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The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Consider a double covering Y of \({\mathbb{P}}^ 3\) branched at a quartic A containing no straight lines. If A is a smooth family of straight lines on the covering Y is parametrized by a surface. \textit{A. S. Tikhomirov} proved that A and consequently Y are recovered from that surface [Math. USSR, Izv. 16, 373-397 (1981); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 44, 415-442 (1980; Zbl 0434.14023)].
In this paper the same is shown for the desingularization X of Y when A has a single double point, after giving a suitable definition of straight lines on X. desingularization of covering of projective 3-space; branching of covering Coverings in algebraic geometry, Projective techniques in algebraic geometry, \(3\)-folds, Ramification problems in algebraic geometry Reconstructing a special double covering of \({\mathbb{P}}^ 3\) from a family of straight lines on it | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(l\) be a line in a projective space \(\mathbb{P}^n\). We consider the blowing up \(\mathbb{P}^n(l)\) of \(\mathbb{P}^n\) along \(l\). Assume that \(X\) is a smooth closed subvariety of \(\mathbb{P}^n\). If the strict transform of \(X\) in \(\mathbb{P}^n(l)\) has a splitting tangent sequence and dim \(X\) is at least 2, then \(X\) is a linear subspace of \(\mathbb{P}^n\). blowing up; tangent sequence; splitting Projective techniques in algebraic geometry, Coverings in algebraic geometry, Complex Lie groups, group actions on complex spaces, Vector bundles on surfaces and higher-dimensional varieties, and their moduli Submanifolds of \(\mathbb{P}^n(l)\) with splitting tangent sequence | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 integers n, \(p\geq 0\), \(n\geq 2p-1\), let C be a complex projective plane curve of degree \(2(n+p-1)\) with \(3(n+2p-2)\) cusps and \(2(n-2)(n- 3)+2p(n+p-7)\) nodes. Such maximal cuspidal curves were considered already by \textit{O. Zariski} [Am. J. Math. 59, 335-358 (1937; Zbl 0016.32502)] and more recently by \textit{I. Dolgachev} and \textit{A. Libgober} [in Algebraic geometry, Proc. Conf. Chicago Circle 1980, Lect. Notes Math. 862, 1-25 (1981; Zbl 0475.14011)]. -- Let \(H_{n,p}\) denote the fundamental group \(\pi_ 1(P^ 2\setminus C)\). Zariski has given a finite presentation of \(H_{n,p}\) for \(p=0,1\). The paper under review treats the general case by a similar method, using the Reidemeister-Schreier procedure. Zariski problem; fundamental group DOI: 10.2206/kyushumfs.39.133 Coverings in algebraic geometry On the fundamental group of the complement to a maximal cuspidal plane 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 [For the entire collection see Zbl 0628.00007.]
A well-known conjecture says that the universal covering space of any smooth projective algebraic variety is holomorphically convex. The author verifies that any unramified covering of a projective elliptic surface S with \(\chi\) (S,\({\mathcal O}_ S)>0\) is holomorphically convex. Note that the latter implies that for any irreducible curve C on S with \(C^ 2>0\) the normal subgroup generated by the image of the fundamental group \(\pi_ 1(\bar C)\) of the normalization of C in the fundamental group \(\pi_ 1(S)\) is of finite index. According to a conjecture of M. Nori this should be true for every surface. holomorphically convex covering space; unramified covering of a projective elliptic surface; fundamental group Rajendra Vasant Gurjar, Coverings of algebraic varieties, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 179 -- 183. Coverings in algebraic geometry, Holomorphically convex complex spaces, reduction theory, Elliptic surfaces, elliptic or Calabi-Yau fibrations Coverings 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 Let \(G\) be a finite group. A Galois cover \(\bar{\omega}: X\to Y\) with Galois group \(G\) is called a versal \(G\)-cover if any \(G\)-cover is induce from \(\bar{\omega}: X\to Y\). In this paper, we prove that, if the essential dimension of \(G\) is equal to 2, then there exists a versal \(G\)-cover \(\bar{\omega}: X\to Y\) such that
(i) \(X\) is a smooth rational surface,
(ii) \(G\) is a finite automorphism group of \(X\),
(iii) the action of \(G\) on \(X\) is minimal, and
(iv) \(Y=X/G\).
We also give some examples of finite groups which have non-conjugate embeddings into the Cremona group \(\text{Cr}_2(C)\). versal \(G\)-cover; rational surface; Cremona group Hiro-o Tokunaga, Two-dimensional versal \?-covers and Cremona embeddings of finite groups, Kyushu J. Math. 60 (2006), no. 2, 439 -- 456. Coverings in algebraic geometry, Rational and ruled surfaces, Group actions on varieties or schemes (quotients) Two-dimensional versal \(G\)-covers and Cremona embeddings of finite 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 When Zariski gave his celebrated example of the two types of 6-cusped sextics he also proved that the irregularities of the corresponding branched cyclic covers of \(\mathbb{P}^ 2\) were different. Here a formula is given for the irregularity of the branched cyclic cover of any plane curve. This involves terms coming from combinatorial invariants of the resolution, and a cokernel which can be regarded as an index of speciality. Most of the proof uses standard techniques, but the final step uses the Alexander polynomials defined by \textit{A. Libgober} [Duke Math. J. 49, 833-851 (1982; Zbl 0524.14026)]. As well as Zariski's example, and the closely related one of the union of a smooth cubic and 3 inflexional tangents, the author gives a detailed discussion of sextic curves with an \(A_{17}\) singularity and shows that there are four types three of which are distinguished by his invariant. irregularity of the branched cyclic cover of a plane curve; 6-cusped sextics Artal-Bartolo E.: Sur les couples de Zariski. J. Algebraic Geom. 3, 223--247 (1994) Coverings of curves, fundamental group, Classical real and complex (co)homology in algebraic geometry, Coverings in algebraic geometry On Zariski pairs | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The object of this paper is the Grothendieck tame fundamental group \(\pi_ 1^ D(A)\) of an effective reduced divisor D on an abelian variety A over an algebraically closed field of arbitrary characteristic. This group fits into the exact sequence \(0\to I\to \pi_ 1^ D(A)\to \pi_ 1(A)\to 0\) with subgroup I called the inertia subgroup. The main result (theorem 5.3) is the calculation of the abelianization of I (i.e. the quotient by the closure of the commutator subgroup). This abelianization is found together with the natural structure of a module over the Tate module T(A) of A which comes from the identification (due to Serre and Lang) of \(\pi_ 1(A)\) with T(A) and the exact sequence above. More precisely if \({\mathbb{Z}}^{\vee}\) is the completion of \({\mathbb{Z}}\) by subgroups of index prime to \(p\) then the abelianization of the inertia subgroup is isomorphic to the sum over all irreducible components \(D_ i\) of the divisor D of the modules \({\mathbb{Z}}^{\vee}[[T(A/stab^ 0_ AD_ i)/T_ i]]\) where \(stab^ 0_ AD_ i\) is the connected component of the identity stabilizer of the component \(D_ i\) and \(T_ i\) is a closed subgroup of the Tate module \(T(A/stab^ 0_ AD_ i)\). Moreover if the codimension of the stabilizer is one then the corresponding subgroup \(T_ i\) of the Tate module is trivial. If the codimension of the stabilizer is greater than one and \(D_ i\) is geometrically unibranched then \(T_ i=T(A/stab^ 0_ AD_ i)\). As part of the proof the author has a series of results on the abelianized inertia subgroup for divisors on arbitrary smooth projective schemes.
These results are used for clarification of interrelationship between the following two conjectures. Conjecture B [E. Bombieri and \textit{S. Lang}, Bull. Am. Math. Soc. 14, 159-205 (1986; Zbl 0602.14019)]: Let V be a projective variety over an algebraic number field k. Then there is a non empty open subscheme U of V for which \(U(k)=\emptyset\). - Conjecture A [\textit{S. Lang}, Publ. Math., Inst. Hautes Étud. Sci. 6, 27-43 (1960; Zbl 0112.134)]: Let us call a subset S of k-rational points on A a D- integral with respect to a domain R which is a finitely generated \({\mathbb{Z}}\)-algebra, if there is \(d\in R\) such that for a basis \(x_ i\) of \(\Gamma\) (X,\({\mathcal O}(nD))\) one has \(x_ i(P)\in (1/d)R\). If D is an ample divisor on an abelian variety A/k then every D-integral subset with respect to a domain R of the group of rational points A(k) is finite. The author shows that conjecture B implies conjecture A. For this he needs coverings of the abelian variety A branched over D which will be varieties of general type. Existence of these coverings is derived from the aforementioned main result of this work. Grothendieck tame fundamental group; effective reduced divisor; abelian variety; abelianization; Tate module; integral subset with respect to a domain; group of rational points Brown, M.L.: The tame fundamental group of an abelian variety and integral points. Compos. Math.72, 1--31 (1989) Homotopy theory and fundamental groups in algebraic geometry, Arithmetic ground fields for abelian varieties, Rational points, Coverings in algebraic geometry The tame fundamental group of an abelian variety and integral points | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In characteristic zero, \textit{B. Elias} and the second author [Ann. Math. (2) 180, No. 3, 1089--1136 (2014; Zbl 1326.20005)] showed that all degree-zero endomorphisms of indecomposable Soergel bimodules are isomorphisms. In characteristic \(p\), this is known to no longer be true.
The article under review proves that in characteristic 2 there is an indecomposable Soergel bimodule (for \(S_{15}\)) that has a non-zero endomorphism of negative degree, or equivalently that there is an indecomposable parity sheaf on the flag variety \(GL_{15}/B\) that is not perverse. This also proves that the `degree bound' for Kazhdan-Lusztig polynomials can in general fail `by more than one' for \(p\)-Kazhdan-Lusztig polynomials. Soergel bimodules Hecke algebras and their representations, Reflection and Coxeter groups (group-theoretic aspects), Modular representations and characters, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Grassmannians, Schubert varieties, flag manifolds A non-perverse Soergel bimodule in type \(A\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems ``Die vorliegende Arbeit befaßt sich mit der Klassifikation von Automorphiesummanden und -faktoren auf verschiedenen Klassen komplexer Räume, insbesondere im Hinblick auf die Lösbarkeit von Hauptteilverteilungen und Divisoren auf dem universellen Überlagerungsraum. Einerseits werden dabei neuere Resultate aus anderen Bereichen der komplexen Analysis aufgegriffen und zu neuen Anwendungen der bestehenden Theorie verwendet. Andererseits werden die bekannten Techniken weiterentwickelt und Verallgemeinerungen aufgezeigt''. automorphy summands; automorphy factors; Cousin distributions; complex spaces; universal covering spaces General theory of automorphic functions of several complex variables, Stein spaces, Divisors, linear systems, invertible sheaves, Coverings in algebraic geometry, Differential geometry of homogeneous manifolds, Holomorphic bundles and generalizations Automorphy summands, automorphy factors and multivalued solution functions of Cousin distributions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 concept of geometric Lorenz attractor was introduced to understand the dynamics of Lorenz attractors. Such attractors possess a singularity, which is expanding in the central directions, and provide an example of a singular-hyperbolic set.
A Rovella attractor [\textit{A. Rovella}, Bol. Soc. Bras. Mat., Nova Sér. 24, No. 2, 233--259 (1993; Zbl 0797.58051)] relies on a special technical construction, aimed to obtain a centrally dissipative singularity. This compact invariant set for a vector field \(X_0\) shares several properties with geometric Lorenz attractors, but is not singular-hyperbolic. As central result of this paper, the authors establish that the Rovella attractor is asymptotically sectorial-hyperbolic. This weakens the features of geometric Lorenz attractors while still preserving some of them. In addition, it is shown that for generic two-parameter families of vector fields containing \(X_0\), asymptotic sectorial-hyperbolicity is an almost 2-persistent property. flow; attractor; asymptotically sectional-hyperbolic Attractors and repellers of smooth dynamical systems and their topological structure, Generic properties, structural stability of dynamical systems, Dynamical systems with hyperbolic orbits and sets, Strange attractors, chaotic dynamics of systems with hyperbolic behavior, Coverings in algebraic geometry, Set-valued and variational analysis The Rovella attractor is asymptotically sectional-hyperbolic | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 finite group and R a commutative ring. A 1-connected G-space X is said to realize an RG-module M (in dimension \(n>0)\) if \(\tilde H_ n(X;R)=M\) and \(\tilde H_ i(X;R)=0\), \(i\neq n\). Steenrod's problem asks whether every M is so realizable. Early work on this problem, by R. Swan and J. Arnold, concentrated on an additional finiteness question but also gave an affirmative answer for G cyclic primary. Since 1980, with work of G. Carlsson, P. Kahn, and J. Smith, the emphasis has been negative: various examples of nonrealizable M have been produced. An exception has been P. Vogel's theorem which extends the Swan-Arnold result to all G of squarefree order.
The main contribution of the present paper is to provide a criterion whereby large numbers of nonrealizable M may be constructed. Furthermore, the criterion allows us for the first time to obtain a glimpse of the general, global situation (as opposed to a more narrow focus on this or that example or computation). For example, it leads to the observation (not at all obvious from definitions or from earlier work) that nonrealizable modules are in general more plentiful than realizable ones.
To describe (a weak version of) the criterion, choose a field k of characteristic \(p>0\) and let \(X_ G\) be the affine variety \(Spec(H^.(G,k))\), where \(\cdot\) ranges over all indices \(\geq 0\) when \(p=2\) and only the even ones when \(p>2\). If \(H\leq G\), the restriction \(H^.(G,k)\to H^.(H,k)\) induces a map \(t_{H,G}: X_ H\to X_ G\). The authors associate with every kG-module V a certain ideal \(I(V)\subseteq H^.(G,k)\) and then define the variety \(X_ G(V)\subseteq X_ G\) to be \(Spec(H^.(G,k)/I(V)).\)
Now let \(I\subseteq R\) be a maximal ideal, and let k be a field of characteristic \(p>0\) containing R/I. Suppose that V is a realizable RG- module. The authors then show that \(X_ G(V \otimes_ R k)\) is of the form \(\cup_{E}t_{E,G}(X_ E)\), where E ranges over some collection of elementary abelian subgroups of G.
The promised (weak) nonrealizability criterion for an RG-module V, then, is simply that \(X_ G(V \otimes_ R k)\) not has the very special form above.
To make this criterion useful for generating nonrealizable modules, the authors restrict to the case \(R={\mathbb{Z}}\) and show that any homogeneous subvariety \(X\subseteq X_ G\) which is closed under the Frobenius map is of the form \(X_ G(V \otimes_{{\mathbb{Z}}} k)\), for some \({\mathbb{Z}}\)-free \({\mathbb{Z}}G\)-module V. Thus, nonrealizable modules are (at least) as numerous as Frobenius-closed homogeneous subvarieties of \(X_ G\) not of the above form.
An additional useful contribution of the paper is a sequence of results which show that the realizability of a module depends only on its cohomological equivalence class. finite group; G-space; Steenrod's problem; realizable RG-module; Frobenius-closed homogeneous subvarieties; cohomological equivalence class DOI: 10.1016/0022-4049(87)90013-2 Equivariant homotopy theory in algebraic topology, Applied homological algebra and category theory in algebraic topology, Modular representations and characters, Schemes and morphisms, Connections of group theory with homological algebra and category theory, Homological algebra in category theory, derived categories and functors Varieties for modules and a problem of Steenrod | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper is devoted to the study of the period map for abelian covers of smooth projective varieties of dimension \(n\geq 2\). Our viewpoint is very close to that of \textit{M. L. Green} [Compos. Math. 55, 135--156 (1985; Zbl 0588.14004)], namely we look for results that hold for abelian covers of an arbitrary variety whenever certain ampleness assumptions on the building data defining the cover are satisfied. We focus on two questions: the infinitesimal and the variational Torelli problems.
Infinitesimal Torelli for the periods of \(k\)-forms holds for a smooth projective variety \(X\) if the map \(H^1(X,T_X) \to \bigoplus_p\Hom (H^p(X,\Omega_X^{k-p})\), \(H^{p+1}(X,\Omega_X^{k-p-1}))\), expressing the differential of the period map for \(k\)-forms, is injective. This is expected to be true as soon as the canonical bundle of \(X\) is ``sufficiently ample''. Here the author continues her work on abelian covers [\textit{R. Pandini} in: Problems in the theory of surfaces and their classification. Meeting Sc. Super. Cortona 1988, Symp. Math. 32, 247--257 (1991; Zbl 0827.14006)] and proves:
Theorem 1. Let \(G\) be an abelian group and let \(f:X\to Y\) be a \(G\)-cover, with \(X,Y\) smooth projective varieties of dimension \(n\geq 2\). If some special properties (A) and (B) for \(Y\) are satisfied, then infinitesimal Torelli for the periods of \(N\)-forms holds for \(X\).
Properties (A) and (B) amount to the vanishing of certain cohomology groups and are certainly satisfied if the building data of the cover are sufficiently ample. If \(Y\) is a special variety (e.g., \(Y=\mathbb{P}^n)\), then theorem 1 yields an almost sharp statement. The variational Torelli problem asks whether, given a flat family \(\mathcal X\to B\) of smooth polarized varieties the map associating to a point \(b\in B\) the infinitesimal variation of Hodge structure of the fibre \(X_b\) is generically injective, up to isomorphism of polarized varieties. Exploiting the variational Torelli result of \textit{M. L. Green} (loc. cit.) the author is able to obtain, under analogous assumptions, a positive answer to the variational Torelli problem for a large class of abelian covers. Pardini, R., On the period map for abelian covers of projective varieties, Ann. sc. norm. super. Pisa, cl. sci., 26, 4, 719-735, (1998) Torelli problem, Coverings in algebraic geometry, Variation of Hodge structures (algebro-geometric aspects), Period matrices, variation of Hodge structure; degenerations On the period map for Abelian covers of projective varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For a Tate curve \(X\) over a \(p\)-adic field, we give a structure theorem of the ``class group'' in the view of the class field theory for \(X\). elliptic curve; Tate curve; Tate local duality theorem Geometric class field theory, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Class field theory; \(p\)-adic formal groups, Local ground fields in algebraic geometry, Class field theory, Coverings in algebraic geometry Galois symbol map for a Tate curve | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper is a short survey on the impact of the two seminal papers, ``The spectrum of the equivariant cohomology ring. I, II.'' of \textit{D. Quillen} [Ann. Math. (2) 94, 549-572, 573-602 (1971; Zbl 0247.57013)]. Those papers had significant impact in algebraic topology, as well as on the development of the cohomology and modular representation theory of finite groups, which is the focus of this article.
The author begins with a brief introduction to the contents of those papers, wherein Quillen worked over a field of prime characteristic \(p\). This includes a discussion of equivariant cohomology, the Atiyah-Swan Conjecture on the Krull dimension of the cohomology ring of the classifying space of a finite group, the maximal ideal spectrum of the cohomology ring of a finite group, and Quillen's Stratification Theorem relating the spectrum of group cohomology to that over elementary Abelian \(p\)-subgroups.
The author then gives a brief discussion of the general theory of cohomological support varieties and rank varieties for non-trivial (finite dimensional) modules over finite groups that developed out of Quillen's original work. Thirdly, the author gives an overview of further developments in support and rank variety theories for other structures such as restricted Lie algebras, infinitesimal group schemes, and more generally, finite group schemes, which includes the original context of finite groups. equivariant cohomology; elementary Abelian subgroups; group cohomology; cohomological support varieties; rank varieties; finite group schemes; cohomology rings; modular representations doi:10.1017/is011012012jkt206 Cohomology of groups, Modular representations and characters, Cohomology theory for linear algebraic groups, Representation theory for linear algebraic groups, Modular Lie (super)algebras, Cohomology of Lie (super)algebras, Group schemes Spectrum of group cohomology 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 We present a canonical proof of both the strict and weak Positivstellensatz for rings of differentiable and smooth functions. Our construction is explicit, preserves definability in expansions of the real field, and it works in definably complete expansions of real closed fields as well as for real-valued functions on Banach spaces. Positivstellensatz; differentiable function; smooth function; definably complete structure; Banach space A. Fischer, Positivstellensätze for differentiable functions, Positivity. DOI: 10.1007/s11117-010-0077-5 Model theory of ordered structures; o-minimality, Semialgebraic sets and related spaces, Constructive real analysis, Rings and algebras of continuous, differentiable or analytic functions Positivstellensätze for differentiable functions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In previous work, the author found a bitangency theorem for surfaces in \(\mathbb{R}^4\), specifically a formula relating the number of bitangencies to the number of double points, the normal Euler number of the surface, and a ``diagonal contribution'' -- the intersection value of a degenerate intersection in the bitangency map. This diagonal contribution is written as \(E_{\mathbf v}+ \frac 12 P_{\mathbf v}\), where \({\mathbf v}\) is a generic unit vector, \(E_{\mathbf v}\) is a number determined by the elliptic points of the surface and \(P_{\mathbf v}\) is a number determined by the flecnodal normal vector field on the parabolic curve on the surface. Outside of the bitangency formula, it is not obvious that these two quantities should be related. Taken separately, both terms depend on the choice of \({\mathbf v}\), while their sum is an invariant of the surface.
In this paper we show, without use of the bitangency formula, that the diagonal contribution is in fact independent of the choice of \({\mathbf v}\) and that it is an invariant of the geometry of the surface. Dreibelbis, D.: Invariance of the diagonal contribution in a bitangency formula. Contemporary mathematics 354, 45-55 (2004) Global theory of singularities, Coverings in algebraic geometry, Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces, Critical points of functions and mappings on manifolds Invariance of the diagonal contribution in a bitangency formula | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper deals with developing classical scheme theory. Namely, the authors consider scheme-theoretical background subject -- relations between spectralization and topology.
The authors show that for a field \(K\) and \(n\geq 1\), the soberification \(\mathcal{S}(\mathbb{A}^n(K))\) of the affine \(n\)-space \(\mathbb{A}^n(K)\) over \(K\) is homeomorphic to its spectralization \(\mathcal{BS}(\mathbb{A}^n(K))\), and it can be embedded into the spectrum \(\mathrm{Spec}(K[X_1,\dots,X_n])\). Moreover, if the field \(K\) is algebraically closed, then there are homeomorphisms \(\mathcal{S}(\mathbb{A}^n(K)) \approx\mathcal{BS}(\mathbb{A}^n(K))\approx \mathrm{Spec}(K[X_1,\dots,X_n])\). They also show that for a space \(X\), the subspace \(z\mathrm{Spec}(C(X))\subseteq\mathrm{Spec}(C(X))\) of prime z-ideals of the ring \(C(X)\) of real-valued continuous functions on \(X\) is homeomorphic to the space \(z\mathcal{SR}(X)\) of prime \(z\)-filters with an appropriate topology and there is a homeomorphism \(\mathcal{BS}(X)\approx z\mathrm{Spec}(C(X))\) provided \(X\) is perfectly normal. (Notions of \textit{Sober space} and \textit{Soberification} deals with space improving. They are defined at pages 514 and 515 resp.)
The paper is self-contained elementary and easy for reader. affine space; perfectly normal space; reflective subcategory; ring of continuous functions; sober space; spectral space; spectrum of a ring; Zariski topology; \(z\)-ideal Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem), Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.), Rings and algebras of continuous, differentiable or analytic functions, Special constructions of topological spaces (spaces of ultrafilters, etc.) On the spectralization of affine and perfectly normal 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 In this article a new relation between Prym varieties of arbitrary morphisms of algebraic curves and integrable systems is discovered. The action of maximal commutative subalgebras of the formal loop algebra of \(GL_n\) defined on certain infinite-dimensional Grassmannians is studied. H. Morikawa's question is: What are the finite-dimensional orbits of these Heisenberg flows, and what kind of geometric objects do they represent?
Theorem A. A finite-dimensional orbit of the Heisenberg flow defined on the Grassmannian of vector valued functions corresponds to a covering morphism of algebraic curves, and the orbit itself is canonically isomorphic to the Jacobian variety of the curve upstairs. Moreover, the action of the traceless elements of the Borel subalgebra (the traceless Heisenberg flows) produces the Prym variety associated with this covering morphism as an orbit.
Let us define the Grassmannian quotient \(Z_n(0)\) as the quotient space of \(\text{Gr}_n(0)\) by the diagonal action of \((1+\mathbb C[[z]]z)^{\times n}.\) The traceless \(n\)-component KP system is defined by the action of the traceless diagonal matrices with entries in \(\mathbb C[z^{-1}]\) on \(Z_n(0)\). Since this system is a special case of the traceless Heisenberg flows, every finite-dimensional orbit of this system is a Prym variety. Conversely, an arbitrary Prym variety associated with a degree n covering morphism of algebraic curves can be realized as finite-dimensional orbit.
Theorem B. An algebraic variety is isomorphic to the Prym variety associated with a degree n covering of an algebraic curve if and only if it can be realized as a finite-dimensional orbit of the traceless \(n\)-component KP system defined on the Grassmannian quotient \(Z_n(0)\).
Theorem D. Every object of the category \(C(n)\) with a smooth curve \(C_n\) and a line bundle \(F\) on \(C_n\) satisfying the cohomology vanishing condition \(H^0(C_n,F)= H^1(C_n,F)=0\) gives rise to a maximal commutative algebra consisting of ordinary differential operators with coefficients in \(n\times n\) matrix valued functions.
Theorem E. The big-cell of the Grassmannian \(\text{Gr}_n(0)\) is canonically identified with the group of monic invertible pseudodifferential operators with matrix coefficients. Prym varieties; integrable systems; morphisms of algebraic curves; formal loop algebra; infinite-dimensional Grassmannians; Heisenberg flows; traceless \(n\)-component KP system; cohomology vanishing condition; algebra of ordinary differential operators Chernov, A., Talalaev, D.: Quantum spectral curves, quantum integrable systems and the geometric Langlands correspondence (2006). [arXiv:hep-th/0604128 [hep-th]] Jacobians, Prym varieties, Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Coverings in algebraic geometry, Ramification problems in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds Prym varieties and integrable systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems By a result due to \textit{R. Miranda} [Am. J. Math. 107, 1123--1158 (1985: Zbl 0611.14011)] it is known that a triple covering \(p:X \to Y\) of algebraic varieties is determined by a locally free rank \(2\) sheaf \(E\) over \(Y\) and a map \(S^3(E) \rightarrow \Lambda^2(E)\), and vice versa.
In the paper under review the authors give a necessary and sufficient criterion such that \(X\) from above is realized as a subvariety of a \(\mathbb{P}^1\)-bundle over \(Y\) equipped with its natural projection onto \(Y\).
Although their result is not new, a more general one has been given by \textit{G. Casnati} and \textit{T. Ekedahl} [J. Algebr. Geom. 5, No. 3, 439--460 (1996; Zbl 0866.14009)], it has the advantage of being more geometric and one could hope it will provide some insight into the more general case. triple covering; subvariety of a \(\mathbb{P}^1\)-bundle Faenzi, Daniele; Stipins, Janis, A small resolution for triple covers in algebraic geometry, Matematiche (Catania), 56, 2, 257-267 (2003), (2001) Coverings in algebraic geometry, Coverings of curves, fundamental group A small resolution for triple covers in algebraic geometry | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The representation theory of a connected smooth affine group scheme over a field \(k\) of characteristic \(p>0\) is faithfully captured by that of its family of Frobenius kernels. Such Frobenius kernels are examples of infinitesimal group schemes, affine group schemes \(G\) whose coordinate (Hopf) algebra \(k[G]\) is a finite-dimensional local \(k\)-algebra. This paper presents a study of the cohomology algebra \(H^* (G,k)\) of an arbitrary infinitesimal group seheme over \(k\).
We provide a geometric determination of the ``cohomological support variety''
\[
|G|\equiv\text{Spec} H^{ev} (G,k)
\]
analogous to that given by \textit{D. Quillen} for the cohomology of finite groups [Ann. Math., II. Ser. 94, 549-572, 573-602 (1971; Zbl 0247.57013)]. We further study finite-dimensional rational \(G\)-modules \(M\) for arbitrary infinitesimal group schemes \(G\) over \(k\). In a manner initiated by \textit{J. L. Alperin} and \textit{L. Evens} [J. Pure Appl. Algebra 22, 1-9 (1981; Zbl 0469.20008)] and \textit{J. F. Carlson} [J. Algebra 85, 104-143 (1983; Zbl 0526.20040)] for finite groups, we consider the variety \(|G|_M\subset |G|\) of the ideal \(I_M=\text{ker} \{H^{ev}(G,K) \to\text{Ext}^*_G (M, M)\}\) and provide a geometric description of this variety which is analogous to that given by \textit{G. S. Avrunin} and \textit{L. L. Scott} for finite-dimensional modules for finite groups [Invent. Math. 66, 277-286 (1982; Zbl 0489.20042)].
This paper is a continuation of our recent work establishing the finite generation of \(H^*(G,k)\) [\textit{E. M. Friedlander} and \textit{A. Suslin}, Invent. Math. 127, No. 2, 209-270 (1997; Zbl 0945.14028)] and investigating the infinitesimal 1-parameter subgroups of \(G\) [\textit{A. Suslin}, \textit{E. M. Friedlander} and \textit{C. P. Bendel}, J. Am. Math. Soc. 10, No. 3, 693-728 (1997; see the preceding review Zbl 0960.14023)]. Earlier work by E. Friedlander and B. Parshall and J. Jantzen concerning the cohomology of restricted Lie algebras are forerunners of the results presented here: Finite-dimensional restricted Lie algebras are in 1-1 correspondence with infinitesimal group schemes of height \(\leq 1\). Our main theorems (theorems 5.2 and 6.7 below) when restricted to infinitesimal group schemes of height \(\leq 1\) significantly strengthen previously known cohomological information for restricted Lie algebras.
An interesting aspect of our work is the extent to which infinitesimal 1-parameter subgroups \(\nu:\mathbb{G}_{a (r)}\to G\) for infinitesimal group schemes \(G\) of height \(\leq r\) play the role of elementary abelian \(p\)-subgroups (and their generalizations, shifted subgroups) for finite groups. Indeed, much of our effort is dedicated to proving that cohomology classes are detected (modulo nilpotence) by such 1-parameter subgroups.
The proof of the detection theorem for arbitrary infinitesimal group schemes over \(k\) relies upon a generalization of a spectral sequence introduced by \textit{H. H. Andersen} and \textit{J. C. Jantzen} [Math. Ann. 269, 487-525 (1984; Zbl 0529.20027)]. Our generalized spectral sequence is presented in \S 3, enabling the proof in \S 4 of the general detection theorem (theorem 4.3). -- The detection theorem demonstrates the essential injectivity of the natural map considered in the preceding paper cited above
\[
\psi:H^{ev} (G,k)\to k\bigl[V_r (G)\bigr],
\]
where \(V_r(G)\) is the scheme of infinitesimal 1-parameter subgroup schemes \(\nu: \mathbb{G}_{a(r)}\to G\) of an infinitesimal group scheme \(G\) over \(k\) of height \(\leq r\). The essential surjectivity of \(\psi\) (more precisely, surjective onto \(p^r\)-th powers) is a main result of our preceding paper reviewed above. This is formalized in theorem 5.2 which presents a geometric, non-cohomological description of the cohomological support variety \(|G|\) of \(G\). Corollary 6.8 gives a similarly geometric, non-cohomological identification of \(|G|_M\subset |G|\) for any finite-dimensional rational \(G\)-module \(M\). We conclude in \S 7 with a few applications of these descriptions, applications analogous to results obtained previously for the cohomology of finite groups. infinitesimal group schemes; cohomological support variety; characteristic \(p\); representation theory of a connected smooth affine group scheme; Frobenius kernels; cohomology algebra; restricted Lie algebras; infinitesimal 1-parameter subgroups A. Suslin, E. M. Friedlander and C. P. Bendel, Support varieties for infinitesimal group schemes, J. Amer. Math. Soc. 10 (1997), no. 3, 729-759. Group schemes, Cohomology theory for linear algebraic groups, Modular representations and characters, Homological methods in Lie (super)algebras, Cohomology of Lie (super)algebras Support varieties for infinitesimal group schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The aim of this survey article is to bring together recent advances concerning the fundamental groups of join-type curves. Though the paper is of purely expository nature, we do also announce a new result. join-type curves; fundamental groups C. Eyral and M. Oka, On the fundamental groups of non-generic R-join-type curves, Experimental and Theoretical Methods in Algebra, Geometry and Topology, Springer Proceedings in Mathematics & Statistics. (to appear). Coverings of curves, fundamental group, Coverings in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry, Singularities of curves, local rings, Local rings and semilocal rings, Singularities in algebraic geometry, Special algebraic curves and curves of low genus, Plane and space curves Fundamental groups of join-type curves -- achievements and perspectives | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 notion of \textit{tame ramification} comes from algebraic number theory. Let \(A \subseteq B\) be a finite extension of discrete valuation rings with maximal ideals \(\mathfrak{m}\) and \(\mathfrak{n}\), respectively. The ramification index \(e\) is the unique positive integer such that \(\mathfrak{m}B = \mathfrak{n}^e\). The extension \(A \subseteq B\) is called \textit{tamely ramified} if the extension of residue fields \(A / \mathfrak{m} \subseteq B/ \mathfrak{n}\) is separable and if the characteristic of \(A/\mathfrak{m}\) does not divide \(e\). (If the residue characteristic is zero, \(A \subseteq B\) is also called tamely ramified.) In particular, if we have a finite separable morphism of \(1\)-dimensional regular schemes \(Y \to X\), then we can ask about the ramification behavior of the induced local homomorphisms on their various local rings. The two main examples are when \(Y\) and \(X\) are curves over a field \(k\), or else arithmetic curves (i.e., spectra of rings of integers in number fields).
It is not a priori clear how to extend the above definition beyond the case of 1-dimensional schemes without adding some extra hypotheses. In the article under review, the author proposes a canonical and intrinsic extension of this definition, shows that it has several first properties one might demand, and then compares it to two previous definitions of tame ramification due to A. Grothendieck and to A. Schmidt. They are denoted (TR1) and (TR2), respectively. Although (TR1) and (TR2) are explicitly described by the author, we avoid giving them here for the sake of brevity. It suffices to say that the main issue with (TR1) and (TR2) is that they are not intrinsic: in order to study tamely ramified covers of a variety \(X\), one must choose a regular completion \(X \subseteq \bar X\). Even worse, they actually depend on the embedding.
Let \(Y \to X\) be any étale cover of varieties (over some field \(k\)) or arithmetic schemes (i.e., integral separated schemes that are flat and of finite type over \(\text{Spec} \mathbb{Z}\)). For purposes of the following definition, a curve is either an arithmetic curve or a variety of dimension~1.
(TR3) For every curve \(C \subseteq X\) with normalization \(\tilde{C}\), the induced cover \(Y \times_X \tilde{C} \to \tilde{C}\) is tamely ramified (outside \(\tilde{C}\)).
The author proves two main results. The first (Proposition~1) has three parts: (a) property (TR3) is stable under finite type base extension; (b) for a fixed connected scheme \(X\), finite étale covers of \(X\) satisfying (TR3) form a Galois category in the sense of Grothendieck; and (c) given a geometric point \(x_0 \to X\), there is a corresponding ``tame fundamental group'' \(\pi_1^t(X, x_0)\) classifying covers satisfying (TR3). The second result (Theorem~2) compares the three definitions (TR1), (TR2), and (TR3) of tame ramification in the setting where they all apply. They are all equivalent if \(X\) is regular and can be embedded in a regular scheme in such a way that its complement is a normal crossings divisor.
A recent preprint of \textit{M. Kerz} and \textit{A. Schmidt} [On different notions of tameness in arithmetic geometry, \url{arXiv:0807.0979v1}] addresses further the equivalence between the myriad definitions of tame ramification, including (TR3). tame ramification; higher dimensional varieties; arithmetic schemes; ramified covers; tame fundamental group Riemann, B.: \textit{Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse}. Inauguraldissertation, Göttingen (1851) Arithmetic varieties and schemes; Arakelov theory; heights, Coverings in algebraic geometry, Ramification problems in algebraic geometry Tamely ramified covers of varieties and arithmetic schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Given a vector bundle \(E\), on an irreducible projective variety \(X\), we give a necessary and sufficient criterion for \(E\) to be a direct image of a line bundle under a surjective étale morphism. The criterion in question is the existence of a Cartan subalgebra bundle of the endomorphism bundle \(\mathrm{End}(E)\). As a corollary, a criterion is obtained for \(E\) to be the direct image of the structure sheaf under an étale morphism. The direct image of a parabolic line bundle under any ramified covering map has a natural parabolic structure. Given a parabolic vector bundle, we give a similar criterion for it to be the direct image of a parabolic line bundle under a ramified covering map. Auffarth, R.; Biswas, I., Direct images of line bundles, J. Pure Appl. Algebra, 222, 1189-1202, (2018) Coverings in algebraic geometry, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Group schemes Direct image of parabolic line 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 \(X\) be a projective manifold of dimension \(n \geq 2\) and \(Y \to X\) be an infinite covering space. Embed \(X\) into projective space by sections of a sufficiently ample line bundle. We prove that any holomorphic function of sufficiently slow growth on the preimage of a transverse intersection of \(X\) by a linear subspace of codimension \(< n\) extends to \(Y\). The proof uses a Hausdorff duality theorem for \(L_2\) cohomology. We also show that every projective manifold has a finite branched covering whose universal covering space is Stein. projective manifold; covering space; Shafarevich conjecture; extension of holomorphic function; slow growth; \(L_ 2\) cohomology; duality; vanishing theorem F. Lárusson, An extension theorem for holomorphic functions of slow growth on covering spaces of projective manifolds , J. Geom. Anal., 5 (1995), no. 2, 281--291. Continuation of analytic objects in several complex variables, Coverings in algebraic geometry, Vanishing theorems An extension theorem for holomorphic functions of slow growth on covering spaces of projective manifolds | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(G\) be a finite abelian group and let \(X,Y\) be algebraic varieties. An abelian cover \(X\) of \(Y\), with group \(G\) is a (finite) morphism \(\pi:X\to Y\), together with a faithful action of \(G\) on \(X\), commuting with \(\pi\), such that \(Y=X/G\). We will focus our attention on the case in which \(X\) is normal and \(Y\) is smooth. The theory of cyclic covers of algebraic surfaces was studied first by A. Comessatti. Then F. Catanese studied smooth abelian covers in the case \((\mathbb{Z}_2)^2\) and R. Pardini analyzed the general case. M. Manetti investigated the property to be Gorenstein for \((\mathbb{Z}_2)^n\)-covers. Recently, R. Vakil used the \((\mathbb{Z}_p)^n\)-covers to show the existence of badly-behaved deformation spaces. The first purpose of this paper is to establish combinatorial conditions so that a normal \(G\)-cover \(X\) of a smooth variety \(Y(\pi:X\to Y)\) is Gorenstein.
Theorem. Let \(X\) be a normal \(G\)-cover of a smooth variety \(Y\) and \(\{(H_i, \psi_i)]_{i=1,\dots,s}\) its combinatorial data at the point \(y\) in \(Y\). Then the points of \(X\) over \(y\) are Gorenstein points of \(X\) if and only if there exists a character \(\chi\) of \(G\) such that
\[
\chi|_{H_i}=\psi_i,\quad\forall 1\leq i \leq s.
\]
The second purpose of this paper is to prove the following
Theorem. Let \(\pi:X\to Y\) be a normal flat \((\mathbb{Z}_p)^n\)-cover with \(Y\) smooth and \(p\) prime. Then the following conditions are equivalent:
(i) \(\pi\) is locally simple;
(ii) \(X\) is locally complete intersection.
This theorem improves a result due to \textit{M. Manetti} [Invent. Math. 143, 29--76 (2001; Zbl 1060.14520)]. doi:10.1016/j.jalgebra.2006.01.025 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Coverings in algebraic geometry Local structure 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 Classical Hurwitz numbers count branched covers of a Riemann surface of genus \(g\) with specified ramifications. Monotone Hurwitz numbers counts a certain subset of these covers, related to the so-called Harish-Chandra-Itzykson-Zuber integral in random matrix theory.
In this paper the authors study monotone single Hurwitz numbers and obtain a precise formula for \(g=1\) and a generating function and polynomiality behavior for \(g \geq 2\). monotone Hurwitz numbers; generating function I. P. Goulden, M. Guay-Paquet, and J. Novak 2013 Polynomiality of monotone Hurwitz numbers in higher genera \textit{Adv. Math.}238 1--23 Exact enumeration problems, generating functions, Coverings in algebraic geometry, Random matrices (algebraic aspects) Polynomiality of monotone Hurwitz numbers in higher genera | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 classical Galois structure theory one considers finite tame Galois extensions \(L/K\) of number fields. To generalize this, we recall in \S2 Grothendieck's and Murre's concept of a tamely ramified \(G\)-cover \(f:X\mapsto Y\) of schemes over a Noetherian ring \(A\), where \(G\) is a finite group. We then discuss the author's results in ``Galois structure of de Rham cohomology of tame covers of schemes'' (to appear) concerning Euler characteristics in Grothendieck groups of \(A[G]\)-modules of suitable complexes of sheaves of \(G\)-modules on \(X\). --- In \S3 we define via de Rham complexes an invariant \(\Psi(X/Y)\) which generalizes the stable isomorphism class of the ring of integers of \(L\) in the classical case. We then discuss a conjectural generalization of Martinet's conjecture when the ground ring \(A\) is a finitely generated \(\mathbb{Z}[1/m]\)- module for some integer \(m\) prime to the order of \(G\). The main result of the author's cited paper, which is summarized in \S4, is a precise counterpart for smooth projective varieties over a finite field of Fröhlich's conjecture concerning rings of integers. One consequence of this result is that the generalization of Martinet's conjecture discussed in \S3 holds if \(A\) is a finite field. Galois structure of de Rham cohomology; tamely ramified cover of schemes; Euler characteristic in Grothendieck groups; rings of integers Chinburg, T.: Galois module structure of de Rham cohomology. J. Théorie Nr. Bordx. 4, 1--18 (1991) de Rham cohomology and algebraic geometry, Integral representations related to algebraic numbers; Galois module structure of rings of integers, Coverings in algebraic geometry, Coverings of curves, fundamental group Galois structure of de Rham 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 Dans un article de 1931 [Ann. Math., II. Ser. 32, 485-511 (1931; Zbl 0001.40301)], \textit{O. Zariski} calcule l'irrégularité d'un revêtement cyclique du plan projectif complexe \(\mathbb{P}^2_\mathbb{C}\) ramifié au dessus d'une courbe \(C\) ayant comme seules singularités des points doubles ordinaires et des points cuspidaux. Il peut en déduire des résultats d'annulation pour certains groupes de cohomologie de faisceaux associés à la courbe \(C\). -- Récemment, en utilisant une construction différente plusieurs auteurs ont généralisé l'étude des revêtements cycliques (en particulier \textit{H. Esnault} et \textit{E. Viehweg}, \textit{F. Loeser} et l'A.). Mais si l'utilisation de théorèmes puissants déduits essentiellement de la théorie de Hodge permet d'obtenir des résultats plus généraux, la beauté de la démonstration originale a disparu.
Dans cet article je poursuis d'une part l'étude des revêtements cycliques et surtout jemontre comment il est possible d'utiliser les méthodes originales de Zariski pour démontrer ces résultats. irregularity of cyclic covering Vaquié, Michel, Irrégularité des revêtements cycliques, (), 383-419, MR 1295085 (95f:14030) Coverings in algebraic geometry Irregularity of 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 investigate the automorphism groups of Galois coverings induced by pluricanonical generic coverings of projective spaces. In dimensions one and two, it is shown that such coverings yield sequences of examples where specific actions of the symmetric group \( S_d\) on curves and surfaces cannot be deformed together with the action of \( S_d\) into manifolds whose automorphism group does not coincide with \( S_d\). As an application, we give new examples of complex and real \( G\)-varieties which are diffeomorphic but not deformation equivalent. generic coverings of projective lines and planes; Galois group of a covering; Galois extensions; automorphism group of a projective variety Kulikov, Vi.k S.; Kharlamov, V. M., Automorphisms of Galois coverings of generic m-canonical projections, Izv. Ross. Akad. Nauk, Ser. Mat., 73, 121-156, (2009) Coverings in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry, Moduli, classification: analytic theory; relations with modular forms, Surfaces of general type, Topology of real algebraic varieties, Deformations of complex structures, Symplectic and contact topology in high or arbitrary dimension, Differential topological aspects of diffeomorphisms Automorphisms of Galois coverings of generic \( m\)-canonical projections | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 deals with the problem of the parameterization by radicals (of degree prime to \(q(q-1)\))\, of algebraic curves of small genus (mainly genus 2) defined over a finite field \(\mathbb{F}_q\),\, of characteristic different from 2 and 3. A such parameterization would allow an efficient deterministic encodings into those curves.
Encoding algorithms for elliptic curves are already known, see [\textit{T. Icart}, Lect. Notes Comput. Sci. 5677, 303--316 (2009; Zbl 1252.94075)], but for genus 2 curves only partial results were known, results only valid for a negligible proportion of all genus 2 curves, see Kammerer, Lercier and Renault [\textit{J.-G. Kammerer} et al., Lect. Notes Comput. Sci. 6487, 278--297 (2010; Zbl 1290.94100)].
Section 2 begins remembering the basic notions of parameterizations, encoding and torsors. Then Tartaglia-Cardan formulae is formulated in the language of torsors and the proposed method of parametrization is showed (2.7 and 2.8). This general method is illustrated in Section 3 for elliptic curves.
Section 4 studies the case of genus 2 curves having two rational points whose difference has order 3 in the associated Jacobian variety. This allows to parameterize by 3-radicals a positive proportion of all genus 2 curves. An example (over \(\mathbb{F}_{83}\)) is worked in detail. Finally the paper studies other parameterizations by \(l\)-radicals, in particular for \(l=5\). algebraic curves; parameterizations; radicals; finite fields; deterministic algorithms; encodings; torsors J.-M. Couveignes and R. Lercier, \textit{The geometry of some parameterizations and encodings}, Adv. Math. Commun., 8 (2014), pp. 437--458. Special algebraic curves and curves of low genus, Coverings in algebraic geometry, Curves over finite and local fields, Galois cohomology, Galois theory The geometry of some parameterizations and encodings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Main theorem. Let X be a compact complex manifold which can be represented as a k-fold cyclic covering \(f: X\to Z\) of a homogeneous rational manifold Z of rank one. Denote by
\[
\begin{matrix} X &&\overset {g} \hookrightarrow && X \\ &f\searrow && \swarrow \pi \\ &&Z \end{matrix}
\]
the embedding of f into the total space \(Y={\mathbb{V}}(F^{-1})\) of the line bundle \(F={\mathcal O}_ Z(m)\in Pic(Z)\) whose dual \(F^{-1}\) generates the \({\mathcal O}_ Z\)-algebra \(f_*{\mathcal O}_ X\). Then for every complex germ S both of the maps \(Def(X/Y)(S)\to Def(X/Z)(S)\to Def(X)(S)\) are surjective, hence any deformation of X can be represented as a covering of Z which is embeddable into Y, in each of the following cases: i) dim \(Z\geq 3\) and \(K_ Z\otimes F\geq 0\) where \(K_ Z\) is the canonical bundle, ii) dim Z\(=2\) i.e. \(Z={\mathbb{P}}_ 2\), and X is not a K3-surface iii) \(Z={\mathbb{G}}(r,N)\) is a Grassmannian with \(2\leq r\leq N/2\) and (r,N)\(\neq (2,4)\), or \(Z={\mathbb{P}}_ n\), \(n\geq 3\). holomorphic mapping; Torelli theorem; complex manifold; cyclic covering; deformation Joachim Wehler, Cyclic coverings: deformation and Torelli theorem, Math. Ann. 274 (1986), no. 3, 443 -- 472. Deformations of special (e.g., CR) structures, Sheaves and cohomology of sections of holomorphic vector bundles, general results, Coverings in algebraic geometry, Vanishing theorems, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Grassmannians, Schubert varieties, flag manifolds, Holomorphic mappings and correspondences, Complex manifolds Cyclic coverings: Deformation and 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 The moduli space of curves of genus three has dimension \(6\), and its hyperbolic locus has dimension \(5\). Introduced in the earlier works of one of the authors is a database for curves of genus \(2\) in terms of coefficients, and in this work under review, the authors present a database of hyperelliptic curves of genus \(3\) in terms of invariants of binary forms, called Shioda invariants \(J_k\). \textit{T. Shaska} proved in [Commun. Algebra 42, No. 9, 4110--4130 (2014; Zbl 1427.14056)] that the set of ordered tuples \([J_2:J_3:\cdots:J_8]\) in the weighted projective space that satisfy a certain polynomial equation forms the moduli space of hyperelliptic curves of genus \(3\). The technique for the case of genus \(2\) became useful for the case of genus \(3\) as the concept of weighted moduli height for weighted moduli spaces was introduced by Shaska. The paper presents a computational overview of finding the tuples in the weighted moduli spaces with bounded weighted moduli height, and is concluded with the database for heights in between \(1\) and \(1.5\). invariants; binary forms; genus 3; algebraic curves Coverings of curves, fundamental group, Rings and algebras of continuous, differentiable or analytic functions On hyperelliptic curves of genus 3 | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This is mainly a didactical paper and, except for the results of the last section, there is no claim at all for originality. Perhaps the only original thing in it is just the definition of folding and branched folding. These concepts generalize, respectively, coverings and branched coverings.
Under the framework of Fox spreads and its completions a theory that generalizes coverings (folding covering theory) and a theory that generalizes branched coverings (branched folding theory) is defined and some properties are proved. Two applications to 3-manifold theory are given. A problem is stated. knots; links; 3-manifolds; branched coverings; foldings; strings; spreads; Cantor sets Relations of low-dimensional topology with graph theory, Coverings in algebraic geometry, Knots and links in the 3-sphere, Wild embeddings, Dehn's lemma, sphere theorem, loop theorem, asphericity, Topology of Euclidean 2-space, 2-manifolds, Topology of general 3-manifolds, Knots and links (in high dimensions) [For the low-dimensional case, see 57M25] Branched folded coverings and 3-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 [For part I see the authors, J. Algebra 118, No. 1, 63-84 (1988; Zbl 0664.14019)]
A projective normal surface \(\overline V\) is a Gorenstein log Del Pezzo surface if it has only rational double singularities and the canonical divisor is ample. In the paper under review, the fundamental group of the smooth part of such a surface is determined.
Main result: Let \(f : V \to \overline V\) be a minimal resolution and let \(D : = f^{-1} (\text{Sing} (\overline V))\), where \(\text{Sing}(\overline V)\) is the singular locus. Let \(V^ 0 : = \overline V -\text{Sing} (\overline V)\). Then the fundamental group \(\pi_ 1 (V^ 0)\) of the smooth part \(V^ 0\) is an abelian group of order \(\leq 9\).
In the paper are also classified Gorenstein log Del Pezzo surfaces, \(\overline V\), under the assumption that \(\overline V\) is relatively minimal. Existence results are also given. quasi-universal covering; abelian fundamental group; projective normal surface; fundamental group M. Miyanishi and D. Q. Zhang, ''Gorenstein Log del Pezzo Surfaces. II,'' J. Algebra 156(1), 183--193 (1993). Special surfaces, Homotopy theory and fundamental groups in algebraic geometry, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Coverings in algebraic geometry Gorenstein log del Pezzo surfaces. 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 present an improved version of the cyclic covering trick ([\textit{H. Esnault} and \textit{E. Viehweg}, Lectures on vanishing theorems. Notes, grew out of the DMV-seminar on algebraic geometry, held at Reisensburg, October 13-19, 1991. Basel: Birkhäuser Verlag (1992; Zbl 0779.14003)], Section 3), which works inside the category of toroidal embeddings. cyclic covers; toroidal embeddings; Zariski-Steenbrink differential forms Coverings in algebraic geometry, Vanishing theorems in algebraic geometry Cyclic covers and toroidal embeddings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 study the kernel and the image of the natural restriction map \(\pi^*:B(U)\to B(V)\) between the Brauer groups of \(U\) and \(V\) where \(S\) is a nonsingular surface over \(k\), \(\text{char} k=0\), \(Z\) is an effective irreducible divisor over \(S\) linearly equivalent with an even multiple of an irreducible curve \(C\), \(C\subset S\), \(\pi:X\to S\) the double cover morphism ramified over \(Z\), \(U=S\backslash Z\) and \(V=\pi^{-1}(U)\). First, the kernel \(\ker \pi^*=B(V/U)\) is studied in order to find bounds for its rank. The result is given in the theorem 1.1: If \(\rho\) is the Picard number of the open set \(X'\) of regular points of \(X\) then there exists a positive integer \(r\), \(\rho-2\leq r\leq\rho\), such that there is the exact sequence
\[
0\to\breve B(V/U)\to B(V/U)\to\mathbb{Z}/2^{(r)}\to 0,
\]
\(\breve B(V/U)\) being the image of a morphism \(H^ 1(U,\mu_ 2)\to{_ 2B}(U)\). Particularly, if \(H^ 0(S,G_ m)=k^*\) then \(B(V/U)@>\sim>>\mathbb{Z}/2^{(r)}\).
The second part deals with the image of \(\pi^*\) in the following hypothesis: \(X\) and \(S\) are nonsingular surfaces, \(\pi:X\to S\) is a nontrivial double cover which ramifies only over the nonsingular irreducible curve \(Z\subset S\), \(U=S\backslash Z\), \(V=\pi^{-1}(U)\), \(U\) and \(V\) are affine and \(\text{Pic} X=\pi^*\text{Pic} S=\mathbb{Z}\). Then there is an exact sequence
\[
0\to B(U)\to B(V)^ G\to\mathbb{Z}/2\to 0,
\]
\(G=\mathbb{Z}/2\) being the Galois group of \(\pi:V\to U\). Azumaya algebra; ramified double covers of surfaces; Brauer group; Galois group Ford, T. J.: The Brauer group and ramified double covers of surfaces. Comm. algebra 20, 3793-3803 (1992) Brauer groups of schemes, Surfaces and higher-dimensional varieties, Picard groups, Étale and other Grothendieck topologies and (co)homologies, Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.), Coverings in algebraic geometry The Brauer group and ramified double covers 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 In the early 70's, Shafarevich conjectured that the universal cover \(\widetilde{X}\) of a smooth projective variety \(X\) over \(\mathbb{C}\) is holomorphically convex, that is, there is a proper holomorphic map \(f: \widetilde{X} \to Y\), such that \(f_* \mathcal{O}_{\widetilde{X}} \cong \mathcal{O}_Y\) and \(Y\) is Stein (Stein manifolds loosely speaking, but not completely precisely, correspond to affine varieties in the complex analytic setting). In this paper the authors prove that if \(X\) is a normal projective variety over \(\mathbb{C}\), then assuming the abundance conjecture the following are equivalent
(a) the universal cover \(\widetilde{X}\) of \(X\) is biholomorphic to a quasi-projective variety,
(b) \(\widetilde{X}\) is biholomorphic to \(F \times \mathbb{C}^m\), where \(F\) is a projective simply connected variety and
(c) there is a finite étale cover \(X' \to X\) that is a fiber bundle over an Abelian variety with simply connected fibers.
This statement can be viewed both as a (conditional) positive answer to Shafarevich's conjecture in a special case, and also as a characterization of when \(\widetilde{X}\) is biholomorphic to a quasi-projective variety. Further note, that in fact only a special case of the abundance conjecture is assumed: if \(X\) is a smooth projective variety over \(\mathbb{C}\) such that \(K_X\) is nef, then \(K_X\) is semi-ample.
The proof consists of two main steps. First, it is shown that the fundmental group of \(X\) is almost abelian, that is, it contains a subgroup of finite index, which is abelian. To this end the authors consider a minimal dimensional quasi-projective variety \(Y\) that has an infinite étale quasi-projective \(\Gamma\)-Galois cover \(\widetilde{Y}\) with \(\Gamma\) not almost abelian and study the effect of running the Minimal Model Program on \(Y\). The main idea is that this yields smaller dimensional examples as above, which is a contradiction. The fundamental group of \(X\) being almost abelian yields a finite Galois étale cover \(X' \to X\) with the fundamental group of \(X'\) being free, abelian. The second, more straightforward, part deals with showing that the Albanese morphism of \(X'\) is a locally trivial holomorphic fiber bundle. The assumption on the abundance conjecture is only used in the first part. universal cover; fundamental group; quasi-projective variety; Shafarevich conjecture Topological properties in algebraic geometry, Transcendental methods, Hodge theory (algebro-geometric aspects), Coverings in algebraic geometry, Minimal model program (Mori theory, extremal rays), Homotopy theory and fundamental groups in algebraic geometry Algebraic varieties with quasi-projective universal cover | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems It is known that the combinatorial data of a plane curve (as the number of its irreducible components, the degree and the types of singularities of each irreducible component and the intersection data of the irreducible components) do not determine the topology of the plane curve. A \(k\)-plet \((C_1,\ldots, C_k)\) of plane curves is a \textit{Zariski \(k\)-plet} if the curves have the same combinatorial data, but there exist no homeomorphisms \(h_{ij}:\mathbb{P}^2\to\mathbb{P}^2\) satisfying \(h_{ij}(C_j) = C_i\) for any \(i\neq j\). For \(k=2\), a Zariski 2-plet is called a \textit{Zariski pair}.
A Zariski pair \((C_1, C_2)\) such that the fundamental groups of the complements of \(C_1\) and \(C_2\) are isomorphic is called a \textit{\(\pi_1\)-equivalent Zariski pair}.
Let \(n\) and \(m\) be positive integers such that \(b:=mn\geq 3\). According to \textit{I. Shimada} [Vietnam J. Math. 31, No. 2, 193--205 (2003; Zbl 1069.14036)], a projective plane curve \(R \subset \mathbb P^2\) is said to be of \textit{type \((b,m)\)} if \(R\) consists of two irreducible non-singular components \(B\) and \(E\) of degree \(b\) and \(3\) respectively, such that the set-theoretical intersection of \(B\) and \(E\) consists of \(3n\) points, and \(B\) and \(E\) intersect with multiplicity \(m\) at each intersection point. Let \(\mathcal{F}_{b,m}\subset \mathbb{P} H^0(\mathbb P^2,\mathcal{O}(b+3))\) be the family of all curves of type \((b,m)\).
In the paper under review the author defines the \textit{splitting number} of an irreducible subvariety \(C\not \subset B\) in a smooth variety \(Y\) for a finite Galois \(G\)-cover \(\phi: X\to Y\) branched at the reduced divisor \(B\) as the number of irreducible components of \(\phi^* C\). This number is invariant under homeomorphisms from the ambient space to itself which do not interchange the subvariety and any component of the branch locus and keep the Galois cover.
Using these ingredients the author achieves the following statement. Let \(b\geq 4\) be an integer, \(m\) be a divisor of \(b\), and \(R_1\) and \(R_2\) be plane curves of type \((b,m)\). Then \((R_1,R_2)\) is a Zariski pair if and only if \(R_1\) and \(R_2\) are in distinct connected components of \(\mathcal{F}_{b,m}\).
As consequence of this theorem and Theorem 1.2 in Shirada's paper, we obtain: for any integer \(k\geq 2\), there exists a \(\pi_1\)-equivalent Zariski \(k-\)plet. Zariski pair; \(\pi_1\)-equivalent Zariski \(k\)-plet; Galois cover; splitting curve Shirane, T., A note on splitting numbers for Galois covers and \(\pi_1\)-equivalent Zariski \(k\)-plets, Proc. Amer. Math. Soc., 145, 3, 1009-1017, (2017) Coverings in algebraic geometry, Topological properties in algebraic geometry, Coverings of curves, fundamental group, Plane and space curves, Low-dimensional topology of special (e.g., branched) coverings A note on splitting numbers for Galois covers and \(\pi_1\)-equivalent Zariski \(k\)-plets | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We explain how to compute the equations of the abelian coverings of any curve defined over a finite field. Then we describe an algorithm which computes curves with many rational points with respect to their genus. The implementation of the algorithm provides seven new records over \(\mathbb F_2\). explicit class field theory; Kummer theory; Witt vectors; curves with many points; equations of abelian coverings Virgile Ducet, Claus Fieker, Computing equations of curves with many points, 2012, preprint. Curves over finite and local fields, Coverings in algebraic geometry, Computer solution of Diophantine equations Computing equations of curves with many points | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems It is a classical result that the invariants of a minimal surface of general type satisfy \(K^2\geq 2p_g-4\) and that surfaces satisfying \(K^2=2p_g-4\) are mapped by the canonical map \(2-\)to\(-1\) onto a surface of minimal degree in \(\mathbb{P}^{p_g-1}\).
Analogously, for \(n\geq 3\) if \(X\) is a normal \(n-\)dimensional variety of general type with \(K_X\) nef, then \(K_X^n\geq 2(p_g(X)-n)\), provided that the canonical map of \(X\) is generically finite. Moreover, if equality holds, then the canonical map of \(X\) is \(2-\)to\(-1\) onto a variety of minimal degree in \(\mathbb{P}^{p_g-1}\).
The author studies the above situation in the following cases:
a) \(K_X^n=2(p_g(X)-n)=2,4\), \(n\geq 3\); b) \(n=3\) and \(K^3_X=2p_g(X)-6\).
In case a) she proves that \(|K_X|\) is free and the canonical map of \(X\) is a double cover of \(\mathbb{P}^n\), when \(K_X^n=2\), or of a quadric, when \(K_X^n=4\).
In case b), the canonical map of \(X\) is of degree 2 onto a variety \(W\) of minimal degree in \(\mathbb{P}^{p_g-1}\). When \(W\) is the cone over the Veronese surface, there exists a \(2-\)dimensional family of examples (\(p_g(X)=7\), in this case). In general, for every value of \(k\geq 3\) the author constructs a family of examples with \(p_g(X)=k+3\) and \(W\) a rational scroll. canonical map; threefold of general type; variety of general type; double cover \(3\)-folds, Coverings in algebraic geometry On threefolds with \(K^3=2p_q-6\) | 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 We discuss the irregularity of cyclic coverings of the projective plane. We prove that if the degree \(n\) of the covering is a power of a prime number, then the irregularity is less than or equal to \((n - 1) (r - 1)/2\), where \(r\) is the number of the irreducible components of the branch curve. This is a generalization of an old result of \textit{O. Zariski} [Proc. Natl. Acad. Sci. USA 15, 494-501 (1929)] to the case in which the branch curve is reducible. In the proof, the Alexander polynomial of the branch curve plays an important role. irregularity; cyclic coverings of the projective plane; irreducible components of the branch curve; Alexander polynomial F. SAKAI, On the irregularity of cyclic coverings of the projective plane, Classification o Algebraic Vaeties, Contemp. Math., 162, Amer. Math. Soc, 1994, 359-369. Coverings in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Milnor fibration; relations with knot theory, Hypersurfaces and algebraic geometry On the irregularity of cyclic coverings of the 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 X and Y be topological spaces such that Y is obtained from X by glueing \(p+1\) distinct points of X. The author shows that \(\pi_ 1(Y)\simeq \pi_ 1(X)*L_ p,\) where * denotes the free product and \(L_ p\) is the free group on p generators. - Moreover, if X and Y are algebraic varieties, the author shows that \(\pi_ 1^{alg}(Y)\simeq(\pi_ 1^{alg}(X)*L_ p)^ 1,\) where 1 denotes the completion with respect to a topology which can be canonically defined. algebraic fundamental group; topological fundamental group Homotopy theory and fundamental groups in algebraic geometry, Coverings in algebraic geometry, Homotopy groups, general; sets of homotopy classes, Topological properties in algebraic geometry Glueing of closed points and algebraic and topological 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 \(V\) be a nonsingular projective algebraic variety of dimension \( n\). Suppose there exists a very ample divisor \(D\) such that \(D^n = 6\) and \(\dim H^0 (V, \mathcal O(D))= n+ 3\). Then, \((V, D)\) defines a \(D_6\)-Galois embedding if and only if it is a Galois closure variety of a smooth cubic in \({\mathbb P^{n+1}}\) with respect to a suitable projection center such that the pull back of hyperplane of \({\mathbb P^n}\) is linearly equivalent to \(D\). Galois embedding; Galois closure variety; sextic variety Arithmetic ground fields for surfaces or higher-dimensional varieties, Hypersurfaces and algebraic geometry, Coverings in algebraic geometry Sextic variety as Galois closure variety of smooth cubic | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 compute the fundamental group of an irreducible rational quintic curve applying Cremona transformations and covering methods. We find a non-abelian group which is also the fundamental group of a \(\mathbb{Q}\)-homology sphere of dimension 3''. fundamental group; projective plane curves Artal-Bartolo E.: A curve of degree five with non-abelian fundamental group. Topol. Appl. 83, 13--29 (1997) Fundamental group, presentations, free differential calculus, Rational and birational maps, Coverings in algebraic geometry, Singularities of curves, local rings A curve of degree five with non-abelian 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 As pointed out to us by Rolf Farnsteiner, the results presented in our paper [ibid. 127, No. 2, 379-420 (2005; Zbl 1072.20009)] require a modified definition of ``Abelian \(p\)-point''. With this modified definition (functionally equivalent to one which we implicitly use), all of the results of our paper become valid. We make explicit this modified definition as well as those arguments which require this new definition and the modification of one proof (of Theorem 4.8) which is required. finite group schemes; cohomology rings; cohomological support varieties; projectivizations; elementary Abelian \(p\)-groups Friedlander, E., Pevtsova, J.: Erratum: Representation-theoretic support spaces for finite group schemes. Am. J. Math. 128, 1067--1068 (2006) Modular representations and characters, Group schemes, Representation theory for linear algebraic groups, Cohomology theory for linear algebraic groups, Cohomology of groups Erratum to ``Representation-theoretic support spaces for finite group schemes'' | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For part I see ibid. 1, No. 1, 137--180 (1994; Zbl 0841.14017).]
A log del Pezzo surface is a normal projective surface \(S/ \mathbb C\) with at worst quotient singularities such that the anticanonical divisor \(-K_S\) is ample. In this second part of a 2-part paper we complete the proof of the following result.
Theorem. The fundamental group of \(S\), \(\text{Sing}\,S\) is finite. (\(\text{Sing}\,S\) denotes the singular locus of \(S\)).
Let \(\widetilde S\) be the minimal resolution of singularities of \(S\). Then \(\widetilde S\) is a smooth rational surface. In part I of this paper we proved the existence of a ``minimal'' exceptional curve of the 1st kind \(C\) on \(\widetilde S\). The proof of the theorem is completed in this part II by making a detailed analysis of how \(C\) meets the exceptional divisor \(D\) for the map \(\widetilde S \to S\) and the nature of \(D\). In this paper, \(C\) meets exactly two irreducible components of \(D\), transversally and once.
This theorem has also been proved by \textit{A. Fujiki}, \textit{R. Kobayashi} and \textit{S. Lu} by differential geometric methods [cf. Saitama Math. J. 11, 15--20 (1993; Zbl 0798.14009)]. quotient singularities; log del Pezzo surface; anticanonical divisor; fundamental group; resolution of singularities Gurjar, R. V.; Zhang, D.-Q., \({\pi}\)1 of smooth points of a log del Pezzo surface is finite. II, J. Math. Sci. Univ. Tokyo, 2, 165-196, (1995) Singularities of surfaces or higher-dimensional varieties, Homotopy theory and fundamental groups in algebraic geometry, Coverings in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Rational and ruled surfaces \(\pi_ 1\) of smooth points of a log del Pezzo surface is finite. 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 Consider a wildly ramified \(G\)-Galois cover of curves \(\varphi:Y \to\mathbb{P}^1_k\) branched at only one point over an algebraically closed field \(k\) of characteristic \(p\). For any \(p\)-pure group \(G\) whose Sylow \(p\)-subgroups have order \(p\), the author shows the existence of such a cover with small conductor. The proof uses an analysis of the semi-stable reduction of families of covers.
For part II of this paper see: \textit{R. J. Pries}, ibid. 485-487 (2002; the following review Zbl 1030.14013). Galois cover of curves; characteristic \(p\); families of covers Pries R. (2002). Conductors of wildly ramified covers, i. C. R. Acad. Sci. Paris Sér. I Math. 335(1): 481--484 Coverings of curves, fundamental group, Local ground fields in algebraic geometry, Arithmetic ground fields for curves, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Coverings in algebraic geometry Conductors of wildly ramified covers. I | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let (X,L) be a smooth polarized threefold such that L is spanned by its global sections. The aim of this paper is to classify all (X,L)'s such that \(c_ 1(L)^ 3=3\), sectional genus \(3\) and \(h^ 0(X,L)=4\) (or equivalently, all triple covers of \({\mathbb{P}}^ 3\) of sectional genus \( 3).\) The methods used rely heavily of the adjunction theory. sectional genus; polarized varieties; spanned line bundles; threefold; triple covers; adjunction \(3\)-folds, Projective techniques in algebraic geometry, Coverings in algebraic geometry Triple solids with sectional genus three | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(G\) be a finite group. Here the author considers Galois \(G\)-covers of complex normal projective varieties. He says that a \(G\)-cover \(\tau : X \to M\) is versal if for any \(G\)-cover \(\pi : Y \to Z\) there is a Zariski open subset \(U\) of \(Z\) and a morphism \(f: U \to M\) such that \(\pi ^{-1}(U)\) is birational to \(U\times _U\;X\) over \(U\). His aim is to find tractable versal \(G\)-covers via a bottom-to-top construction. He considers the case \(G = S_4\) and uses as a base some elliptic surfaces. versality; versal \(G\)-cover; Galois cover of complex normal projective varieties Tokunaga, H; Brasselet, J-P (ed.); Suwa, T (ed.), \(2\)-dimensional versal \(S_4\)-covers and rational elliptic surfaces, No. 10, 307-322, (2005), Paris Coverings in algebraic geometry, Elliptic surfaces, elliptic or Calabi-Yau fibrations 2-dimensional versal \(S^4\)-covers and 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 From the authors' abstract: We give an infinite dimensional generalized Weierstrass representation for space-like constant mean curvature (CMC) surfaces in Minkowski 3-space \(\mathbb R^{2,1}\). The formulation is analogous to that given by Dorfmeister, Pedit and Wu for CMC surfaces in Euclidean space, replacing the group \(SU_{2}\) with \(SU_{1,1}\). The non-compactness of the latter group, however, means that the Iwasawa decomposition of the loop group, used to construct the surfaces, is not global. We prove that it is defined on an open dense subset, after doubling the size of the real form \(SU_{1,1}\), and prove several results concerning the behavior of the surface as the boundary of this open set is encountered. We then use the generalized Weierstrass representation to create and classify new examples of spacelike CMC surfaces in \(\mathbb R^{2,1}\). In particular, we classify surfaces of revolution and surfaces with screw motion symmetry, as well as studying another class of surfaces for which the metric is rotationally invariant. differential geometry; surface theory; loop groups; integrable systems Brander, D; Rossman, W; Schmitt, N, Holomorphic representation of constant Mean curvature surfaces in Minkowski space: consequences of non-compactness in loop group methods, Adv. Math., 223, 949-986, (2010) Differential geometry of immersions (minimal, prescribed curvature, tight, etc.), Coverings in algebraic geometry, Minimal surfaces in differential geometry, surfaces with prescribed mean curvature, Non-Euclidean differential geometry Holomorphic representation of constant mean curvature surfaces in Minkowski space: consequences of non-compactness in loop group methods | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author reformulates the main result contained in his previous paper [Invent. Math. 69, 103-108 (1982; Zbl 0505.14017)] in order to correct an error in the description of the relations defining the fundamental group of the complement of a real arrangement of lines in \({\mathbb{C}}{\mathbb{P}}^ 2\). fundamental group of the complement of a union of complex; hyperplanes; fundamental group of the complement of a real; arrangement of lines in projective 2-space R. Randell, ``The Fundamental Group of the Complement of a Union of Complex Hyperplanes.'' Invent. Math. 80 (1985): 467--468. Homotopy theory and fundamental groups in algebraic geometry, Coverings in algebraic geometry, Homotopy groups, general; sets of homotopy classes, Homological methods in Lie (super)algebras The fundamental group of the complement of a union of complex hyperplanes: Correction | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Hurwitz numbers count branched covers of the Riemann sphere with specified ramification data, or equivalently, transitive permutation factorizations in the symmetric group with specified cycle types. Monotone Hurwitz numbers count a restricted subset of these branched covers related to the expansion of complete symmetric functions in the Jucys-Murphy elements, and have arisen in recent work on the the asymptotic expansion of the Harish-Chandra-Itzykson-Zuber integral.
In this paper we begin a detailed study of monotone Hurwitz numbers. We prove two results that are reminiscent of those for classical Hurwitz numbers. The first is the monotone join-cut equation, a partial differential equation with initial conditions that characterizes the generating function for monotone Hurwitz numbers in arbitrary genus. The second is our main result, in which we give an explicit formula for monotone Hurwitz numbers in genus zero. Hurwitz numbers; matrix models; enumerative geometry Goulden, I. P. and Guay-Paquet, Mathieu and Novak, Jonathan, Monotone {H}urwitz numbers in genus zero, Canadian Journal of Mathematics. Journal Canadien de Mathématiques, 65, 5, 1020-1042, (2013) Exact enumeration problems, generating functions, Coverings in algebraic geometry, Random matrices (algebraic aspects) Monotone Hurwitz numbers in genus 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 Similarly to the case of surfaces, it is known (see Theorem 1.1 and references therein) that for \(X\) a nonsingular minimal complex projective \(3\)-fold of general type, if the canonical map \(\phi:X \dashrightarrow \mathbb{P}^{p_g(X)-1}\) is generically finite then, either \(p_g(\phi(X))=0\), or \(p_g(X)=p_g(\phi(X))\) and the canonical map of \(\phi(X)\) is of degree one (\(p_g\) stands for the geometric genus). Naturally, the question of bounding the degree of the possible canonical maps appears. The main result of the paper under review (see Theorem 1.2) is that if \(M\) is a nonsingular model of \(\phi(X)\) then: (1) if \(p_g(\phi(X))=0\) and \(K_M\) is pseudoeffective then \(\deg(\phi)\leq 108\); (2) if \(p_g(X)=p_g(\phi(X))\) then \(\deg(\phi) \leq 86\) and moreover if \(p_g(X) \geq 111\) then \(\deg(\phi) \leq 32\). projective threefold; general type; canonical map; canonical degrees \(3\)-folds, Coverings in algebraic geometry On the canonical maps of nonsingular threefolds of general type | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper is an introduction to theory and application of \(D_{2n}\)-covers. We start with some elementary examples of Galois covers and survey how we treat \(D_{2n}\)-covers and apply them through examples. Tokunaga, H.: Introduction to branched Galois covers. Vietnam J. Math. \textbf{33}, 97-107 (2005) (Special Issue) Coverings in algebraic geometry, Ramification problems in algebraic geometry, Coverings of curves, fundamental group Introduction to branched 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 Given a normal complete variety \(Y\), distinct irreducible effective Weil divisors \(D_1,\dots, D_n\) of \(Y\) and positive integers \(_1,\dots, d_n\), we spell out the conditions for the existence of an Abelian cover \(X\to Y\) branched with order \(d_i\) on \(D_i\) for \(i= 1,\dots,n\).
As an application, we prove that a Galois cover of a normal complete toric variety branched on the torus-invariant divisors is itself a toric variety. Alexeev, V; Pardini, R, On the existence of ramified abelian covers, Rend. Semin. Mat. Univ. Politec. Torino, 71, 307-315, (2013) Coverings in algebraic geometry, Group actions on varieties or schemes (quotients) On the existence of ramified Abelian covers | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper is a version of the lecture I gave at the conference on ``Representation Theory, Homological Algebra and Free Resolutions'' at MSRI in February
2013, expanded to include proofs. My goals in this lecture were to explain to an audience of commutative algebraists why a finite group representation
theorist might be interested in zero dimensional complete intersections, and to give a version of the Orlov correspondence in this context that is well suited
to computation. In the context of modular representation theory, this gives an equivalence between the derived category of an elementary abelian \(p\)-group of rank \(r\), and the category of (graded) reduced matrix factorisations of the polynomial \(y_1 X_1^p+\dots+y_r X_r^p\). Finally, I explain the relevance to some recent joint work with Julia Pevtsova on realisation of vector bundles on projective space from modular representations of constant Jordan type. Derived categories and commutative rings, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Modular representations and characters Modules for elementary abelian groups and hypersurface 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 Let \(L_k\) be the affine line over an algebraically closed field \(k\), with char \(k = p \neq 0\). In a previous paper it has been conjectured by the first named author [Am. J. Math. 79, 825-856 (1957; Zbl 0087.036)] that the algebraic fundamental group \(\pi_A (L_k)\) (i.e. the set of finite Galois groups of unramified coverings of \(L_k)\) is the set \(Q(p)\) of all quasi \(p\)-groups (i.e. finite groups which are generated by all their \(p\)-Sylow subgroups), and it was shown that \(\pi_A (L_k) \subseteq Q(p)\). -- In particular, the conjecture would imply that \(\pi_A (L_k) \supseteq A_n\), the alternating group, for all \(n \geq p\) when \(p > 2\), and for all \(n \neq 3,4\) when \(p = 2\).
Later this fact has been proved for \(p > 2\); this paper deals with the case \(p = 2\) and \(n \neq 3,4,6,7\). Explicit equations of the required unramified coverings of \(L_k\), are given, and they have equation \(Y^n - XY^t + 1 = 0\), with \(n = q + t\), \(q\) is a positive power of \(p\) and \(t\) is a positive integer nondivisible by \(p\). -- The proof is based mainly on a particular form of Jacobson's criterion which works in every characteristic. Several variations of this criterion are given which can be of interest not only for this problem. algebraic fundamental group; Galois groups of unramified coverings S. S. Abhyankar, J. Ou and A. SathayeAlternating group coverings of the affine line for characteristic two, Discrete Mathematics133 (1994), 25--46. Coverings of curves, fundamental group, Coverings in algebraic geometry Alternating group coverings of the affine line for characteristic two | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a complete nonsingular irreducible algebraic curve of genus \(g \) defined over an algebraically closed field \(k\) of characteristic \(p > 0\). Let \(\pi_1 (X)\) be its algebraic fundamental group, and \(\Gamma_g\) the topological fundamental group of (any) compact Riemann surface of genus \(g\). It is well known since Grothendieck, that a finite group \(G\) of order prime to \(p\) can be realized as the Galois group of an unramified cover of \(X\) if and only if it is a quotient of \(\Gamma_g\). Other information on the structure of \(\pi_1 (X)\) comes from the Hasse-Witt invariant of \(X\), i.e. the \(\mathbb{F}_p\)-dimension of the \(p\)-torsion subgroup of the Jacobian variety of \(X\).
In this paper the author counts the number of unramified Galois coverings of \(X\) whose Galois group is isomorphic to an extension of a group \(G\) of order prime to \(p\) by a finite group \(H\), which is an irreducible \(\mathbb{F}_p [G]\)-module. This counting depends on some invariants attached to a Galois cover \(Y \to X\) of group \(G\), called generalized Hasse-Witt invariants, arising in the canonical decomposition of the \(p\)-torsion space of the Jacobian variety \(J_Y\) of \(Y\). -- Some particular cases had already been treated by \textit{S. Nakajima} [in: Galois groups and their representations, Proc. Symp., Nagoya 1981, Adv. Stud. Pure Math. 2, 69-88 (1983; Zbl 0529.14016)] and \textit{H. Katsurada} [J. Math. Soc. Japan 31, 101-125 (1979; Zbl 0401.14004)], whose results are generalized in the present paper. algebraic curve; algebraic fundamental group; topological fundamental group; Hasse-Witt invariant; Jacobian variety; number of unramified Galois coverings Pacheco, A., Unramified Galois coverings of algebraic curves, J. number theory, 53, 211-228, (1995) Coverings of curves, fundamental group, Inverse Galois theory, Coverings in algebraic geometry Unramified Galois coverings of 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 prove that the only separable commutative ring-objects in the stable module category of a finite cyclic \(p\)-group \(G\) are the ones corresponding to subgroups of \(G\). We also describe the tensor-closure of the Kelly radical of the module category and of the stable module category of any finite group. separable; étale; ring-object; stable category Modular representations and characters, Étale and other Grothendieck topologies and (co)homologies, Derived categories, triangulated categories Separable commutative rings in the stable module category of cyclic 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 \(K\) be a complete discrete valued field of unequal characteristic \((0,p)\). The aim of this paper is to describe the semi-stable models for covers \(\mathbb{P}_K^1\to \mathbb{P}_K^1\) of degree \(p\), unramified outside \(r\leq p\) points and totally ramified above one of them, under the assumption that the ramification locus has a particular reduction type (which always occurs if \(r\leq 4\)). We are principally concerned with the minimal semi-stable models which separate the ramified fibers. L. Zapponi, ''Specialization of polynomial covers of prime degree,'' Pacific J. Math., 214, No. 1, 161--183 (2004). Coverings of curves, fundamental group, Coverings in algebraic geometry Specialization of polynomial covers of prime 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 In this note, we construct three new infinite families of surfaces of general type with canonical map of degree 2 onto a surface of general type. For one of these families the canonical system has base points. surfaces of general type; canonical maps; abelian covers Surfaces of general type, Coverings in algebraic geometry Some infinite sequences of canonical covers of degree 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 For a smooth finite cyclic covering over a projective space of dimension greater than one, we show that its group of automorphisms faithfully acts on its cohomology except for a few cases. In characteristic zero, we study the equivariant deformation theory and groups of automorphisms for complex cyclic coverings. The proof uses the decomposition of the sheaf of differential forms due to Esnault and Viehweg. In positive characteristic, a lifting criterion of automorphisms reduce the faithfulness problem to characteristic zero. To apply this criterion, we prove the degeneration of the Hodge-de Rham spectral sequences for a family of smooth finite cyclic coverings, and the infinitesimal Torelli theorem for finite cyclic coverings defined over an arbitrary field. finite cyclic covering; automorphism groups; cohomology Automorphisms of surfaces and higher-dimensional varieties, Coverings in algebraic geometry Remarks on automorphism and cohomology of finite cyclic coverings of projective 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 Let \(X\) be an integral projective variety, \(G\) a finite group acting effectively on \(X\). The author investigates if for every \(L\in\text{Pic}(X)\) and every \(t>0\) the restriction map \(\rho_{L,GS}: H^0(X,L) \to H^0(GS,L| GS)\) has maximal rank for a general \(S\subset X\) with \(\text{card} (S)=t\). Usually, the answer is negative, but in some special cases it is affirmative. Most of the results are for the case in which \(X\) is a smooth curve. finite group action; covering of curves Coverings in algebraic geometry, Automorphisms of curves, Automorphisms of surfaces and higher-dimensional varieties, Group actions on varieties or schemes (quotients) Automorphisms of projective varieties and postulation of zero-dimensional schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper contains some examples of surfaces of general type such that the image of the canonical map is a surface of general type and the degree of the canonical map is greater \(than\quad 2.\) The existence of such surfaces had been doubtful for several years. All the examples are abelian covers of the projective plane. surfaces of general type; image of the canonical map; abelian covers of the projective plane Pardini, R, Canonical images of surfaces, J. Reine Angew. Math., 417, 215-219, (1991) Surfaces of general type, Coverings in algebraic geometry Canonical images 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 Let \(X\) be a compact algebraic variety with a fixed ample line bundle \(H\), and \(x \in X\) a base point. The object of this paper is an investigation of semi-simple representations of the fundamental group \(\rho \in \Hom (\pi_1 (X,x), G)^{ss}\), where \(G \subset SL_r (\mathbb{C})\) is a semisimple Lie-group. The quotient space \(\Hom (\pi_1 (X,x), G)^{ss}/G\) is called the Betti space. A representation \(\rho\) is called rigid if \(\rho\) is an isolated point in the Betti space; \(\rho\) is called étale-reducible, if there exists a finite étale covering \(\varphi : Z \to X\), so that the pull back of \(\rho\) to the universal covering of \(X\) splits as a nontrivial direct sum of simple- representations; \(\rho\) is called fibration-reducible, if there is a surjective morphism \(f : X \to B\) with connected fibres, so that the restriction representation of \(\rho\) to a generic fibre splits as a nontrivial direct sum of simple representations, and \(\rho\) is called unitary fibration-reducible, if each component in the direct sum is a unitary representation. -- The following result are obtained:
Theorem 1: Let \(G = SL_r (\mathbb{C})\) and the Higgs bundle corresponding to \(\rho\) is not nilpotent. Then:
(1) either \(\rho\) is étale-reducible, if \(c_1(I) H^{\dim H - 1} = 0\), where \(I\) is the image sheaf corresponding to the Higgs structure; or
(2) there exists a finite étale covering \(\varphi : Z \to X\) so that the pull back \(\varphi^* \rho\) is unitary fibration-reducible, and the base space is an algebraic curve, if \(c_1 (I) H^{\dim H - 1} > 0\), and rank \(I = 1\). If \(r = 2\), then \(\rho\) in (2) is unitary fibration- reducible.
Theorem 2: If \(G = SL_2 (\mathbb{C})\), then there are only three possibilities: either \(\rho\) is rigid, or \(\rho\) is étale-rigid, in fact, a double covering, or \(\rho\) is unitary fibration-reducible and the base space is an algebraic curve.
Theorem 3: If \(G = SL_3 (\mathbb{C})\), there are only three possibilities: either \(\rho\) is rigid, or \(\rho\) is étale-reducible, or \(\rho\) is fibration-reducible, and the base space is an algebraic curve or a surface. representations of the fundamental group; Betti space Zuo, K.: Some structure theorems for semi-simple representations of \({\pi}1\) of algebraic manifolds. Math. ann. 295, 365-382 (1993) Coverings in algebraic geometry Some structure theorems for semi-simple representations of \(\pi_ 1\) of algebraic manifolds | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This is a fabulous book, linking differential geometry (smooth manifolds) and algebraic geometry in a direct way. To a smooth manifold \(X\), there is an associated structure on a set \(C^\infty\) called a \(C^\infty\)-ring. These rings satisfies specific properties, making up a category of \(C^\infty\)-rings. This book builds algebraic geometry over \(C^\infty\)-rings, replacing commutative rings with \(C^\infty\)-rings. The basic concepts of algebraic geometry are covered.
The basic idea is the following: Given a smooth manifold \(X\) with its ring of smooth functions \(C^\infty(X).\) This ring has more than a pure algebraic structure. Let \(f:\mathbb R^n\rightarrow\mathbb R\) be a smooth function. Then \(\Phi_f:C^\infty(X)\rightarrow C^\infty(X)\) given by \((\Phi_f(c_1,\dots,c_n))(x)=(f(c_1(x),\dots,c_n(x))\) satisfy several identities, giving rise to the general definitions:
I. A \(C^\infty\)-ring is a set \(\mathfrak C\) together with operations \(\Phi_f:\mathfrak C^n\rightarrow\mathfrak C\) for all \(n\geq 0\) and smooth functions \(f:\mathbb R^n\rightarrow\mathbb R\) such that: Suppose \(m,n\geq 0,\) and \(f_i:\mathbb R^n\rightarrow R\) for \(1,\dots,m\) and \(g:\mathbb R^n\rightarrow\mathbb R\) are smooth. Define smooth \(h:\mathbb R^n\rightarrow\mathbb R\) by \(h(x_1,\dots,x_n)=g(f_1(x_1,\dots,x_n),\dots,f_m(x_1,\dots,x_n))\) for all \((x_1,\dots,x_n)\in\mathbb R^n.\) Then for all \((x_1,\dots,x_n)\in\mathfrak C^n\) we have \(\Phi_h(c_1,\dots,c_n)=\Phi_g(\Phi_{f_1}(c_1,\dots,c_n),\dots,\Phi_{f_m}(c_1,\dots,c_n)).\)
It is also required that for \(1\leq j\leq n,\) defining \(\pi_j:\mathbb R^n\rightarrow\mathbb R\) by \(\pi_j:(x_1,\dots,x_n)\mapsto x_j,\) we have \(\Phi_{\pi_j}(c_1,\dots,c_n)\in\mathfrak C^n.\) The morphisms of \(C^\infty\)-rings are the maps \(\mathfrak C\rightarrow\mathfrak D\) commuting naturally with \(\Phi_f\).
II. Let \(\mathbf{Man}\) be the category of manifolds, \(\mathbf{Euc}\) the full subcategory of Euclidean spaces \(\mathbb R^n.\) A category-teoretic \(C^\infty\)-ring is a product preserving functor \(F:\mathbf{Euc}\rightarrow\mathbf{Sets}.\) Also, it is required that \(F\) preserves the empty product.
Now, with the basis in the category \(\mathbf{C^\infty Sch}\) of \(C^\infty\)-rings, this book considers algebraic geometry over \(C^\infty\)-rings. It includes the study of \(C^\infty\)-schemes and Deligne-Mumford \(C^\infty\)-stacks. These are geometric spaces generalizing manifolds and orbifolds respectively.
The author states the following appearance of \(C^\infty\)-rings: The theory is a part of synthetic differential geometry where one is interested in \(C^\infty\)-schemes as a category \(\mathbf{C^\infty Sch}\) of geometric objects including smooth manifolds and certain infinitesimal objects. Synthetic differential geometry concerns proving theorems about manifolds using synthetic reasoning involving infinitesimals, so they need a model, that is, a category of geometric spaces including manifolds and infinitesimals where the synthetic arguments can be applied. Once there exists one object with the wanted properties, the job of synthetic differential geometry is done. Thus there are no further development of \(C^\infty\)-schemes in synthetic differential geometry.
More recently, \(C^\infty\)-rings and \(C^\infty\)-ringed spaces appeared in the theory of derived differential algebraic geometry, the differential geometric analogue of derived algebraic geometry by \textit{B. Toën} and \textit{G. Vezzosi} [Homotopical algebraic geometry. II: Geometric stacks and applications. Providence, RI: American Mathematical Society (AMS) (2008; Zbl 1145.14003)] and \textit{J. Lurie} [Derived Algebraic Geometry V: Structured spaces, \url{arXiv:0905.0459}]. This theory studies derived smooth manifolds and derived smooth orbifolds. In particular, Lurie sketched the definition of an \(\infty\)-category of derived \(C^\infty\)-schemes, including derived manifolds, and based on this, using \textit{D. I. Spivak}'s definition [Duke Math. J. 153, No. 1, 55--128 (2010; Zbl 1420.57073)] of the \(\infty\)-category of derived manifolds, \text{D. Borisov} and \textit{J. Noel} [``Simplicial approach to derived differential geometry'', \url{arXiv:1112.0033}] and the present author extended this theory. They defined the notion of \(d\)-manifolds which are built using the theory of \(C^\infty\)-schemes with quasicoherent sheaves, forming a 2-category. An enhancement of this is the orbifold versions of the theory, which are built using the theory in this book of Deligne-Mumford \(C^\infty\)-stacks and their quasicoherent sheaves.
Symplectic geometry includes the study of \(J\)-holomorphic curves in symplectic manifolds, which are made into Kuranishi spaces in the framework of Fukaya, Oh, Ohta and Ono [\textit{K. Fukaya} et al., Lagrangian intersection Floer theory. Anomaly and obstruction. I. Providence, RI: American Mathematical Society (AMS); Somerville, MA: International Press (2009; Zbl 1181.53002)]. The author argues that Kuranishi spaces are derived orbifolds, and gives a new definition of a 2-category of Kuranishi spaces \(\mathbf{Kur}\) equivalent to the 2-category of \(d\)-orbifolds \(\mathbf{dOrb}\). Thus derived differential geometry will have important applications in symplectic geometry.
The objective of the book is stated verbatim by the following: ``To set up our theory of d-manifolds and d-orbifolds requires a lot of preliminary work on \(C^\infty\)-schemes and \(C^\infty\)-stacks, and quasicoherent sheaves upon them. That is the purpose of this book''.
The book starts with the (abstract) definition of \(C^\infty\)-rings and the category of such, \(\mathbf{C^\infty Rings}\). This is a clear and self content theory, which includes a localization concept. For further generalizations, the concepts of fair \(C^\infty\)-ring, flat ideal, and pushout, are introduced. Then the \(C^\infty\)-rings \(C^\infty(X)\) of manifolds \(X\) are studied, and they are proved to satisfy the essential properties. On this basis, \(C^\infty\)-ringed spaces implies \(C^\infty\)-schemes via sheaves on topological spaces. This is defined by the correct category theoretic principle using (inverse) limits, imitating (squeezing the juice from) the theory from Hartshorne's classical book on algebraic geometry. There is even a section giving a criterion for affine \(C^\infty\)-schemes. From these schemes, quotients by finite groups are introduced, leading to the later chapters studying orbifolds and their like.
The way to classification of orbits go through modules over \(C^\infty\)-rings and \(C^\infty\)-schemes. Sheaves of modules and their functorial properties are studied and established, in particular the cotangent sheaves of \(C^\infty\)-schemes. This more or less concludes what can be done by glueing on topological spaces.
Solving further classification problems, the \(C^\infty\)-schemes invites to \(C^\infty\)-stacks. Grothendieck topologies, cites and descent are introduced and studied. The concept Deligne-Mumford \(C^\infty\)-stack is given, and it is proved that orbifolds can be defined as a 2-subcategory of such as soon as they are given a characterization.
Sheaves on Deligne-Mumford \(C^\infty\)-stacks is defined and developed to give the final object of study which is orbifold strata of \(C^\infty\)-stacks.
In addition to giving the theory of \(C^\infty\)-rings and their application, this book makes a framework for developing new algebraic theories that includes differential geometry (or analytical principles in general). It represents a big step in the development of algebraic geometry. $C^\infty$-ring; smooth manifold; category-theoretic $C^\infty$-ring; $C^\infty$-scheme; Deligne-Mumford $C^\infty$-stack; synthetic differential geometry; infinitesimal objects; $C^\infty$-ringed spaces; derived differential algebraic geometry; derived smooth manifolds; derived smooth orbifolds; derived $C^\infty$-schemes; $C^\infty$-stacks; 2-category; Kuranishi spaces; $d$-orbifolds; symplectic geometry fair $C^\infty$-ring;flat ideal;pushout of $C^\infty$-ring Research exposition (monographs, survey articles) pertaining to algebraic geometry, Fundamental constructions in algebraic geometry involving higher and derived categories (homotopical algebraic geometry, derived algebraic geometry, etc.), Generalizations (algebraic spaces, stacks), Rings and algebras of continuous, differentiable or analytic functions Algebraic geometry over \(C^\infty \)-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 In this paper the author investigate properties of a new notion of hypercovering introduced by himself in a previous paper [``Is Alexander property étale local?'' (preprint)]: A scheme valued ordered system is a contravariant functor \(F\) from the category of finite strictly ordered sets to the category of schemes.
Given a scheme valued ordered system \(F\), the author defines, for every collection \(\{A_1, \ldots A_n\}\) of finite subsets of \(\mathbb N\), a closed subscheme \(F[A_1,\ldots,A_n]\) of \(F(A_1) \times \ldots \times F(A_n)\). Set theoretically, a geometric point of \(F[A_1,\ldots ,A_n]\) is a point \((x_1,\ldots,x_n) \in F(A_1) \times \ldots \times F(A_n)\), such that any two components are compatible in \(F (A_i \cap A_j)\). -- If for all \(A_i=\{1,\ldots, n\}-\{i\}\), then \(F[A_1,\ldots , A_n]\) is said to be the cosquelton \(\cos k_n(F)\).
A scheme valued ordered system is said to be a hypercovering if the canonical morphism \(F(\{1,\ldots ,n\}) \rightarrow\cos k_n(F)\) belongs to a given Grothendieck topology. The main result of this paper is that, given a hypercovering \(F\) and finite subsets of \(\mathbb N\), \(A_1, \ldots, A_n, B_1, \ldots B_m\), such that for all \(i\) there is a \(j\) with \(B_i \subset A_j\), then there is a natural map \(F[A_1,\ldots, A_n]\rightarrow F[B_1,\ldots B_s]\), and it still belongs to the given Grothendieck topology.
As explained by the author in the last section of the paper, this research should help in investigating the following conjecture:
Let \(F\) be a hypercovering with all \(F(\{1,\ldots,n\})\) Alexander. Then for any bivariant sheaf \(\mathcal F\) on \(F(\emptyset)\), the Čech cohomology \(H^i({\mathcal F},F)\) vanishes for all \(i>0\). hypercovering; cosquelton; scheme valued ordered system; vanishing of Cech cohomology Coverings in algebraic geometry, Vanishing theorems in algebraic geometry On hypercoverings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 proves Abhyankar's conjecture on the fundamental group of affine curves \(U\) over algebraically closed fields of characteristic \(p\). Let \(\pi_{\text{A}} (U)\) denote the set of finite quotients of \(\pi_ 1 (U)\); this is the set of Galois groups of finite unramified Galois covers of \(U\). If \(U\) is a curve of genus \(g\) with \(r>0\) points deleted, and \(G\) is a finite group, then the conjecture states that \(G\) is in \(\pi_{\text{A}} (U)\) if and only if its maximal prime-to-\(p\) quotient is in \(\pi_{\text{A}}\) of a complex curve of genus \(g\) with \(r\) points deleted. The case of \(U = \mathbb{A}^ 1\) was previously shown by \textit{M. Raynaud} [Invent. Math. 116, No. 1-3, 425-462 (1994; Zbl 0798.14013)], using rigid methods. The present paper uses Raynaud's result and methods of formal patching of schemes. It shows moreover that the asserted \(G\)- Galois cover can be chosen so that its smooth completion has the property that only a single, predetermined, branch point is wildly ramified. characteristic \(p\); formal schemes; Galois cover of algebraic curve; Abhyankar's conjecture; fundamental group of affine curves David Harbater, ``Abhyankar's conjecture on Galois groups over curves'', Invent. Math.117 (1994) no. 1, p. 1-25 Coverings of curves, fundamental group, Representations of groups as automorphism groups of algebraic systems, Coverings in algebraic geometry Abhyankar's conjecture on Galois groups over curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The main result of this fundamental article is: Let \(X\) be a compact Kähler manifold with nef tangent bundle \(T_X\). Moreover, let \(\widetilde X\) be a finite étale cover of \(X\) of maximum irregularity \(q = q (\widetilde X) = h^1 (\widetilde X, {\mathcal O}_{\widetilde X})\). Then: \(\pi_1 (\widetilde X) \cong \mathbb{Z}^{2q}\).
The albanese map \(\alpha : \widetilde X \to A (\widetilde X)\) is a smooth fibration over a \(q\)-dimensional torus with nef relative tangent bundle.
The fibres of \(\alpha\) are Fano manifolds with nef tangent bundles.
Here a line bundle \(L\) on a compact complex manifold \(X\) with a fixed hermitian metric \(\omega\) is nef if, for every \(\varepsilon > 0\), there exists a smooth hermitian metric \(h_\varepsilon\) on \(L\) such that the curvature satisfies \(\Theta_{h_\varepsilon} \geq - \varepsilon \omega\). A bundle \(E\) on \(X\) is nef if the line bundle \({\mathcal O}_E (1)\) on \(\mathbb{P} (E)\) is nef. -- Many other interesting and important results are contained in the article. It is proved that:
Let \(E\) be a vector bundle on a compact Kähler manifold \(X\).
If \(E\) and \(E^*\) are nef, then \(E\) admits a filtration whose graded pieces are hermitian flat.
If \(E\) is nef, then \(E\) is numerically semi-positive.
Moreover, algebraic proofs are given for the result:
Any Moisheson manifold with nef tangent bundle is projective.
A compact Kähler \(n\)-fold with \(T_X\) nef and \(c_1 (X)^n > 0\) is Fano.
Further the two following classification results are given:
The smooth non-algebraic compact complex surfaces with nef tangent bundles are:
non-algebraic tori; Kodaira surfaces; Hopf surfaces.
Let \(X\) be a non-algebraic three-dimensional compact Kähler manifold. Then \(T_X\) is nef if and only if \(X\), up to a finite étale cover, is either a torus or of the form \(\mathbb{P} (E)\), where \(E\) and \(E^*\) are nef rank-2 vector bundles over a two-dimensional torus. fundamental group; nef line bundle; Albanese map; Moisheson manifold; three-dimensional compact Kähler manifold Demailly J.-P., Peternell T. and Schneider M., Compact complex manifolds with numerically effective tangent bundles, J. Algebraic Geom. 3 (1994), no. 2, 295-345. \(3\)-folds, Divisors, linear systems, invertible sheaves, Compact complex \(3\)-folds, Homotopy theory and fundamental groups in algebraic geometry, Global differential geometry of Hermitian and Kählerian manifolds, Coverings in algebraic geometry Compact complex manifolds with numerically effective tangent 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 Here we consider the existence (in positive characteristics) of integral non-degenerate curves \(C\subset\mathbb{P}^n\) with one of the Mathieu groups as monodromy group of their general hyperplane section. Here we prove the non existence of such curve \(C\) for \(M_{11}\) and \(M_{23}\) if \(n > 4\) and for \(M_{12}\) and \(M_{23}\) for \(n > 5\). projective curve; inverse problem of Galois theory; monodromy group of the general hyperplane section; positive characteristic; Mathieu group Algebraic functions and function fields in algebraic geometry, Inverse Galois theory, Coverings in algebraic geometry, Finite ground fields in algebraic geometry On the monodromy group of the general hyperplane section of a curve in char.\(p\) and the Mathieu 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 For a finite group scheme, the subadditive functions on finite-dimensional representations are studied. It is shown that the projective variety of the cohomology ring can be recovered from the equivalence classes of subadditive functions. Using Crawley-Boevey's correspondence between subadditive functions and endofinite modules, we obtain an equivalence relation on the set of point modules introduced in our joint work with \textit{S. B. Iyengar} and \textit{J. Pevtsova} [J. Am. Math. Soc. 31, No. 1, 265--302 (2018; Zbl 1486.16011)]. This corresponds to the equivalence relation on \(\pi \)-points introduced by \textit{E. M. Friedlander} and \textit{J. Pevtsova} [Duke Math. J. 139, No. 2, 317--368 (2007; Zbl 1128.20031)]. subadditive function; endofinite module; stable module category; finite group scheme Group schemes, Cohomology of groups, Representations of associative Artinian rings, Modular representations and characters, Cohomology theory for linear algebraic groups The variety of subadditive functions for finite group schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The aim of this paper is to study some properties of linear systems and the locus of linear systems on a complex projective algebraic curve which is a covering of another curve.
In section 1, we prove the irreducibility of the \(W^1_d(X)\) for all \(d\geq g-h+1\) on a curve \(X\) of genus \(g\) which is a double covering of a general curve \(C\) of genus \(h>0\). And this result is sharp. In the proof of theorem 1.1, we use the equivalence of the irreducibility of \(W_d^1(X)\) and the connectivity of \(W_d^1(X)\), if \(W^1_d(X)\) has the positive expected dimension and is non-singular in codimension one [cf. \textit{W. Fulton} and \textit{R. Lazarsfeld}, Acta Math. 146, 271-283 (1981; Zbl 0469.14018)]. We also use the so-called Castelnuvo-Severi inequality for a double covering \(X\) of genus \(g\) over a curve \(C\) of genus \(h\); every base-point-free \(g^1_n\) on \(X\) is a pull-back of a \(g^1_{n/2}\) on \(C\) for any \(n\leq g-2h\) [cf. \textit{R. D. M. Accola}, ``Topics in the theory of Riemann surfaces'', Lect. Notes Math. 1595 (1994; Zbl 0820.30002), chapter 3].
In section 2, we consider a problem of base-point-free pencils of certain degree on a \(k\)-gonal curve as well as on a curve which is a double covering of a genus two curve. In proving the main results of section 2, we use enumerative methods and computatios in \(H^*(C_\alpha, \mathbb{Q})\) of various sub-loci of the symmetric product \(C_\alpha\) of the given curve \(C\). Specifically, we compare the fundamental class of \(C^1_\alpha: =\{D\in C_\alpha: \dim|D|\geq 1\}\) with the class of all irreducible components of \(C^1_\alpha\) whose general elements correspond to pencils on \(C\) with base points. This argument works because the latter components are all induced from the base curve of the covering and \(C^1_\alpha\) has the expected dimension. Throughout, we work over the field of complex numbers. double covering curves; linear systems; base-point-free pencils E. Ballico and C. Keem, Variety of linear systems on double covering curves, J. Pure Appl. Algebra 128 (1998), no. 3, 213--224. Special divisors on curves (gonality, Brill-Noether theory), Divisors, linear systems, invertible sheaves, Coverings in algebraic geometry Variety of linear systems on double covering curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper deals with the computation of the flat sections of the tangent bundle to a complex vector space, carrying a flat connection. This is a general question arising from a work of Givental [\textit{A. Givental}, Equivariant Gromov-Witten invariants, Int. Math. Res. Not. 1996, No. 13, 613-663 (1996; Zbl 0881.55006)], where equivariant Gromov-Witten invariants are used to obtain flat sections of the Dubrovin connection on the tangent bundle to the even cohomology of a quintic hypersurface in the projective space \(\mathbb{P}^3\).
Let \(H\) be a complex vector space with an associative, and commutative product denoted by \(*\). On \(TH\) the Dubrovin connection is defined by: \(\nabla_Y X=dX (Y)+ iY*X\), with connection 1-form \(\omega(Y) (X)=iY*X\). The de Rham cohomology of \(H\), computed with respect to the exterior derivative coupled to \(\nabla\), vanishes and since \(H\) is contractible, the moduli space (under gauge transformations) of flat connections reduces to a point. Thus \(\nabla\) is globally gauge equivalent to the trivial connection \(d\), given by \(d_YX=dX(Y)\), and finding flat sections is equivalent to solving: \(g^{-1}dg= \omega\) for \(g: \mathbb{C}^m \to\text{GL}(m,\mathbb{C})\), \(m\) being the dimension of \(H\). In coordinates such equation reads as a PDE system.
The authors compute the gauge transformation mentioned and the corresponding flat sections, using systematically the Frobenius integrability theorem. They find a basis of the flat sections for the Dubrovin connection for the small quantum product, a deformation in the \(H^2(M;\mathbb{C})\) directions of the cup product on the even cohomology \(H=H^{\text{ev}}(M;\mathbb{C})\), considering \(M\) as a Fano variety. Then, they compute flat sections for \(\mathbb{P}^m\) and in this case the calculation reduces to exponentials of ordinary matrices, whereas, in general, it involves exponentiating an infinite matrix with matrices as entries. They recover all the Gromov-Witten invariants. The paper contains also some comparisons with the result obtained by \textit{R. Pandharipande} [Rational curves on hypersurfaces (after A. Givental), Séminaire Bourbaki, Volume 1997/98. Exposés 835-849. Paris: Soc. Math. de France, Astérisque 252, 307-340, Exp. No. 848 (1998; Zbl 0932.14029)].
Finally, there are some comments for the big quantum product and for the quantum cohomology coupled to gravity in genus zero, which will be treated in a future work. Dubrovin connection; quantum product; flat section Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Quantum groups and related algebraic methods applied to problems in quantum theory, Algebraic properties of function spaces in general topology, Applications of global differential geometry to the sciences, Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills), Gauge theory techniques in quantum 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 The article deals with the structure of fundamental groups of compact Kähler manifolds and can be seen as an extension of the paper [Bull. Soc. Math. Fr. 126, No. 4, 483-506 (1998; Zbl 0942.32020)] by the same author. Let \(\mathcal G\) be a set of isomorphy classes of (finitely generated) groups. \(\mathcal G\) is called stable if it is invariant with respect to group extensions. A normal compact complex space \(X\) is called \(\mathcal G\)-connected, if for any generic \(x,y\in X\) there exists a connected compact analytic subset \(Z\) of \(X\) such that \(x,y\in Z\) and for any irreducible component \(Z_{i}\) of \(Z\) with normalization \(\widetilde{Z_{i}}\) it is true that the image of \(\pi_{1}(\widetilde{Z_{i}})\) in \(\pi_{1}(X)\) is an element of \( \mathcal G\).
As a main result it is shown that if \(X\) is a \(\mathcal G\)-connected Kähler manifold, then \(\pi_{1}\in \mathcal G\). Here the Kähler assumption is essential. But if \(\mathcal G\) is stable and \(X\) is \(\mathcal G\)-connected, then \(\pi_{1}(X)\in \mathcal G\) for every normal compact complex space \(X\) in class \(\mathcal C\). If \(X\) is rationally connected (any two generic points in \(X\) can be connected by a chain of rational curves) and in class \(\mathcal C\), then \(X\) is Moishezon and \(\pi_{1}(X)\) is finite. Finally it is shown that any finite group is the fundamental group of some normal rationally connected projective threefold.
The main tools for the proofs are Hodge theory, mixed Hodge structures and Chow schemes. fundamental groups; Kähler manifolds; \(\mathcal G\)-connected normal complex spaces CAMPANA (F.) . - g-connectedness of compact Kähler manifolds , I, Soumis aux Proceedings of the Conference ``Hirzebruch 70'', Warsaw 1998 , (Szurek, Wisniewski éd., Pragacz), Amer. Math. Soc. Zbl 0965.32021 Compact Kähler manifolds: generalizations, classification, Parametrization (Chow and Hilbert schemes), Coverings in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry, Group actions on varieties or schemes (quotients), Analytic subsets and submanifolds, Transcendental methods of algebraic geometry (complex-analytic aspects), (Equivariant) Chow groups and rings; motives, Mixed Hodge theory of singular varieties (complex-analytic aspects), Transcendental methods, Hodge theory (algebro-geometric aspects), Kähler manifolds \(\mathcal G\)-connectedness of compact Kähler manifolds. I | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We present a number of finiteness results for algebraic tori (and, more generally, for algebraic groups with toric connected component) over two classes of fields: finitely generated fields and function fields of algebraic varieties over fields of type (F), as defined by J.-P. Serre. Group schemes, Arithmetic varieties and schemes; Arakelov theory; heights, Galois cohomology of linear algebraic groups, Coverings in algebraic geometry, Arithmetic aspects of modular and Shimura varieties Finiteness theorems for algebraic tori over function fields | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A covering \(S\) of a smooth projective surface \(\Sigma\) is \(n\)-dihedral when the dihedral group \({\mathcal D}_{2n}\) is the Galois group of the function field extension \(K(S)\) over \(K(\Sigma)\). The author studies divisors \(B\) of \(\Sigma\) which arise as the branch locus of dihedral coverings. In particular, in the paper under review, the case of coverings \(S\to\Sigma\) branched along \(B\) with ramification index \(2\) are considered. The author proves that these divisors \(B\) are also the branch locus of some finite double cover \(Z\to\Sigma\).
Starting with a double cover \(Z\to\Sigma\) branched along \(B\) and assuming that the Néron-Severi group of a canonical resolution of \(Z\) is torsion free, the author provides conditions on the singularities of \(B\) which imply that it is also the branch locus of a dihedral covering (with ramification index \(2\)). For instance, when \(\Sigma={\mathbb P}^2\) and \(B\) is a plane curve with only \(a\) nodes and \(b\) cusps as singularities, then there exists a \(6\)-dihedral covering branched along \(B\) as soon as \(2a+6b>2(\deg(B))^2-6(\deg(B))+6\).
As a main application, branch loci \(B\) of dihedal coverings provide examples of divisors such that the fundamental group of the complement \(\pi_1(\Sigma-B)\) is non-abelian. ramification index; dihedral coverings; Néron-Severi group; canonical resolution; branch locus Tokunaga, H.: Dihedral coverings of algebraic surfaces and their application. Trans. amer. Math. soc. 352, No. 9, 4007-4017 (2000) Coverings in algebraic geometry, Ramification problems in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects) Dihedral coverings of algebraic surfaces and their application | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We prove the following theorem:
Let \(X\) be an \(n\)-dimensional algebraic variety and \(x\in X\) be a smooth point on \(X\). Then there is a Zariski open neighborhood \(U_x\subset X\) of \(x\) which is isomorphic to a closed smooth hypersurface in \(\mathbb{C}^{n+1}\).
In particular it implies that every \(n\)-dimensional smooth algebraic variety \(X\) can be covered by Zariski open subsets \(U_i\) which are isomorphic to closed smooth hypersurfaces in \(\mathbb{C}^{n+1}\).
As an application of the theorem above we give a characterization of components of the set \(S_f\) of points at which a polynomial mapping \(f:\mathbb{C}^n \to\mathbb{C}^m\) is not proper. Let us recall that \(f\) is not proper at a point \(y\) if there is no neighborhood \(U\) of \(y\) such that \(f^{-1} (\text{cl} (U))\) is compact. We showed [\textit{Z. Jelonek}, Math. Ann. 315, No. 1, 1-35 (1999; Zbl 0946.14039)] that the set \(S_f\) (if non-empty) has pure dimension \(n-1\) and it is \(\mathbb{C}\)-uniruled, i.e., for every point \(x\in S_f\) there is an affine parametric curve through this point. In this paper we show that, conversely, for every \(\mathbb{C}\)-uniruled \((n-1)\)-dimensional variety \(X \subset\mathbb{C}^m\) (where \(2\leq n\leq m)\), there is a generically-finite (even quasi-finite) polynomial mapping \(F:\mathbb{C}^n \to\mathbb{C}^m\) such that \(X\subset S_F\). This gives (together with the cited paper) a full characterization of irreducible components of the set \(S_f\). polynomial mapping Z. Jelonek. \textit{Local characterization of algebraic manifolds and characterization of components of the set \(S_f\).} Ann. Polon. Math. \textbf{75} (2000) 7--13 Local structure of morphisms in algebraic geometry: étale, flat, etc., Jacobian problem, Rational and birational maps, Coverings in algebraic geometry, Birational automorphisms, Cremona group and generalizations Local characterization of algebraic manifolds and characterization of components of the set \(S_f\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 contains some results in Grothendieck's dessins d'enfant theory over arbitrary fields. The definition of a Belyi pair in positive characteristic and primes of bad reduction are given. Considering the graph \(K_{3,3}\), it is shown that it corresponds to three different dessins. For each dessin, the authors find the Belyi pair and the positive characteristics for which the pair exists and describe the set of primes of bad reduction. dessins d'enfant; Belyi pairs Dremov, V. A.; Vashevnik, A. M., On Belyi pairs over arbitrary fields, Fund. Prikl. Matem., 12, 3, (2006) Coverings of curves, fundamental group, Arithmetic ground fields for curves, Coverings in algebraic geometry On Belyi pairs over arbitrary fields | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author considers a smooth branched covering \(f:\quad X\to {\mathbb{P}}^ n\) that is étale outside a union of smooth hypersurfaces of \({\mathbb{P}}^ n\) meeting transversally and proves that for such a covering the maps induced on cohomology \(f^*:\quad H^ i({\mathbb{P}}^ n,{\mathbb{C}})\to H^ i(X,{\mathbb{C}})\) are isomorphisms for \(i\neq n\). The proof consists of reducing to a special class of coverings, the so-called multicyclic coverings, and proving that for a smooth multicyclic covering f:X\(\to Y\) with k-ample branch divisors the maps \(f^*:\quad H^ q(Y,\Omega^ p_ Y)\to H^ q(X,\Omega^ p_ X)\) are isomorphisms for \(| p+q- n| >k\). multicyclic coverings; branch divisors Ramification problems in algebraic geometry, Coverings in algebraic geometry, Classical real and complex (co)homology in algebraic geometry A result on coverings of \({\mathbb{P}}^ n\) branched along a hypersurface with only simple normal crossings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems One studies the period map for weighted projective hypersurfaces. The main results concern two types of such hypersurfaces: k-sheeted branched coverings of \({\mathbb{P}}^ r\) and hyperelliptic fiber spaces over \({\mathbb{P}}^ r\) (equivalently, Veronese double cones in a sense given by the author in the paper). One shows firstly the existence of the associated moduli space M, one defines a period map \(p: M\to G\setminus D,\) one proves the existence of regular values for p and the local Torelli theorem. As main result, one proves the weak global Torelli theorem: the period map p has degree one onto its image (except of some special cases). The paper extends the work of \textit{R. Donagi} [Compos. Math. 50, 325-353 (1983; Zbl 0598.14007)] which considers hypersurfaces in projective spaces. period map for weighted projective hypersurfaces; branched coverings; local Torelli theorem; weak global Torelli theorem Saitō, M.-H., Weak global Torelli theorem for certain weighted projective hypersurfaces, Duke Math. J., 53, 1, 67-111, (1986), MR 835796 Moduli, classification: analytic theory; relations with modular forms, Complex-analytic moduli problems, Transcendental methods, Hodge theory (algebro-geometric aspects), Coverings in algebraic geometry Weak global Torelli theorem for certain weighted projective hypersurfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The article starts with some introductory material about resolution graphs of normal surface singularities (definitions, topological/homological properties, etc.). Then we discuss the problem of \(N\)-cyclic coverings \((X_{f,N},0) \to(X,0)\) of \((X,0)\), branched along \((\{f=0\}, 0)\), where \(f(X,0)\to(\mathbb{C},0)\) is the germ of an analytic function. We present non-trivial examples in order to show that from the embedded resolution graph \(\Gamma(X,f)\) of \(f\) it is not possible to recover the resolution graph of \((X_{f,N},0)\). The main results are the construction of a ``universal covering graph'' of \(\Gamma(X,f)\) from the topology of the germ \(f\), and the completely combinatorial construction of the resolution graph of \((X_{f,N},0)\) from this universal graph of \(f\) and the integer \(N\). For this we also prove some classification theorems of ``graph coverings'', results which are purely graph-theoretical. In the last part, we connect the properties of the universal covering graph with the topological invariants of \(f\), e.g. with the nilpotent part of its algebraic monodromy.
[For part II of this paper see \textit{A. Némethi} and \textit{Á. Szilárd}, same collection, Contemp. Math. 266, 129-164 (2000; see the following review Zbl 1017.14013)]. germ of an analytic function; cyclic coverings; graphs of normal surface singularities; embedded resolution graph; universal covering graph; monodromy A. Némethi, Resolution graphs of some surface singularities, I, Cyclic coverings, Singularities in algebraic and analytic geometry (San Antonio, 1999), Contemp. Math., 266, pp. 89--128, Amer. Math. Soc., Providence, RI, 2000. Singularities of surfaces or higher-dimensional varieties, Complex surface and hypersurface singularities, Global theory and resolution of singularities (algebro-geometric aspects), Coverings in algebraic geometry, Modifications; resolution of singularities (complex-analytic aspects), Germs of analytic sets, local parametrization, Applications of graph theory Resolution graphs of some surface singularities. I: 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 In this important paper the author describes and gives numerous applications of the algorithm which allows one to decide whether two 3-manifolds obtained by plumbing according to a graph of \(S^ 1\)-bundles over compact surfaces (possibly with boundary) are homeomorphic. Namely the manifolds associated with two graphs are the same if and only if there is a sequence of moves (which belong to 8 types) which transforms one graph into the other. Practical applications are based on the use of normal forms to which any graph can be reduced and such that manifolds corresponding to different graphs are distinct. As the author points out, this calculus is implicit in \textit{F. Waldhausen}'s work [Invent. Math. 3, 308-333; ibid. 4, 87-117 (1967; Zbl 0168.445)] on classification of graph manifolds. The author earlier used this calculus to define an integral invariant of plumbed homology spheres [Lect. Notes Math. 788, 125-144 (1980; Zbl 0436.57002)]. Closely related to the author's calculus is Bonahon and Siebenmann's census of oriented diffeomorphism types of manifolds arising from weighted trees. This technique is applied to the analysis of two types of 3-manifolds naturally appearing in algebraic geometry. The first type is the singularity links, i.e., the boundaries of regular neighbourhoods of isolated singularities of complex surfaces.
The main result is that with two exceptions the fundamental group of the singularity link determines the genera, normal bundles and intersection numbers of a minimal good resolution of the singularity (i.e., the resolution in which all intersections are normal crossings, no more than two curves intersect in one point and there are no \(CP^ 1\)'s with self-intersection \(-1\)). The exceptional singularity links which do not determine the resolution are the lens spaces \(L(p,q)\) and the torus bundles over circles with monodromy \(A\in SL_ 2({\mathbb{Z}})\) such that trace \(A\) is \(\geq 3\). Among other facts the author shows that the singularity links are irreducible 3-manifolds (Problem 3.20 in \textit{R. Kirby}'s list [Proc. Symp. Pure Math. 32, Part 2, 273-312 (1978; Zbl 0394.57002)]) and that the resolution of a singularity is star-shaped provided the singularity link is a Seifert manifold. Another interesting result is that a complex surface V is topologically the suspension of a closed 3-manifold if and only if it is homeomorphic to an Inoue surface [\textit{M. Inoue}, Complex Anal. algebr. Geom. 91-106 (1977; Zbl 0365.14011)].
The second type of 3-manifold arising from algebraic geometry is the link of families of curves. The latter are the manifolds of the form \(\pi^{- 1}(\partial D)\), where \(\pi: W\to D\) is an analytic map of a complex surface on the unit disk such that all fibres except the one over the origin are nonsingular complete algebraic curves. The main result is that the fundamental group of the link of a minimal good family of curves defines the numerical type of the family, i.e., genera, normal bundles and intersection numbers of components. plumbing of circle bundles over compact surfaces; 3-manifolds; classification of graph manifolds; boundaries of regular neighbourhoods of isolated singularities of complex surfaces; fundamental group of the singularity link; lens spaces; torus bundles over circles; Seifert manifold; Inoue surface; link of families of curves Neumann W., A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves, Trans. Amer. Math. Soc. 268 (1981), 299-344. Topology of general 3-manifolds, Local complex singularities, Global theory and resolution of singularities (algebro-geometric aspects), Fundamental group, presentations, free differential calculus, Algebraic topology on manifolds and differential topology, \(3\)-folds, Special surfaces, Transcendental methods of algebraic geometry (complex-analytic aspects), Families, moduli of curves (analytic), Singularities in algebraic geometry, Coverings in algebraic geometry A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex 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 smooth projective minimal surface of general type, let \(\pi : X \rightarrow C\) be a genus two fibration over a smooth projective curve and assume \(K_X\) relatively ample.
In this paper it is proved that, if moreover \(\pi\) is not a \(C^{\infty}\)-locally trivial fibre bundle, the fundamental group of a general fibre maps trivially on \(\pi_1(X)\). An analogous result for elliptic surfaces was proved by \textit{R. V. Gurjar} and \textit{A. R. Shastri} [Compos. Math. 54, 95--104 (1985; Zbl 0583.14013)].
As a consequence, the authors prove that the Shafarevich conjecture is true for \(X\) (as in [loc. cit.] for the elliptic surfaces). Further corollaries are that \(\pi_2(X)\) is a free abelian group and the proof of a conjecture of \textit{M. V. Nori} [Ann. Sci. Éc. Norm. Super., IV. Sér. 16, 305--344 (1983; Zbl 0527.14016)]. Shafarevich conjecture; genus 2 fibrations Demailly, J.-P.: Monge-Ampère operators, Lelong numbers and intersection theory. In: Complex analysis and geometry, Univ. Ser. Math., pp. 115-193. Plenum, New York (1993) Surfaces of general type, Compact complex surfaces, Coverings in algebraic geometry On the Shafarevich conjecture for genus-2 fibrations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems See the review in Zbl 0662.14004. fundamental group of the complement of a plane algebraic curve; nodal algebraic curves; computer algorithm; Zariski problem 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 this valuable paper, the author describes the monodromy method and its application to several related problems: (1) describing pairs \(f,g\) of polynomials of one variable such that the variables separated polynomial \(f(x)-g(y)\) is reducible; (2) describing polynomials \(f\) of one variable and natural numbers \(d\) such that the degree \(d\) reducibility set \({\mathcal R}_f(d)\) of \(f\), defined to be the set of elements \(z_0\) of degree no more than \(d\) over the ground field such that \(f(x)-z_0\) is reducible over the ground field with \(z_0\) adjoined, is the union of the degree \(d\) value set \(\mathcal V_f(d)\) of \(f\) with a finite number of additional elements; (3) describing pairs of polynomials \(f, g\) which have the same value sets \(\mathcal V_f\) for almost every finite residue field (Davenport's problem); and (4) describing the simple groups that occur as composition factors of geometric monodromy groups of some \(f(x)-z\) over \({\mathbb P}^1_z\), where \(f\) is a rational function of one variable (the genus \(0\) problem).
The monodromy method studies covers of the projective line \({\mathbb P}^1\) of a field \(K\) through the action of the geometric monodromy group (the Galois group of the Galois closure of the cover over \(\overline K\)) on the points lying over branch points, or through the finer action given by the arithmetic monodromy group (the Galois group over \(K\)). Covers with common Galois group \(G\), number of branch points \(r\), and distinguished conjugacy classes of inertial groups for the branch points are naturally parametrized by Hurwitz spaces, and this paper shows how a close analysis of these spaces leads to explicit results on the set of exceptions to a specific Diophantine outcome. The survey highlights the difference in the natures of the problems over characteristic \(0\) and over characteristic \(p\), as well as the roles played by doubly transitive representations and the classification of finite simple groups in solutions Fried, M.: Variables separated polynomials. Number theory in progress, vol. 1 1, 169-228 (1999) Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Algebraic field extensions, Coverings in algebraic geometry Variables separated polynomials, the genus 0 problem and moduli 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 Let \(X\) and \(Y\) be smooth projective varieties over \({\mathbb{C}}\) of dimension \(n \geq 2\) and \(f:Y \to X\) be an abelian cover. We get a central extension \(0 \rightarrow K \rightarrow \pi_1 (Y)\rightarrow \pi_1 (X) \rightarrow 1\), and the aim of this article is to show how the kernel \(K\) and the cohomology class \(c(f) \in \text{H}^2 (\pi_1 (X) , K)\) can be computed from some data associated to the cover \(f:Y \to X\). Using this result the authors construct covers of the same variety \(X\) with the same Galois group, branch locus and inertia subgroups, which are not homeomorphic.
An abelian cover \(f:Y \to X\) is characterized by the following geometrical data: To any irreducible component \(D_j\) of the branch locus \(D\) of \(f\) we can associate a cyclic subgroup \(G_j\) of \(G\), the inertia subgroup, and a faithful representation \(\psi_j\) of \(G_j\); the action of \(G\) on \(f_* ({\mathcal O}_Y)\) induces a splitting: \(f_* ({\mathcal O}_Y) = \bigoplus_{\chi \in G^*} L^{-1}_{\chi}\), where \(L^{-1}_{\chi}\) is a line bundle on \(X\) such that \(G\) acts on \(L ^{-1}_{\chi}\) via the character \(\chi\).
We consider an abelian cover \(f:Y \to X\) which is totally ramified, i.e. such that the inertia subgroups of the components \(D_1 , \ldots , D_k\) of the branch locus \(D\) generate \(G\), and such that all the components \(D_j\) are ample. Let \(\pi:\widetilde X \to X\) be the universal covering of \(X\) and \(\widetilde D = \pi ^{-1} (D)\), and \(\widetilde D_j = \pi ^{-1} (D_j)\) is connected for every \(j\); then the kernel \(K\) is isomorphic to \(\text{ker} (\bigoplus G_j \to G) / \text{Im} (\sigma \circ \rho)\), where \(\rho\) is the restriction \(H^{2n-2}_{\text{c}} (\widetilde X) \to H^{2n-2}_{\text{c}}\bigl (\widetilde D)\) and \(\sigma\) is the map \(H^{2n-2}_{\text{c}} (\widetilde D) \simeq \bigoplus {\mathbb{Z}} \widetilde D_j \to \bigoplus G_j\).
To compute the cohomology class \(c(f)\) the authors introduce the divisors \(M_1, \ldots , M_q\), such that the subgroup of \(\text{Pic}(X)\) generated by \(D_1, \ldots , D_k\) and \(L_{\chi}\) is \(A = \bigoplus_{l=1} ^q \langle M_l\rangle\), and the matrix \(C = (c_{jl})\) with integral coefficients such that: \( D_j = C M_l\) . Then each column \((c_{jl})\) represents an element of \(N = \text{ker} (\bigoplus G_j \to G)\) and the matrix \(C\) defines a map \(\Theta:{\mathbb{Z}}^q \to K\). Let \(\Theta_*:\bigoplus ^q \text{H}^2 (X,{\mathbb{Z}}) \to \text{H}^2 (X,K)\) be the map induced on cohomology; we can give the principal result of this article:
Let \(f:Y \to X\) be a totally ramified abelian cover of smooth projective varieties such that the components \(D_j\) of the branch locus are ample, then the cohomology class in \(\text{H}^2 (X,K)\) of the central extension \(0 \rightarrow K \rightarrow \pi _1 (Y) \rightarrow \pi _1 (X) \rightarrow 1\), is \(c(f) = \Theta _* ([M_1] , \ldots , [M_q])\).
The authors can recover a previous result:
The image of \(c(f)\) in \(\text{H}^2 (X, \widetilde G)\), where \(\widetilde G = (\bigoplus G_j) / \text{Im} (\sigma \circ \rho)\), is \(i_* (c(f)) = \Phi _* ([ D_1 ] , \ldots , [ D_k ])\), where \(\Phi\) is the map \({\mathbb Z} ^k \to \widetilde G\).
The cohomology class \(c(f)\) of the extension depends on the choice of the \(\{ L _{\chi}\}\), once the branch divisors \((D_j)\) and the covering structure \((G;G_j)\) are fixed; then we can get covers with the same \(D_j\), \(G_j\) and \(G\), and with different fundamental groups. For any \(m\), the authors construct non homeomorphic covers \(X_1 , \ldots , X_m\) of the same variety \(X\) with the same branch divisors \(D_j\), inertia subgroups \(G_j\) and Galois group \(G\). abelian cover; fundamental group; non homeomorphic coverings Homotopy theory and fundamental groups in algebraic geometry, Coverings in algebraic geometry On the fundamental group of an abelian cover | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(F\) denote the finite field \(\mathbb F_q\) with \(q\) elements for \(q\) a power of a prime \(p\), and let \(G\) denote the general linear group \(\text{GL}_2(F)\). Let \(V=F^2\) denote the natural two-dimensional \(FG\)-module. Further, for a non-negative integer \(k\), let \(V_k\) denote the \(k\)-th symmetric power of \(V\) as an \(FG\)-module. The author first considers a map \(e\otimes V_{k-(q+1)}\to V_k\) for \(k>q\) where \(e\) denotes the character determinant. It is shown that the cokernel of this map (having dimension \(q+1\)) is isomorphic to the reduction mod \(p\) of a principal series representation.
The main focus of the paper is on a map \(D\colon V_k\to V_{k+(q-1)}\) defined by Serre. The main result is an identification of the cokernel of \(D\) for \(q>2\), \(2\leq k\leq p-1\), \(k\neq\frac{q+1}{2}\). Precisely, it is shown that the cokernel of \(D\) is isomorphic to the reduction mod \(p\) of an integral model of a cuspidal representation for \(\overline{\mathbb Q}_pG\) (where \(\overline{\mathbb Q}_p\) is the algebraic closure of the \(p\)-adic field). The proof makes use of a short exact sequence involving the cokernel of \(D\). This short exact sequence is identified with a short exact sequence in crystalline cohomology for the projective curve \(XY^q-X^qY-Z^{q+1}=0\) due to \textit{B. Haastert} and \textit{J.~C. Jantzen} [J. Algebra 132, No. 1, 77-103 (1990; Zbl 0724.20030)].
Lastly, in the case \(q=p>3\), the author applies his results to modular forms over \(G\). The map \(D\) discussed above is used to extend a cohomological analogue of the Hasse invariant operator constructed by \textit{B. Edixhoven} and \textit{C. Khare} [Doc. Math., J. DMV 8, 43-50 (2003; Zbl 1044.11030)] on the cohomology of spaces of mod \(p\) modular forms for \(\text{GL}_2\). modular representations of finite groups; congruences for mod \(p\) modular forms; general linear groups; principal series representations; cuspidal representations; symmetric powers; crystalline cohomology Modular representations and characters, Congruences for modular and \(p\)-adic modular forms, \(p\)-adic cohomology, crystalline cohomology, de Rham cohomology and algebraic geometry, Representations of finite groups of Lie type, Representation theory for linear algebraic groups, Linear algebraic groups over finite fields Reduction mod \(p\) of cuspidal representations of \(\text{GL}_2(\mathbb F_{p^n})\) and symmetric powers. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Denote by \(D_{2n}\) the dihedral group of order \(2n\) and by \(H_{n}<D_{2n}\) the cyclic subgroup of order \(n\). A dihedral cover of a smooth complex surface \(Y\) is a finite morphism \(\pi\colon X\to Y\) where \(X\) is a normal surface and the map \(\pi\) is Galois with Galois group \(D_{2n}\). Set \(D(X/Y):=X/H_n\). Then \(\pi\) is the composition of the \(H_n\)-cover \(\beta_1(\pi)\colon X\to D(X/Y)\) and of the double cover \(\beta_2(\pi)\colon D(X/Y)\to Y\). The author calls \(\pi\) an elliptic dihedral cover if the surface \(D(X/Y)\) has an elliptic fibration with a section \(O\). The elliptic dihedral cover is said to be of torsion type if the elliptic fibration on \(D(X/Y)\) pulls back to an elliptic fibration on \(X\). In earlier work, the author has investigated a special class of \(D_{2n}\)-elliptic covers of torsion type, using the existence of such covers to obtain information on the fundamental group of the complement of certain plane curves of low degree. Here he starts the study elliptic \(D_{2p}\)-covers not of torsion type, with \(p\neq 2\) a prime, with special attention to the case where \(D(X/Y)\) is rational. In particular, he analyzes the case in which \(Y\) is the complex projective plane and the branch locus of the dihedral cover is a plane quintic. dihedral cover; elliptic surface; rational surface; plane quintic Tokunaga, H.: Dihedral covers and an elementary arithmetic on elliptic surfaces. J. Math. Kyoto Univ. \textbf{44}, 255-270 (2004) Coverings in algebraic geometry, Elliptic surfaces, elliptic or Calabi-Yau fibrations Dihedral covers and an elementary arithmetic on 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 The paper under review develops a theory of polynomial representations of the super general linear group \(\mathrm{GL}(m|n,A)\), defined over an arbitrary commutative superalgebra \(A\). The methods used adapt and parallel Green's approach to the usual Schur algebra via comodules, as presented in \S 2 of [\textit{J. A. Green}, Polynomial representations of \(\mathrm{GL}_n\). With an appendix on Schensted correspondence and Littelmann paths by K. Erdmann, J. A. Green and M. Schocker. 2nd corrected and augmented edition. Berlin: Springer (2007; Zbl 1108.20044)]. Thus, rather than work directly with super modules for \(\mathrm{GL}(m|n,A)\), the author first defines, for each suitable sub-super coalgebra \(D\) of the super algebra of finitary functions \(\mathrm{GL}(m|n,A)\), a module category \(\mathrm{mod}_D(A\mathrm{GL}(m|n,A))\) of representations of the super group algebra \(A\mathrm{GL}(m|n,A)\) such that the coefficient functions of the representing matrices lie in \(D\). There is an equivalence of categories
\[
\mathrm{mod}_D(A\mathrm{GL}(m|n,A)) \simeq \mathrm{com}(D)
\]
between this category and the category of super \(D\)-comodules. Taking \(D\) to be the super coalgebra of polynomial functions \(\mathrm{GL}(m|n,A)\) of degree \(r\) and dualizing, the author obtains a superalgebra \(S_A(m|n,r)\) that is the super analogue of the usual Schur algebra. A key result on the Schur algebra is that if \(F\) is an infinite field then category of polynomial representations of \(\mathrm{GL}(n,F)\) of polynomial degree \(r\) is equivalent to the category of representations of the Schur algebra \(S_F(n,r)\), defined over the field \(F\). The author proves the analogous result for representations of \(S_A(m|n,r)\) in Proposition 4.2.
The main result of this paper, described as `super Schur duality' is Theorem 5.1. In it \(A\) is a commutative superalgebra over an infinite field \(F\) and \(E_A = A^{m|n}\) is a free rank \(m|n\) \(A\)-module, generated by \(m\) even elements and \(n\) odd elements. The free \(A\)-module \(E_A^{\otimes r}\) is acted on by \(\mathrm{GL}(m|n,A)\) with coefficient functions of polynomial degree \(r\). It may therefore be regarded as a representation of \(S_A(m|n,r)\). The symmetric group \(S_r\) acts on \(E_A^{\otimes r}\) by permuting the factors (with signs coming from the super structure). Theorem 5.1 states that the natural action map
\[
S_A(m|n,r) \rightarrow \mathrm{End}_A(E_A^{\otimes r}) \tag{\(\star\)}
\]
is injective, and its image is precisely those \(A\)-endomorphisms of \(E_A^{\otimes r}\) which commute with the action of the symmetric group \(S_r\). This is the super version of Schur's theorem (see [loc. cit., Theorem 2.6c]) that \(S_F(n,r) \cong \mathrm{End}_{S_r}(V^{\otimes r})\), where \(V\) is the natural \(\mathrm{GL}_n(F)\)-module. As is the case with Schur's theorem, the main force of the result is that \emph{every} \(S_r\) endomorphism of the tensor algebra comes from an element of the super Schur algebra, and so is induced by the action of a suitable linear combination of elements in the super group algebra \(A\mathrm{GL}(m|n,A)\). An important corollary is that if \(F\) has infinite characteristic or prime characteristic \(p > r\) then \(S_A(m|n,r)\) is semisimple.
The author begins with a brief but useful survey of other approaches to Schur-Weyl duality, emphasising that the main novel feature in his paper is to work with the supergroup \(\mathrm{GL}(m|n)\) rather than its super Lie algebra \(\mathfrak{gl}(m|n)\). In this connection we mention that modules for \(S_A(m|n,r)\) are direct sums of the special class of covariant \(\mathfrak{gl}(m|n)\)-modules: see Chapter 3 of Moens' Ph.D.~thesis [Supersymmetric Schur functions and Lie superalgebra representations. Universiteit Gent (2007)] for an excellent introduction. In general, and in contrast to the case for \(S_A(m|n,r)\), modules for \(\mathfrak{gl}(m|n)\) are not completely reducible. Another reference one might add to the author's list is [\textit{D. Benson} and \textit{S. Doty}, Arch. Math. 93, No. 5, 425--435 (2009; Zbl 1210.20039)], which shows (amongst other results) that the Schur algebra analogue of (\(\star\)) holds over any field \(F\) such that \(|F| > r\).
The paper under review includes all the results needed to perform \(p\)-modular reduction on the category of representations of the Super Schur algebra \(S(m|n,r)\). The author remarks that `It seems to us such a modular theory is needed for a geometric theory'. modular representations; supergroups; \(\mathrm{GL}(m|n)\); Schur's duality; super Schur algebra; finitary maps; comodules Modular Lie (super)algebras, Superalgebras, Group actions on varieties or schemes (quotients), Representations of finite symmetric groups, Modular representations and characters, Lie bialgebras; Lie coalgebras, Graded Lie (super)algebras, Vector and tensor algebra, theory of invariants Polynomial representations of \(\mathrm{GL}(m|n)\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let k be an algebraically closed field, \({\mathbb{P}}^ n\) the projective space over k of dimension \(n\) and X a smooth complete algebraic variety over k. One proves the following: if \(f: {\mathbb{P}}^ n\to X\) is a surjective separable morphism, then necessarily \(X\cong {\mathbb{P}}^ n\) and f is a finite morphism. The proof is based on the characterization due to Mori of the projective space by the ampleness of the tangent bundle.
Reviewer's remark: the result was also obtained independently by \textit{R. Lazarsfeld} [in Complete intersections, Lect. 1st Sess. C.I.M.E., Acireale/Italy 1983, Lect. Notes Math. 1092, 29-61 (1984; Zbl 0547.14009)]. characterization of the projective space; finite morphism; ampleness of the tangent bundle Projective techniques in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Coverings in algebraic geometry A certain consequence from Mori's theorem on tangent bundle | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 context of uniformisation problems, we study projective varieties with klt singularities whose cotangent sheaf admits a projectively flat structure over the smooth locus. Generalising work of Jahnke-Radloff, we show that torus quotients are the only klt varieties with semistable cotangent sheaf and extremal Chern classes. An analogous result for varieties with nef normalised cotangent sheaves follows. Bogomolov-Gieseker inequality; abelian variety; klt singularities; Miyaoka-Yau inequality; stability; projective flatness; uniformisation Uniformization of complex manifolds, Notions of stability for complex manifolds, Coverings in algebraic geometry, Minimal model program (Mori theory, extremal rays), Projective connections Projectively flat klt 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 carries out a program outlined in the author's paper in J. Am. Math. Soc. 1, No.4, 867-918 (1988; Zbl 0669.58008): extending to noncompact algebraic curves X the known correspondence between Higgs bundles and local systems. The main problem is the behaviour of certain singularities at the punctures of X. It builds on the theorem of Narasimhan-Seshadri on stable vector bundles, the theory of harmonic maps of Riemann surfaces, variations of Hodge structures, and differential equations with regular singularities. All together, here is a marvellous example of interrelationships between algebraic/differential geometry and analysis. Higgs bundle; \({\mathcal D}_ X\)-modules; noncompact algebraic curves; theorem of Narasimhan-Seshadri; stable vector bundles; harmonic maps of Riemann surfaces; variations of Hodge structures C. T. Simpson, \textit{Harmonic bundles on noncompact curves}, J. Amer. Math. Soc. \textbf{3} (1990), no. 3, 713-770. Harmonic maps, etc., Coverings in algebraic geometry, Variational problems concerning extremal problems in several variables; Yang-Mills functionals, Variation of Hodge structures (algebro-geometric aspects), Hodge theory in global analysis, Riemann surfaces Harmonic bundles on noncompact curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The classical detection theorem for finite groups (due to \textit{D. Quillen} and \textit{B. Venkov} [Topology 11, 317--318 (1972; Zbl 0245.18010)]) tells that if \(\pi\) is a finite group and \(\Lambda\) is a \(\mathbb{Z}/p[\pi]\)-algebra then a cohomology class \(z\in H^*(\pi,\Lambda)\) is nilpotent iff for every elementary abelian \(p\)-subgroup \(\pi_0\) the restriction of \(z\) to \(\pi_0\) is nilpotent. This theorem is very useful for the identification of the support variety of the group \(\pi\) and for the identification of support varieties of \(\pi\)-modules. In the detection theorem for infinitesimal group schemes \(G\) [due to \textit{A. Suslin}, \textit{E. M. Friedlander} and \textit{C. P. Bendel}, J. Am. Math. Soc. 10, No. 3, 729--759 (1997; Zbl 0960.14024)] the role of elementary abelian \(p\)-subgroups is played by one parameter subgroups \(\mathbb{G}_{a(r)}\) of \(G\).
In the present paper the author proves a general detection theorem for finite group schemes. This theorem generalizes both the classical detection theorem and the detection theorem for infinitesimal group schemes and looks as follows.
Theorem. Let \(G/k\) be a finite group scheme, let further \(\Lambda\) be a unital associative rational \(G\)-algebra and let \(z\in H^*(G,\Lambda)\) be a cohomology class. Assume that for any field extension \(K/k\) and any closed subgroup scheme \(i: \pi_0\times \mathbb{G}_{a(r)}\to G_K\) the restriction \(i^*(z_K)\) of \(z_K\) to \(\pi_0\times \mathbb{G}_{a(r)}\) is nilpotent then \(z\) is nilpotent itself. finite group scheme; detection theorem A. Suslin, The detection theorem for finite group schemes, to appear in J. Pure Appl. Algebra. Group schemes, Modular representations and characters, Linear algebraic groups over the reals, the complexes, the quaternions, Cohomology theory for linear algebraic groups, Modular Lie (super)algebras Detection theorem for finite group schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A correction is provided to a result of the author's [Isr. J. Math. 118, 317--355 (2000; Zbl 1032.12007)] concerning embedding problems with \(p\)-group kernel. It is also shown that every such embedding problem for a Laurent series field in characteristic \(p\) has a proper solution. Harbater, D.: Correction and addendum to ''embedding problems with local conditions'', Israel J. Math. 162, 373-379 (2007) Inverse Galois theory, Coverings in algebraic geometry, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Fundamental groups and their automorphisms (group-theoretic aspects) Correction and addendum to ``Embedding problems with local conditions'' | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Here the author constructs a cyclic covering \(\pi:X\to E\) with \(X\) a smooth curve of genus \(g\geq 7\), \(E\) an elliptic curve, \(\pi\) a cyclic covering ramified at exactly two points, say \(P\) and \(Q\), which are totally ramified and such that (seeing them as Weierstrass points on \(X)\) their gap sequence is \(\{1,\dots,g-2,g,2g-1\}\) (characteristic 0). total ramification point; cyclic covering; smooth curve; Weierstrass points; gap sequence Riemann surfaces; Weierstrass points; gap sequences, Elliptic curves, Elliptic curves over global fields, Coverings of curves, fundamental group, Coverings in algebraic geometry Cyclic coverings of an elliptic curve with two branch points and the gap sequences at the ramification points | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The aim of the paper under review is to develop a theory of triple covers in algebraic geometry. One of the most important general result obtained says that a triple cover \(X\to Y\) (with X and Y irreducible varieties over an algebraically closed field) is determined by a rank-two vector bundle E and a map \(S^ 3E\to \bigwedge^ 2E\), and conversely.
Among the numerous corollaries of this theory we mention the following: (1) The general triple cover in dimension \(\geq 2\) has singular branch locus. (2) The general triple cover in dimension \(\geq 4\) is singular. (3) The moduli space of trigonal curves of genus \(g\) is connected, unirational, of dimension \(2g+1\). (4) An approach to construct surfaces of general type X with \(K^ 2\) arbitrarily close to 3e(X). Many other interesting results are also obtained. triple covers; rank-two vector bundle; singular branch locus; trigonal curves R. MIRANDA, Triple covers in algebraic geometry, Amer. J. Math. 107 (1985), 1123-1158 JSTOR: Coverings in algebraic geometry, Coverings of curves, fundamental group Triple covers in algebraic geometry | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems From the Introduction: ``Andrei Suslin (1950--2018) was both
friend and mentor to us. This article discusses
some of his many mathematical achievements, focusing on the
role he played in shaping aspects of algebra and algebraic
geometry. We mention some of the many important results
Andrei proved in his career [\dots] Andrei was deeply involved
in both the formulation and the solution of many of the most
important questions in algebraic \(K\)-theory. [\dots] Towards the
end of his career Andrei made important contributions to the
modular representation theory of finite group schemes. [\dots]
Time and again, Andrei introduced new techniques and
structures in order to solve challenging problems [\dots].''
The authors explain contributions of Andrei Suslin to the
following topics: projective modules; \(K_2\) of fields and the Brauer group;
Milnor \(K\)-theory versus algebraic K-theory;
\(K\)-theory and cohomology theories;
motivic cohomology and K-theories;
modular representation theory.
In each case, there is a helpful description of the context. algebraic \(K\)-theory; motivic cohomology; modular representation theory Representation theory for linear algebraic groups, Modular representations and characters, Cohomology theory for linear algebraic groups, History of group theory, History of \(K\)-theory, History of algebraic geometry, History of mathematics in the 20th century, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Motivic cohomology; motivic homotopy theory, Biographies, obituaries, personalia, bibliographies The mathematics of Andrei Suslin | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 propose a theory relying on the theories of Kummer and Artin- Schreier. They prove the universality of an isogeny of group schemes for cyclic étale coverings of degree \(p\) over a base of mixed characteristic 0 and \(p\). isogeny of group schemes; cyclic étale coverings T. Sekiguchi and N. Suwa , ' Théorie de Kummer-Artin-Schreier ', C.R. Acad. Sci. Paris, 312 (1991) 417-420. Isogeny, Group schemes, Coverings in algebraic geometry Théorie de Kummer-Artin-Schreier. (Kummer-Artin-Schreier theory) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 log-terminal projective variety of dimension n defined over an algebraically closed field of characteristic zero and \(f: X\to {\mathbb{P}}^ n\) a branched covering. Then it is proved that the ramification divisor R of f is an ample \({\mathbb{Q}}\)-Cartier divisor unless f is an isomorphism. This result generalizes theorem 1 of a paper by \textit{L. Ein} [Math. Ann. 261, 483-485 (1982; Zbl 0519.14005)]. log-terminal projective variety; branched covering; ramification divisor Maeda, H., Ramification divisors for branched coverings of \(\mathbb{P}_n\), Math. Ann., 288, 2, 195-199, (1990) Ramification problems in algebraic geometry, Coverings in algebraic geometry, Projective techniques in algebraic geometry Ramification divisors for branched coverings of \({\mathbb{P}}^ n\) | 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 A \(G\)-principal Higgs bundle over a variety \(X\) (with values in an arbitrary line bundle on \(X)\) determines a family of spectral covers \(\widetilde X_ \rho\) of \(X\), one for each irreducible representation \(\rho\) of \(G\). We show that each of the \(\text{Pic} (\widetilde X_ \rho)\) is isogenous to the sum, with multiplicities, of a finite collection of abelian varieties, obtained as isotypic pieces for the action of the Weyl group \(W\) on \(\text{Pic} (\widetilde X)\), where \(\widetilde X\) is the cameral, or \(W\)-Galois, cover of \(X\), independent of \(\rho\). The piece \(\text{Prym} (\widetilde X)\), corresponding to the reflection representation of \(W\), is distinguished: it occurs in \(\text{Pic} (\widetilde X_ \rho)\) for each \(\rho\) (this characterizes Prym for classical \(G\) but not for exceptional groups such as \(G_ 2\), \(E_ 6)\), and is essentially the moduli space of Higgs bundles with spectral data \(\widetilde X\). Various Prym identities are recovered as in the case \(X = \mathbb{P}^ 1\), \(G\) simply laced, studied previously by \textit{V. Kanev} [in: Theta functions, Proc. 35th Summer Res. Inst. Borodoin Coll., Brunswick 1987, Proc. Symp. Pure Math. 49, Part 1, 627-645 (1989; Zbl 0711.14026)]. Prym variety; Picard variety; principal Higgs bundle; spectral covers; moduli space of Higgs bundles Donagi, R., Decomposition of spectral covers, Astérisque, 218, 145-175, (1993) Picard schemes, higher Jacobians, Coverings in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Group actions on varieties or schemes (quotients), Picard groups Decomposition of spectral covers | 0 |
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