) in terms of the properties of Y and of the building data, the computation of the direct images via \(\pi\) of the tangent, cotangent and canonical sheaves of X and a definition of ``natural deformations'' of an abelian cover. abelian covers of smooth algebraic varieties R. Pardini, Abelian covers of algebraic varieties. \textit{J. Reine Angew. Math.}\textbf{417} (1991), 191-213. MR1103912 Zbl 0721.14009 Coverings in algebraic geometry Abelian covers of algebraic varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A reduced plane curve \(B\) of even degree \(2n\) gives a branched double cover of the projective plane \(\phi :S_{B} \to \mathbb{P}^{2}\). An irreducible plane curve \(C\) of degree \(d\) is simple contact curve of \(B\) if all the intersection points of \(B\) and \(C\) are smooth on both of \(B\) and \(C\) and the intersection multiplicity of each intersection point is \(2\). In the paper under review, the author gives a criterion of splitting for a simple contact curve of a reduced plane quartic curve \(B\) with at most simple double points (Proposition 3.3). When \(C_{1}\) and \(C_{2}\) are two splitting simple contact curve of B, the inverse image \(\phi ^{*}(C_{i})\) has two components \(C_{i} ^{\pm}\). The pair of intersection numbers \((C_{1}^{+} \cdot C_{2}^{+} , C_{1}^{+} \cdot C_{2}^{-})\) is called the splitting type of the triple \((C_{1}, C_{2};B)\). For a reduced plane quartic curve with at most simple double points, the splitting type of every possible two splitting simple contact curves can be calculated. (Proposition 3.4)
Using this result, the author constructs new examples of pairs of plane curves \(\mathcal{C}_{1}, \mathcal{C}_{2}\) such that the combinatorial types of \(\mathcal{C}_{1}\) and \(\mathcal{C}_{2}\) are equal but \((\mathbb{P}^{2}, \mathcal{C}_{1})\) is not homeomorphic to \((\mathbb{P}^{2}, \mathcal{C}_{2})\). elliptic surfaces; contact curves; splitting curves; Zariski pairs Bannai, S., A note on splitting curves of plane quartics and multi-sections of rational elliptic surfaces, Topology Appl., 202, 428-439, (2016) Elliptic surfaces, elliptic or Calabi-Yau fibrations, Coverings in algebraic geometry A note on splitting curves of plane quartics and multi-sections of rational elliptic 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 Finite covers of \(K3\) surfaces are classically a source of examples of surfaces with interesting geometry. The case of double covers is comprehensively studied by \textit{A. Garbagnati} [Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 19, No. 1, 345--386 (2019; Zbl 1419.14059)], but much less is known for covers of higher degree, except for sporadic examples.
The goal of the present article is to classify the numerical invariants of triple covers of \(K3\) surfaces. Following \textit{R. Miranda} [Am. J. Math. 107, 1123--1158 (1985; Zbl 0611.14011)], to a triple cover \(f:X\rightarrow S\) is naturally associated the \textit{Tschirnhausen vector bundle} \(\mathscr{E}\) such that \(f_*\mathcal{O}_X = \mathcal{O}_S \oplus \mathscr{E}\). The classification of triple covers is then divided into three cases: when \(f\) is Galois, when \(\mathscr{E}\) is decomposable but \(f\) is not Galois (the \textit{split non-Galos} case), and when \(\mathscr{E}\) is indecomposable (the \textit{non-split} case).
If \(f:X\rightarrow S\) is a normal triple cover of a \(K3\) surface \(S\), the Kodaira dimension of \(X\) is \(0\), \(1\) or \(2\). For the values \(\kappa(X)\in \{1,2\}\), the authors provide an example of a triple cover \(f\) in the Galois, in the split non-Galois and in the non-split case. When instead \(\kappa(X)=0\), then \(X\) is either a \(K3\) or an abelian surface, and the authors provide an example in the Galois and in the split non-Galois case.
In the case of a Galois triple cover \(f:X\rightarrow S\), the authors directly relate the Kodaira dimension of \(X\) with numerical properties of the branch locus \(D\). More precisely, denoted by \(\Lambda_D\) the lattice spanned by the irreducible components of \(D\), it holds \(\kappa(X)=0\) (resp. \(\kappa(X)=1\) or \(\kappa(X)=2\)) if and only if \(\Lambda_D\) is negative definite (resp. degenerate or hyperbolic). Finally, when \(X\) is of general type, they explicitly compute the numerical invariants of the minimal model of \(X\).
The main results of the paper rely on a careful analysis of the singular locus of the covers and of the configuration of \((-1)\)-curves on them, in order to understand explicitly their minimal model. \(K3\) surface; triple covers; vector bundles Coverings in algebraic geometry, \(K3\) surfaces and Enriques surfaces, Elliptic surfaces, elliptic or Calabi-Yau fibrations, Surfaces of general type Triple covers 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 A visualization of Blaschke product mappings can be obtained by treating them as canonical projections of covering Riemann surfaces and finding fundamental domains and covering transformations corresponding to these surfaces. A working tool is the technique of simultaneous continuation we introduced in previous papers. Here, we are refining this technique for some particular types of Blaschke products, for which colouring pre-images of annuli centred at the origin allow us to describe the mappings with a high degree of fidelity. Additional graphics and animations are provided on the website of the project (\url{http://math.holycross.edu/~cballant/complex/complex-functions.html}). Möbius transformations; visualization of complex functions; covering Riemann surface; covering transformations 4. Ballantine, C., Ghisa, D.: Colour visualization of Blaschke product mappings. Complex Var. Elliptic Equat. 55 , 201-217 (2010) Blaschke products, Coverings in algebraic geometry Colour visualization of Blaschke product mappings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 is the published version of a series of lectures delivered by the author at the C.I.M.E. Summer School 2003 in Cetraro, Italy. The aim of these lectures was to illustrate some fundamental developments in the understanding of the connectedness properties of the moduli spaces of complex algebraic surfaces of general type, thereby emphasizing those results that were obtained in the course of the past 20 years, that is, after the last C.I.M.E. course on this topic held by \textit{F. Catanese} [in: Moduli of algebraic surfaces. Theory of moduli, Lect. 3rd Sess. Cent. Int. Mat. Estivo, Montecatini Terme/Italy 1985, Lect. Notes Math. 1337, 1--83 (1988; Zbl 0658.14017)]. In this context,the focus of the present lecture notes is set on the deformation-theoretic and degeneration aspects, especially on smoothings of singularities of surfaces, whereas the parallel course ``Differentiable and deformation type of algebraic surfaces, real and symplectic structures'' by \textit{F. Catanese} published in the same volume [Lecture Notes in Mathematics 1938, 55--167 (2008; Zbl 1145.14001)] has a somewhat broader and more general content. In fact, these two treatises effectively complement each other and may be regarded as a unit.
Also, the results described systematically in the current notes are largely due to F. Catanese and the author himself, but up to now were scattered in a long series of (independent) research articles by both authors.
As for the precise contents, the present work consists of six chapters, each of which comes with several sections.
Chapter 1 serves as an introduction to the topic, including some motivating remarks and a number of basic facts on deformation types of compact complex surfaces. Moreover, the author explains the statements of two of his main theorems on deformation types of diffeomorphic minimal surfaces of general type [Invent. Math. 143, No. 1, 29--76 (2001; Zbl 1060.14520)]. Actually, the main goal of the following chapters is to explain the principal ideas and the crucial techniques used in the proofs of these theorems. In order to avoid too many technicalities, and to make the underlying geometric ideas as transparent as possible, the author has chosen to restrict his exposition to significant examples of such surfaces.
Chapter 2 discusses the deformation equivalence of surfaces with respect to their rational singularities (and other quotient singularities) and their (relative) canonical models. This requires the existence of semiuniversal and minimal versal deformations (à la Kodaira-Spencer and Kuranishi), which is also briefly touched upon.
Chapter 3 turns to moduli spaces of canonical surfaces by means of the approaches of D. Gieseker and S. Tankeev, with a special emphasis on strategies for constructing their connected components.
Chapter 4 describes the technique of smoothings of normal surface singularities (after Brieskorn, Milnor, Mumford, Looijenga, Schlessinger, and others). In particular, the author depicts \(\mathbb{Q}\)-Gorenstein smoothings and introduces the notion of deformation \(T\)-equivalence of algebraic surfaces, together with an instructive, highly non-trivial example.
Chapter 5 illustrates (partially with proofs) the author's approach via double and multi-double covers of normal surfaces, whereas Chapter 6 focuses then on the author's stability criteria for flat double covers. This is done by means of the instructive working example of the class of the so-called \(abc\)-surfaces, which were introduced and investigated very recently by \textit{F. Catanese} and \textit{B. Wajnryb} [J. Differ. Geom. 76, No. 2, 177--213 (2007; Zbl 1127.14039)]. Together with the author's previous ideas, methods, and results (as mentioned in the introctory Chapter 1 of the present paper), this leads to various new examples of connected components of moduli spaces of surfaces of general type, and to further counterexamples to the ``DEF = DIFF'' problem of Friedman and Morgan in addition. All in all, these notes provide a masterly written and utmost informative survey on the present state of art regarding this central topic in both the theory of complex algebraic surfaces and the theory of symplectic four-manifolds. survey (algebraic geometry); surfaces of general type; fibrations; moduli of algebraic surfaces; coverings; singularities; deformations; symplectic geometry M. Manetti: Smoothings of singularities and deformation types of surfaces ; in Symplectic 4-Manifolds and Algebraic Surfaces, Lecture Notes in Math. 1938 , Springer, Berlin, 2008, 169-230. Research exposition (monographs, survey articles) pertaining to algebraic geometry, Surfaces of general type, Moduli, classification: analytic theory; relations with modular forms, Fibrations, degenerations in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Local deformation theory, Artin approximation, etc., Coverings in algebraic geometry, Symplectic manifolds (general theory) Smoothings of singularities and deformation types of surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The Shafarevich conjecture asserts that the universal cover of a smooth projective variety \(X\) is holomorphically convex. This conjecture holds true for curves (by the Poincaré uniformization theorem). The paper under review is a survey on the recent developments in understanding the Shafarevich conjecture. The author discusses first the results of Kollár and Campana concerning the birational existence of the Shafarevich maps. Then one presents certain approximations of the Shafarevich conjecture for special groups, e.g. a theorem of Jost-Yau and Siu which says that if the fundamental group of a curve of genus \(g\) is a quotient of \(\pi_1(X)\), then \(X\) has a morphism onto a curve of genus at least \(g\). Next one considers the nonabelian Hodge theory (Corlette, Hitchin, Simpson) and its \(p\)-adic analogue (Gromov, Schoen). As an application, one proves that the Shafarevich conjecture holds true for surfaces with linear fundamental group. In the last part the author proposes an example which reduces the Shafarevich conjecture to some difficult unsolved group-theoretic questions. universal cover; Shafarevich maps; fundamental group; nonabelian Hodge theory; Shafarevich conjecture L. Katzarkov, On the Shafarevich maps, Algebraic geometry --- Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 173 -- 216. Coverings in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry, Coverings of curves, fundamental group, Variation of Hodge structures (algebro-geometric aspects), Transcendental methods, Hodge theory (algebro-geometric aspects) On the Shafarevich maps | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems At the beginning of the present paper, the authors pose two general questions about families of algebraic schemes:
(1) Does every one-parameter flat family of reduced subschemes of \(\mathbb{P}^n\) have a well-defined flat completion whose special fiber is a reduced scheme admitting a finite morphism to \(\mathbb{P}^n\)?
(2) Is there a universal substitute for the Hilbert scheme parametrizing reduced schemes \(X\) equipped with a finite morphism \(f: X\to\mathbb{P}^n\) (instead of subschemes of \(\mathbb{P}^n\) with fixed Hilbert polynomial)?
In the course of the paper, they answer both of these questions positively in the framework of so-called ``branchvarieties''. More precisely, if \(Y\) is a projective scheme over a field \(k\), then a branchvariety of \(Y\) is a scheme \(X\) of finite type over \(k\) such that \(X:= X\otimes_k\overline k\) is reduced, together with a finite morphism \(f: X\to Y\).
Such a branchvariety \(X\) of \(Y\) admits a well-defined Hilbert polynomial \(h\) (defined via \(f^*{\mathcal O}_Y(1)\)) and therefore a well-determined degree, too. Then, for a fixed Noetherian base scheme \(C\), the functor of branchvarieties
\[
\text{Branch}^b_{h, Y}: (C\text{-schemes})\to(\text{Sets})^{\text{opp}}
\]
is defined by associating to each \(C\)-scheme \(S\) the set \(\text{Branch}^b_{h,Y}(S)\) of proper families \(f: X\to Y\times_C S\) such that \(X\) is flat over \(S\) and every fiber \(f_s: X_s\to Y_s:= Y\times_Ck(s)\), \(s\in S\), is a branchvariety of \(Y_s\) with prescribed Hilbert polynomial \(h\) and degree sequence \(b= (b_0,\dots, b_{\dim X})\) defined by the degrees of the \(i\)-dimensional irreducible components of \(X\).
Also, the authors define a stack in groupoids \(\underline{\text{Branch}}^b_{h,Y}\) by associating to each \(C\)-scheme \(S\) a category whose objects are the families \(f: X\to Y\times_C S\) as above, and morphisms are isomorphisms of such families intertwining the structure morphisms.
In this context, the authors' main result (Theorem 0.4.) states the following:
The stack in groupoids \(\underline{\text{Branch}}^b_{h,Y}\) is an Artin stack of finite type with a finite diagonal. Moreover, this stack has a coarse moduli space, also denoted by \(\text{Branch}^b_{h,Y}\), which is a proper algebraic space. In particular, the algebraic space \(\text{Branch}^b_{h, Y}\) has finitely many connected components. In contrast with the usual Hilbert scheme, which is known to be connected for each Hilbert polynomial \(h\), the ``moduli space'' of branchvarieties \(\text{Branch}^b_{h, Y}\) is not connected for most triples \((h,b,Y)\).
As the authors show in the sequel, different connected components of \(\text{Branch}^b_{h,Y}\) may often be distinguished by certain graphs associated to branchvarieties. These new invariants are called ``labeled rooted forests'', and they turn out to be crucial refinements of both the Hilbert polynomial and the degree sequence of a branchvariety.
In the final section of their work, the authors discuss the relations of the moduli spaces of branchvarieties to other classifying spaces, mainly with a view toward classical Hilbert schemes and Chow varieties.
The usefulness and power of the branch moduli space \(\text{Branch}^b_{h,Y}\) are reflected by the fact that it has been successfully applied, in the meantime, to several constructions of other moduli spaces, in particular by the first author (V. Alexeev) himself. Moreover, there are recent stacky versions of the authors' branchvariety approach by \textit{M. Lieblich} [Int. Math. Res. Not. 2006, No. 11, Article ID 75273 (2006; Zbl 1108.14003)] and by \textit{J. Starr} [cf.: Artin's axioms, composition and moduli spaces, Preprint, \url{arXiv:math.AG/0602646}]), who extended some of the present constructions to the more general case of an Artin stack \(Y\) as the target object. algebraic stacks; algebraic spaces; coarse moduli spaces; branched coverings; Hilbert schemes; Chow varieties; branchvarieties Alexeev V. and Knutson A., Complete moduli spaces of branchvarieties, J. reine angew. Math. 639 (2010), 39-71. Generalizations (algebraic spaces, stacks), Algebraic moduli problems, moduli of vector bundles, Fine and coarse moduli spaces, Stacks and moduli problems, Coverings in algebraic geometry, Parametrization (Chow and Hilbert schemes) Complete moduli spaces of branchvarieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 examples of algebraic surfaces which are desingularizations of singular \(({\mathbb Z}/2\mathbb{Z})^2\)-covers of \({\mathbb P}^1\times{\mathbb P}^1\). One series of examples are simple canonical surfaces with \(p_g=4\) (the minimal possible value for a canonical surface) and \(K^2=11,...,28\), that disproves a prediction of Enriques about \(24\) as the maximum value of \(K^2\) in this case. Among other examples one finds surfaces with \(p_g=q=1\) and \(K^2=4,5\), and an infinite series of surfaces whose canonical map is composed of a pencil of curves of genus \(2\) and \(3\), with non-constant moduli. simple canonical algebraic surfaces; bidouble Galois covers; resolution of singularities; moduli of surfaces of general type F. Catanese, Singular bidouble covers and the construction of interesting algebraic surfaces, in Algebraic Geometry: Hirzebruch 70, eds. P. Pragacz \textit{et al.}, Proc. Algebraic Geometry Conference in honor of F. Hirzebruch's 70th birthday, \textit{Contempory Mathematics}, Vol. 241 (Springer, 1999), pp. 97-120. Coverings in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Special surfaces, Singularities of surfaces or higher-dimensional varieties, Surfaces of general type, Families, moduli, classification: algebraic theory Singular bidouble covers and the construction of interesting 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 The author defines a class of nodal algebraic curves in \({\mathbb{C}}^ 2\), the so called ``negative to infinity'' curves. This class includes the curves of the form \(C\setminus L\subset {\mathbb{P}}^ 2\setminus L\), where C is a nodal curve in \({\mathbb{P}}^ 2\) and L is a general line. This includes also the general curve parametrized by two polynomials of one variable of fixed degrees or the general curve having a prescribed Newton polygon. Let \(K\subset {\mathbb{C}}^ 2\) be a curve as above. The main result asserts that one can associate to each irreductible component of K an element in \(\pi_ 1({\mathbb{C}}^ 2\setminus K)\) in such a way that these elements generate \(\pi_ 1({\mathbb{C}}^ 2\setminus K)\) and commute with each other when the corresponding components intersect. As the inclusion \(({\mathbb{P}}^ 2\setminus L)\setminus (C\setminus L)\hookrightarrow {\mathbb{P}}^ 2\setminus C\) induces a surjection of fundamental groups, one refinds the result of Fulton and Deligne on the commutativity of \(\pi_ 1({\mathbb{P}}^ 2\setminus C)\). A computer algorithm to calculate \(\pi_ 1({\mathbb{C}}^ 2\setminus K)\) in terms of coefficients of the equations defining the curve is also included. fundamental group of the complement of a plane algebraic curve; nodal algebraic curves; computer algorithm S. Yu. Orevkov, ''The fundamental group of the complement of a plane algebraic curve,''Mat. Sb. [Math. USSR-Sb.],137 (179), No. 2, 260--270 (1988). Coverings in algebraic geometry, Singularities of curves, local rings, Software, source code, etc. for problems pertaining to algebraic geometry, Surfaces and higher-dimensional varieties The fundamental group of the complement of a plane algebraic curve | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In the sixties A. Grothendieck developed some conceptual tools to handle the general question of descent: fibered categories, gerbes, nonabelian cohomology, etc. In the seventies, M. Fried's moduli approach to the arithmetic of covers revealed two significant descent questions: descent to the field of moduli of a cover and existence of Hurwitz families above a moduli space of covers. Many works have since been devoted to these two questions.
However it is only recently that Grothendieck's conceptual framework was (re)introduced in this topic. Our goal in the paper is to join these two branches of the theory: that is, we wish to recast the work done about covers within Grothendieck's perspective, use this new light on the subject to measure the progress achieved and show how new developments already came out of this unified viewpoint. P. Dèbes: Descent theory for algebraic covers ; in Arithmetic Fundamental Groups and Noncommutative Algebra (Berkeley, CA, 1999), Proc. Sympos. Pure Math. 70 , Amer. Math. Soc., Providence, RI, 3-25, 2002. Arithmetic ground fields (finite, local, global) and families or fibrations, Coverings in algebraic geometry, Fibered categories Descent theory for algebraic 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 The authors' aim is to classify the pairs \((X,L)\) where \(X\) is a projective manifold and \(L\) a very ample line bundle on \(X\) such that the linear system \(|L |\) contains a smooth divisor \(A\) which admits a finite morphism \(\pi : A \to \mathbb{P}^n\) of degree 3. One assumes \(n \geq 2\). The following examples are given:
\((\text{a}_n)\) \((\mathbb{P}^{n+1}, {\mathcal O}_{\mathbb{P}^{n + 1}} (3))\);
\((\text{b}_n)\) a cubic hypersurface in \(\mathbb{P}^{n + 2}\) with hyperplane section;
\((\text{c}_n)\) \((X,3L)\) where \((X,L)\) is del Pezzo, i.e. \(- K_X = nL\) is ample, with \(L^{n + 1} = 1\).
The authors show that for \(n \geq 4\) the above examples are the only ones. In case \(n = 3\) the examples are the only ones if one makes the additional assumption that the cover \(\pi : A \to \mathbb{P}^3\) is cyclic. The case \(n = 2\) is more complicated; it is studied under the additional requirement \(K_A = \pi^* {\mathcal O}_{\mathbb{P}^2} (k)\) (for some integer \(k)\) and some results are given. triple covers; very ample line bundle; linear system LANTERI A., PALLESCHI M. and SOMMESE A.J., ''On triple covers of \(\mathbb{P}\) n as very ample divisors'', inClassification of Algebraic Varieties, Proceedings L'Aquila, 1992, ed. by C. Ciliberto, E.L. Livorni and A.J. Sommese, Contemp. Math. 162 (1994), 277--292. Divisors, linear systems, invertible sheaves, Coverings in algebraic geometry, Projective techniques in algebraic geometry On triple covers of \(\mathbb{P}^ n\) as very ample divisors | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 central extension \(U\) of a group \(G\) is called universal if any central extension \(E\) of \(G\) factors uniquely through a homomorphism \(U\to E\) over \(G\); such an extension exists if and only if \(G\) is equal to its commutator subgroup \(G'=(G,G)\) and then it is unique up to a \(G\)-isomorphism. In this case, such a universal central extension is described from a free group \(F\) with image \(G\) as the commutator group of \(F/(K,F)\) with \(K\) the kernel of \(F\to G\), and the kernel of the projection of this extension onto \(G\) is naturally the second homology group \(H_2(G,{\mathbb{Z}})\). When \(G\) is a topological group, one has to consider the topological central extensions; when the group is second countable, the classes of central extensions of \(G\) by a commutative group are classified by the group \(H^2(G,C)\) of measurable classes of 2-cocycles on \(G\) with values in \(C\). For the group \(G=SL_ 2(F)\) over a field \(F\), the restriction to \(F^{\times}\) embedded in \(G\) by the co-root \(t\mapsto\left(\begin{smallmatrix} t&0\\ 0&t^{-1}\end{smallmatrix}\right)\) of a 2-cocycle on \(G\) gives a 2-cocycle on \(F^{\times}\), and using the presentation given by R. Steinberg, C. Moore and H. Matsumoto described all the co-cycles obtained, using a short co-root and also those coming from the case when \(F\) is a local field and the extensions are topological. For a non-archimedean local field, this gives \(H_ 2(G,{\mathbb{Z}})\) as the group \(\mu(F)\) of roots of unity in \(F\), and \(H^ 2(G,{\mathbb{R}}/{\mathbb{Z}})\) as its dual \(\mu(F)^{\wedge}\) sent in \(H^ 2(G,{\mathbb{R}}/{\mathbb{Z}})\) through the norm residue symbol and the above restriction from \(G\) to \(F^{\times}\). H. Matsumoto proved that the same result holds when \(G\) is the group of \(F\)-points, \(F\) a non-archimedean local field, of a connected, simply connected, simple, split algebraic group over \(F\), and the case of quasi-split groups was done by V. V. Deodhar and P. Deligne. These methods are based on the presentation given by R. Steinberg of the Chevalley groups.
In this article, the authors deal with the group \(G\) of rational points in a non-archimedean local field \(F\) of a connected, simply-connected, absolutely simple, isotropic algebraic group over \(F\). They use the Bruhat-Tits building of this group on which it acts simplicially to reduce the computation of the cohomology of \(G\) to the one of the parahoric subgroups. They observe that \(H^ 2(G,{\mathbb{R}}/{\mathbb{Z}})\) is in fact \(H^ 2(G,{\mathbb{Q}}/{\mathbb{Z}})\) with the discrete topology on \({\mathbb{Q}}/{\mathbb{Z}}\), and they decompose \({\mathbb{Q}}/{\mathbb{Z}}\) in its \(p\)-primary component -- \(p\) is the residual characteristic of the field -- and its \(p'\)-primary component. This latter gives \(H^ 2(G,({\mathbb{Q}}/{\mathbb{Z}})_{p'})\) as the dual of \(\mu (F)_{p'}\), the group of roots of unity of order prime to \(p\); the former leads to \(H^ 2(G,({\mathbb{Q}}/{\mathbb{Z}})_ p)\) as the dual of \(\mu(F)_ p\) or to a subgroup of index two in \(\mu(F)^{\wedge}_ p\), and this part is technically the most difficult one of the paper.
As a consequence of the determination of \(H^ 2(G,{\mathbb{R}}/{\mathbb{Z}})\), the authors prove that \(G\) is equal to its commutator subgroup, that \(G\) admits a universal central extension in the topological sense with fundamental group the dual of \(H^ 2(G,{\mathbb{R}}/{\mathbb{Z}})\). They also sketch a proof of the Kneser-Tits conjecture, proved by V. P. Platonov by a case by case check, which asserts that \(G\) is generated by the unipotent radicals of its \(F\)-parabolic subgroups. universal central extensions; topological central extensions; local field; roots of unity; norm residue symbol; quasi-split groups; rational points; non-archimedean local fields; connected, simply-connected absolutely simple isotropic algebraic groups; Bruhat-Tits buildings; cohomology; parahoric subgroups; Kneser-Tits conjecture; unipotent radicals; parabolic subgroups G. Prasad and M. S. Raghunathan, ''Topological central extensions of semisimple groups over local fields,'' Ann. Math. 119, 143--201, 203--268 (1984). Linear algebraic groups over local fields and their integers, Group schemes, Cohomology theory for linear algebraic groups, General properties and structure of other Lie groups, Coverings in algebraic geometry Topological central extensions of semi-simple groups over local fields. 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 This paper does what it says in the title, providing a representation-theoretic description of the homology of the loop space on the Bousfield-Kahn \(p\)-completion of the classifying space of a finite group \(G\). This description is sufficiently explicit to allow computer calculation of the homology and cohomology of \(\Omega BG_{p}^{\wedge}\) and it also provides information about when the homology ring exhibits polynomial growth and when the growth is exponential. classifying space; loop space; completion; representation D.J. Benson, ``An algebraic model for chains on \({\Omega}\)\textit{BG}\textit{\(\wedge\)}\textit{p}'', \textit{Trans. Am. Math.} \textit{Soc. }361(4) (2009), 2225--2242. Loop spaces, Classifying spaces of groups and \(H\)-spaces in algebraic topology, Modular representations and characters, Localization and completion in homotopy theory, Cohomology of groups, Linkage, complete intersections and determinantal ideals, Complete intersections An algebraic model for chains on \(\Omega BG^{\wedge }_p\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The article gives a description of the author's results on the fundamental group of the complement of a plane curve \(C\) obtained by studying the global and local Alexander invariants attached to it. Globally this is the product of the invariants of the module \(H\), first homology group (with coefficients in a field) of the infinite cyclic covering of \(\mathbb{P}^2\) branched along \(C\) and a line \(L\) in general position. Locally they are the characteristic polynomials of the local monodromies around the singularities of \(C\). The author establishes a divisibility relation between the two. Applying it to cuspidal curves he obtains some informations on a torsion of \(H\) (which can be deduced also from the paper of \textit{R. Randell} [in; Topology, Proc. Symp., Siegen 1979, Lect. Notes Math. 788, 145--164 (1980; Zbl 0433.14020)]) and on the irregularity of a finite cyclic cover of \(\mathbb{P}^2\) branched along \(C\) and \(L\) (see also {\S} 3, lemma 10 of the reviewer's paper in [Invent. Math. 68, 477--496 (1982; Zbl 0489.14009)]). Finally he gives examples where the Alexander invariants are computable in terms of superabundance of suitable linear systems.
[For the entire collection see Zbl 0509.00008.] singularities; fundamental group of the complement of a plane curve; Alexander invariants A. Libgober, ''Alexander invariants of plane algebraic curves,'' In:Proc. Symp. Pure. Math., Vol. 40 (1983), pp. 29--45. Coverings in algebraic geometry, Singularities in algebraic geometry, Covering spaces and low-dimensional topology, Topological properties in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry, Special algebraic curves and curves of low genus Alexander invariants of plane algebraic curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We construct explicitly a finite cover of the moduli stack of compact Riemann surfaces with a given group of symmetries by a smooth quasi-projective variety. Families, moduli of curves (algebraic), Stacks and moduli problems, Coverings in algebraic geometry, Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) Smooth covers of moduli stacks of Riemann surfaces with symmetry | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems To any tensor triangulated category \(\mathcal{T}\), that is, a triangulated category equipped with a symmetric monoidal structure \(\otimes\), one can associate a locally ringed space which is called the spectrum and denoted by \(\text{Spec}(\mathcal{T})\), see \textit{P.\ Balmer} [J. Reine Angew.\ Math.\ 588, 149--168 (2005, Zbl 1080.18007)]. For instance, if \(\mathcal{T}\) is the triangulated category of perfect complexes on a quasi-compact and quasi-separated scheme \(X\) (which is the usual bounded derived category of coherent sheaves if \(X\) is smooth and projective), then the spectrum is precisely \(X\). If \(\mathcal{T}\) is the stable category of finitely generated \(kG\)-modules, where \(G\) is a finite group and \(k\) is a field of finite characteristic, then the spectrum turns out to be the associated projective support variety \(\mathcal{V}_G\). Since tensor triangulated categories appear in several areas of mathematics, this spectrum allows one to transport ideas from one area to another.
If \(G\) is a finite group and \(k\) is an algebraically closed field of positive characteristic, then \textit{J.\ Rickard} [J.\ Lond.\ Math.\ Soc., II.\ Ser.\ 56, No.~1, 149--170 (1997, Zbl 0910.20034)] tells us how, given a closed subset \(W\) of \(\mathcal{V}_G\), one can associate to it two (possibly infinite-dimensional) \(kG\)-modules \(E(W)\) and \(F(W)\) satisfying \(E(W)\otimes E(W)\cong E(W)\) and \(F(W)\otimes F(W)\cong F(W)\) modulo projectives. These modules are called tensor idempotents and they satisfy additional properties. The idea of the current article is to generalise this construction to arbitrary compactly generated tensor triangulated categories replacing the projective support variety by the spectrum. To be more precise, if \(\mathcal{T}\) is such a category and a Thomason subset \(Y\) in \(\text{Spec}(\mathcal{T}^c)\), where \(\mathcal{T}^c\) is the category of compact objects in \(\mathcal{T}\), is given, then the authors construct objects \(e(Y)\) and \(f(Y)\) in \(\mathcal{T}\) which satisfy similar properties as Rickard's tensor idempotents. As a corollary the authors give a quick proof that the telescope conjecture holds for the derived category of a noetherian scheme by reducing it to the affine case. triangulated category; monoidal category; Bousfield localization; telescope conjecture; Mayer-Vietoris exact triangle P. Balmer and G. Favi, Generalized tensor idempotents and the telescope conjecture, Proc. Lond. Math. Soc. (3) 102 (2011), no. 6, 1161-1185. Derived categories, triangulated categories, Modular representations and characters, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Generalized tensor idempotents and the telescope conjecture | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author makes simple general remarks in connection with the coverings and the fundamental group of a complex analytic manifold. Riemann surfaces; coverings; fundamental group Coverings in algebraic geometry, Homotopy groups of special spaces A criterion for restricted ideal sheaves and on regular covering 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 This is the first of two papers in which we determine the spectrum of the cohomology algebra of infinitesimal group schemes over a field \(k\) of characteristic \(p>0\). Whereas our following paper [\textit{A. Suslin}, \textit{E. M. Friedlander} and \textit{C. P. Bendel}, J. Am. Math. Soc. 10, No. 3, 729-759 (1997; see the following review Zbl 0960.14024)] is concerned with detection of cohomology classes, the present paper introduces the graded algebra \(k[V_\tau(G)]\) of functions on the scheme of infinitesimal 1-parameter subgroups of height \(\leq r\) on an affine group scheme \(G\) and demonstrates that this algebra is essentially a retract of \(H^{ev}(G,k)\) provided that \(G\) is an infinitesimal group scheme of height \(\leq r\). This work is a continuation of a paper by \textit{E. M. Friedlander} and \textit{A. Suslin} [Invent. Math. 127, No. 2, 209-270 (1997; Zbl 0945.14028)] in which the existence of certain universal extension classes was established, thereby enabling the proof of finite generation of \(H^*(G,k)\) for any finite group scheme \(G\) over \(k\). The role of the scheme of infinitesimal 1-parameter subgroups of \(G\) was foreshadowed in a paper by \textit{E. M. Friedlander} and \textit{B. J. Parshall} [Am. J. Math. 108, 235-253 (1986; Zbl 0601.20042)] where \(H^*(G_{(1)}, k)\) was shown to be isomorphic to the coordinate algebra of the scheme of \(p\)-nilpotent elements of \(g=\text{Lie}(G)\) for \(G\) a smooth reductive group, \(G_{(1)}\) the first Frobenius kernel of \(G\), and \(p=\text{char} (k)\) sufficiently large. Indeed, \(p\)-nilpotent elements of \(g\) correspond precisely to infinitesimal 1-parameter subgroups of \(G_{(1)}\). Much of our effort in this present paper involves the analysis of the restriction of the universal extension classes to infinitesimal 1-parameter subgroups. infinitesimal group schemes; characteristic \(p\); spectrum of the cohomology algebra; infinitesimal 1-parameter subgroups; universal extension classes Demazure, M., Gabriel, P.: Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs, Masson & Cie, Éditeur, Paris; North-Holland Publishing Co., Amsterdam (1970) Group schemes, Homological methods in Lie (super)algebras, Cohomology theory for linear algebraic groups, Modular representations and characters, Cohomology of Lie (super)algebras, Étale and other Grothendieck topologies and (co)homologies Infinitesimal 1-parameter subgroups and cohomology | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We describe CW complexes for complex projective algebraic surfaces in the context of practical computation of topological invariants. CW complexes; cohomology of algebraic surfaces; computation of topological invariants Kresch, A.: CW complexes for complex algebraic surfaces. Exp. Math. 19(4), 413--419 (2010) Topological aspects of complex manifolds, Coverings in algebraic geometry, Algebraic topology on manifolds and differential topology CW complexes for complex 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 Let \(X\) be a complex projective manifold that admits an endomorphism, that is a finite morphism \(f: X \rightarrow X\) of degree at least two. If \(X\) is rationally connected and has Picard number one a classical conjecture claims that \(X\) is a projective space. This conjecture is known in dimension up to three but completely open in higher dimension. In the paper under review the author considers the more general problem whether a rationally connected threefold admitting an endomorphism is always rational. He gives a positive answer for smooth Fano threefolds and for certain terminal threefolds that are Mori fibre spaces. In general one would like to reduce the problem to Mori fibre spaces by running a MMP
\[
X \dashrightarrow X'
\]
such that \(f\) descends to an endomorphism \(f': X' \rightarrow X'\). While the existence of such an \(f\)-equivariant MMP is not known in general the author establishes this property under the assumption that the anticanonical divisor \(-K_X\) is big. endomorphism; rationally connected variety; rational varieties Coverings in algebraic geometry, Fano varieties, Rationality questions in algebraic geometry, Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables Rationality of rationally connected threefolds admitting non-isomorphic endomorphisms | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Spectral curves arose historically out of the study of differential equations of Lax type. Following Hitchin's work [\textit{N. Hitchin}, Duke Math. J. 54, 90-114 (1987; Zbl 0627.14024)], they have acquired a central role in understanding the moduli spaces of vector bundles and Higgs bundles on a curve. Simpson's work [\textit{C. T. Simpson}, ``Moduli of representations of the fundamental group of a smooth projective variety''. I. II., Publ. Math., Inst. Hautes Étud. Sci. 79, 47-129 (1994) and 80, 5-79 (1995)] suggests a similar role for spectral covers \(\widetilde{S}\) of higher-dimensional varieties \(S\) in moduli questions for bundles on \(S\). The purpose of these notes is to combine and review various results about spectral covers, focusing on the decomposition of their Picards (and the resulting Prym identities) and the interpretation of a distinguished Prym component as parameter space for Higgs bundles. Much of this is modeled on Hitchin's system, which we recall in section 1, and on several other systems based on moduli of Higgs bundles, or vector bundles with twisted endomorphisms, on curves. By peeling off several layers of data that are not essentially for our purpose, we arrive at the notions of an abstract principal Higgs bundle and a cameral (roughly, a principal spectral) cover. Following \textit{R. Donagi}'s preprint: ``Abelianization of Higgs bundles'', this leads to the statement of the main result, theorem 12, as an equivalence between these somewhat abstract `Higgs' and `spectral' data, valid over an arbitrary complex variety and for a reductive Lie group \(G\). Several more familiar forms of the equivalence can then be derived in special cases by adding choices of representation, value bundle and twisted endomorphism. This endomorphism is required to be regular, but not semisimple. Thus the theory works well even for Higgs bundles that are everywhere nilpotent. After touching briefly on the symplectic side of the story in section 6, we discuss some of the issues involved in removing the regularity assumption, as well as some applications and open problems, in section 7. Picard group; vector bundles; spectral covers; Prym component; abstract principal Higgs bundle Ron Donagi, Spectral covers, Current topics in complex algebraic geometry (Berkeley, CA, 1992/93) Math. Sci. Res. Inst. Publ., vol. 28, Cambridge Univ. Press, Cambridge, 1995, pp. 65 -- 86. Vector bundles on curves and their moduli, Picard groups, Determinantal varieties, Coverings in algebraic geometry Spectral 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 [For the entire collection see Zbl 0747.00024.]
The following generalization of the well-known result of Fulton-Deligne on the abelianness of the fundamental group of the complement of a nodal curve in \(\mathbb{P}^ 2\) is proved.
Theorem. Let \(\Gamma_ 1\), \(\Gamma_ 2\) be (not necessarily irreducible) curves in \(\mathbb{P}^ 2\), \(E\subset\mathbb{P}^ 2\) a line and \(\mathbb{C}^ 2=\mathbb{P}^ 2-E\). Suppose \(\Gamma_ 1\), \(\Gamma_ 2\) intersect transversally and \(\Gamma_ 1\cap\Gamma_ 2\cap E=\varphi\). Then \(\pi_ 1(\mathbb{C}^ 2-\Gamma_ 1\cup\Gamma_ 2)=\pi_ 1(\mathbb{C}^ 2- \Gamma_ 1)\times\pi_ 1(\mathbb{C}^ 2 -\Gamma_ 2)\).
The important point is that \(\Gamma_ i\) is not assumed to intersect \(E\) transversally. Since \(\Gamma_ 1\), \(\Gamma_ 2\) are allowed to be reducible, the above theorem extends easily to finitely many curves \(\Gamma_ 1\), \(\Gamma_ 2,\ldots,\Gamma_ n\) with appropriate assumption. -- The proof uses linear automorphisms of \(\mathbb{C}^ 2\) to deform one of the curves and Zariski's method of pencils. plane curve; complement of a nodal curve in \(\mathbb{P}^ 2\); Zariski problem; abelianness of the fundamental group Homotopy theory and fundamental groups in algebraic geometry, Curves in algebraic geometry, Coverings in algebraic geometry Fundamental group of the complement of affine plane curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For the entire collection see Zbl 0583.00010.]
Let X be an n-dimensional compact Kähler manifold with nonnegative holomorphic bisectional curvature and let \(\tilde X\) be its universal covering. The main theorem of the paper says that \(\tilde X\) is isometrically biholomorphic to \({\mathbb{C}}^ k\times {\mathbb{P}}^{N_ 1}\times...\times {\mathbb{P}}^{N_ r}\times M_ 1\times...\times M_ p,\) where \(M_ i\) are some irreducible compact Hermitian symmetric spaces of rank \(\geq 2\) with the canonical metrics, \({\mathbb{C}}^ k\) has its Euclidean metric and \({\mathbb{P}}^{N_ i}\) have some Kähler metrics with nonnegative holomorphic bisectional curvature (and k, r, p are some nonnegative integers).
By using the splitting theorem of Howard-Smyth-Wu, the author reduces the proof to a proof of a theorem saying that if a compact Kähler manifold X has nonnegative holomorphic bisectional curvature such that the Ricci curvature is positive at one point and if \(b_ 2(X)=1\) then X is biholomorphic to \({\mathbb{P}}^ n\) or is isometrically biholomorphic to an irreducible compact Hermitian symmetric space of rank \(\geq 2\). The main idea of the proof is a unification of the methods of algebraic geometry (the study of rational curves on X) and differential geometry (various forms of the maximum principle on tensors) which were used by other authors who have proved special cases of the main theorem. The paper contains only a sketch of the proof of the main theorem. uniformization in n dimensions; Mori's theory of rational curves; parabolic Einstein equation; holonomy; compact Kähler manifold; holomorphic bisectional curvature; Hermitian symmetric spaces Differential geometry of symmetric spaces, Global differential geometry of Hermitian and Kählerian manifolds, Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects), Coverings in algebraic geometry Compact Kähler manifolds of nonnegative holomorphic bisectional curvature | 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.