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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 See the preview in Zbl 0546.14006. analytic coverings; branching sets; fundamental groups; Milnor fibers Dimca, A., Dimiev, S.: On analytic covering of weighted projective spaces. Bull. Lond. Math. Soc.17, 234-238 (1985) Coverings in algebraic geometry, Singularities in algebraic geometry, Projective techniques in algebraic geometry On analytic coverings of weighted 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 The object of this note is to sketch a theory of infinitesimal deformations of cyclic covers of projective varieties, somewhat different to that of \textit{E. Horikawa} [Invent. Math. 31, 43-85 (1975; Zbl 0317.14018)]. We obtain some cohomological criteria for the smoothness of the infinitesimal deformation functors, both absolute and relative, of a cyclic cover. In the last section this theory is applied to cyclic projective spaces of dimension greater than one. We prove that the absolute deformation functor of these varieties is always smooth and that, with the sole exception of those covers which are K 3 surfaces all deformations are obtained by deforming the covering map. These are essentially the results of Horikawa (loc. cit.) but some unnecessary restrictions (on the degree of the cover, degree of the branch locus, and dimension) have been removed. Our results are valid under the sole hypothesis that the degree of the cover is not divisible by the characteristic of the base field.
The paper is organised as follows. In {\S} 2 a theory of relative deformations of branched covers is developed, subject to hypotheses of tame ramification and non-singularity; {\S} 3 specialises this to the case of cyclic covers; {\S} 4 applies the specialised theory to cyclic projective spaces. Most of the results of {\S} 2 and 3 could be deduced, with some pain, from general machineries such as those developed by Illusie and Laudal (c.f. in particular, the spectral sequence of {\S} 2). However, it seems worthwhile to develop the theory for our special cases ab initio, since it admits an elementary and intuitive treatment, and, though restricted, covers many cases of importance in classical algebraic geometry. infinitesimal deformations of cyclic covers; smoothness of the infinitesimal deformation functors; deformations of branched covers Berry, TG, Infinitesimal deformations of cyclic covers, Acta Cient. Venezolana, 35, 177-183, (1984) Coverings in algebraic geometry, Ramification problems in algebraic geometry, Formal methods and deformations in algebraic geometry, Coverings of curves, fundamental group, Infinitesimal methods in algebraic geometry Infinitesimal deformations of cyclic covers | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This lecture is a continuation of the lecture of \textit{A. Chambert-Loir} [same proceedings, Prog. Math. 187, 233-248 (2000; see the preceding review Zbl 0978.14019); see also this review for the statements of Abhyankar's conjecture]. In the lecture under review one explains the part of Raynaud's proof of Abhyankar's conjecture for the affine line which makes use of the of the semi-stable geometry of curves [see \textit{M. Raynaud}, Invent. Math. 116, 425-462 (1994; Zbl 0798.14013)]. semi-stable curves; fundamental group; Abhyankar's conjecture; Galois cover Saı\ddot{}di, M.: Raynaud's proof of abhyankar's conjecture for the affine line. Progr. math. 187, 249-265 (2000) Rigid analytic geometry, Coverings in algebraic geometry, Coverings of curves, fundamental group, Galois theory Abhyankar's conjecture. II: The use of semi-stable 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 long and subtle paper stands within a series of notes devoted to the intimate relations between theta constants with rational characteristics, meromorphic modular forms for the principal congruence subgroup \(\Gamma(k)\) of the modular group \(\Gamma = \mathbb{P}\text{SL} (2,\mathbb{Z})\) and the geometry of the punctured Riemann surface \(\mathbb{H}^2/ \Gamma(k)\). These connections are established by associating the theta characteristics to the cusps of the group which leads in turn to new theta constant identities and explicit constructions of covering maps.
In the present paper this approach is refined and extended to the larger group \(\Gamma_0(k)\). For this let \(V(k)\) be the finite-dimensional Hilbert space of holomorphic functions on the upper half plane \(\mathbb{H}^2\) spanned by the modified theta constants \(\varphi_\ell : \tau\mapsto \theta [\chi_\ell] (0,k. \tau)\) where \(\chi_\ell\) varies over the classes of rational characteristics. Then for \(k\) odd the full modular group operates on \(\mathbb{P} V(k) \simeq \mathbb{P} \mathbb{C}^{(k-3)/2}\) and induces a holomorphic map of the compactification \(\overline{\mathbb{H}^2/ \Gamma(k)}\) into \(\mathbb{P}\mathbb{C}^{(k-3)/2}\). Moreover for \(k\) prime this map reveals the automorphism group \(\Gamma/ \Gamma(k)\) of the surface as a group of projective transformations hence of isometries. It is this fact which yields a lot of geometric insight into the surface. Since the \({k-1 \over 2}\) ``distinguished'' punctures of the surface are permuted by the group \(\Gamma_0 (k)\) and support the divisor of \(\varphi_\ell/ \varphi_\ell\), two explicit generators of the function field \(K\overline {(\mathbb{H}^2/ \Gamma(k)})\) are constructed in terms of \(\varphi_1/ \varphi_0\) and \(\varphi_2/ \varphi_1\) for \(k\) prime \(> 7\). In general this field is a Galois extension of \(K\overline {(\mathbb{H}^2/ \Gamma)}\) generated by \(\varphi_\ell/ \varphi_{k-3 \over 2}\), \(\ell= 0,1, \dots, {k-5 \over 2}\) with Galois group \(\Gamma/ \Gamma(k)\).
Finally quartic relations on the theta constants are given. It seems to be unknown whether these relations already determine the image of the curve \(\overline {\mathbb{H}^2/ \Gamma (k)}\) in \(\mathbb{P}\mathbb{C}^{(k-3)/2}\). Nevertheless for certain \(k\)'s the defining equations are found. uniformisation; modular curves; theta constants with rational characteristics; meromorphic modular forms; principal congruence subgroup; geometry of the punctured Riemann surface; explicit constructions of covering maps Farkas, H., Kopeliovich, Y., Kra, I.: Uniformizations of modular curves. Commun. Anal. Geom. 4(2), 207--259 (1996) Holomorphic modular forms of integral weight, Compact Riemann surfaces and uniformization, Theta series; Weil representation; theta correspondences, Unimodular groups, congruence subgroups (group-theoretic aspects), Coverings in algebraic geometry Uniformizations of modular curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(f:X\to Y\) be a branched covering of compact complex nonsingular surfaces with ramification divisor \(B\subset X\) whose irreducible components \(B_ 1,\dots, B_ k\) are smooth. The author proves that the cotangent bundle of \(X\) is numerically effective provided the same is true for \(Y\) and \(B_ i^ 2\leq 0\) for each \(i\). The latter condition is satisfied if \(C_ i^ 2\leq 0\) for any irreducible component of the branch divisor \(C\subset Y\). Recall that a vector bundle \(E\) over a smooth variety \(V\) is called numerically effective (nef) if the tautological line bundle \({\mathcal O}(1)\) on the projectivization \(\mathbb{P}(E)\) is numerically effective. The result is applied to the situation where \(f: X\to Y\) is a Galois covering branched over an arrangement of lines. numerically effective cotangent bundle; branched covering; ramification divisor; arrangement of lines Spurr, Michael~J., Nef cotangent bundles of branched coverings, Proc. Amer. Math. Soc., 118, 1, 57-66, (1993) Coverings in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Divisors, linear systems, invertible sheaves, Special surfaces, Projective techniques in algebraic geometry, Ramification problems in algebraic geometry Nef cotangent bundles of branched coverings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\pi:X\to X'\) be a finite surjective morphism of complex connected projective manifolds of dimension \(k\) factoring through an embedding of \(X'\) into the total space of an ample line bundle \({\mathcal L}\) on \(X\). Due to a Lefschetz-type theorem the Betti numbers of \(X\) and \(X'\) are equal except possibly the middle ones, for which the following inequality holds: \((*)\) \(b_ k(X')\geq b_ k(X)\).
The author proves the following facts:
(a) If \({\mathcal L}\) is spanned and \(\deg\pi\geq k\), then \((*)\) is a strict inequality;
(b) If \(X=\mathbb{P}^ k\) equality in \((*)\) can occur only for \(k\) odd, \(\pi\) being the double cover of \(\mathbb{P}^ k\) by a smooth hyperquadric;
(c) If \(X\) is a smooth hyperquadric and \(k\geq 3\), equality can never occur in \((*)\).
This completely settles a problem posed by the reviewer and \textit{D. C. Struppa} [Can. J. Math. 41, No. 3, 462-479 (1989; Zbl 0699.14019)] in the setting of cyclic coverings. The result is obtained by means of an interesting estimate of the difference \(b_ k(X')-b_ k(X)\) in terms of the cohomology groups \(H^ p(\Omega_ X^ q\otimes{\mathcal L}^{\otimes i})\). branched covering; ample line bundle; \(n\)-section; Betti numbers Jarosław A. Wiśniewski, On topological properties of some coverings. An addendum to a paper of A. Lanteri and D. C. Struppa: ''Topological properties of cyclic coverings branched along an ample divisor'' [Canad. J. Math. 41 (1989), no. 3, 462 -- 479; MR1013464 (90i:14021)], Canad. J. Math. 44 (1992), no. 1, 206 -- 214. Topological properties in algebraic geometry, Coverings in algebraic geometry, Transcendental methods, Hodge theory (algebro-geometric aspects) On topological properties of some coverings. An addendum to a paper of Lanteri and Struppa | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let X/\({\mathbb{F}}\) be a smooth projective geometrically integral variety over a finite field \({\mathbb{F}}\) (char \({\mathbb{F}}=p)\). Let \(A_ 0(X)\) denote the (torsion) group of degree 0 zero-cycles on X modulo rational equivalence and let \(\pi_ 1^{geom}(X)\) denote the abelian geometric fundamental group of X which classifies abelian étale covers of X which do not arise from extending \({\mathbb{F}}\). There is a natural surjective map \(\phi: A_ 0(X)\to \pi_ 1^{geom}(X)\) [\textit{S. Lang}, Ann. Math., II. Ser. 64, 285-325 (1956; Zbl 0089.262)] and the target group has long been known to be finite. The main theorem of unramified class field theory for X, due to \textit{K. Kato} and \textit{S. Saito} [Ann. Math., II. Ser. 118, 241-275 (1983; Zbl 0562.14011)], claims that \(\phi\) is an isomorphism (of finite groups). The case dim X\(=1\) is classical and the case dim X\(>2\) reduces to the crucial case dim X\(=2.\)
Use of the Merkur'ev-Suslin results [\textit{A. S. Merkur'ev} and \textit{A. A. Suslin}, Math. USSR, Izv. 21, 307-340 (1983); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 46, No.5, 1011-1046 (1982; Zbl 0525.18008)], of the Bloch-Kato-Gabber results [\textit{S. Bloch} and \textit{K. Kato}, Publ. Math., Inst. Haut. Étud. Sci. 63, 107-152 (1986)] and of the Weil conjectures for étale cohomology of varieties over finite fields, as proved by Deligne, has already led to another proof of this result [\textit{J.-L. Colliot-Thélène}, \textit{J.-J. Sansuc}, and \textit{C. Soulé}, Duke Math. J. 50, 763-801 (1983; Zbl 0574.14004)] for the prime-to-p part; \textit{M. Gros}, Commun. Algebra 13, 2407-2420 (1985; Zbl 0591.14003) for the p-part].
The present note uses the same tools to give a simpler proof based on a counting argument: for dim X\(=2\), one produces an injective homomorphism from \(A_ 0(X)\) to \(\pi_ 1^{geom}(X)\). This certainly implies that \(\phi\) is an isomorphism. reciprocity law for surfaces over finite fields; group of degree 0 zero- cycles; rational equivalence; abelian geometric fundamental group; unramified class field theory; K-theory; Chow groups Jean-Louis Colliot-Thélène & Wayne Raskind, ``On the reciprocity law for surfaces over finite fields'', J. Fac. Sci. Univ. Tokyo Sect. IA Math.33 (1986) no. 2, p. 283-294 Finite ground fields in algebraic geometry, Coverings in algebraic geometry, Algebraic cycles, Parametrization (Chow and Hilbert schemes), Homotopy theory and fundamental groups in algebraic geometry, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Arithmetic theory of algebraic function fields On the reciprocity law for surfaces over finite 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 For a component \(X_0\) of a nonsingular compact real algebraic surface \(X\), the author defines the complex genus of \(X_0\), denoted by \(g_\mathbb{C}(X_0)\), and uses this to prove the nonexistence of nonzero degree entire rational maps \(f:X_0\to Y\) provided that \(g_\mathbb{C} (Y)< g_\mathbb{C}(X_0)\), analogous to the topological category. He constructs connected real surfaces of arbitrary topological genus with zero complex genus. complexification; complex genus real algebraic surfaces; algebraic homology; entire rational maps Ozan, Y.: On entire rational maps of real surfaces, J. Korean Math. Soc. 39 (2002), 77-89. Topology of real algebraic varieties, Rational and birational maps, Coverings in algebraic geometry, Topological properties in algebraic geometry On entire rational maps of real 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 \(E\) be an elementary Abelian \(p\)-group of rank \(r\) with generators \(g_i\) and let \(k\) be an algebraically closed field of characteristic \(p\). Let \(M\) be a finitely generated \(kE\)-module. For each \(r\) tuple \(\alpha=(\lambda_1,\dots,\lambda_r)\) of elements of \(k\) consider the \(k\)-linear transformation \(X_\alpha=\sum\lambda_i(g_i-1)\) on \(M\). Its Jordan blocks have maximal size \(p\) (since \(X_\alpha^p=0\)); let \(a_i\) be the number of blocks of size \(i\). The expression \([p]^{a_p}\dots[1]^{a_1}\) (with terms suppressed when \(a_i=0\)) is called the Jordan type of \(X_\alpha\). If this expression is the same for all \(\alpha\neq 0\) then \(M\) is said to have constant Jordan type, and if so then \([p-1]^{a_{p-1}}\dots[1]^{a_1}\) is called the stable Jordan type of \(M\).
This paper concerns stable Jordan types \([a][b]\) and \([a]^2\). In the first case, the author shows that if \(p\geq 5\) and \(r\geq 4\) then either (1) \(a=p-b\); (2) \(a=p-b\pm1\); or (3) \(a^2+b^2-ab-1\equiv 0\pmod p\).
In the second case, the author shows that if \(p\geq 7\), \(r\geq 5\), and \(2\leq a\leq p-2\) then either \(a=(p-1)/2\) or \(a=(p+1)/2\).
Benson and Pevtsova have constructed functors from \(kE\)-modules of constant Jordan type to vector bundles on projective \(r-1\) space over \(k\). These have Chern polynomials. For a given bundle the product of the Chern polynomials of the \(0\)-th through \((p-1)\)-th twists is congruent to \(1\) modulo \(p\) up to degree \(p-1\). The author calculates these products for the Benson-Pevtsova functors and uses this congruence to obtain his results. elementary Abelian groups; modules of constant Jordan type; Chern characters; vector bundles Baland S.: Modules of constant Jordan type with two non-projective blocks. J. Algebra 346, 343--350 (2011) Modular representations and characters, Group rings of finite groups and their modules (group-theoretic aspects), Vector bundles on surfaces and higher-dimensional varieties, and their moduli Modules of constant Jordan type with two non-projective blocks. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Suppose \(\overline C\) is a smooth covering of the complex projective line, branched over a finite set of points, and suppose \(S\) is the line with these points removed. This paper considers the relationship between the (co)homology of \(\overline C\) and various twisted (co)homologies of \(S\). The main result is that the rational homology of \(\overline C\) is isomorphic to the homology of \(S\) with coefficients in local systems related to divisors of the total degree of the covering. The paper also relates intersection pairings of the two spaces. comparison of homologies; covering spaces DOI: 10.2206/kyushujm.48.111 Classical real and complex (co)homology in algebraic geometry, Other homology theories in algebraic topology, Coverings in algebraic geometry Comparison of (co)homologies of branched covering spaces and twisted ones of base spaces. 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 This survey retraces the author's talk at the Workshop \textit{Birational geometry of surfaces}, Rome, January 11--15, 2016. We consider various birational invariants extending the notion of gonality to projective varieties of arbitrary dimension, and measuring the failure of a given projective variety to satisfy certain rationality properties, such as being uniruled, rationally connected, unirational, stably rational or rational. Then we review a series of results describing these invariants for various classes of projective surfaces. Bastianelli, F., Irrationality issues for projective surfaces, Boll. unione mat. ital., (2017), in press Families, moduli, classification: algebraic theory, Special surfaces, Rational and ruled surfaces, Elliptic surfaces, elliptic or Calabi-Yau fibrations, \(K3\) surfaces and Enriques surfaces, Surfaces of general type, Hypersurfaces and algebraic geometry, Rational and birational maps, Coverings in algebraic geometry, Rational and unirational varieties, Rationally connected varieties Irrationality issues for projective surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The Lyashko-Looijenga mapping associates to a polynomial (i.e. its coefficients) the set of critical values. This map is holomorphic and maps real points to real points.
Let \(X \subset \mathbb{C}^ \mu\) be the subset of the base of the miniversal deformation of a simple singularity corresponding to polynomials with \(\mu\) distinct critical values. The restriction of the Lyashko-Looijenga map to \(X\) is a finite-sheeted covering. The multiplicity of the Lyashko- Looijenga covering restricted to a connected component of the complement in \(\mathbb{R}^ \mu\) is expressed in terms of invariants of this component.
For \(A_ \mu\)-singularities an algorithm to compute these invariants is described. simple singularities; miniversal deformation; Lyashko-Looijenga mapping Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants, Singularities in algebraic geometry, Differentiable maps on manifolds, Theory of singularities and catastrophe theory, Coverings in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Deformations of complex singularities; vanishing cycles, Coverings of curves, fundamental group, Local complex singularities On the real preimages of a real point under the Lyashko-Looijenga covering for simple singularities | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We start with a small paradigm shift about group representations, namely the observation that restriction to a subgroup can be understood as an extension-of-scalars. We deduce that, given a group \(G\), the derived and the stable categories of representations of a subgroup \(H\) can be constructed out of the corresponding category for \(G\) by a purely triangulated-categorical construction, analogous to étale extension in algebraic geometry.
In the case of finite groups, we then use descent methods to investigate when modular representations of the subgroup \(H\) can be extended to \(G\). We show that the presheaves of plain, derived and stable representations all form stacks on the category of finite \(G\)-sets (or the orbit category of \(G\)), with respect to a suitable Grothendieck topology that we call the \textit{sipp topology}.
When \(H\) contains a Sylow subgroup of \(G\), we use sipp Čech cohomology to describe the kernel and the image of the homomorphism \(T(G)\to T(H)\), where \(T(-)\) denotes the group of endotrivial representations. restrictions of representations; extensions of representations; stacks; modular representations; finite groups; ring objects; descent; endotrivial representations; categories of finite \(G\)-sets; derived categories; stable categories P. Balmer, Stacks of group representations. J. Eur. Math. Soc. (JEMS) 17 (2015), no. 1, 189--228.MR 3312406 Zbl 06419400 Modular representations and characters, Generalizations (algebraic spaces, stacks), Grothendieck topologies and Grothendieck topoi, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects), Derived categories, triangulated categories, Étale and other Grothendieck topologies and (co)homologies Stacks of group representations. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors give a geometric proof of the following fact: Let \(F\) be an extension of \(\mathbb C\) of transcendental degree 2, then (for all \(m>>0\)) \(F\) admits a subfield \(L\simeq {\mathbb C}(x_1,x_2)\) such that \(F:L\) is an algebraic extension of degree \(m\) and the group \(M(F,L)\) is the alternating group \(A_m\). Precisely, starting from a smooth complex surface \(S\) with field of rational functions \({\mathbb C}(S)\simeq F\), they construct a complex smooth surface \(X\) birationally equivalent to \(S\) and a generically finite surjective morphism \(f: X\to Y\) of degree \(m\) into a smooth complex rational ruled surface \(Y\), such that the monodromy group \(M(f)\) is \(A_m\).
The proof is based on several ingenious constructions. In particular, using a suitable pencil \(\{ F_t\}\) of curves on \(S\) and a family of meromorphic functions on \(F_t\) with even monodromy (i.e. whose monodromy group is a subgroup of the alternating group), the authors obtain a family \(p: H \to {\mathbb P^1}\) of normal rationally connected projective varieties, which (according to a recent result of \textit{T. Graber, J. Harris} and \textit{J. Starr} [J. Am. Math. Soc. 16, 57--67 (2003; Zbl 1092.14063)] and \textit{A. J. de Jong} and \textit{J. Starr} [Am. J. Math. 125, 567--580 (2003; Zbl 1063.14025)]), has a section. This section allows them to produce the required morphism \(f: X\to Y\). S. Brivio and G. Pirola. Alternating groups and rational functions on surfaces, preprint. Coverings of curves, fundamental group, Coverings in algebraic geometry, Inverse Galois theory, Special surfaces Alternating groups and rational functions on surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Branched Galois covers has been playing important roles in the study of algebraic varieties. Double covers have been intensively used to construct algebraic surfaces having prescribed Chern invariants, and cyclic covers have been used to investigate the topology of the complements to plane curves. On the other hand, there seem to be few systematic methods for non-abelian covers which are as useful as those for abelian covers; and there do not seem to be many results by using non-abelian Galois covers. Therefore it seems worthwhile to make a study of non-abelian Galois covers even for elementary non-abelian groups. \textit{H. Tsuchihashi} [Osaka J. Math. 36, 615--639 (1999; Zbl 0958.32028)] studied singularities which appear in Galois covers with Galois groups, \(D_{2n}\), \({\mathfrak A}_4\) and \({\mathfrak S}_4\). In this paper, in order to describe our conditions for constructing \({\mathfrak S}_4\) covers, we use rather global language: divisors and their linear equivalences. Both Tsuchihashi's approach and ours are based on Galois theory for \({\mathfrak A}_4\) and \({\mathfrak S}_4\). We try to understand Lagrange's method to solve quartic equations by geometric language. This is our goal for the first half of this article.
In Part II, we apply the results for \({\mathfrak S}_4\) covers in Part I to studying the topology of the complements to plane sextic curves. Let be \(G(R_x)\) and \(G(R)\) \(\sum_{x\in\text{Sing}(Z')}G(R_x)\). Denote by \(A_n\) the linear Dynkin graph. Our main results are:
Theorem. Let \(B\) be a reduced sextic curve with at most simple singularities, and let \(f:Z'\to\mathbb{P}^2\) be the double cover branched along \(B\). If there exists an \({\mathfrak S}_4\)-cover \(\pi:S\to \mathbb{P}^2\) of \(\mathbb{P}^2\) such that (i) \(\pi\) is branched at \(2B\) and (ii) \(\pi\) factors \(f:Z'\to\mathbb{P}^2\). Then \(G(R)\) contains a subgraph either \(A_2^{\oplus 9}\) or \(A_2^{\oplus 6}\oplus A_1^{\oplus 4}\).
Theorem. Suppose that \(G(R)\) contains \(A_2^{\oplus 6}\oplus A_1^{\oplus 4}\) such that \(A_1^{\oplus 4}\) is an invariant block under the involution induced by the covering transformation. Then there exists an \({\mathfrak S}_4\)-cover of \(\mathbb{P}^2\) such that (i) \(\pi\) is branched at \(2B\) and (ii) \(\pi\) factors \(f:Z'\to\mathbb{P}^2\). Tokunaga H. (2002) Galois covers for S 4 and A 4 and their applications. Osaka. J. Math. 39, 621--645 Coverings in algebraic geometry, Ramification problems in algebraic geometry Galois covers for \({\mathfrak S}_4\) and \({\mathfrak A}_4\) and their applications. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a connected variety over a field \(k\), and denote by \(\pi_1(X,\overline x)\) its algebraic fundamental group with respect to a base point \(\overline x\in X\). As it is known, \(\pi_1(X,\overline x)\) is a pro-finite group depending on the choice of a base point \(\overline x\in X\) only up to conjugation. Let \(\overline k\) be a fixed algebraic closure of the ground field \(k\), \(G_k = \Aut(\overline k/k)\) its absolute Galois group, and assume that \(X_k =X\times_k\overline k\) is still a connected variety. Then, according to the results of A. Grothendieck and his school, there is a natural outer Galois representation
\[
\rho_X: G_k\to \text{Out}(\pi_1(X_{\overline k}))
\]
of \(G_k\) in the group of outer automorphisms of \(\pi_1(X_{\overline k},\overline x)\), which is independent of the choice of the base point.
Anabelian geometry, as proposed by A. Grothendieck, studies the problem of figuring out the geometric information about \(X/k\) that is encoded in the group-theoretical structure of \(\pi_1(X,\overline x)\) or in the associated Galois representation \(\rho_X\). A variety is called anabelian, in a broadly accepted understanding, if it possesses a ``very non-abelian'' fundamental group. This philosophy of Grothendieck's, his so-called ``yoga of anabelian geometry'' declares certain varieties over fields of absolutely finite type to be anabelian, and there are basically three kinds of Grothendieck conjectures of general nature in this context [\textit{H. Nakamura}, \textit{A. Tamagawa} and \textit{S. Mochizuki}, Sugaku Expo. 14, No. 1, 31--53 (2001; Zbl 0943.14014)]: the Hom-form, the Isom-form, and the section conjecture.
The doctoral dissertation under review investigates the ``anabelian phenomenon'' in the case of hyperbolic curves, i.e., smooth curves with negative Euler characteristic, which are indeed subject to Grothendieck's conjectures in anabelian geometry. More precisely, as the affine case (in characteristic zero or over finite-fields) has been affirmatively settled by the more recent work by \textit{A. Tamagawa} [Compos. Math. 109, No. 2, 135--194 (1997; Zbl 0899.14007)] and by \textit{S. Mochizuki} [J. Math. Sci., Tokyo 3, No. 3, 571--627 (1996; Zbl 0889.11020) and Invent. Math. 138, No. 2, 319--423 (1999; Zbl 0935.14019)], the author turns his attention to the anabelian geometry of projective hyperbolic curves in positive characteristic over finitely generated field extensions of the prime field \(\mathbb{F}_p\). His main result is a complete proof of the Grothendieck conjecture for such curves in Isom-form. The precise result reads as follows:
Let \(k\) be a finitely generated field of characteristic \(p\geq 0\) with algebraic closure \(\overline k\), and let \(X\) and \(X'\) be two smooth, hyperbolic, geometrically connected curves over \(k\). Assume that at least one of these curves is not isotrivial. Then the functor \(\pi_1\) induces a natural bijection
\[
\text{Isom}_{k,F^{-1}_k}(X,X')\widetilde\rightarrow \text{Isom}_{G_k}(\pi_1(X_{\overline k}), \pi_1(X_{\overline k}'))
\]
of finite sets, where \(F^{-1}_k\) stands for the formal inverse of the geometric Frobenius morphism.
The proof of this profound theorem in anabelian geometry is utmost subtle, both strategically and methodologically innovating, and technically very refined. The basic idea is to suitably modify \textit{A. Tamagawa's} results for affine curves over finite fields [loc. cit.], which requires quite a lot of new ingredients from the theory of algebraic fundamental groups, logarithmic geometry, descent theory for log-étale covers, and from the theory of moduli stacks of projective genus-\(g\) curves. One of the most important novel tools is an abstract-algebraic version of the Van Kampen theorem in the framework of logarithmic geometry, apart from a clever specialization trick helping to make Tamagawa's earlier (affine) results applicable.
Basically, the whole thesis is divided into two main parts. Anabelian geometry, including the author's main theorem, is the topic of the second part, whereas the first part is merely of auxiliary nature, at least in view of the main objective of this doctoral dissertation. Actually, Part 1 provides the (old and new) preliminaries from logarithmic geometry needed in the sequel, including the above-mentioned abstract version of the Van Kampen theorem and the necessary ingredients from the descent theory for log-étale covers. Two appendices, at the end of the treatise, compile the utilized facts about algebraic \(K(\pi,1)\)-spaces and about isotriviality, respectively.
All in all, this thesis represents a highly substantial contribution to the development of Grothen\-dieck's fundamental, pioneering idea of anabelian geometry, with many auxiliary results that are interesting and important in their own right. Due to its depth, completeness, profundity, comprehensiveness, and clarity, this work must be seen as a decisive source for further research in this extremely advanced, yet widely unexplored area of algebraic geometry. algebraic curves; fundamental groups; coverings; logarithmic geometry; theory of descent; anabelian geometry Stix J., Projective anabelian curves in positive characteristic and descent theory for log étale covers, Bonner Math. Schriften 354, Universität Bonn, Bonn 2002. Coverings of curves, fundamental group, Coverings in algebraic geometry, Families, moduli of curves (algebraic), Finite ground fields in algebraic geometry, Varieties over finite and local fields Projective anabelian curves in positive characteristic and descent theory for log-étale covers | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(Y\) be a smooth variety with \(h^0(\omega_Y)=0\) and let \(\pi: X\to Y\) be a double cover such that \(X\) is smooth and of general type. The projection formula for double covers shows that the canonical map \(\phi\) of \(X\) factors through \(\pi\); if, in addition, the branch locus of \(\pi\) is sufficiently positive, then \(\phi\) is the composition of \(\pi\) with an embedding of \(Y\) into projective space, and we say that \(X\) is a canonical double cover of \(Y\).
Assume now that \(Y\) is one of the following rational varieties of dimension \(\geq 2\): projective space, a smooth quadric, a projective space bundle over \({\mathbb P}^1\). In this paper it is proven that, given a canonical double cover \(X\to Y\) and a \(1\)-parameter deformation \(X_t\) of \(X\), then for \(t\) general the variety \(X_t\) is a canonical double cover of a deformation of \(Y\). double cover; canonical map; deformation; surface of general type; variety of general type Gallego, F. J.; González, M.; Purnaprajna, B. P.: Deformations of canonical double covers. J. algebra 463, 23-32 (2016) Surfaces of general type, Surfaces and higher-dimensional varieties, Coverings in algebraic geometry Deformations of canonical double covers | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Fix integers \(n,m,d,t\) such that \(n\geq m\geq 1\), \(n\geq 2\), \(d\geq n!(n+1)-n\) and \(t\geq mn+1\). Here we prove that the Segre-Veronese embedding of \(\mathbb{P}^n\times\mathbb{P}^m\) induced by the complete linear system \(|{\mathcal O}+{\mathbb P^n\times \mathbb P^m}|\) is neither defective nor weakly defective. secant variety; Horace lemma Projective techniques in algebraic geometry, Coverings in algebraic geometry On the non-defectivity and non weak-defectivity of Segre-Voronese embeddings of products of \(\mathbb P^n\times\mathbb P^m\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 Galois extension \(F/K\) and an algebraic cover \(f:X\to B\) a priori defined over \(F\). The cover \(f\) may have several models (and possibly none) over each given subfield \(E\) of \(F\). How do these models compare to each other? Are there better models than others? We establish here a structure result for the set of all various models which can be used to investigate these questions. The structure, which is of cohomological nature, yields an interesting arithmetical tool: \(K\)-covers can be `twisted' to provide other \(K\)-models with possibly better properties. One application is concerned with the Beckmann-Black problem. E. Black conjectures that each Galois extension \(E/K\) is the specialization of a Galois branched cover of \(\mathbb{P}^1\) defined over \(K\) with the same Galois group \(G\). We show the conjecture holds when \(G\) is abelian and \(K\) is an arbitrary field; this was known for number fields from results of S. Beckmann (1992) and E. Black (1995). Other applications include discussions of existence for a given cover, of a ``good'' model, a stable model, a model with a totally rational fiber etc. Finally, we continue our study of local-global principles for covers: If a cover is defined over each \(\mathbb{Q}_p\), does it follow it is defined over \(\mathbb{Q}\)? Here we consider the case of Galois covers of a general base space \(B\). Galois extension; Beckmann-Black problem; local-global principles; Galois covers P. Dèbes , Some arithmetic properties of algebraic covers , In: '' Aspects of Galois Theory '', Cambridge University Press , (to appear). MR 1708602 | Zbl 0977.14009 Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Coverings in algebraic geometry, Separable extensions, Galois theory Some arithmetic properties of algebraic covers | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For a connected reduced scheme X over an arbitrary field k the fundamental group scheme is constructed and studied (which is a group scheme over k) and its connection with principal G-fiberings, where G is a finite (or nilpotent) group scheme, and finite linear fiberings on X is investigated.
For X and its fixed k-point \(\chi_ 0:\quad Spec k\to X\) a triple \((P,\pi (X,\chi_ 0),*)\) is constructed (where \(\pi (X,\chi_ 0)\) is the group k-scheme which is the projective limit of finite group schemes, P is the principal \(\pi (X,\chi_ 0)\) fibering over X with marked point * lying over \(\chi_ 0)\) which has the universality property with respect to principal G-fiberings on X, where G is a finite group k-scheme with marked point lying over \(\chi_ 0\). The group k-scheme \(\pi (X,\chi_ 0)\) is called the fundamental group scheme of the scheme X and possesses many natural properties.
The case of a complete scheme X over a perfect field k was considered by the author earlier and is connected with finite and essentially finite vector bundles over X which are multidimensional generalizations of one- dimensional bundles corresponding to the finite order points on the Jacobian of a curve [the author, Compos. Math. 33, 29-41 (1976; Zbl 0337.14016)]. Owing to the consideration of non-complete schemes the author carries over these results to linear, parabolic (defined by Seshadri) essentially finite bundles over a smooth projective curve X with a finite set S of points deleted. Such bundles are constructed by k- linear representations of the group \(\pi (X-S,\chi_ 0)\), where \(\chi_ 0\not\in S\), which are passed through representations of finite group schemes over k.
The last chapter is devoted to the study of a nilpotent fundamental group-scheme \(U(X,\chi_ 0)\) (it is constructed if \(\Gamma\) (X,\({\mathcal O}_ X)=k)\). If char k\(>0\) and \(\dim H^ 1(X,{\mathcal O}_ X)<\infty\) then \(U(X,\chi_ 0)\) is a factor of \(\pi (X,\chi_ 0)\). The connection of \(U(X,\chi_ 0)\) with Pic X is studied. In the case of complete reduced curves and \(p=char k>0\), the computation of \(U(X,\chi_ 0)\) leads to non-commutative formal groups which are computed by the author for rational curves with the simplest singularities. As a corollary, an old result of I. R. Shafarevich is reproved: For a complete curve X with \(\Gamma\) (X,\({\mathcal O}_ X)=k\), the maximal p-factor of the étale fundamental group is a free pro-p-group in characteristic p.
The main means used by the author are the equivalence between the Tannaka category and the category of finite-dimensional representations of a certain affine group scheme [see \textit{N. R. Saavedra}, ''Catégories tannakiennes'', Lect. Notes Math. 265 (1972; Zbl 0241.14008)]. finite-dimensional representations of affine group scheme; fundamental group scheme; non-commutative formal groups; characteristic p Nori, M. V., \textit{the fundamental group-scheme}, Proc. Indian Acad. Sci. Math. Sci., 91, 73-122, (1982) Coverings in algebraic geometry, Group schemes, Homotopy theory and fundamental groups in algebraic geometry, Schemes and morphisms The fundamental group-scheme | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The purpose of this paper is to construct a complex which computes the homology of Hurwitz spaces of branched covers of \(\mathbb{P}^1\). We also compute some of the low dimensional homology groups of a compactification of the Hurwitz space, and report on computer calculations performed in specific examples. Picard group; homology of Hurwitz spaces of branched covers of projective line Diaz, S.; Edidin, D., Towards the homology of Hurwitz spaces, J. Diff. Geom., 43, 66-98, (1996) Coverings in algebraic geometry, Classical real and complex (co)homology in algebraic geometry, Picard groups Towards the homology of Hurwitz spaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper contains more or less trivial remarks concerning the relationship between the commutator subgroup of the fundamental group of a connected complex analytic manifold and its associated covering space. fundamental group; covering Coverings in algebraic geometry Homology group and generalized Riemann-Roch theorem | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems With each meromorphic function (which is assumed to be -- except at critical points -- an \(m\)-to-one holomorphic mapping of a compact connected Riemann surface onto the Riemann sphere, with a total of \(n\) finite critical values), an edge labelled graph on \(m\) vertices and \(n\) edges is associated. The graph itself reflects in a natural way the ramification of the meromorphic function; the edge labelling is a 1-1 mapping from the edge set to \(\{ 1,2,\dots,n\}\). The Coxeter permutation of such a labelled graph is the permutation of its vertex set defined as the product \(\sigma_1\dots\sigma_n\) where \(\sigma_i\) is the transposition which permutes the endvertices of the edge labelled \(i\).
The author surveys some of the connections between meromorphic functions, their edge labelled graphs, and the corresponding Riemann surfaces. Laurent polynomials are then studied in detail. Typical results: The manifold of classes of topological equivalence of Laurent polynomials with poles of order \(p\) and \(q\) and with \(p+q\) distinct critical values is naturally covered by the manifolds of bipolynomials of bidegree \((p,q)\) with \(p+q\) distinct critical values and naturally covers the complement of the swallow tail in the space \(C^{p+q-1}\) with multiplicity \(M(p,q)=p^pq^q(p+q-1)!/p!q!\). The same manifold is an Eilenberg-MacLane space \(K(\pi,1)\) for the corresponding subgroup of index \(M(p,q)\) in the braid group on \(n\) strings. Results of the above type have implications in combinatorics and graph theory, because the topological classification of the above mappings is equivalent to the enumeration of the connected graphs on \(n\) vertices with \(n\) labelled edges (i.e., edge labelled unicyclic graphs of order \(n\)). For example, it follows that the number of edge labelled \((n,n)\) graphs whose Coxeter permutation consists of two cycles of lengths \(p\) and \(q\) is equal to \(M(p,q)\). meromorphic function; Riemann surface; Laurent polynomial; edge labelled graph; Coxeter permutation V. I. Arnold, ''Topological classification of complex trigonometric polynomials and the combinatorics of graphs with an identical number of vertices and edges,'' Funktsional. Anal. i Prilozhen. 30(1), 1--17 (1996) [Functional Anal. Appl. 30 (1), 1--14 (1996)]. Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants, Enumeration in graph theory, Coverings in algebraic geometry, Braid groups; Artin groups Topological classification of trigonometric polynomials and combinatorics of graphs with an equal number of vertices and edges | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We study the intersection of the Torelli locus with the Newton polygon stratification of the modulo \(p\) reduction of certain Shimura varieties. We develop a clutching method to show that the intersection of the open Torelli locus with some Newton polygon strata is non-empty. This allows us to give a positive answer, under some compatibility conditions, to a question of Oort about smooth curves in characteristic \(p\) whose Newton polygons are an amalgamate sum. As an application, we produce infinitely many new examples of Newton polygons that occur for smooth curves that are cyclic covers of the projective line. Most of these arise in inductive systems that demonstrate unlikely intersections of the open Torelli locus with the Newton polygon stratification in Siegel modular varieties. In addition, for the 20 special Shimura varieties found in Moonen's work, we prove that all Newton polygon strata intersect the open Torelli locus (if \(p\gg 0\) in the supersingular cases). Modular and Shimura varieties, Arithmetic aspects of modular and Shimura varieties, Coverings in algebraic geometry, Formal groups, \(p\)-divisible groups, Toric varieties, Newton polyhedra, Okounkov bodies Newton polygon stratification of the Torelli locus in unitary Shimura 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 shall give, in an optimal form, a sufficient numerical condition for the finiteness of the fundamental group of the smooth locus of a normal \(K3\) surface. We shall moreover prove that, if the normal \(K3\) surface is elliptic and the above fundamental group is not finite, then there is a finite covering which is a complex torus. Catanese, F; Keum, J; Oguiso, K, Some remarks on the universal cover of an open \(K3\) surface, Math. Ann., 325, 279-286, (2003) \(K3\) surfaces and Enriques surfaces, Homotopy theory and fundamental groups in algebraic geometry, Coverings in algebraic geometry Some remarks on the universal cover of an open \(K3\) surface | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(K\) be a field of characteristic \(p\)\(\neq 0\), \(\overline {K}\) be an algebraic closure of \(K\), \(v\) be a rank 1 valuation of \(\overline {K}\) and \(K^h\) be the decomposition field of \(v\) in the separable closure of \(K\) (in \(\overline {K}\)). Let \(S\) be a finite closed subset of \(\mathbb{P}^1_K\) and \(\overline {S}= S\times_K \overline {K}= \{s_1, \dots, s_m\}\) with \(m=2n\) even. The author says that \(S\) is pairwise \(v\)-adjusted if the elements of \(\overline {S}\) can be organized in pairs \(p_k= (x_k, y_k)\), \(1\leq k\leq n\), which are permuted between themselves by \(G^h= \text{Gal} (\overline {K}/ K^h)\) and satisfy \(v(x_k- y_k)> v(x_k- x_{k'})\) for all \(k\neq k'\). Let \(\Pi\) be the profinite group defined (by generators and relations) as follows: \(\Pi= \{g_{x_1}, h_{x_1}, \dots, g_{x_n}, h_{x_n}|g_{x_k} h_{x_k}=1\), \(g^s_{x_k} =1\) for all \(k\}\) where \(s\) is a topological generator of \(\widehat {\mathbb{Z}}\) and \(g^s_{x_k}=1\) is a topological relation; \(\Pi\) is endowed with a right \(G^h\) action by \((\xi\) is the cyclotomic character of \(G^h)\): \(g^\sigma_{x_k}= g^{\xi( \sigma^{-1})}_{\sigma^{-1} x_k}\), \(\sigma\in G^h\).
Main theorem. Let \(\mathbb{U}= \mathbb{P}^1_K- S\), \(\overline {\mathbb{U}}= \mathbb{U}\times_K \overline {K}\), \(\mathbb{U}^{\mathbf h}= \mathbb{U}\times_K K^h\). Suppose that \(S\) is pairwise \(v\)-adjusted for a rank 1 valuation \(v\) of \(\overline {K}\). Then there exists \(\Pi\) as before, an exact sequence of groups with \(G^h\) actions
\[
1\to \Pi\to G^h \propto \Pi\to G^h\to 1 \tag{1}
\]
which is canonically a quotient of the canonical exact sequence
\[
1\to \pi_1 (\overline {\mathbb{U}})\to \pi_1 (\mathbb{U}^{\mathbf h})\to G^h\to 1. \tag{2}
\]
Moreover, the generators \(g_{x_k}\) and \(h_{x_k}\) are inertia elements associated to \(x_k\) and \(y_k\) respectively for all \(k\).
Clairly, this result is closed to the classical Riemann existence theorem. It is interesting because \(\text{char} (K)>0\): the canonical sequence (2) is split, but one does not know in general the structure of \(\pi_1 (\overline {\mathbb{U}})\), on the contrary, in the sequence (1), the kernel of \(G^h\propto \Pi\to G^h\) is known. A variant of the main theorem in unequal characteristics gives a ``\({1\over 2}\) Riemann existence theorem'' (theorem 2). profinite group as fundamental group; characteristic \(p\); Riemann existence theorem Pop, F.: 12 Riemann existence theorem with Galois action. Algebra and number theory (1994) Local ground fields in algebraic geometry, Coverings in algebraic geometry, Fundamental groups and their automorphisms (group-theoretic aspects) \(1\over 2\) Riemann existence theorem with Galois action | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 Riemannian manifold (respectively a complex space) \(X\) is called Liouville if it carries no nonconstant bounded harmonic (respectively holomorphic) function. It is called Carathéodory, or Carathéodory hyperbolic, if bounded harmonic (respectively holomorphic) functions separate the points of \(X\). A Galois covering \(X\) with Galois group \(G\) is called a \(G\)-covering.
In this paper, the authors are interested in the following problem: When is a \(G\)-covering over a Liouville base \(Y\) Liouville, or at least, when is it not Carathéodory hyperbolic? The paper contains a survey of known results together with some new results on the subject. Previous results of Lyons-Sullivan and of the first author show that to ensure the Liouville property of \(X\) one has to impose strong conditions on \(Y\), and for this reason the authors assume here that \(Y\) is compact, or at least, that \(Y\) does not carry non-constant bounded subharmonic (respectively plurisubharmonic) functions. The authors show in particular, as a consequence of theorems of Varopoulos and of Lyons-Sullivan, that a \(G\)-covering over a compact Riemannian (respectively Kähler) manifold is Liouville if \(G\) is an extension of an almost nilpotent group by \(\mathbb{Z}\) or \(\mathbb{Z}^2\). They give then examples of non-Liouville and, especially, of Carathéodory hyperbolic cocompact coverings with relatively small Galois groups. Using again a construction of Lyons-Sullivan, they produce, on any compact Riemann surface of genus \( \geq 2\), a Carathéodory hyperbolic two-step solvable covering. The authors consider then the universal covering \(X\to {\mathcal I}\) of an Inoue surface. (These surfaces were introduced by \textit{M. Inoue} in [Invent. Math. 24, 269-310 (1974; Zbl 0283.32019)]). The surface \({\mathcal I}\) considered here is a non-Kähler compact complex surface with a polycyclic fundamental group, and \(X\) is neither Liouville nor Carathéodory. The authors show that \(X\) admits bounded holomorphic functions which are nonconstant on the orbits of suitable infinite conjugacy classes in \(\pi_1(G)\). Liouville manifold; harmonic function; Carathéodory hyperbolic; \(G\)-space; Kähler manifold; covering space Vladimir Ya. Lin and Mikhail Zaidenberg, Liouville and Carathéodory coverings in Riemannian and complex geometry, Voronezh Winter Mathematical Schools, Amer. Math. Soc. Transl. Ser. 2, vol. 184, Amer. Math. Soc., Providence, RI, 1998, pp. 111 -- 130. Picard-type theorems and generalizations for several complex variables, Coverings in algebraic geometry, Local differential geometry of Hermitian and Kählerian structures Liouville and Carathéodory coverings in Riemannian and complex 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 In this paper we construct families of algebraic nonsingular 3-folds \(X\) of general type having minimal \(K^3_X\) for their \(p_g\) in order that the canonical morphism is finite of degree 3. For such 3-folds it is \(K^3_X= 3(p_g-3)\) and \(p_g\) is odd. geometric genus; triple cover; 3-folds of general type; canonical morphism \(3\)-folds, Coverings in algebraic geometry, Projective techniques in algebraic geometry Triple covers of 3-folds as canonical maps | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(k\) be an algebraically closed field of characteristic \(p>0\). Let \(G\) be a semi-direct product of the form \((\mathbb Z/\ell\mathbb Z)^b\rtimes\mathbb Z/p\mathbb Z\) where \(b\) is a positive integer and \(\ell\) is a prime distinct from \(p\). In this paper, we study Galois covers \(\psi:Z\to\mathbb P_k^1\) ramified only over \(\infty\) with Galois group \(G\). We find the minimal genus of a curve \(Z\) which admits a covering map of this form and we give an explicit formula for this genus in terms of \(\ell\) and \(p\). The minimal genus occurs when \(b\) equals the order \(a\) of \(\ell\) modulo and \(b\) and we also prove that the number of curves \(Z\) of this minimal genus which admit such a covering map is at most \((p-1)/a\) when \(p\) is odd. Coverings of curves, fundamental group, Coverings in algebraic geometry, de Rham cohomology and algebraic geometry Semi-direct Galois covers of the affine line | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We show that there exists a Lamé operator \(L_n\) with projective octahedral monodromy for each \(n\in{\frac12(\mathbb{N}+\frac12)}\cup{\frac13(\mathbb{N}+\frac12)}\), and with projective icosahedral monodromy for each \(n\in{\frac13(\mathbb{N}+\frac12)}\cup{\frac15(\mathbb{N}+\frac12)}\). To this end, we construct Grothendieck's dessins d'enfants corresponding to the Belyi morphisms which pull back hypergeometric operators into Lamé operators \(L_n\) with the desired monodromies. Structure of families (Picard-Lefschetz, monodromy, etc.), Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms, Special ordinary differential equations (Mathieu, Hill, Bessel, etc.), Coverings in algebraic geometry Lamé operators with projective octahedral and icosahedral monodromies | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper we mainly consider surfaces with a ``representation'' as a family of projective curves over a curve. We use a base change construction combined with a finite ramified covering to associate a collection of surfaces with infinite fundamental groups to a given surface. Most of these fundamental groups were not previously known to be fundamental groups of smooth projective surfaces. We also analyze the universal coverings of these surfaces in the context of the Shafarevich's conjecture which states the universal covering of a smooth complex projective variety must be holomorphically convex. \textit{E. Zelmanov} conjectured that certain explicit quotients of free groups and of surface groups are non-residually finite groups. If the above conjecture is true then the base change construction provides us with many new simple examples of nonresidually finite fundamental groups of smooth projective surfaces.
The authors also raise the question whether there are certain types of finite (resp. infinite) quotients of free (resp. surface) groups given by explicit presentations. If the answer to this question is affirmative then one gets a counterexample to the Shafarevich conjecture. fundamental group; Shafarevich conjecture; curves over a curve; surfaces; covering F. Bogomolov and L. Katzarkov, ''Complex projective surfaces and infinite groups,'' Geom. Funct. Anal., vol. 8, iss. 2, pp. 243-272, 1998. Homotopy theory and fundamental groups in algebraic geometry, Rational and ruled surfaces, Fundamental groups and their automorphisms (group-theoretic aspects), Coverings in algebraic geometry Complex projective surfaces and infinite 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 $X$ be a smooth complex projective variety, $x$ be a point of $X$ and $\rho : \pi _1 (X, x) \rightarrow\text{SL} (3, {\mathbb C})$ be a rigid irreducible representation. Then $\rho$ and its associated local system $V_{\rho}$ are called integral if the traces of $\rho ( \gamma)$ are algebraic initegers for all $\gamma \in \pi _1 (X, x)$. The monodromy representtion $\rho$ and its corresponding local system $V_{\rho}$ are of geometric origin if there is a Zariski open subset $U \subseteq X$ and a smooth projective family $f: Z \rightarrow U$ of algebraic varieties, such that $V_{\rho} \vert _U$ is a direct factor of some higher direct image $R^{i} f_* ( {\mathbb C}_Z)$ of the constant sheaf ${\mathbb C}_Z$ on $Z$. The article under review establishes that any rigid integral irreducible representation $\rho : \pi _1 (X, x) \rightarrow \text{SL} (3, {\mathbb C})$ is of geometric origin. \par By a result of Corlette, any rigid local system $V_{\rho}$ underlies a ${\mathbb C}$-variation of Hodge structure. An article of \textit{K. Corlette} and \textit{C. Simpson} [Compos. Math. 144, No. 5, 1271--1331 (2008; Zbl 1155.58006)] implies that the ${\mathbb C}$-variation of Hodge structure, associated with a rigid irreducible local system $V_{\rho}$ of rank $3$ is either of weight $1$ or of weight $2$ and with Hodge numbers $(1,1,1)$. The polarized ${\mathbb Z}$-variations of Hodge structure of weight $1$ are known to arise from smooth algebraic families of abelian varieties. That is why, the main result of the article reduces to the characterization of the irreducible local systems of rank $3$, underlying a ${\mathbb C}$-variation of Hodge structure of type $(1,1,1)$. An irreducible representation $\rho : \pi _1 (X, x) \rightarrow \text{GL} (n, {\mathbb C})$ projectively factors through an orbicurve $C$ if there exist a surjective morphism $f : X \rightarrow C$ with connected fibres and a representation $\tau : \pi _1 (C, f(x)) \rightarrow \text{PGL} (n, {\mathbb C})$, such that the projectivization $\overline{\rho} : \pi _1 (X, x) \rightarrow \text{PGL} (n, {\mathbb C})$ of $\rho$ has factorization $\overline{\rho} = \tau f_*$. The authors show that if an irreducible representation $\rho : \pi _1 (X, x) \rightarrow \text{SL} (3, {\mathbb C})$ is a ${\mathbb C}$-variation of Hodge structure of type $(1,1,1)$ then either $\rho$ projectively factors through an orbicurve or there exist a finite covering $\pi : Y \rightarrow X$ by a smooth projective variety $Y$, a rank $1$ local system $W_1$ on $X$ and a rank $2$ local system $W_2$ on $Y$, such that $\pi ^* V_{\rho} = \pi ^* W_1 \otimes \text{Sym} ^2 (W_2)$ and $W_1 ^{\otimes 3}$ is trivial. More precisely, let $X$ be a smooth quasi-projective variety with a smooth projective compactification $\overline{X}$ by a divisor $D = \overline{X} \setminus X$ with simple normal crossings and $A$ be a fixed very ample divisor on $\overline{X}$. Then any logarithmic system $(E = E^{2,0} \oplus E^{1,1} \oplus E^{0,2}, \theta)$ of Hodge bundles of type $(1,1,1)$ is associated with rank $1$ subsheaves $L_1 := E^{2,0} \otimes ( E^{1,1}) ^* \hookrightarrow \Omega _{\overline{X}} ( \log D)$ and $L_2 := E^{1,1} \otimes ( E^{0,2}) ^* \hookrightarrow \Omega _{\overline{X}} ( \log D)$. If $(E, \theta)$ is slope $A$-stable and $E$ has vanishing rational Chern classes then the classes of $L_1$ and $L_2$ in $H^2 ( \overline{X}, {\mathbb Q})$ are shown to be on one and a same line. Moreover, there exist effective divisors $B_1$, $B_2$, which provide a common saturation $M := L_1 (B_1) = L_2 (B_2)$ of $L_1$ and $L_2$ in $\Omega_{\overline{X}} ( \log D)$ with $L_i. M =0$ and $L_i. B_j =0$ for all $i,j$. For any generic smooth irreducible complete intersection $\overline{Y}$ in $\overline{X}$, exactly one of the restrictions $L_1 \vert _{\overline{Y}}$ or $L_2 \vert _{\overline{Y}}$ is nef and has $(L_i \vert _{\overline{Y}})^2=0$, $\deg (L_i \vert _{\overline{Y}}) >0$. \par Let $\rho : \pi _1 (X, x) \rightarrow G$ be an irreducible representations of the fundamental group $\pi _1 (X,x)$ of a smooth complex projective variety $X$ in a connected semi-simple Lie group $G$ over ${\mathbb C}$, $f : Z \rightarrow X$ be a finite surjective morphism and $G^{o} _{\rho f_*}$ be the identity component of the Zariski closure $G_{\rho f_*}$ of $\rho f_* (\pi _1 (Z, z))$, $z \in f^{-1} (x)$ in $G$. The article shows that if for any finite etale covering $f : Z \rightarrow X$ the subgroup $G_{\rho _f*}$ of $G$ is not contained in a proper maximal parabolic subgroup of $G$ and $G_{\rho _f*}$ has a proper normal Zariski closed connected subgroup of positive dimension then $\rho f_* : \pi _1 (Z, z) \rightarrow G$ factors through a representation $\tau : \pi _1 (Z, z) \rightarrow H_1 \times \ldots \times H_m$ in the direct product of the minimal Zariski closed connected subgroups $H_i$ of $G^{o} _{\rho _f*}$ of positive dimension. The aforementioned result enables to characterize the projective factorization through an orbicurve for an irreducible representation . In order to formulate precisely, the authors introduce the notion of an alteration, which is a proper surjective geometrically finite morphism $f : Z \rightarrow X$ of a smooth variety $Z$. A representation $\rho : \pi _1 (X, x) \rightarrow \text{SL} (n, {\mathbb C})$ is virtually reducible if there is an alteration $f: Z \rightarrow X$, such that the local system $V_{\rho f_*}$, associated with $\rho f_* : \pi _1 (Z, z) \rightarrow \text{SL} (n, {\mathbb C})$ decomposes into a direct sum of non-zero local systems. Similarly, $\rho$ is virtually tensor decomposable if $V_{\rho f_*}$ is a tensor product of two local systems of rank $\geq 2$ for some alteration $f : Z \rightarrow X$. A representation $\rho$ virtually projectively factors through an orbicurve if there is an alteration $f: Z \rightarrow X$, such that $\rho f_* : \pi _1 (Z, z) \rightarrow \text{SL} (n, {\mathbb C})$ projectively factors through an orbicurve. If an irreducible representation $\rho: \pi _1 (X, x) \rightarrow \text{SL} (n, {\mathbb C})$, $n \geq 2$ is not virtually reducible, nor virtually tensor decomposable then the virtual projective factorization of $\rho$ through an orbicurve is shown to be equivalent to the projective factorization of $\rho$ through an orbicurve. Moreover, this holds exactly when there exist an alteration $f : Z \rightarrow X$ and a morphism $g : Z \rightarrow C$ to an orbicurve $C$, such that the restriction of $\rho f _*$ to the fundamental group $\pi _1 ( g^{-1} (c))$ of a fibre $g^{-1} (c) \subset Z$ of $g$ is reducible. In particular, if $\rho : \pi _1 (X,x) \rightarrow \text{SL} (3, {\mathbb C})$ is an irreducible representation, underlying a variation of Hodge structure with at least two non-zero Hodge numbers, then $\rho$ is not virtually reducible, nor tensor decomposable and there holds the aforementioned characterization for projective factorization of $\rho$ through an orbicurve. \par The article under review discusses also some properties of the local systems $V$ of geometric origin. More precisely, a direct sum $V_1 \oplus V_2$ of local systems is shown to be of geometric origin if and only if $V_1$ and $V_2$ are of geometric origin. If $V_1$ and $V_2$ are of geometric origin then $V_1 \otimes V_2$ is of geometric origin. A rank $1$ local system is of geometric origin exactly when it is of finite order. If $f : Y \rightarrow X$ is a generically surjective morphism of irreducible varieties and $V$ is a semi-simple local system on $X$ then $f^* (V)$ is of geometric origin if and only if $V$ is of geometric origin. Irreducible representations $\rho _i : \pi _1 (X,x) \rightarrow \text{SL} (n, {\mathbb C})$, $1 \leq i \leq 2$ with isomorphic projectivizations and determinants of finite order are proved to be simultaneously of geometric origin. \par The characterization of the irreducible local systems of rank $3$, which are ${\mathbb C}$-variation of Hodge structure of type $(1,1,1)$ allows to study those integral representations $\rho : \pi _1 (X, x) \rightarrow \text{SL} (3, K)$ over a number field $K$ and their associated local systems $V$, for which all embeddings $\sigma : K \rightarrow {\mathbb C}$ induce ${\mathbb C}$-variations of Hodge structure $V_{\sigma}$ with Zariski dense $\rho ( \pi _1 (X, x))$ in $\text{SL} (3, \overline{K})$. For any such $V$, either $\rho$ projectively factors through an orbicurve or all $V_{\sigma}$ are direct factors of the monodromy of a family of abelian varieties. Combining with some results of [Zbl 1155.58006], the article establishes that if an integral irreducible representation $\rho : \pi _1 (X, x) \rightarrow \text{GL} (3, {\mathbb C})$ with Zariski closure $G_{\rho}$ of $\rho ( \pi _1 (X, x))$ in $\text{Gl} (3, {\mathbb C})$ is rigid as a representation $\rho : \pi _1 (X, x) \rightarrow G_{\rho}$ and if the determinant $\det (V_{\rho})$ of the associated local system $V_{\rho}$ is of finite order then $\rho$ and $V_{\rho}$ are of geometric origin. As far as all rigid local systems have determinant of finite order, that suffices for any rigid integral irreducible representation $\rho : \pi _1 (X, x) \rightarrow \text{SL} (3, {\mathbb C})$ to be of geometric origin. \par As a corollary of the main result of the article, the standard representation of any integral uniform torsion-free lattice $\Gamma$ of $\text{PU} (2,1)$ turns to be of geometric origin. fundamental group; complex varieties; rigid representation; complex variation of Hodge structure Homotopy theory and fundamental groups in algebraic geometry, Fibrations, degenerations in algebraic geometry, Variation of Hodge structures (algebro-geometric aspects), Coverings in algebraic geometry Rank 3 rigid representations of projective fundamental groups | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For part I see \textit{G. Casnati} and \textit{T. Ekedahl}, J. Algebr. Geom. 5, No. 3, 439-460 (1996; Zbl 0866.14009).]
In part I of this article there were studied general Gorenstein covers. By a result of part I the author gives first the structure of a Gorenstein scheme \(X\) of dimension 0 and degree 5. More precisely there is an embedding \(i: X \hookrightarrow {\mathbb{P}} _k ^3\) and the minimal resolution of the homogeneous ring \(S_X\) of \(i(X)\) as a module over \(S = k[x_0, \dots,x_3]\) is:
\[
0 \longrightarrow S(-5) \longrightarrow S(-3) ^{\oplus 5} \buildrel \delta \over \longrightarrow S(-2) ^{\oplus 5}\longrightarrow S \longrightarrow S_X \longrightarrow 0,
\]
where the matrix \(\Delta\) of \(\delta\) is alternating and \(X\) is defined by the vanishing of the pfaffians of order 2 of \(\Delta\). Let \(\rho: X \to Y\) be a Gorenstein cover of degree 5 with \(Y\) integral, and let \({\mathbb{P}}\) be the projective bundle \(\pi: {\mathbb{P}} ({\mathcal E}) \to Y\), where \({\mathcal E}\) is the locally free sheaf of rank 4 on \(Y\) which is the dual of the cokernel of \(\rho ^{\sharp}: {\mathcal O} _Y \to \rho _* {\mathcal O}_X\). By part I of this paper there is an exact sequence
\[
0 \longrightarrow {\mathcal N}_3 (-5) \buildrel {\alpha _2} \over \longrightarrow {\mathcal N}_2 (-3) \buildrel {\alpha _1} \over \longrightarrow {\mathcal N}_1 (-2) \longrightarrow {\mathcal O} _{\mathbb{P}} \longrightarrow {\mathcal O} _X \longrightarrow 0,
\]
where \({\mathcal N}_3 \simeq \pi ^* \text{det} {\mathcal E}\) and \(\text{rk} {\mathcal N}_2 = \text{rk} {\mathcal N}_1 =5\). By the self-duality and uniqueness of this presentation of \({\mathcal O}_X\), we can assume that this exact sequence is:
\[
0 \longrightarrow \pi ^* \text{det} {\mathcal E} (-5)\longrightarrow \breve {\mathcal N} \otimes \text{det} {\mathcal E} (-3) \buildrel \delta \over \longrightarrow {\mathcal N} (-2) \longrightarrow {\mathcal O} _{\mathbb{P}} \longrightarrow {\mathcal O} _X \longrightarrow 0,
\]
where \({\mathcal N}: = \pi ^* {\mathcal F}\) and \(\delta \in {\text{H}} ^0 ({\mathbb{P}} , \wedge ^2 {\mathcal N} \otimes \text{det} {\mathcal E} ^{-1} (1))\). From the isomorphism
\[
{\text{H}} ^0 ({\mathbb{P}} , \wedge ^2 {\mathcal N}\otimes \text{det} {\mathcal E} ^{-1} (1)) \simeq \text{Hom} _Y ( \wedge ^2 \breve {\mathcal F} \otimes \text{det} {\mathcal E} , {\mathcal E}),
\]
we get a map \(\eta: \wedge ^2 \breve {\mathcal F} \otimes \text{det} {\mathcal E} \to {\mathcal E}\) and we say that the cover \(\rho\) is regular at \(y \in Y\) if \(\eta\) is surjective, and is special if not. For a locally free \({\mathcal O}_Y\)-sheaf \({\mathcal F}\) of rank 5 and \({\mathcal G} (2, {\mathcal F}) \hookrightarrow \bar {\mathbb{P}} = {\mathbb{P}} ( \wedge ^2 \breve {\mathcal F}) \buildrel p \over \to Y\) the Plücker embedding of the Grassmann bundle, the author gives an exact sequence:
\[
\begin{multlined} 0 \longrightarrow p ^* \text{det } {\mathcal F} ^{-2} (-5) \buildrel {\breve \gamma _Y} \over \longrightarrow p ^* (\breve {\mathcal F} \otimes \text{det } {\mathcal F} ^{-1}) (-3) \buildrel {G_Y} \over \longrightarrow p ^* ({\mathcal F} \otimes \text{det } {\mathcal F} ^{-1}) (-2) \buildrel {\gamma _Y} \over \longrightarrow\\ {\mathcal O} _{\overline {\mathbb{P}}} \longrightarrow {\mathcal O} _{{\mathcal G} (2, {\mathcal F})} \longrightarrow 0.\end{multlined}
\]
By comparing this exact sequence with the previous one, the author shows that the set of special points of the cover \(\rho\) is \(D_3 (\eta)\), where \(D_r (\varphi)\) is, for any morphism \(\varphi\) between locally free sheaves \({\mathcal F}\) and \({\mathcal G}\) on a scheme \(X\), the closed subscheme of \(X\) defined by the \((r+1) \times (r+1)\)-minors of the matrix of \(\varphi\), i.e. \(| D_r (\varphi)| =\{ x \in X\mid \text{rk } \varphi _x \leq r \}\). Moreover if \(\rho\) is regular at each \(y \in Y\), the author gets \(X = D_2 (\delta)\) and \(\det{\mathcal F} \simeq \det{\mathcal E} ^2\). Then the author gives a complete description of covers of degree 5, with an extra condition: \(\eta\) has the right codimension at \(y \in Y\) if \(\dim D_2 (\delta _y) = 0\) and \(D_0 (\delta _y) = \emptyset\).
Theorem: Any regular cover \(\rho:X \to Y\) of degree 5 of an integral scheme \(Y\) with \(\breve {\mathcal E} \simeq\) coker\(\rho ^{\#}\), determines a locally free \({\mathcal O}_Y\)-sheaf \({\mathcal F}\) of rank 5 such that det \({\mathcal F} \simeq\) det \({\mathcal E}^2\) and a section \(\eta \in {\text{H}}^0 (Y , \wedge ^2 {\mathcal F} \otimes {\mathcal E} \otimes \text{det } {\mathcal E} ^{-1})\) having the right codimension at each \(y \in Y\). -- Conversely given a locally free sheaf \({\mathcal F}\) of rank 5 with det \({\mathcal F} \simeq\) det \({\mathcal E}^2\) and \(\eta \in {\text{H}}^0 (Y , \wedge ^2 {\mathcal F} \otimes {\mathcal E} \otimes \text{det } {\mathcal E} ^{-1})\) having the right codimension at each \(y \in Y\), the restriction \(\rho\) of \(\pi:{\mathbb{P}} ({\mathcal E}) \to Y\) at \(X = D_2 (\Phi _5 (\eta))\) is a regular cover of degree 5 such that \(\breve {\mathcal E} \simeq\) coker \(\rho ^{\#}\) and \({\mathcal F} \simeq\) ker \(({\mathcal S}^2 {\mathcal E} \to \rho _* \omega _{X| Y})\).
For a curve or a surface \(Y\) over \({\mathbb{C}}\), the author gives a Bertini type theorem:
Theorem: If \(Y\) is smooth, projective, \(\dim (Y) \leq 2\), over \({\mathbb{C}}\) and if the sheaf \({\mathcal H} := \wedge ^2 {\mathcal F} \otimes {\mathcal E} \otimes \text{det } {\mathcal E} ^{-1}\) is globally generated, any generic section \(\eta \in{\text{H}}^0 (Y,{\mathcal H})\) defines a cover \(\rho:X \to Y\) of degree 5 which is regular at every point \(y \in Y\) with \(X\) smooth.
The proof of this result is standard, we have to compute the dimension of some incidence subvariety in a Hilbert scheme associated to the Gorenstein subschemes of the Grassmannian \({\mathcal G}(2,5)\). If we have a cover \(\rho:X \to Y\) of degree 5, with \(X\) and \(Y\) smooth, connected, projective surfaces we can compute the invariants of \(X\) in terms of the invariants of \(Y\) and of the Chern classes of \({\mathcal E}\) and \({\mathcal F}\). It is a direct consequence of the isomorphisms \(\breve {\mathcal E} \simeq \text{coker } \rho ^{\#}:{\mathcal O} _Y \to \rho _* {\mathcal O}_X\) and \({\mathcal F} \simeq\text{ker} ({\mathcal S}^2 {\mathcal E} \to \rho _* \omega {X| Y})\). By using these results the author can give a description of the canonical model \(X\) of a minimal surface \(S\) of general type with \(K_S ^2 =5\), \(p_g (S) =3\), whose canonical system is base-point-free. More generally he shows that for any \(d= 3,\dots,9\), there exist minimal surfaces \(S\) of general type with \(K_S ^2 =d\), \(p_g (S) =3\), \(q(S)=0\) whose canonical system is base-point-free and not composed with a pencil; and the canonical map of such surfaces is a Gorenstein cover of degree \(d\) of \({\mathbb{P}} ^2 _k\).
[Part III of this paper by \textit{G. Casnati} appeared in Trans. Am. Math. Soc. 350, No. 4, 1359-1378 (1998)]. caonical model of a minimal surface of general type; Gorenstein cover Casnati G.: Covers of algebraic varieties II. Covers of degree 5 and construction of surfaces. J. Algebraic Geom. 5, 461--477 (1996) Coverings in algebraic geometry, Special surfaces, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Covers of algebraic varieties. II: Covers of degree 5 and construction 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 \(k\) be a field of finite characteristic \(p\), and let \(G\) be a finite group scheme whose order is a multiple of \(p\). Let \(kG\) be the dual of the coordinate algebra \(k[G]\). In previous work the authors, along with others, have created \(\pi\)-points, flat \(K\)-algebra maps \(\alpha_K\colon K[t]/(t^p)\to KG\) for \(K\) an extension of \(k\), as an aide in studying support varieties of \(kG\)-modules. The scheme of equivalence classes of \(\pi\)-points is denoted \(\Pi(G)\). Each finite dimensional \(kG\)-module \(M\) gives rise to a closed subset \(\Pi(G)_M\) of \(\Pi(G)\), and every closed subset is of this form. However, this is not a one-to-one correspondence, i.e., it is possible for \(M_1\) and \(M_2\) to be nonisomorphic \(kG\)-modules which give the same closed subset of \(\Pi(G)\): examples can be constructed using the observation that \(\Pi(G)_M\) is empty whenever \(M\) is a projective \(kG\)-module. The non-maximal subvariety \(\Gamma(G)_M\) provides some refinement to the support variety \(\Pi(G)_M\).
In the work under review, the authors introduce a new family of invariants which they call ``generalized support varieties''. For a finite dimensional \(kG\)-module \(M\) and \(1\leq j<p\) they define \(\Gamma^j(G)\) to be the subsets of \(\Pi(G)\) which satisfy a certain non-maximal rank condition which depends on both \(j\) and \(M\). The \(\Gamma^j(G)\) are called non-maximal rank varieties. The collection \(\{\Gamma^j(G)_M\}\) is finer than \(\Pi(G)\), and each \(\Gamma^j(G)_M\) is a proper closed subset of \(\Pi(G)\). Other properties are proved, such as \(\Gamma^j(G)_M\) is empty if and only if \(M\) have constant \(j\)-rank, these varieties do not differentiate between stably isomorphic \(kG\)-modules nor modules in the same component of the stable Auslander-Reiten quiver, and the union of the \(\Gamma^j(G)_M\) is equal to \(\Gamma(G)_M\).
Another class of invariants is introduced when \(M\) is of constant rank, i.e., then the rank of the operator \(M_K\to M_K\) induced from a \(\pi\)-point is independent of the choice of \(\pi\)-point. A cohomology class \(\zeta\in H^1(G,M)\) gives rise to an extension \(E_\eta\) of \(k\) by \(M\). In an effort to generalize the zero locus in \(\text{Spec}(H^\bullet(G,k))\) a subset \(Z(\zeta)\) is constructed. Here \(Z(\zeta)\) is shown to be \(\Pi(G)\) if the extension with \(E_\zeta\) as above is locally split, otherwise \(Z(\zeta)=\Gamma^1(G)_{E_\zeta}\), establishing that \(Z(\zeta)\) is necessarily closed in \(\Pi(G)\). This construction is then generalized to extension classes \(\zeta\in\text{Ext}_G^n(M,N)\) where \(M\) and \(N\) are \(kG\) modules of constant Jordan type. support varieties; finite group schemes; \(\pi\)-points; coordinate algebras; modules of constant Jordan type; modular representations Eric M. Friedlander and Julia Pevtsova, Generalized support varieties for finite group schemes, Doc. Math. Extra vol.: Andrei A. Suslin sixtieth birthday (2010), 197 -- 222. Representation theory for linear algebraic groups, Group schemes, Modular representations and characters, Cohomology theory for linear algebraic groups, Representations of associative Artinian rings Generalized support varieties 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 Kodaira fibred surfaces are remarkable examples of projective classifying spaces, and there are still many intriguing open questions concerning them, especially the slope question. The topological characterization of Kodaira fibrations is emblematic of the use of topological methods in the study of moduli spaces of surfaces and higher dimensional complex algebraic varieties, and their compactifications. Our tour through algebraic surfaces and their moduli (with results valid also for higher dimensional varieties) deals with fibrations, questions on monodromy and factorizations in the mapping class group, old and new results on Variation of Hodge Structures, especially a recent answer given (in joint work with Dettweiler) to a long standing question posed by Fujita. In the landscape of our tour, Galois coverings, deformations and rigid manifolds (there are by the way rigid Kodaira fibrations), projective classifying spaces, the action of the absolute Galois group on moduli spaces, stand also in the forefront. These questions lead to interesting algebraic surfaces, for instance remarkable surfaces constructed from VHS, surfaces isogenous to a product with automorphisms acting trivially on cohomology, hypersurfaces in Bagnera-de Franchis varieties, Inoue-type surfaces. algebraic surfaces; Kähler manifolds; moduli; deformations; topological methods; fibrations; Kodaira fibrations; Chern slope; automorphisms; uniformization; projective classifying spaces; monodromy; fundamental groups; variation of Hodge structure; absolute Galois group; locally symmetric varieties Pencils, nets, webs in algebraic geometry, Transcendental methods, Hodge theory (algebro-geometric aspects), Fibrations, degenerations in algebraic geometry, Variation of Hodge structures (algebro-geometric aspects), Fine and coarse moduli spaces, Coverings in algebraic geometry, Modular and Shimura varieties, Coverings of curves, fundamental group, Surfaces of general type, Automorphisms of surfaces and higher-dimensional varieties, Kähler-Einstein manifolds, Uniformization of complex manifolds, Transcendental methods of algebraic geometry (complex-analytic aspects), Topological aspects of complex manifolds, Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects), General theory of automorphic functions of several complex variables, Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Period matrices, variation of Hodge structure; degenerations, Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions) Kodaira fibrations and beyond: methods for moduli 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 In this nicely written survey the author relates moduli spaces of coverings of the projective line plus ramification data, the so-called Hurwitz spaces, to arithmetic problems such as the regular inverse Galois problem for \(\mathbb{Q}(T)\) and the Hilbert-Siegel theorem [cf. \textit{M. Fried}, Commun. Algebra 5, 17-82 (1977; Zbl 0478.12006)]. This intrinsic approach allows one to classify ``equations'' abstracting their structural properties. The diophantine flavour of the questions is still present. One idea behind this work is that group theory via the description of coverings by their monodromy groups controls the arithmetic of the coverings. Solving problems such as the regular inverse Galois problem for \(\mathbb{Q}(T)\) is equivalent to finding rational points in the appropriate Hurwitz space. Finally, the survey ends by briefly describing the modular towers introduced by \textit{M. Fried} [Contemp. Math. 186, 111-171 (1995)] and relate them to classical modular towers. The same type of moduli problem may be searched for this situation. moduli of curves; modular towers; regular inverse Galois problem; moduli spaces of coverings; Hurwitz spaces; Hilbert-Siegel theorem Pierre Dèbes, Arithmétique et espaces de modules de revêtements, Number theory in progress, Vol. 1 (Zakopane-Kościelisko, 1997) de Gruyter, Berlin, 1999, pp. 75 -- 102 (French, with French summary). Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Families, moduli of curves (algebraic), Coverings of curves, fundamental group, Inverse Galois theory, Arithmetic aspects of modular and Shimura varieties, Coverings in algebraic geometry Arithmetic and moduli spaces of 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 This paper is devoted to study the properties of dualizing coverings of the plane which are associated to plane curves. In particular, the author gives a complete description of the set of plane curves for which all singularities of the Galoisation of the associated dualizing covering are quotient singularities.
Let \(C\subset \mathbb{P}^2\) be an irreducible reduced curve of degree \(\deg(C)=:d\geq 2\). Let \(\widehat{\mathbb{P}}^2\) denote the dual plane and \(I:=\{(P,l) \in \mathbb{P}^2 \times \widehat{\mathbb{P}}^2 \mid P\in l \} \) the incident variety. Let \(\mathrm{pr}_1: \mathbb{P}^2 \times \widehat{\mathbb{P}}^2 \to \mathbb{P}^2 \) be the projection onto the first factor and define \(X':=\mathrm{pr}_1^{-1}(C)\cap I\). Let \(\nu': X \to X'\) be its normalization and \(h': X' \to \widehat{\mathbb{P}}^2\) the restriction of the projection onto the second factors. The morphism \(h:= h' \circ \nu': X \to \widehat{\mathbb{P}}^2 \) is a covering of degree \(d\), called the \textit{dualizing covering} associated to the curve \(C\).
In the first part of the paper the author studies the properties of dualizing coverings; in particular, he shows that \(X\) is a smooth ruled surface and provides a description of the ramification locus \(\overline{R}\), of the branch locus \(\overline{B}\) and of the restriction \(h_{|\overline{R} }\).
In the second part the author recalls some facts about finite coverings of surfaces. Then he introduces the concept of \textit{passport of singularities} of the curve \(C\) with respect to a line \(l\in \widehat{\mathbb{P}}^2\), which keeps trace of the local behaviour of \(C\) at the singulars points of \(l\) with respect to \(C\).
In Theorem 2 he determines the local monodromy groups of dualizing coverings at a point \(l\in\widehat{\mathbb{P}}^2\): this group is the direct product of symmetric and cyclic groups.
Finally he states the main theorem, which gives a complete description of those plane curves for which the Galoisation of the associated dualizing covering has only quotient singularities.
The Galoisation \(f: Z \to \widehat{\mathbb{P}}^2 \) of the dualizing cover \(h: X \to \widehat{\mathbb{P}}^2 \) associated to an irreducible reduced curve \(C\subset \mathbb{P}^2\) has only quotient singularities if and only if \(C\) satisfies the following.
(i) If an irreducible germ \((C_i,P)\) of the curve \(C\) is singular at \(P\), then the singularity is either of type \(A_2\), or of type \(E_6\). Moreover, if there are several singular germs through \(P\), then they have the same type.
(ii) The passport of singularities of the curve \(C\) with respect to a line \(l\in \widehat{\mathbb{P}}^2\) belongs to a prescribed list, which reports also the singularities of \(Z\) over the point \(l\). dualizing coverrings; quotient singularities Kulikov, Vik. S., Dualizing coverings of the plane, Izv. Ross. Akad. Nauk Ser. Mat., 79, 5, 163-192, (2015) Coverings in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Families, moduli, classification: algebraic theory Dualizing coverings of the plane | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Using a new description of the surfaces discovered by Keum and later investigated by Naie, and of their fundamental group, we prove the following main result. Let \(S\) be a smooth complex projective surface which is homotopically equivalent to a Keum-Naie surface. Then \(S\) is a Keum-Naie surface. The connected component of the Gieseker moduli space corresponding to Keum-Naie surfaces is irreducible, normal, unirational of dimension 6. algebraic surfaces; moduli spaces; homotopy type; fundamental groups Bauer, I., Catanese, F.: The moduli space of Keum-Naie surfaces. Group Geom. Dyn. 5(2), 231--250 (2011) Surfaces of general type, Special surfaces, Families, moduli, classification: algebraic theory, Fine and coarse moduli spaces, Coverings in algebraic geometry, Fundamental groups and their automorphisms (group-theoretic aspects), Deformations of complex structures, Uniformization of complex manifolds The moduli space of Keum-Naie surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We revisit a statement of Birch that the field of moduli for a marked three-point ramified cover is a field of definition. Classical criteria due to \textit{P. Dèbes} and \textit{M. Emsalem} [J. Algebra 211, No. 1, 42--56 (1999; Zbl 0934.14019)] can be used to prove this statement in the presence of a smooth point, and in fact these results imply more generally that a marked curve descends to its field of moduli. We give a constructive version of their results, based on an algebraic version of the notion of branches of a morphism and allowing us to extend the aforementioned results to the wildly ramified case. Moreover, we give explicit counterexamples for singular curves. descent; field of moduli; field of definition; marked curves; Belyĭ maps Arithmetic aspects of dessins d'enfants, Belyĭ theory, Coverings in algebraic geometry, Algebraic moduli problems, moduli of vector bundles On explicit descent of marked curves and maps | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Nonabelian cohomology, more precisely the Grothendieck-Giraud theory of gerbes, is used by \textit{P. Dèbes} and \textit{J.-C. Douai} [Commun. Algebra 27, No. 2, 577-594 (1999; Zbl 0917.18008)] to study some questions related to fields of definition of Galois covers. The paper under review continues the same ideas, by studying the existence of Hurwitz families (i.e., of families of coverings of the Riemann sphere) parametrized by a (possibly coarse) moduli space of such coverings (i.e., a Hurwitz space). Given an irreducible component \({\mathcal H}\) of a Hurwitz space, the authors construct a gerbe \({\mathcal G}\) above the étale site \({\mathcal H}_{\text{ét}}\) of \({\mathcal H}\). The class of \({\mathcal G}\) in \(H^2({\mathcal H}_{\text{ét}},-)\), of equivalence classes of gerbes (of given band) encodes the obstruction to the existence of Hurwitz families above \({\mathcal H}\) (\S 3). Applications are given in \S 4, e.g. concrete criteria for the existence of Hurwitz families or determination of the function field of \({\mathcal H}\). Hurwitz families; covers; nonabelian cohomology; coarse moduli space; gerbes; Hurwitz space; étale site P. Dèbes, J.-C. Douai, and M. Emsalem, Families de Hurwitz et cohomologie non abélienne, Ann. Inst. Fourier (Grenoble) 50 (2000), no. 1, 113 -- 149 (French, with English and French summaries). Nonabelian homological algebra (category-theoretic aspects), Coverings in algebraic geometry, Structure of families (Picard-Lefschetz, monodromy, etc.), Fine and coarse moduli spaces, Coverings of curves, fundamental group Hurwitz families and nonabelian 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 A variety \(V\) defined over a number field \(F\) has potential density (or, the set of rational points of \(V\) is potentially dense) if there is a finite extension \(K\) of \(F\) such that \(V(K)\) is Zariski-dense in \(V\). The present paper investigates the potential density property of sextic double solids, that is, double covers of \({\mathbb P}^3\) ramified over a sextic surface. The study is motivated by the result, proved by \textit{F. A. Bogomolov} and \textit{Yu. Tschinkel} [J. Reine Angew. Math. 511, 87--93 (1999; Zbl 0916.14008)] that double covers of \({\mathbb P}^2\) branched over a singular sextic curve have potential density, except when the branch curve is six concurrent lines. (The exception is pointed out in the present paper.) This follows because such surfaces have an elliptic or rational fibration coming from lines through a singular point of the branch locus.
The principal result of the present paper is that the double solid branched over a singular sextic surface has potential density, except when the branch surface is a cone over a non-singular sextic curve or six planes with a line in common. This is analogous to the two-dimensional result, but the authors have proved that in \(\text{char}> 5\) these sextic double solids do \textit{not} carry elliptic fibrations (reference is made to [\textit{I. Cheltsov}, J. Math. Sci., New York 102, No. 2, 3843--3875 (2000; Zbl 1016.14005)] and to a preprint) so the generalization is not straightforward. Once the 3-dimensional case is proven, a similar result for double covers of \(\mathbb P^r\), \(r>3\), branched over sextic hypersurfaces, follows by induction. To complete the picture, sextic double solids in char 5 are also discussed. An example is given of one such solid which has an elliptic fibration, and the final section gives a general study, using the theory of the moving boundary introduced by the first author in his already cited paper. It is proven that the type of elliptic fibration given in the example is the only type that can occur. elliptic fibration; number field; perfect field of characteristic 5; potential density Cheltsov, Ivan; Park, Jihun, Two remarks on sextic double solids, J. Number Theory, 0022-314X, 122, 1, 1-12, (2007) Rational points, Rational and birational maps, Finite ground fields in algebraic geometry, \(3\)-folds, Fano varieties, Coverings in algebraic geometry Two remarks on sextic double solids | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We define real parabolic structures on real vector bundles over a real curve. Let \((X, \sigma_{X})\) be a real curve, and let \(S \subset X\) be a non-empty finite subset of \(X\) such that \({\sigma}_{X}(S) = S\). Let \(N \geq 2\) be an integer. We construct an \(N\)-fold cyclic cover \(p : Y\to X\) in the category of real curves, ramified precisely over each point of \(S\), and with the property that for any element \(g\) of the Galois group \(\Gamma\), and any \(y \in Y\), one has \(\sigma_Y(gy) = g^{-1}\sigma_Y(y)\). We established an equivalence between the category of real parabolic vector bundles on \((X, {\sigma}_{X})\) with real parabolic structure over \(S\), all of whose weights are integral multiples of \(1/N\), and the category of real \(\Gamma\)-equivariant vector bundles on \((Y, \sigma_Y)\). real parabolic bundles; real curve Vector bundles on curves and their moduli, Real algebraic and real-analytic geometry, Coverings in algebraic geometry Real parabolic vector bundles over a real curve | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(f:X \to Y\) be a cover of geometrically irreducible \(K\)-schemes such that \(f_{K_s}\) is Galois, and suppose further that there exists some Galois \(K\)-form \(\bar f : \bar X \to Y\) of \(f\). The author investigates the minimal extension field of \(K\) over which \(f\) becomes Galois and the existence of \(K\)-rational points. Finally, the author gives a set-up where for a finite separable extension \(L/K\) there is some \(L\)-form of \(f_L\) which does not descend to a cover of \(Y\). Galois cover; twisting lemma Hasson, H, Minimal fields of definition for Galois action, J. Pure Appl. Algebra, 220, 3327-3331, (2016) Coverings of curves, fundamental group, Group actions on varieties or schemes (quotients), Hilbertian fields; Hilbert's irreducibility theorem, Coverings in algebraic geometry Minimal fields of definition for Galois action | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(Y\) be a smooth Fano \(n\)-fold with \(n\geq 3\), \(b_2(Y)=1\) and of index \(n-1\). Let \(A\) be a smooth \(n\)-fold which is a finite double cover of \(Y\). Suppose that \(A\) is embedded in a smooth projective manifold \(X\) as a very ample divisor. This paper gives a precise classification of such \(X\) for each pair \((A, Y)\). Note that \({\text{Pic}}(Y)\cong {\text{Pic}}(A)\cong {\text{Pic}}(X)\cong {\mathbb Z}\) by the Lefschetz theorem. Moreover, if \(h\) and \(H\) are the ample generators of \({\text{Pic}}(Y)\) and \({\text{Pic}}(X)\) respectively, then \(h_A=H_A\) and this is the ample generator of \({\text{Pic}}(A)\). The branch locus of \(A\to Y\) is a member of \(|2bh|\) for some natural number \(b\) and \(A\) is a member of \(|aH|\) on \(X\) for some natural number \(a\). The authors enumerate all the possible values of \((a,b)\) for each \(d=d(Y,h)=h^n\) (note that \(d\leq 5\) by the classification theory of Del Pezzo manifolds), and describe the structure of \(X\) for each possible case. In particular \(X\) does not exist when \(d=1\).
The proof depends on various results in the theory of polarized manifolds, including those on cases of \(\Delta\)-genus two. The assumption that \(A\) is very ample (i.e., not merely ample) is essential and makes the argument very subtle in several cases.
\{ Reviewer's remark: The authors seem to assert lemma 2.2 under a weaker assumption that \(A\) is merely ample. But this is not true. The claim ``\(\phi\) factors through \(\pi\)'' is false. In fact, when \(d=1\) and \(b=1\), \(A\) becomes a double cover of \(\mathbb{P}^n\), which can always be embedded as an ample divisor. However, one can easily show that such \(A\) cannot be a very ample divisor, so the main result can be justified\}. Picard group; double cover of Fano \(n\)-fold; embedded as a very ample divisor; branch locus; Del Pezzo manifolds; polarized manifolds; \(\Delta\)-genus Fano varieties, Coverings in algebraic geometry, Divisors, linear systems, invertible sheaves, \(n\)-folds (\(n>4\)), \(3\)-folds Double covers of some Fano manifolds as hyperplane sections | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(\pi: \tilde C\to C\) be an unramified double cover of a nonsingular complete curve C of genus \(g\geq 2\) on an algebraically closed field K of characteristic \(\neq 2\). Let \(Pic_ d(C)\) be the variety of isomorphism classes of invertible sheaves of degree d on C, and \(\sigma \in Pic_ 0(C)\) be the 2-division point associated with \(\pi\). We will say that a curve is k-gonal if it has a \(g_ k^ 1\) (i.e. a linear series of degree k and dimension 1), but not a \(g^ 1_{k'}\) with \(k'<k\) (hence a \(g^ 1_ k\) on a k-gonal curve must be complete and without fixed points). Let us denote by \(W_ d\) the subset of \(Pic_ d(C)\) of invertible sheaves associated to effective divisors. We prove the following:
(1) if C is k-gonal then \(\tilde C\) is k-gonal if and only if k is even and \(W_{k/2}\cdot (W_{k/2}+\sigma)\neq \emptyset.\)
(2) if C is a general point in \({\mathcal M}^ 1_{g,k}\) (the moduli space of curves of \(genus\quad g\) with a \(g^ 1_ k)\) and \(g\geq 4k-5\) then \(\tilde C\) is 2k-gonal.
A curve C is said to be elliptic-hyperelliptic if there exists a degree two morphism \(\epsilon: C\to E\) onto an elliptic curve E. Applying (1), together with the description of the Prym-canonical map, we get the following:
(3) if C is elliptic-hyperelliptic then \(\tilde C\) is elliptic- hyperelliptic if and only if \(\sigma = \epsilon^*(\eta)\) where \(\eta\) is a 2-division point in \(Pic_ 0(E).\)
At the end, we show that given an elliptic-hyperelliptic curve \(\epsilon: C\to E,\) we can build on it, in a natural way, a ''tower'' of unramified double covers, having on each ''floor'' an elliptic-hyperelliptic curve. k-gonal curve; double cover of a nonsingular complete curve; linear series Del Centina, A.: \(g{\kappa}1\) on an unramified double cover of a k-gonal curve and applications. Rend. sem. Mat. Torino 41, 53-63 (1983) Coverings of curves, fundamental group, Divisors, linear systems, invertible sheaves, Special algebraic curves and curves of low genus, Coverings in algebraic geometry \(g_ k^ l\)'s on an unramified double cover of a k-gonal curve and applications | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For the entire collection see Zbl 0518.00005.]
Let k be an algebraically closed field of characteristic zero, and let \(\sigma_{n,d}\) be a maximal irreducible system, defined over k, of plane curves of degree n such that the general curve of the system has d nodes. An irreducible curve of degree n has no more than \((n-1)(n-2)/2\) nodes. In Section 11 of Anhang F of the book by \textit{F. Severi} [''Vorlesungen über algebraische Geometrie'' (Leipzig-Berlin 1921)] it is claimed that if \(d\leq (n-1)(n-2)/2\) and the general curve \(C^*\) of \(\sigma_{n,d}\) is irreducible, then \(\sigma_{n,d}\) is unique. The argument which supports this claim is not convincing: this is the well- known gap in Anhang F. The author refers to the uniqueness of \(\sigma_{n,d}\), with the stated assumptions, as Basic Conjecture I.
In this interesting paper the author shows that Basic Conjecture I is implied by a quite different assertion, called Basic Conjecture II. Let \(f(X,Y)=\sum A_{ij}X^ iY^ j=0\)\ be the equation of \(C^*\), where \(C^*\) is assumed irreducible. Let R be the ring generated over k by the ratios of the \(A_{ij}\), and K the quotient field of R. It is clear that the coordinates of the d nodes \(Q^*_ 1,...,Q^*_ d\) of \(C^*\) are algebraic over K. Basic Conjecture II asserts that the d nodes \(Q^*_ i\) form a complete set of conjugate algebraic points over K.
The proof depends on a basic lemma, which has as a corollary that any system of curves of the above kind contains all the n-gons of the plane (curves consisting of n lines). Severi had pointed out that this implies Basic Conjecture I, but it is exactly here that the gap in Anhang F occurs.
A recent reference is a paper by the author [Am. J. Math. 104, 209-226 (1982; Zbl 0516.14023)]. A proof of Basic Conjecture II would imply, by a well-known argument of the author, that the fundamental group of the complement of a nodal plane curve is abelian. irreducibility of system of irreducible plane curves; abelian; fundamental group of the complement of a nodal plane curve; degree n Oscar Zariski, On the problem of irreducibility of the algebraic system of irreducible plane curves of a given order and having a given number of nodes, Arithmetic and geometry, Vol. II, Progr. Math., vol. 36, Birkhäuser Boston, Boston, MA, 1983, pp. 465 -- 481. Families, moduli of curves (algebraic), Singularities of curves, local rings, Enumerative problems (combinatorial problems) in algebraic geometry, Coverings in algebraic geometry On the problem of irreducibility of the algebraic system of irreducible plane curves of a given order and having a given number of nodes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper investigates the depth of modular rings of invariants for certain finite groups. Although the ring of invariants of a non-modular representation of a finite group is Cohen-Macaulay, and hence has depth equal to the dimension of the representation, the ring of invariants of a modular representation may fail to be Cohen-Macaulay. In this case computing the depth of the ring of invariants can be a difficult problem.
In this paper the authors introduce the concept of a shallow representation and prove that the depth of the ring of invariants of a shallow representation, \(V\), is given by \(\min(\dim (V^P)+2, \dim(V))\) where \(P\) is any \(p\)-Sylow subgroup. For an abelian group with a cyclic \(p\)-Sylow subgroup, every representation is shallow. Thus the paper provides a relatively elementary proof of the well known result of \textit{G. Ellingsrud} and \textit{T. Skjelbred} [Compos. Math. 41, 233-244 (1980; Zbl 0438.13007)]. Additional examples of shallow representations are given in the paper and the observation is made that the direct sum of \(m\) copies of a shallow representation is still shallow. The final section of the paper gives an alternate proof of the fact, originally due to \textit{H. Nakajima} [Tsukuba J. Math. 3, 109-122 (1979; Zbl 0418.20041)], that the ring of invariants of a \(p\)-group is Cohen-Macaulay if and only if it is Buchsbaum. The authors then conjecture that this should hold for any finite linear group acting on a polynomial algebra. The conjecture has since been proven by one of the authors [see \textit{G. Kemper}, ``Loci in quotients by finite groups, pointwise stabilizers and the Buchsbaum property'' (preprint, Univ. Heidelberg 2000)]. \(p\)-Sylow subgroups; Cohen-Macaulay ring; Buchsbaum ring; modular rings of invariants; shallow representation Campbell H.E.A., Hughes I.P., Kemper G., Shank R.J., Wehlau D.L.: Depth of modular invariant rings. Transform. Groups 5(1), 21--34 (2000) Actions of groups on commutative rings; invariant theory, Modular representations and characters, Dimension theory, depth, related commutative rings (catenary, etc.), Geometric invariant theory Depth of modular invariant 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 On établit dans cet article le theoreme suivant:
Théorème. Soient \(k\) un corps fertile [``large field'', voir \textit{F. Pop}, Ann. Math. (2) 144, No. 1, 1--34 (1996; Zbl 0862.12003)], \(\Gamma\) une courbe projective, lisse et connexe sur \(k\), \(\varepsilon\in\Gamma(k)\) un point rationnel, \(G\) un \(k\)-schéma en groupes fini étale, \(X\) un \(G\)-torseur sur \(k\). Alors il existe:
(i) une \(k\)-courbe projective, lisse et géométriquement connexe \(C\), et un \(k\)-morphisme \(\pi:C\to\Gamma\); (ii) une action fidèle de \(G\) sur \(C\), faisant de \(\pi\) un \(G\)-torseur au-dessus d'un ouvert de \(\Gamma\) contenant \(\varepsilon\);
(iii) un isomorphisme \(\pi^{-1}(\varepsilon)@>\sim>> X\) de \(G\)-torseurs. rational point; \(G\)-torsor; homogeneous space Moret-Bailly, L.: Construction de revêtements de courbes pointées. J. algebra 240, 505-534 (2001) Coverings of curves, fundamental group, Homogeneous spaces and generalizations, Coverings in algebraic geometry, Singularities of curves, local rings Construction of coverings of pointed 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 central object of study is a formal power series that we call the content series, a symmetric function involving an arbitrary underlying formal power series \( f\) in the contents of the cells in a partition. In previous work we have shown that the content series satisfies the KP equations. The main result of this paper is a new partial differential equation for which the content series is the unique solution, subject to a simple initial condition. This equation is expressed in terms of families of operators that we call \( \mathcal {U}\) and \( \mathcal {D}\) operators, whose action on the Schur symmetric function \( s_{\lambda }\) can be simply expressed in terms of powers of the contents of the cells in \( \lambda \). Among our results, we construct the \( \mathcal {U}\) and \( \mathcal {D}\) operators explicitly as partial differential operators in the underlying power sum symmetric functions. We also give a combinatorial interpretation for the content series in terms of the Jucys-Murphy elements in the group algebra of the symmetric group. This leads to an interpretation for the content series as a generating series for branched covers of the sphere by a Riemann surface of arbitrary genus \( g\). As particular cases, by suitable choice of the underlying series \( f\), the content series specializes to the generating series for three known classes of branched covers: Hurwitz numbers, monotone Hurwitz numbers, and \( m\)-hypermap numbers of Bousquet-Mélou and Schaeffer. We apply our pde to give new and uniform proofs of the explicit formulas for these three classes of numbers in genus 0. generating functions; transitive permutation factorizations; symmetric functions; Jucys-Murphy elements; contents of partitions Exact enumeration problems, generating functions, Permutations, words, matrices, Symmetric functions and generalizations, Combinatorial aspects of partitions of integers, Coverings in algebraic geometry Contents of partitions and the combinatorics of permutation factorizations in genus 0 | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 continuation of the author's previous paper [ibid. 34, No. 2, 225-228 (1990; Zbl 0716.14014)] with the same title.
Let \(X\) be a smooth proper connected algebraic curve defined over an algebraic number field \(K\). Let \(\pi_1 (\overline X)_l\) be the pro-\(l\) completion of the geometric fundamental group of \(\overline X = X \otimes_K \overline K\). Let \({\mathfrak p}\) be a prime of \(K\), which is coprime to \(l\). Assuming that \(X\) has bad reduction at \({\mathfrak p}\) and the Jacobian variety of \(X\) has good reduction at \({\mathfrak p}\), we describe the action of the inertia group \(I_{\mathfrak p}\) on the quotient groups of \(\pi_1 (\overline X)_l\) by the higher commutator subgroups. Galois representation of the fundamental group of an algebraic curve Oda, T., \textit{A note on ramification of the Galois representation on the fundamental group of an algebraic curve. II}, J. Number Theory, 53, 342-355, (1995) Coverings of curves, fundamental group, Galois theory, Coverings in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry A note on ramification of the Galois representation on the fundamental group of an algebraic curve. II | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper, the author gives geometric constructions associated to suitable representations of a finite group \(\tau\), working over a field \(k\) of characteristic \(p\). These are compared with analogous constructions for infinitesimal group schemes (see [\textit{E. M. Friedlander} and \textit{J. Pevtsova}, Duke Math. J. 139, No. 2, 317--368 (2007; Zbl 1128.20031)], for example): for \(\mathbb{G}\) a linear algebraic group, he compares the constructions for the finite group \(\tau = \mathbb{G} (\mathbb{F}_p)\) and for the \(r\)th Frobenius kernel \(\mathbb{G}_{(r)}\).
The author's basic ingredient is the universal \(p\)-nilpotent operation \(\Theta_E \in S^\bullet (J_E^*) \otimes kE\), where \(kE\) is the group ring of the elementary abelian \(p\)-group \(E\) and \(S^\bullet (J_E^*)\) the symmetric algebra on the dual of the radical \(J_E \subset kE\). By naturality, for \(\tau\) a finite group, this gives
\[
\Theta_\tau \in A_\tau \otimes k \tau
\]
by passing to the inverse limit over the poset \(\mathcal{E}(\tau)\) of sub elementary abelian \(p\)-groups of \(\tau\), with \(A_\tau := \lim_{\leftarrow, E\in \mathcal{E}(\tau)} S^\bullet (J_E^*)\). For \(M\) a finite-dimensional \(\tau\)-module, \(\Theta_\tau\) induces \(\Theta_{\tau,M} \in \mathrm{End} (A _\tau \otimes M)\); this is \(p\)-nilpotent.
The author defines \(X_\tau\) to be \(\mathrm{Spec} (A_\tau)\) and, likewise, the closed subscheme \(X_\tau^{(2)}\) built using the algebras \(S^\bullet ((J_E^2)^*)\). The scheme \(Y_\tau\) is built using the algebras \(S^\bullet ((J_E/J_E^2)^*)\), so that there is a \(\tau\)-equivariant map \(X_\tau \rightarrow Y_\tau\) and a \(p\)-isogeny \textit{à la Quillen}
\[
(Y_\tau) /\tau \rightarrow \mathrm{Spec} (H^\bullet (\tau ; k)).
\]
The author proves that, for \(M\) a finite-dimensional \(\tau\)-module of constant \(j\)-rank, the modules \(\ker (\Theta_{\tau, M}^j )\), for \(1 \leq j <p\), define \(\tau\)-equivariant vector bundles on \(X_\tau \backslash X_\tau^{(2)}\) (likewise for the image and cokernel of \(\Theta_{\tau, M}^j \)). There are projective versions of these results, replacing \(\mathrm{Spec} (H^\bullet (\tau ; k))\) by \(\mathrm{Proj} (H^\bullet (\tau ; k))\) etc. and considering \(\mathbb{P}\Theta_{\tau, M}\).
For comparison with the cases arising from the linear algebraic group \(\mathbb{G}\), the author builds on work of [\textit{P. Sobaje}, J. Pure Appl. Algebra 219, No. 6, 2206--2217 (2015; Zbl 1360.20045)] and others providing good exponential maps. For \(\mathbb{G}\) of good exponential type, he exhibits a \(\mathbb{G}(\mathbb{F}_p)\)-equivariant isomorphism \(\mathcal{L} : Y_{\mathbb{G}(\mathbb{F}_p)} \rightarrow Y_\mathfrak{g}\), where \(Y_\mathfrak{g}\) is the variety associated to the Lie algebra \(\mathfrak{g}\) of \(\mathbb{G}\), and hence the equivariant map \( \mathcal{L}_{(r)} : Y_{\mathbb{G}(\mathbb{F}_p)} \rightarrow V_r (\mathbb{G})\) corresponding to the \(r\)th Frobenius kernel \(\mathbb{G}(r)\).
For \(M\) a finite-dimensional rational \(\mathbb{G}\)-module of exponential degree \(< p^r\), the author uses \( \mathcal{L}_{(r)}\) to compare the vector bundles associated to \(\mathbb{G}(\mathbb{F}_p)\) and \(\mathbb{G}(r)\) respectively. He shows that they give the same class in Weibel's homotopy-invariant \(K\)-theory by relating the vector bundles by an \(\mathbb{A}^1\)-homotopy.
The author concludes by extending the constructions to the case of group schemes of the form \(G \rtimes \tau\), where \(G\) is an infinitesimal group scheme and \(\tau\) a finite group. vector bundles associated to modular representations; universal p-nilpotent operator; finite group; linear algebraic group; Frobenius kernel; rational module Group schemes, Modular representations and characters, Representations of finite groups of Lie type, Cohomology of groups Geometric invariants of representations 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 This paper is motivated by [\textit{I. Choe} et al., Topology 44, No. 3, 585--608 (2005; Zbl 1081.14045)]. Let \(X\) be a smooth complex projective curve of genus \(g\) and let \(L\) be a line bundle on \(X\) with \(\operatorname{deg}L>0\). Let \(\mathbf{M}\) be the moduli space of semistable rank \(2\) \(L\)-twisted Higgs bundles with trivial determinant on \(X\). Let \(\mathbf{M}_X=\bigcup_{x\in X}\mathbf{M}_{\mathcal{O}_X{(-x)},L}\) where \(\mathbf{M}_{\mathcal{O}_X{(-x)},L}\) denotes the moduli space of rank \(2\) stable \(L\)-twisted Higgs bundles with determinant \(\mathcal{O}_X(-x)\) on \(X\). The purpose of this paper is to construct a cycle in the product of a stack of rational maps from nonsingular curves to \(\mathbf{M}_X\) and \(\mathrm{Pic}^{2\operatorname{deg}L}(X)\). Here this cycle, which is called a Hecke cycle associated to a stable \(L\)-twisted Higgs bundles, is given by a collection of Hecke modifications of a stable \(L\)-twisted Higgs bundle in \(\mathbf{M}\).
This paper is organized as follows: the first section is an introduction to the subject and a description of the results. In the second section the author reviews some of basics on twisted Higgs bundles, the spectral data associated to them and parabolic modules. The third section deals with the Hecke modification of \(L\)-twisted Higgs bundles. The forth section deals with a Hecke cycle associated to a twisted Higgs bundle. moduli space of twisted Higgs bundles; Hecke modifications of twisted Higgs bundle; Hecke cycles associated to stable twisted Higgs bundles Vector bundles on curves and their moduli, Algebraic cycles, Algebraic moduli problems, moduli of vector bundles, Stacks and moduli problems, Coverings in algebraic geometry, Families, moduli of curves (algebraic), Jacobians, Prym varieties Hecke cycles associated to rank 2 twisted Higgs bundles on a curve | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(f: S\to X\) be a surjective morphism of projective manifolds. \textit{R. Lazarsfeld} has shown that if \(S= \mathbb{P}^n\) then \(X\) must be isomorphic to \(\mathbb{P}^n\) [in: Complete intersections, Lect. 1st Sess. C.I.M.E., Acireale/Italy 1983, Lect. Notes Math. 1092, 29--61 (1984; Zbl 0547.14009)]. Recently, the authors of the paper under review have generalized this result to the case of a rational homogeneous space \(S\) of Picard number one [Invent. Math. 136, No. 1, 209--231 (1999; Zbl 0963.32007)]. On the other hand, O. Debarre studied the question for a simple abelian variety \(A\) and a finite morphism \(f: A\to X\). His result states that if such a morphism has non-empty ramification, then \(X\) must be isomorphic to a projective space \(\mathbb{P}^n\), too [\textit{O. Debarre}, C. R. Acad. Sci., Paris, Sér. I 309, 119--122 (1989; Zbl 0699.14050)].
In the present paper, the authors give a generalization of Debarre's result to arbitrary abelian varieties. Their main theorem reads as follows: Let \(f: A\to X\) be a finite morphism with non-empty ramification from an abelian variety \(A\) to a projective manifold \(X\). Then \(X\) is a holomorphic bundle of projective spaces over a projective algebraic manifold \(Y\) of smaller dimension. Moreover, there exists a finite morphism from an abelian variety onto \(Y\) and there is a finite sequence of surjective morphisms \(g_i: X_{i-1}\to X_i\) between projective manifolds, \(1\leq i\leq N\), \(X_0= X\), such that each \(g_i\) is a holomorphic projective bundle and \(X_N\) admits a finite unramified cover by an abelian variety. In particular, if \(X\) has Picard number 1, then \(X\) is a projective space \(\mathbb{P}^n\).
The proof is based on a refined study of deformations of minimal rational curves on the manifold \(X\) together with an analysis of the structure of multi-valued distributions defined by the linear span of varieties of minimal rational tangents. morphisms of projective varieties; coverings; ramification problems; rational curves Hwang, J.M., Mok, N.: Projective manifolds dominated by abelian varieties. Math. Z. 238, 89--100 (2001) Ramification problems in algebraic geometry, Coverings in algebraic geometry, Algebraic theory of abelian varieties, Varieties and morphisms Projective manifolds dominated by abelian varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Sei C eine ebene projektive Kurve, die sich als Durchschnitt einer Hyperfläche und (n-2) Hyperebenen im n-dimensionalen projektiven Raum schreiben läßt. Das Ergebnis dieser Note ist die Berechnung der Fundamentalgruppe des Komplementes von C. Dazu wird der Kurve C, oder genauer gesagt dem Komplement von C, auf kanonische Weise eine Hyperfläche F im \((n+1)\)-dimensionalen affinen Raum zugeordnet. Die Fundamentalgruppe des Komplementes von V ist dann gleich einer wohlbestimmten Erweiterung der Fundamentalgruppe von F mit einer endlichen zyklischen Gruppe. fundamental group of complement of curve; Zariski problem Némethi, A. : On the fundamental group of the complement of certain singular plane curves , Math. Proc. Cambridge Philos. Soc. 102 (1987), 453-457. Coverings in algebraic geometry, Singularities of curves, local rings, Complete intersections On the fundamental group of the complement of certain singular plane curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper builds on the authors' results [in Math. Ann. 318, No. 4, 805-834 (2000; Zbl 0971.20004)]. Let \(G\), \(T\) be finite groups, \(\Gamma\) a profinite group, and \(\pi\colon\Gamma\to T\), \(\lambda\colon G\to T\) surjections. This defines the (proper) embedding problem: does there exist a homomorphism (epimorphism) \(h\colon\Gamma\to G\) with \(\lambda h=\pi\). The first observation reduces the question whether the proper solutions to an embedding problem arise from a versal deformation [compare, e.g., \textit{B. Mazur}, Publ., Math. Sci. Res. Inst. 16, 385-437 (1989; Zbl 0714.11076)] to studying when versal deformations of representations of finite groups are faithful. Let \(V\) be a representation of \(G\) over an algebraically closed field \(k\) with \(\text{char}(k)=p>0\) such that \(\text{End}_{kG}(V)=k\), and let \(K\leq G\) be its kernel. Assume that \(V\) belongs to a cyclic block \(B_{G,V}\) of \(kG\).
Theorem: The universal deformation \(U(G,V)\) is a faithful representation of \(G\) if, and only if, \(K\) is a \(p\)-group, the Brauer tree \(\Lambda(B_{G,V})\) is a star with central exceptional vertex, and, in case \(\Lambda(B_{G,V})\) has more than one edge, \(V\) is not simple.
In the proof it is observed that (i) each finite solvable extension of a given field can be constructed using a finite sequence of versal deformations; (ii) whether a versal deformation of a representation of a finite group is faithful can be detected from versal deformation rings associated to quotients of the group through which the representation factors. versal deformation rings; representations of finite groups; proper embedding problems; Brauer trees Bleher F.M., Chinburg T. (2003). Applications of versal deformations to Galois theory. Comment. Math. Helv. 78:45--64 Group rings of finite groups and their modules (group-theoretic aspects), Inverse Galois theory, Galois theory, Modular representations and characters, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory) Applications of versal deformations to Galois 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 The paper is devoted to the study of slowly growing holomorphic functions on Abelian coverings of projective manifolds. The approach is based on the \(\alpha^2\)-cohomology technique for holomorphic vector bundles on complete Kähler manifolds and on the geometric properties of projective manifolds. regular covering with an abelian transformation group; positive vector bundle; \(\alpha^2\)-cohomology; holomorphic functions Alexander Brudnyi, Holomorphic functions of exponential growth on abelian coverings of a projective manifold, Compositio Math. 127 (2001), no. 1, 69 -- 81. Special families of functions of several complex variables, Coverings in algebraic geometry Holomorphic functions of exponential growth on Abelian coverings of a projective manifold | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(C\) be the Cayley singular cubic surface in the complex projective 3-space \(\mathbb{P}^3\), having as singular set \(\mathrm{Sing}(C)\) four \(A_1\)-singularities.
In this paper it is shown, using the braid monodromy technique, that the fundamental group of the smooth part \(C \setminus \mathrm{Sing}(C)\) is cyclic of order two. fundamental groups; singularities; Cayley cubic surface. Singularities in algebraic geometry, Coverings in algebraic geometry, Coverings of curves, fundamental group, Computational aspects of algebraic surfaces The fundamental group for the complement of Cayley's singularities | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper is devoted to the study of the group \(\pi_1 (X)/N\), where \(X\) is a projective smooth variety, \(Y \subset X\) is a closed subvariety, \(f : Z \to Y\) is surjective, all components of \(Z\) are complete and normal, and \(N \subset \pi_1 (X)\) is the normal subgroup generated by the images of the components of \(Z\). The result is the following theorem:
Assume \(\pi_1 (Y,y) \to \pi_1 (X,y)\) is surjective, or that \(\pi_1^{\text{alg}} (Y,y) \to \pi^{\text{alg}}_1 (X,y)\) is surjective; then for any integer \(n\), there are only finitely many conjugacy classes of representations
\[
\pi_1 (X)/N \to \text{GL} (n, \mathbb{C}).
\]
The proof uses Weil's construction of the set of representations of \(\pi_1 (X)\) modulo conjugacy as an algebraic variety, and the description of its tangent space as an \(H^1\)-group of a local system on \(X\). It also uses Simpson's result on the existence of local systems \(V\) underlying a variation of Hodge structure in each component of this variety, and the fact that the group \(H^1 (\text{End} V)\) for such a local system has a functorial mixed Hodge structure. The proof of the infinitesimal version of the theorem, namely the injectivity of the map \(H^1 (X, \text{End} V) @>f^*>> H^1 (Z, \text{End} V)\), then follows from a weight argument at such a point \(V\). complex local systems; fundamental group; mixed Hodge structure Lasell, B. : Complex local systems and morphisms of varieties. Dissertation , University of Chicago (1994). Coverings in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry Complex local systems and morphisms of varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A triple plane is a finite flat morphism \(f : X \rightarrow \mathbb{P}^2\). In this paper, the authors show that smooth projective surfaces \(X\) over \(\mathbb{C}\) which arise as general triple planes and such that \(p_g(X)=q(X)=0\) belong to at most 12 families which the authors name I, II,\(\ldots\), XII. The authors show that the families of type I to VII exist and they completely classify those of type I to VI. triple plane; Tschirnhausen bundle; Steiner bundle; adjunction theory Coverings in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Triple planes with \(p_g=q=0\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 set of canonical models of surfaces of general type which are double covers of \(\mathbb{P}^2\) branched over a plane curve of degree \(2h\) is a connected component of the moduli space if and only if \(h\) is even. To get a connected component when \(h\) is odd, we must add some special surfaces called degenerate double covers of \(\mathbb{P}^2\). Moreover, we show that the theory of simple iterated double covers [cf. \textit{M. Manetti}, Topology 36, No. 3, 745-764 (1997; Zbl 0889.14014)] ``works'' for every degenerate double cover of \(\mathbb{P}^2\); this allows us to construct many examples of connected components of the moduli space having simple iterated double covers of \(\mathbb{P}^2\) as generic members. canonical models of surfaces of general type; degenerate double covers Manetti, M., \textit{degenerate double covers of the projective plane}, New trends in algebraic geometry, Warwick, 1996, 255-281, (1999), Cambridge University Press, Cambridge Coverings in algebraic geometry, Surfaces of general type, Families, moduli, classification: algebraic theory, Projective techniques in algebraic geometry Degenerate double covers 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 This paper contains results concerning finite groups that occur as Galois groups of unramified covers of projective curves over an algebraically closed field of characteristic \(p>0\). The fundamental groups for such curves are known only for genus zero and one. The author shows that every finite group \(G\) is a Galois group for a curve (as before) of genus \(g\) bounded by the number of generators of \(G\). Moreover, she proves that \(G\) satisfies the same property for curves in a non-empty open subset of the modular scheme of curves of genus \(g\). The method of proofs uses formal patching, which has been introduced by \textit{M. Raynaud} [Invent. Math. 116, No. 1-3, 425-462 (1994; Zbl 0798.14013)] and developed by \textit{D. Harbater} [in: Recent developments in the inverse Galois problem, Summer Res. Conf., Univ. Washington 1993, Contemp. Math. 186, 353-369 (1995; Zbl 0858.14013)]. The author mentions also that some of the results of this paper have been simultaneously and independently found by \textit{M. Saïdi} in his thesis (Bordeaux 1994, see also the preceding review). characteristic \(p\); Galois groups of unramified covers of projective curves K. F. Stevenson, Galois groups of unramified covers of projective curves in characteristic \(p\), Journal of Algebra 44, (To Appear). Coverings of curves, fundamental group, Inverse Galois theory, Coverings in algebraic geometry, Finite ground fields in algebraic geometry Galois groups of unramified covers of projective curves in characteristic \(p\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The object of this paper is to generalize to tame covers of schemes the theory of the Galois module structure of tamely ramified rings of integers. To motivate this generalization, we recall two theorems of Noether and Taylor, respectively. Let \(N/K\) be a finite Galois extension of number fields with group \(G = \text{Gal} (N/K)\). Let \({\mathfrak O}_N\) be the ring of integers of \(N\). Noether proved that \({\mathfrak O}_N\) is a projective \(\mathbb{Z} [G]\)-module if and only if \(N/K\) is at most tamely ramified: we suppose from now on that is the case. Then \({\mathfrak O}_N\) defines a class \(({\mathfrak O}_N)\) in the Grothendieck group \(K_0 (\mathbb{Z} [G])\) of all finitely generated projective \(\mathbb{Z} [G]\)-modules. The class group \(\text{Cl} (\mathbb{Z} [G])\) of \(\mathbb{Z} [G]\) is defined to be the quotient of \(K_0 (\mathbb{Z} [G])\) by the subgroup generated by the class of \(\mathbb{Z} [G]\). Let \(({\mathfrak O}_N)^{\text{stab}}\) be the image of \(({\mathfrak O}_N)\) in \(\text{Cl} (\mathbb{Z} [G])\). In Invent. Math. 63, 41-79 (1981; Zbl 0469.12003), \textit{M. J. Taylor} proved Fröhlich's conjecture that \(({\mathfrak O}_N)^{\text{stab}}\) is equal to another invariant \(W_{N/K}\) in \(\text{Cl} (\mathbb{Z} [G])\) which Cassou-Noguès had defined by means of the root-numbers of symplectic representations of the Galois group \(G\).
To generalize the results above, let \(G\) be an abstract finite group. We consider tame \(G\)-covers \(f : X \mapsto Y\) of schemes of finite type over a Noetherian ring \(A\) with respect to a divisor \(D\) on \(Y\) having normal crossings.
Let \(\text{CT} (A[G])\) be the Grothendieck group of all finitely generated \(A[G]\)-modules which are cohomologically trivial as \(\mathbb{Z}[G]\)- modules. The ``forgetful'' homomorphism \(K_0 (\mathbb{Z} [G]) \mapsto \text{CT} (\mathbb{Z} [G])\) is an isomorphism. The image of \(({\mathfrak O}_N)\) under this isomorphism is \(\Psi (X/Y)\). -- Our main results concern the case in which \(A\) is a finite field of characteristic \(p\).
Theorem: Suppose \(A\) is a finite field of characteristic \(p\). Assume that \(X\) and \(Y\) are projective over \(A\), \(X\) is regular, and that \(D\) has strictly normal crossings. Restriction of operators from \(A[G]\) to \(\mathbb{F}_p [G]\) induces a homomorphism \(\text{Res}_{A \mapsto \mathbb{F}_p} : \text{CT} (A [G]) \mapsto \text{CT} (\mathbb{F}_p [G])\), where \(\mathbb{F}_p\) is the field of order \(p\). The class \(\text{Res}_{A \mapsto \mathbb{F}_p} (\Psi (X/Y))\) in \(\text{CT} (\mathbb{F}_p [G]) = K_0 (\mathbb{F}_p [G])\) both determines and is determined by the set of \(p\)- adic absolute values \(|j_p \varepsilon (Y,V) |_p\) as \(V\) ranges over all of the irreducible complex representations of \(G\).
Our second main result over finite fields is a precise counterpart for regular projective schemes of Taylor's proof of Fröhlich's conjecture concerning the ring of integers. To state this, identify the class group \(\text{Cl} (\mathbb{Z} [G])\) of \(\mathbb{Z} [G]\) with \(\text{CT} (\mathbb{Z} [G])/ \langle (\mathbb{Z} [G]) \rangle\). If \(A\) is a finitely generated \(\mathbb{Z}\)- module, one has a homomorphism \(\text{Res}_{A \mapsto \mathbb{Z}}^{\text{stab}} : \text{CT} (A[G]) \mapsto \text{Cl} (\mathbb{Z} [G])\) induced by restriction of operators from \(A[G]\) to \(\mathbb{Z} [G]\). We define in Cl\((\mathbb{Z}[G])\) a symplectic root-number class \(W_{x/y}\) and a ramification class \(R_{x/y}\) which depends only on root-numbers associated to the restriction of \(f:X\mapsto Y\) to the branch locus of \(f\).
Theorem: Under the same hypotheses as the theorem above:
\[
\text{Res}_{A \mapsto \mathbb{Z}}^{\text{stab}} \bigl( \Psi (X/Y) \bigr) = W_{X/Y} + R_{X/Y} .
\]
The class \(W_{X/Y}\) is determined by the signs at infinity of the (totally real) algebraic numbers \(\varepsilon (Y,V)\) as \(V\) ranges over all of the irreducible symplectic representations of \(G\). In particular, \(W_{X/Y}\) has order 1 or2, and \(W_{X/Y}\) is trivial if \(G\) has no irreducible symplectic representation. tame covers of schemes; Galois module structure T. Chinburg, ''Galois structure of de Rham cohomology of tame covers of schemes,'' Ann. of Math., vol. 139, iss. 2, pp. 443-490, 1994. Coverings in algebraic geometry, Group actions on varieties or schemes (quotients), Integral representations related to algebraic numbers; Galois module structure of rings of integers Galois structure of de Rham cohomology of tame covers of schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The concept of Whitehead group for arbitrary group schemes is introduced (under certain assumptions). In the case of an isotropic absolutely almost simple simply connected algebraic group \(L\) over a field \(k\), Soulé and Margaux gave a description of the Whitehead group of the constant group scheme \(L\times_k{\mathbb{A}}_k^1\) over the affine line \(\mathbb{A}^1_k\). A similar situation is considered replacing the affine line by the punctured affine line \(\text{Spec } k[t,t^{-1}]\). The structure of the Whitehead group of an arbitrary simple simply connected group scheme over \(k[t, t^{-1}]\) is studied. Whitehead groups; loop group schemes; fundamental domain; Lie theory Group actions on varieties or schemes (quotients), Galois cohomology of linear algebraic groups, Coverings in algebraic geometry Whitehead groups of loop group schemes of nullity one | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The isoalgebraic functions may also be called analytic-algebraic functions, or complex Nash functions. They may be defined over any algebraically closed field of characteristic zero, with a chosen real closed subfield, using semialgebraic sections of étale mappings. The paper presents some new results about isoalgebraic functions and spaces. Most proofs are contained in the first author's thesis [``Isoalgebraische Räume'', Univ. Regensburg 1984]. Some of the results of complex analysis in several variables which we are used to think as ``transcendental'' have exact analogs in this purely algebraic context, such as Hartog's Kugelsatz. The paper ends with a paragraph about coverings in the category of (locally) isoalgebraic spaces, which exhibits strange phenomena, such as infinitely many non isomorphic smooth isoalgebraic structures of the torus \(S^ 1\times S^ 1\). isoalgebraic functions; complex Nash functions; coverings; isoalgebraic spaces R. Huber, M. Knebusch: A glimpse at isoalgebraic spaces, Proc. Joensuu 1987, Lecture Notes Math. Springer Verlag, erscheint demnächst Nash functions and manifolds, Real-analytic sets, complex Nash functions, Coverings in algebraic geometry A glimpse at isoalgebraic spaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The main object of this paper is to construct an example of a normal noetherian pseudo-geometric surface which is locally, but not globally, excellent \((*)\).
The authors consider in the rational plane an infinite union of horizontal and vertical lines defined by \(X=a_ i\) and \(Y=b_ i\). In the infinite dimensional space with coordinates \((X,Y,T_ 1,T_ 2,\dots)\) let us define the surface \(G\) given by the equations \(T_ i^{d_ i}=(X-a_ i)(Y-b_ i)\). This is an infinite abelian covering of the plane. When the infinite sequence of points \((a_ i,b_ i)\) is siutably chosen this covering satisfies \((*)\). excellent surface; abelian covering Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Coverings in algebraic geometry, Singularities in algebraic geometry, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) Singular locus of an infinite integral extension | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 surface of degree \(n\), projected onto \(\mathbb{CP}^2\). The surface has a natural Galois cover with Galois group \(S_n\): It is possible to determine the fundamental group of a Galois cover from that of the complement of the branch curve of \(X\): In this paper we survey the fundamental groups of Galois covers of all surfaces of small degree \(n\leq 4\), that degenerate to a nice plane arrangement, namely a union of \(n\) planes such that no three planes meet in a line. We include the already classical examples of the quadric, the Hirzebruch and the Veronese surfaces and the degree 4 embedding of \(\mathbb{CP}^1\times\mathbb{CP}^1\) and also add new computations for the remaining cases: the cubic embedding of the Hirzebruch surface \(F_1\), the Cayley cubic (or a smooth surface in the same family), for a quartic surface that degenerates to the union of a triple point and a plane not through the triple point, and for a quartic 4-point. In an appendix, we also include the degree 8 surface \(\mathbb{CP}^1\times\mathbb{CP}^1\) embedded by the \((2,2)\) embedding, and the degree \(2n\) surface embedded by the \((1,n)\) embedding, in order to complete the classification of all embeddings of \(\mathbb{CP}^1\times\mathbb{CP}^1\) which was begun in [\textit{B. Moishezon} and \textit{M. Teicher}, Invent. Math. 89, 601--643 (1987; Zbl 0627.14019)]. singularities; coverings; fundamental groups; surfaces; mapping class group Singularities in algebraic geometry, Coverings in algebraic geometry, Coverings of curves, fundamental group, Computational aspects of algebraic curves, Computational aspects of algebraic surfaces Classification of fundamental groups of Galois covers of surfaces of small degree degenerating to nice plane arrangements | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(K\) be a finitely generated field with separable closure \(K_s\) and absolute Galois group \(G_K\). By a curve \(C/K\) we always understand a smooth geometrically irreducible projective curve. Let \(F(C)\) be its function field and let \(\Pi(C)\) be the Galois group of the maximal unramified extension of \(F(C)\). We have the exact sequence
\[
1\to\Pi_g (C)\to \Pi(C)\to G_K\to 1 \tag{*}
\]
where \(\Pi_g(C)\) is the geometric (profinite) fundamental group of \(C\times \text{Spec} (K_s)\) (i.e. \(\Pi_g(C)\) is equal to the Galois group of the maximal unramified extension of \(F(C)\otimes K_s)\). This sequence induces a homomorphism \(\rho_C\) from \(G_K\) to \(\text{Out} (\Pi_g(C))\) which is the group of automorphisms modulo inner automorphisms of \(\Pi_g(C)\). It is well known that \(\rho_C\) is an important tool for studying \(C\). For instance, it determines \(C\) up to \(K\)-isomorphisms if the genus of \(C\) is at least 2 and \(K\) is a number field or even a \({\mathfrak p}\)-adic field [see \textit{S. Machizuki}, Invent. Math. 138, No. 2, 319-423 (1999; Zbl 0935.14019)].
So it is of interest to find quotients \(\overline \Pi(C)\) of \(\Pi(C)\) such that the induced map of \(G_K\) is not the identity but the induced representation \(\overline\rho_C\) becomes trivial. We give a geometric interpretation of such quotients. For this we assume that the sequence (*) is split and choose a section \(s\) which induces a homomorphism \(\sigma\) from \(G_K\) to \(\Aut(\Pi_g (C))\). Let \(U\) be a normal subgroup of \(\Pi(C)\) contained in \(\Pi_g(C)\). The representation \(\rho_C\) becomes trivial modulo \(U\) if and only if the map \(\sigma\bmod U\) has its image inside of \(\text{Inn} (\Pi_g(C)/U)\). Let \(Z\) be the center of \(\Pi_g(C)/U\) and \(\overline\Pi =(\Pi_g(C)/U)/Z\). Our condition on \(U\) implies that there is exactly one group theoretical section \(\overline s\) from \(G_K\) to \(\overline \Pi(C)/U)/Z\) inducing the trivial action on \(\overline\Pi\) and so \(\overline\Pi\) occurs as Galois group of an unramified regular extension of \(F(C)\) in a natural way. Thus, to find center free infinite factors of \(\Pi_g\) on which \(\rho_C\) becomes trivial is equivalent with finding infinite regular Galois coverings of \(C\). Choosing as base point a geometric point of \(C\) we can say that ``\(\overline \Pi\) is a factor of the geometric fundamental group of \(C\) over \(K\)''.
Assume that \(C\) has a \(K\)-rational point \(P\) and choose a splitting \(s_P\) of (*) corresponding to \(P\) by identifying \(G_K\) with the decomposition group of an extension of the place corresponding to \(P\) in \(F(C)\) to its separable closure. Now the finite quotients of \(\Pi_g(C)\) on which \(s_P(G_K)\) operates trivially correspond to unramified Galois coverings \(C'\) of \(C\) on which the decomposition group of \(P\) operates trivially. Hence \(P\) has \(K\)-rational extensions to \(C'\) and the choice of \(s_P\) corresponds to the choice of such an extension \(P'\). (If we make another choice, then the corresponding section is replaced by a conjugate in \(\Pi(C).)\) We can regard \(C'\) as étale covering of \(C\) with respect to the base points \(P\) respectively \(P'\). Taking the limit we get the \(K\)-rational geometric fundamental group of \(C\) with base point \(P\):
\[
\Pi_g(C,P): =\Pi_g(C)/ \biggl\langle \bigl(s_P (\sigma)-1\bigr) \Pi_g(C) \biggr \rangle_{\sigma \in G_K}.
\]
We use either quartic coverings of the projective line or quadratic coverings of elliptic curves with enough ramification points to find for every genus \(g\geq 3\) curves with infinite geometric fundamental group defined over \(\mathbb{Q}(i)\) or over \(\mathbb{F}_q(i)\) (where \(i\) is a fourth root of unity) and we find even parametric families of such curves over every ground field containing \(i\). [Remark: We do not have any example of a curve defined over \(\mathbb{Q}\) with infinite \(\mathbb{Q}\)-rational geometric fundamental group.]
We thus have examples of curves of genus \(g\) with an infinite \(K\)-rational geometric fundamental group for every \(g\geq 3\). No such example can exist for curves of genus \(g=1\), as will be explained in the paper. This leaves only the case \(g=2\). Here we find very special curves with an infinite tower of regular unramified Galois coverings but we cannot decide there whether there are rational points in the tower. infinite regular Galois covering curves; rational geometric fundamental group with base point Gerhard Frey, Ernst Kani, and Helmut Völklein, Curves with infinite \?-rational geometric fundamental group, Aspects of Galois theory (Gainesville, FL, 1996) London Math. Soc. Lecture Note Ser., vol. 256, Cambridge Univ. Press, Cambridge, 1999, pp. 85 -- 118. Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Coverings of curves, fundamental group, Coverings in algebraic geometry, Rational points Curves with infinite \(K\)-rational geometric 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 The article consists of two sections. In the first one there are considered totally ramified triple coverings. Any such covering of a simply connected smooth projective variety is cyclic. The study of non- Galois triple coverings is equivalent to the study of Galois coverings whose Galois group is the third symmetric group \(S_ 3\). -- In the second section there are studied finite non-Galois triple coverings of \(\mathbb{P}^ 2\) whose branch locus consists of two smooth curves. The results in this section can be applied to the study of a 6-sheeted Galois covering of \(\mathbb{P}^ 2\). totally ramified triple coverings; Galois covering Coverings in algebraic geometry Two remarks on non-Galois triple 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 See the preview in Zbl 0739.14003. finite fundamental groups; invariants of surfaces of general type B. Moishezon and M. Teicher, Finite fundamental groups, free over \(\mathbb{Z}\)/c\(\mathbb{Z}\), Galois covers of \(\mathbb{C}\)\(\mathbb{P}\) 2, Mathematische Annalen 293 (1992), 749--766. Coverings in algebraic geometry Finite fundamental groups, free over \(\mathbb{Z}/c\mathbb{Z}\), for Galois covers of \(\mathbb{C}\mathbb{P}^2\). | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \({\mathcal M}_{g, n}\) be the moduli space of Riemann surfaces with \(n\) ordered marked points; the permutation group \(S_n\) acts naturally on this space, and the quotient \({\mathcal M}_{g, n}/S_n =: {\mathcal M}_{g, [n]}\) classifies the Riemann surfaces with \(n\) unordered marked points. One may look at \({\mathcal M}_{g, n}\) (respectively, \({\mathcal M}_{g, [n]}\)) as the quotient of \({\mathcal T}_{g, n}\) by the action of the mapping class group \(\Gamma_{g, n}\) (respectively, \(\Gamma_{g, [n]}\)), where \({\mathcal T}_{g, n}\) is the Teichmüller space. Let \(\varphi\) be an element of finite order of the mapping class group, then the image, in the moduli space, of the points of \({\mathcal T}_{g, n}\) fixed by \(\varphi\) is called the special locus of \(\varphi\).
The paper under review presents several results on special loci. The author starts by reviewing results on the moduli spaces of Riemann surfaces and on the geometry of \({\mathcal M}_{g, n}\) and \({\mathcal M}_{g, [n]}\) for \((g, n)\) in \(\{(0,4), (0,5), (1,1), (1,2)\}\). After that, she proves some results describing the special loci in \({\mathcal M}_{0, n}\) and \({\mathcal M}_{0, [n]}\), for arbitrary \(n\). The author then goes on to present many results on the relation of the special locus of \(\varphi\) and the moduli space of the topological quotient \(S/\varphi\). automorphisms of curves Schneps L.: Special Loci in Moduli Spaces of Curves. Mathematical Sciences Research Institute Publications, vol. 41, pp. 217--275. Cambridge University Press, London (2003) Families, moduli of curves (algebraic), Riemann surfaces; Weierstrass points; gap sequences, Automorphisms of curves, Coverings of curves, fundamental group, Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Coverings in algebraic geometry Special loci in moduli spaces of curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(f: X\to Y\) be a tamely ramified finite Galois covering of projective varieties over the field k and let F be a coherent \(G (=Gal(X/Y))\quad sheaf\) on X. The author proves that there is a finite complex of finitely generated projective k[G]-modules whose cohomology groups are \(H^*(X,F)\) (as k[G]-modules). The proof consists basically of annotations to his earlier proof of a similar result in the unramified case [the author, Invent. Math. 75, 1-8 (1984)].
In case X and Y are curves, the author produces k[G]-modules which connect the two cohomology groups \(H^ 0=H^ 0(X,E)\) and \(H^ 1=H^ 1(X,E)\) for locally free E. These modules realize the Brauer character \(ch_ G(H^ 0)-ch_ G(H^ 1)\) and, with a Schanuel lemma argument, show how each of \(H^ 0, H^ 1\) determines the other as k[G]-module, in a sort of ''equivariant Riemann-Roch'' formula. Galois module structure of cohomology groups; Brauer character Shōichi Nakajima, Galois module structure of cohomology groups for tamely ramified coverings of algebraic varieties, J. Number Theory 22 (1986), no. 1, 115 -- 123. Applications of methods of algebraic \(K\)-theory in algebraic geometry, Coverings in algebraic geometry, Coverings of curves, fundamental group Galois module structure of cohomology groups for tamely ramified 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 \(k\) be a field of finite characteristic \(p\), and let \(G\) be a finite group scheme whose order is a multiple of \(p\). Let \(kG\) be the dual of the coordinate algebra \(k[G]\). In the study of \(kG\)-modules one can consider \(\pi\)-points, flat \(K\)-algebra maps \(\alpha_K\colon K[t]/(t^p)\to KG\) for \(K\) an extension of \(k\). Two \(\pi\)-points \(\alpha_K,\beta_L\) are equivalent if for any finite dimensional \(kG\)-module \(M\) the restriction of \(M_K\) along \(\alpha_K\) is a free \(KG\)-module if and only if the restriction of \(M_L\) along \(\beta_L\) is free. If the rank of the linear operator \(\alpha_K(t)\) on \(M_K\) is independent of the choice of \(\alpha_K\) we say \(M\) has constant rank; moreover if the Jordan type of \(\alpha_K(t)\) is independent of the choice of \(\pi\)-point we say \(M\) is of constant Jordan type. This work is a study of \(k(\mathbb Z/p\times\mathbb Z/p)\)-modules of constant Jordan type, paying particular attention to a subcategory of such modules, \(W\)-modules, which are introduced here.
For \(G=(\mathbb Z/p)^r\), \(kG=k[t_1,\dots,t_r]/(t_1^p,\dots,t_r^p)\), a \(\pi\)-point \(\alpha_K\) is equivalent to the image of \(t\in K[t]/(t^p)\) -- denote this point in \(KG\) by \(\ell_{\alpha_K}\), or \(\ell_\alpha\) for short. A finite dimensional \(kG\)-module \(M\) has the equal images property if for any two \(\pi\)-points \(\alpha_K\) and \(\beta_L\) the images of \(\ell_\alpha(M_K)\) and \(\ell_\beta(M_L)\) agree after base change to a field extension \(\Omega\) of both \(K\) and \(L\). There are a number of equivalent formulations of the equal images property, such as for all \(\ell\in\text{Rad}(KG)\setminus\text{Rad}(KG)^2\) we have \(\ell(M_K)=\text{Rad}(M_K)\). As an example, if \(I=\text{Rad}(kG)\) then \(I^j\), a module of constant Jordan type, has the equal images property if and only if \((r-1)(p-1)\leq j\). More generally, if \(M\) has the equal images property, then \(M\) has constant Jordan type. Furthermore, for \(L\subset M\) a \(kG\)-submodule and \(M\) possesses the equal images property, then so does the quotient \(M/L\).
After the formulation of the above property, attention is focused on the case \(G=\mathbb Z/p\times\mathbb Z/p\). Then we may write \(kG=k[t_1,t_2]/(t_1^p,t_2^p)\). Let \(x\) and \(y\) denote the classes of \(t_1\) and \(t_2\) in \(\text{Rad}(kG)\): \(x\) and \(y\) clearly generate this radical. For \(1\leq d\leq n\), \(d\leq p\), let \(W_{n,d}\) be the \(kG\)-module generated by \(\{v_1,\dots,v_n\}\) subject to the relations \(xv_1=yv_n=x^dv_n=0\) and \(x^dv_i=yv_i-xv_{i+1}=0\) for \(1\leq i<n-1\). For \(d>n\) let \(W_{n,d}=W_{n,n}\). Collectively, these are called \(W\)-modules. It is shown that \(W\)-modules have the equal images property and a formula for the (constant) Jordan type is given. For \(n\leq p\) the classes \([W_{n,n}]\) form a minimal generating set of the Grothendieck group \(K_0(\mathcal CW)\) of the category of \(kG\)-modules of the form \(W_{n,d}\); furthermore \(K_0(\mathcal CW)\cong\mathbb Z^p\), the isomorphism arising from composing the map \(K_0(\mathcal CW)\to K_0(\mathcal C)\), where \(\mathcal C\) is the category of modules of constant Jordan type, with the Jordan type mapping.
One reason for focusing on the \(W\)-modules is their prevalence in \(\mathcal C\). For example, if \(\text{Rad}^2(M)\) is trivial, then \(M\) decomposes into a direct sum of \(W\)-modules; furthermore the decomposition is into \(W\)-modules with \(n=1\) or \(2\). Also, any \(kG\) module with the equal images property can be realized as a quotient of \(W_{n,d}\) for some \(n\), where \(\text{Rad}^d(M)\) is trivial.
Suppose for this paragraph that \(k\) is infinite. To any finite dimensional \(kG\)-module \(M\) and any point \(\langle a,b\rangle\in\mathbb P^1(k)\) we can define \(_{\langle a,b\rangle}M=\text{Ker}\{ax+by\colon M\to M\}\); more generally for \(S\subset\mathbb P^1(k)\) we let \(_SM\) be the sum of \(_{\langle a,b\rangle}M\) over all \(\langle a,b\rangle\in\mathbb P^1(k)\). The generic kernel \(\mathfrak R(M)\) is then the intersection of all \(_SM\) with \(S\subset\mathbb P^1(k)\) cofinite. The generic kernel is compatible with extending the field \(k\) in a natural way, a fact needed to prove that \(\mathfrak R(M)\) also has the equal images property. In fact, \(\mathfrak R(M)\) is the maximal submodule of \(M\) with the equal images property, and \(\mathfrak R(M)\) is the maximal submodule of \(M\) arising as the quotient of a \(W\)-module.
Let \(W\) be the generic kernel of \(M\), a module of constant rank. Then an increasing filtration of \(M\) is given, namely by submodules of the form \(x^iW\) for \(1-p\leq i\leq p-1\). In this filtration, \(x^iW\) has the equal images property and \(M/x^iW\) has the equal kernels property.
Finally, it is shown that any cyclic \(kG\)-module of constant Jordan type is a quotient of \(kG\) by a power of the augmentation ideal. This holds regardless of the characteristic of \(k\). finite group schemes; coordinate algebras; modules of constant Jordan type; \(W\)-modules; equal images property; modular representations; Grothendieck groups Carlson, J. F.; Friedlander, E. M.; Suslin, A. A., Modules for \(\mathbb{Z} / p \times \mathbb{Z} / p\), Comment. Math. Helv., 86, 609-657, (2011) Representation theory for linear algebraic groups, Group schemes, Cohomology theory for linear algebraic groups, Modular representations and characters, Representations of associative Artinian rings Modules for \(\mathbb Z/p\times\mathbb Z/p\). | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems \textit{R. Miranda} established [cf. Am. J. Math., 107, 1123-1158 (1985; Zbl 0611.14011)] that there exists a 1-1 correspondence between triple covers of varieties over fields of characteristic not equal to 2 or 3 and sections of certain vector bundles.
In this work the author analyzes this correspondence in the case of characteristic 3. Thus, he studies the ramification, branch locus, local structure of triple covers and the inseparable triple covers in the sections 5 and 6. - Finally in the section 7 and 8 he establishes the problem of lifting a triple cover in characteristic 3 to characteristic 0, and computes the invariants of triple covers of surfaces in the section 7 and 8. triple covers; characteristic 3; ramification Pardini, R., Triple covers in positive characteristic, Ark. Mat., 27, 2, 319-341, (1989) Coverings in algebraic geometry, Finite ground fields in algebraic geometry, Ramification problems in algebraic geometry Triple covers in positive characteristic | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(G\) be a finite group. A \(G-\)cover of projective varieties is a finite map \(f\colon X\to Y\), with \(X\) and \(Y\) projective varieties, such that \(f\) is Galois with Galois group equal to \(G\). \textit{M. Namba} [J. Math. Soc. Japan 41, No. 3, 391--403 (1989; Zbl 0701.14013)] has introduced the notion of versal \(G-\)cover: roughly speaking, a \(G-\)cover \(f\colon X\to Y\) is versal if every \(G-\)cover \(W\to Z\) can be obtained from \(f\) by taking base change with a suitable map \(Z\to Y\) and normalizing. Namba has also shown that the quotient map \(({\mathbb P}^1)^{| G| }\to ({\mathbb P}^1)^{| G| }/G\) is a versal \(G-\)cover for every \(G\). However, in practice it is not always easy to obtain \(G-\)covers of a given variety by applying the above described construction to \(({\mathbb P}^1)^{| G| }\to ({\mathbb P}^1)^{| G| }/G\), since the geometry of \(({\mathbb P}^1)^{| G| }/G\) is quite complicated. In the paper under consideration, the author constructs versal \(G-\)covers for \(G\) in a class of groups that contains in particular the symmetric groups and the dihedral groups. The advantage of this contruction is that the versal covers are of the form \(f: Y\to {\mathbb P}^n\) and therefore it is easier to produce suitable maps \(Z\to {\mathbb P}^n\) for a given projective variety \(Z\). Hiroyasu Tsuchihashi, Galois coverings of projective varieties for dihedral and symmetric groups, Kyushu J. Math. 57 (2003), no. 2, 411 -- 427. Coverings in algebraic geometry Galois coverings of projective varieties for dihedral and symmetric 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 In this corrigendum to the author's previous article [ibid. 104, No.1, 1- 6 (1988; Zbl 0687.14035)] it is pointed out that theorem 3 is incorrect. It relates the projective limit of the system \(G_{um}\) to the abelian fundamental group of U if \(U\subset X\) is an open subset of a proper smooth surface and \((G_{um},\alpha)\) a generalized Albanese pair. abelian fundamental group; Albanese pair Coverings in algebraic geometry, Picard schemes, higher Jacobians, Surfaces and higher-dimensional varieties Corrigendum to: ``On generalized Albanese varieties for surfaces'' | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We consider the existence of Hitchin's connections on complete noncompact Riemannian manifolds. Hitchin's self-duality; complete Riemannian manifolds; Hitchin connection; flat connection Harmonic maps, etc., Coverings in algebraic geometry, Riemann surfaces Hitchin's self-duality equations on complete Riemannian 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 \(Y\) be a complex projective surface and let \(B\subseteq Y\) be a reduced divisor. For \(n\in\mathbb{N}\), \(n\geq 2\), consider the map \(\varphi_ n:H^ 1(Y-B;\mathbb{Z})\to H^ 1(Y-B;\mathbb{Z}/n)\): \(\varphi_ n\) determines an unbranched Galois covering \(W_ n\to Y-B\) and this in turn determines uniquely a normal covering \(X_ n\to Y\) branched along \(B\). The author studies here the behaviour of the first Betti number \(b(n)\) of a desingularization \(\tilde X_ n\) of \(X_ n\) as a function of \(n\). (Remark that \(b(n)\) does not depend on the chosen desingularization.) The main result of the paper can be summarized as follows: under some topological conditions on \(B\), \(b(n)\) is a polynomial periodic function of \(n\), namely there exist an integer \(N\) and polynomials \(q_ 0,\dots,q_{N-1}\) such that if \(n\equiv i\pmod N\) then \(b(n)=q_ i(n)\). Polynomial periodicity for the first Betti number \(b'(n)\) of the unbranched covering \(W_ n\to Y-B\) had already been proven by \textit{Sarnak} [``Betti numbers of congruence groups'' (The Austral. Nat. Univ. Res. Rep., Canberra/Australia 1989)]. The author uses a generalization of Sarnak's method to show that the difference \(b(n)-b'(n)\) is polynomial periodic, too. polynomial periodicity for Betti numbers; branched covering; Betti number of a desingularization [Ha2] E. Hironaka,Polynomial periodicity for Betti numbers of covering surfaces, Inventiones Mathematicae108 (1992), 289--321. Coverings in algebraic geometry, Classical real and complex (co)homology in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Divisors, linear systems, invertible sheaves Polynomial periodicity for Betti numbers of covering 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 purpose of this paper is to find explicitly a polynomial equation \(f(z,w)=0\) such that (z,w)\(\to z\) extends to a map of the Riemann surface defined by this equation on \({\mathbb{P}}^ 1({\mathbb{C}})\) which has to be a covering of degree \(n,\) branched at three points, say \(a_ 1, a_ 2=0\) and \(a_ 3,\) and for which the monodromy permutations corresponding to these points are (2,1,...,1), (n), \((n_ 1)(n_ 2)\), respectively. The equation found has the form \(f(z,w)=w^ n+\alpha zw^{n-1}+\gamma zw^{n-2}+...+\mu z=0\) where the coefficients \(\alpha\),...,\(\mu\) are given explicitly in terms of the branch points \(a_ 1, a_ 2, a_ 3\) and a parameter d. covering of Riemann surface; monodromy permutations Coverings in algebraic geometry Constructin of an algebraic function field corresponding to an n-sheeted covering of a sphere | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This note is the second part of the preceding part I [\textit{T. Nakano} and \textit{T. Kamai}, Osaka J. Math. 33, 347-364 (1996; Zbl 0902.14009)]. We recall that in part I the finite maximal Galois coverings over a complex affine plane \(\mathbb{A}^2\) with branch locus \(B_q:= \{(v,w)\in \mathbb{A}^2 \mid w^2= v^q\}\) with \(q\) odd were studied in some detail, and further, the existence of maximal Galois coverings over a complex projective plane \(\mathbb{P}^2\) with branch locus \(\overline {B_q}\cup l_\infty\) was discussed, where \(\overline {B_q}\) is the projective closure of \(B_q\) and \(l_\infty\) is the infinite line. In this note, we study the finite maximal Galois coverings over \(\mathbb{A}^2\) (respectively \(\mathbb{P}^2)\) with branch locus \(B_q\) (respectively \(\overline {B_q}\cup l_\infty)\) with even \(q\). Our main results are:
(i) the maximal Galois covering of \(\mathbb{A}^2\) with branch locus \(B_q\) exists and is isomorphic to \(\mathbb{A}^2\) if the corresponding maximal Galois group is finite (theorem 4), and
(ii) a criterion for the existence of maximal Galois coverings over \(\mathbb{P}^2\) with branch locus \(\overline {B_q}\cup l_\infty\) (theorem 7).
Our results are easy consequences of the explicit calculations of the maximal Galois groups with some simple presentation. We thus spend many pages on elementary combinatorial group-theoretic computation. The reason for our doing this is twofold:
(i) These explicit descriptions of the maximal Galois groups are essentially used in our main results;
(ii) It will be convenient for the readers who are not accustomed to combinatorial group theory.
In order to calculate the order of finite groups with given presentation, we use Cayley/Magma system, which performs the Todd-Coxeter process on computers. Fermat ramification curve; maximal Galois coverings; branch locus Nakano, T., Nishikubo, H.: On some maximal Galois coverings over affine and projective planes II. Tokyo J. Math. 23(2), 295--310 (2000) Coverings in algebraic geometry, Ramification problems in algebraic geometry On some maximal Galois coverings over affine and projective planes. II | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A one page proof (using the real spectrum) of the following result: Let \(f,g: X\to {\mathbb{R}}\) be a continuous functions on a topological space X, such that f is strictly positive on \(g^{-1}(0)\). Then there exist continuous functions u, v and w such that \((1+v^ 2)f=1+w^ 2+gu\). J. M. Gamboa, ''A positivstellensatz for rings of continuous functions,''J. Pure Appl. Algebra,45, No. 3, 211--212 (1987). Relevant commutative algebra, Real algebraic and real-analytic geometry, Algebraic properties of function spaces in general topology A Positivstellensatz for rings of continuous 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 This paper concerns the liftability of Galois covers of smooth curves, and in particular extends the main result of [\textit{M. Garuti}, Compos. Math. 104, No. 3, 305--331 (1996 Zbl 0885.14011)] to the situation of towers of Galois covers. Let \(R\) be a complete discrete valuation ring of mixed characteristic with algebraically closed residue field \(k\) of characteristic \(p>0\). Suppose that \(X\) is a smooth proper \(R\)-curve, and \(Y_k\rightarrow X_k\) is a Galois (ramified) cover of the special fiber with group \(G\). Garuti proved in [loc. cit.] that after a finite extension \(R' | R\), there exists a Galois cover of normal \(R'\)-curves \(Y'\rightarrow X\times_R R'\) with group \(G\) such that the special fiber \(Y_k'\rightarrow X_k\) is generically Galois with group \(G\). Moreover, \(Y_k'\) has only unibranch singularities, and the original cover \(Y_k\rightarrow X_k\) factors as \(Y_k\rightarrow Y_k'\rightarrow X_k\) where the first map is the normalization and an isomorphism away from the ramification points of \(Y_k\rightarrow X_k\). The present author calls such a cover \(Y'\rightarrow X\times_R R'\) a Garuti lifting of \(Y_k\rightarrow X_k\). The main result of the present paper is the following extension of Garuti's theorem: in the context described above, let \(H\) be a quotient of the group \(G\), corresponding to an intermediate Galois cover \(Z_k\rightarrow X_k\) of \(Y_k\rightarrow X_k\). Suppose that \(Z'\rightarrow X\times_R R'\) is a given Garuti lifting of \(Z_k\rightarrow X_k\). Then there exists a finite extension \(R'' | R'\) and a Garuti lifting \(Y''\rightarrow X\times_R R''\) which dominates \(Z'\times_{R' }R''\rightarrow X\times_R R''\).
The proof of this result follows along similar lines to Garuti's original result, and makes essential use of formal patching in the style of Harbater. Along the way, the present author proves an important new theorem on the structure of a certain pro-\(p\) quotient of the ``geometric Galois group'' of a \(p\)-adic open disc \(\text{Spec}(R[[T]]\otimes_R \text{Frac}(R))\). Namely, the author establishes in Theorem 2.3.1 that this group is a free pro-\(p\) group (he also proves in Theorem 2.4.1 that a certain pro-\(p\) quotient of the ``geometric Galois group'' of the boundary of a \(p\)-adic open disc is free pro-\(p\)). A key ingredient is the analogous result due to Garuti in [loc. cit.] which says that the pro-\(p\) geometric fundamental group of a \(p\)-adic annulus of zero thickness is a free pro-\(p\) group. In Question 2.3.2, the present author poses the interesting open problem of whether the \textit{maximal} pro-\(p\) quotient of the ``geometric Galois group'' of a \(p\)-adic open disc is free pro-\(p\). lifting Galois covers; \(p\)-adic open disc; formal patching Coverings of curves, fundamental group, Coverings in algebraic geometry On a theorem of Garuti | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Ostensibly the main result here is in complex algebraic geometry. Suppose \(U\) is the complement of a reduced simple normal crossings divisor in a smooth connected projective surface s \(X\) over \({\mathbb C}\), and that \(h: X\to C\) is a flat morphism (not necessarily with connected fibres) to a curve. If \(V\to U\) is a finite étale cover, not necessarily connected although this makes no difference later, it extends to a finite map \(\pi: Y\to X\) in a natural way. Letting \(\rho: Y'\to Y\) be the minimal resolution of singularities and writing \(f: Y'\to C\) for the composition \(h\pi\rho\), the authors show that there is an integer \(c\) such that for all such \(\pi\)
\[
|\deg\det R^\cdot f_*{\mathcal O}_{Y'}|\leq c\deg \pi.
\]
Here \(\det R^\cdot f_*{\mathcal O}_{Y'}\) is the determinant of cohomology in the sense of Deligne.
It is of course immediate that some such bound exists, since the fundamental group of \(U\) is finitely generated. The novelty is in the explicit bound and the fact that it is linear.
The authors give two and a half proofs. The main one uses Grothendieck-Riemann-Roch and intersection theory on \(Y'\). For this one needs an analysis of the singularities of \(Y\). They are cyclic quotient singularities and the procedure of resolving them, and the relation with Hirzebruch-Jung continued fractions, is well known, but is summarised here for convenience. A second proof, ascribed by the authors to Esnault and Viehweg, uses Arakelov's inequality. It actually gives more, namely a formula for the constant \(c\), but needs very slightly stronger conditions (the fibres of \(h\) should be semistable). Finally, the need to control \(c_1^2\) in terms of \(\deg \pi\), which is where much of the work occurs, can be sidestepped by using the Bogomolov-Miyaoka-Yau inequality to compare \(c_1^2\) with \(c_2\): Arakelov's inequality can also be replaced in this way, at the cost of losing the formula for \(c\).
Multiple proofs are given in the hope that one of them will give a proof of an arithmetic analogue, stated here as a conjecture, which is perhaps the real point of the paper. The conjecture bounds the Faltings height of the normalisation of \({\mathbb P}^1_{\mathbb Q}\) in the function field of a étale cover \(\pi: V\to U_{{\mathbb Z}[1/\ell]}\) of an open part of \({\mathbb P}^1_{\mathbb Z}\) away from a prime \(\ell\) by \((\deg \pi)^a\ell^b\), where \(a\) and \(b\) are integers depending on \(U\). If true, this conjecture would be expected to yield a polynomial time algorithm for computing mod~\(\ell\) étale cohomology (as Galois modules) of a surface over \({\mathbb Q}\) and hence a method of counting points over \({\mathbb F}_p\) on a surface over \({\mathbb F}_p\) in polynomial time in \(\log p\). Existing algorithms for this last problem have exponential running time. branched covers; intersection theory; heights; Galois representations Edixhoven, B; Jong, R; Schepers, J, Covers of surfaces with fixed branch locus, Int. J. Math., 21, 859-874, (2010) Coverings in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Ramification problems in algebraic geometry, Arithmetic varieties and schemes; Arakelov theory; heights Covers of surfaces with fixed branch locus | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper is a continuation of our paper ``Simply connected algebraic surfaces of positive index'' [Invent. Math. 89, 601-643 (1987; Zbl 0627.14019)]. There we constructed a series of surfaces of positive and zero indices denoted \(\hbox{Gal}(X_{ab})\) for \(a\), \(b\in \mathbb{Z}\), \(a\geq 5\), \(b\geq 6\). For \(a,b\) relatively prime we got \(\pi_ 1(\hbox{Gal}(X_{ab}))=0\). This gave a counterexample to the Bogomolov- Watershed conjecture that surfaces of general type and positive index are not simply connected. In this paper we compute \(\pi_ 1(\hbox{Gal}(X_{ab}))\) for arbitrary \(a\), \(b\) and prove that \(\pi_ 1(\hbox{Gal}(X_{ab}))\) is a finite commutative group free over \(\mathbb{Z}/c\mathbb{Z}\) for \(c=\hbox{g.c.d.}(ab)\) with \(2ab\) generators.
This study is related to the problem of finding new invariants of surfaces of general type distinguishing different components of moduli spaces.
We use all the notations, results and subresults of our paper cited above. finite fundamental groups; invariants of surfaces of general type Boris Moishezon and Mina Teicher, Finite fundamental groups, free over \?/\?\?, for Galois covers of \?\?², Math. Ann. 293 (1992), no. 4, 749 -- 766. Coverings in algebraic geometry Finite fundamental groups, free over \(\mathbb Z/c\mathbb Z\), for Galois covers of \(\mathbb C\mathbb P^2\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a smooth projective geometrically irreducible curve over a local field \(k\). The purpose of this paper is to develop unramified class theory for \(X\), i.e. to try to approximate the abelian fundamental group \(\pi_ 1^{ab}(X)\) which controls the abelian étale covers of \(X\) by means of some group defined ``downstairs''. \textit{S. Bloch} [Ann. Math. (2) 114, 229--265 (1981; Zbl 0512.14009)] devised what this group should be, namely the group \(SK_ 1(X)\) which is the cokernel of the tame residue map going from \(K_ 2\) of the function field \(K=k(X)\) of \(X\) to the direct sum of the multiplicative groups \(k(x)^*\) of the residue fields \(k(x)\) at all closed points \(x\) of \(X\). There is a natural map from \(SK_ 1(X)\) to \(k^*\) induced by the norm maps \(k(x)^*\to k^*\). The kernel of this map is denoted \(V(X)\).
The aim of the paper is to define and study a reciprocity map \(\sigma: SK_ 1(X)\to \pi_ 1^{ab}(X)\) and an induced map \(\tau: V(X)\to T_ G\) where \(T_ G\) is the module of coinvariants of the Tate module of the Jacobian of \(X\), which is known to control the abelian unramified extensions of \(X\) which do not come from \(k\) (\(T_ G\) is the kernel of the surjective map \(\pi_ 1^{ab}(X)\to \text{Gal}(k^{ab}/k))\). The map \(\sigma\) is defined in a rather elaborate manner via the class field theory of two-dimensional local fields [\textit{K. Kato}, J. Fac. Sci., Univ. Tokyo, Sect. IA 26, 303--376 (1979; Zbl 0428.12013); ibid 27, 603--683 (1980; Zbl 0463.12006)] and the reciprocity law for two-dimensional local rings, due to Kato and described in \S1 of the present paper. The map \(\sigma\) then induces the map \(\tau\), first introduced by Bloch for \(X(k)\neq \emptyset\). In the good reduction case, Bloch (loc. cit.) showed that \(\tau\) is a surjection onto a finite group. At least when \(\text{char}(k)=0\), the author shows: (i) the image of \(\tau\) is always finite; (ii) its kernel is the maximal divisible subgroup of \(V(X)\); (iii) the cokernel \(\tau\) is isomorphic to \(\widehat{\mathbb Z}^ r\) (\(\widehat{\mathbb Z}\) is the profinite completion of \(\mathbb Z\)), where \(r\) is some integer which may be computed from the structure of the special fibre of a regular proper model of \(X\) over the ring of integers of \(k\), is at most equal to the genus of \(X\) and is zero if \(X\) has good reduction. Similar results hold for \(SK_ 1(X)\).
In the good reduction case over a \(p\)-adic field \(k\), part (ii) was also obtained by \textit{K. R. Coombes} [``Local class field theory for curves'', in Contemp. Math. 55, part I, 117--134 (1986; Zbl 0601.14025)], and it is shown by the reviewer and \textit{W. Raskind} [Math. Ann. 270, 165--199 (1985; Zbl 0536.14004)] that the kernel of \(\tau\) is uniquely divisible prime-to-\(p\). In view of Bloch's remark that the map \(\tau\) may be viewed as some analogue of the norm residue symbol \(K_ 2k\to \mu (k)\), one may wonder whether in the general case more can be said about torsion of \(\text{ker}\,\tau\).
That the cokernel of \(\tau\) might be non-trivial reflects a quite interesting geometric fact which makes a big difference with classical local class field theory: there may exist étale covers \(Y/X\) which are completely split, i.e. the fibre of \(Y/X\) over any closed point \(x\) of \(X\) consists of copies of \(\text{Spec}\,k(x)\). Such covers are not controlled by the \(K\)-theory of \(X\).
Among the ingredients used for the proof are the higher local class field theory of Kato, results of Y. Ihara and of \textit{H. Miki} [J. Fac. Sci., Univ. Tokyo, Sect. I A 21, 377--393 (1974; Zbl 0301.12003)], and the Merkur'ev-Suslin theorem [\textit{A. S. Merkur'ev} and \textit{A. A. Suslin}, Math. USSR, Izv. 21, 307--340 (1983); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 46, No. 5, 1011--1046 (1982; Zbl 0525.18008)].
Some results of this paper are used, and sometimes given a slightly different approach by \textit{K. Kato} and the author in Ann. Math. (2) 118, 241--275 (1983; Zbl 0562.14011) and Galois groups and their representations, Proc. Symp., Nagoya/Jap. 1981, Adv. Stud. Pure Math. 2, 103--152 (1983; Zbl 0544.12011) -- this last paper also considers ramified abelian coverings of \(X\). Extensions to higher dimensional varieties over a local field k are considered by the author in Ann. Math. (2) 121, 251--281 (1985; Zbl 0593.14001). class field theory for curves over local fields; abelian fundamental group; class field theory of two-dimensional local fields; reciprocity law Saito S.: Class field theory for curves over local fields. J. Number Theory 21(1), 44--80 (1985) 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, Coverings in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry, Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects) Class field theory for curves over local 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 In this paper we prove that the universal covering of a smooth projective variety \(X\) is holomorphically convex if \(\pi_1(X)\) is nilpotent. This is a partial case of the Shafarevich conjecture saying that the universal coverings of smooth projective varieties are holomorphically convex. The technique used in the proof is the strictness property of the mixed Hodge structures. At the end we also exhibit some properties of the solvable linear fundamental groups of smooth projective varieties. holomorphic convexity; nilpotent fundamental groups; universal covering; Shafarevich conjecture L. Katzarkov, ''Nilpotent groups and universal coverings of smooth projective varieties,'' J. Differential Geom., vol. 45, iss. 2, pp. 336-348, 1997. Coverings in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry Nilpotent groups and universal coverings of smooth 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 This is mainly an announcement of results and proofs are either absent or sketched. The authors first outline how to introduce a group structure on an \(n\)-fold covering of a compact connected group in such a way that the covering map becomes a group homomorphism. This is done by approximating compact connected groups by compact Lie groups in the usual way, subsequently using the covering group theory for path connected and locally path connected topological groups already present in \textit{Pontryagin}'s 1938 classical monograph and finally putting everything together in an adequate way.
The algebraicity of all coverings of a compact Abelian group and a criterion for the triviality of all \(n\)-fold coverings of a compact connected Abelian group are then announced as applications. This is further applied to give a criterion for the solvability of algebraic equations with functional coefficients. \(n\)-fold covering; covering group; algebraic covering; triviality of \(n\)-fold coverings; algebraic equation with functional coefficients; dual group; one dimensional Čech cohomology group Grigorian, S. A.; Gumerov, R. N., On a covering group theorem and its applications, Lobachevskii J. Math., 10, 9-16, (2002) Analysis on general topological groups, Coverings in algebraic geometry, Harmonic analysis on general compact groups, Measure algebras on groups, semigroups, etc., Character groups and dual objects, Banach algebras of continuous functions, function algebras, Cohomology of groups, Compact groups, Topological groups (topological aspects), Covering spaces and low-dimensional topology On a covering group theorem and its applications | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a nonsingular complex projective curve and \(\pi:Y\to X\) be an étale covering. The behavior of vector bundles under the direct image functor \(\pi_*\) was carefully studied by \textit{M. C. Narasimhan} and \textit{S. Ramanan} [J. Indian Math. Soc., New Ser. 39, 1--19 (1975; Zbl 0422.14018)]. In this paper, the following result of the cited paper is generalized to holomorphic triples, and related issues concerning moduli spaces are discussed:
Theorem. Let \(\pi:Y\to X\) be a finite, unramified Galois covering over a complete, nonsingular algebraic curve, and let \(m\) be the order of the Galois group \(G\). If \(E\) is a vector bundle over \(Y\), then we have
(1) \(E\) is semistable if and only if \(\pi_*E\) is semistable.
(2) \(E\) is stable if and only if \(\pi_*E\) is stable and no two of the bundles \(s^*E\), \(s\in G\), are isomorphic. semistable vector bundle; behavior of vector bundles under the direct image; Galois covering; algebraic curve Vector bundles on curves and their moduli, Coverings in algebraic geometry Direct images of stable triples | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We give a counterexample to the most optimistic analogue (due to Kleshchev and Ram) of the James conjecture for Khovanov-Lauda-Rouquier algebras associated to simply-laced Dynkin diagrams. The first counterexample occurs in type \(A_5\) for \(p=2\) and involves the same singularity used by Kashiwara and Saito to show the reducibility of the characteristic variety of an intersection cohomology \(D\)-module on a quiver variety. Using recent results of Polo one can give counterexamples in type \(A\) in all characteristics. James conjecture; Khovanov-Lauda-Rouquier algebras; simply-laced Dynkin diagrams; intersection cohomology \(D\)-modules; quiver varieties Williamson, G., On an analogue of the James conjecture, Represent. Theory, 18, 15-27, (2014) Hecke algebras and their representations, Modular representations and characters, Representations of finite symmetric groups, Representations of quivers and partially ordered sets, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) On an analogue of the James conjecture. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We construct a family of plane curves as pull-backs of a conic by abelian coverings of \(\mathbb{P}^2\). If the conic is tangent to the branching lines one obtains a family of curves of degree \(2n\) with \(3n\) singularities of type \(A_{n-1}\). For \(n\) odd, this construction produces an irreducible curve. We calculate the fundamental group for any member of this family and the Alexander polynomial for the irreducible ones. Finally we study also how deformations of the family affect to the fundamental group. cuspidal curves; family of plane curves; pull-backs of a conic; fundamental group; Alexander polynomial Cogolludo-Agustín, J. I., Fundamental group for some cuspidal curves, Bull. Lond. Math. Soc., 31, 2, 136-142, (1999) Singularities of curves, local rings, Coverings of curves, fundamental group, Coverings in algebraic geometry Fundamental group for some cuspidal curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We examine the cohomology and representation theory of a family of finite supergroup schemes of the form \((\mathbb{G}_a^- \times \mathbb{G}_a^-) \rtimes (\mathbb{G}_{a (r)} \times (\mathbb{Z} / p)^s)\). In particular, we show that a certain relation holds in the cohomology ring, and deduce that for finite supergroup schemes having this as a quotient, both cohomology mod nilpotents and projectivity of modules is detected on proper sub-supergroup schemes. This special case feeds into the proof of a more general detection theorem for unipotent finite supergroup schemes, in a separate work of the authors joint with Iyengar and Krause.
We also completely determine the cohomology ring in the smallest cases, namely \((\mathbb{G}_a^- \times \mathbb{G}_a^-) \rtimes \mathbb{G}_{a (1)}\) and \((\mathbb{G}_a^- \times \mathbb{G}_a^-) \rtimes \mathbb{Z} / p\). The computation uses the local cohomology spectral sequence for group cohomology, which we describe in the context of finite supergroup schemes. cohomology; finite supergroup scheme; invariant theory; Steenrod operations; local cohomology spectral sequence Group schemes, Superalgebras, Modular representations and characters, Cohomology of groups, Hopf algebras and their applications, ``Super'' (or ``skew'') structure, Homological methods in Lie (super)algebras Representations and cohomology of a family of finite supergroup 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 studies some properties of totally \(d\) cyclic coverings \(Y\) of a smooth projective surface \(X\) of general type branched along smooth curves \(B\subset X\) that are numerically equivalent to a multiple of the canonical class of \(X\). In particular, the authors prove that the moduli space of \(Y\) consists of at least two connected components under some condition of \(\text{Tor}_d\text{Pic}(X)\).
The paper under review concerns mainly the cases when \(X\) are surfaces of general type with \(p_g=0\) or Miyaoka-Yau surfaces (surfaces of general type with \(c_1^2=3c_2\)). In particular, the authors show that if \(X\) is a Miyaoka-Yau surface with a suitable condition then the number of connected components of \(Y\) is equal to the number of orbits of the action of \(\text{Aut}(X)\) on \(\text{Tor}(X)\). Finally, the authors find new examples of multi-component moduli spaces of surfaces with given Chern numbers and new examples of surfaces that are not deformation equivalent to their complex conjugates.
We mention that there are many interesting examples of surfaces that are not deformation equivalent to their complex conjugates in the paper [Ann. Math. (2) 158, No. 2, 577--592 (2003; Zbl 1042.14011)] by \textit{F. Catanese}. numerically pluricanonical cyclic coverings of surfaces; irreducible components of moduli spaces of surfaces Coverings in algebraic geometry, Surfaces of general type, Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants), Topological aspects of complex manifolds, Families, moduli, classification: algebraic theory, Singularities of surfaces or higher-dimensional varieties On numerically pluricanonical cyclic coverings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors study the analytic equivalences of some cyclic coverings of \({\mathbb{P}}^ 1\) and Kummer branched coverings of \({\mathbb{P}}^ 1\) and \({\mathbb{P}}^ 2\) [which are introduced by \textit{F. Hirzebruch} in Arithmetic and Geometry, Pap. dedic. I. R. Shafarevich, Vol. II: Geometry, Prog. Math. 36, 113-140 (1983; Zbl 0527.14033)] in the case of asymmetric branch locus (if \(B=\{b_ 1,...,b_ r\}\) is a set of r distinct points of \({\mathbb{P}}^ 1\) with \(r\geq 4\), B is asymmetric if \(Aut({\mathbb{P}}^ 1,B)=\{\phi \in Aut({\mathbb{P}}^ 1);\quad \phi (B)=B\}=id).\) automorphisms; cyclic coverings; Kummer branched coverings; asymmetric branch locus Ramification problems in algebraic geometry, Coverings in algebraic geometry, Coverings of curves, fundamental group Equivalence problem and automorphisms of some abelian branched coverings of the Riemann sphere | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In the paper under review, the author gives a concise survey of recent developments on embeddings of orthogonal Grassmannians. After a brief review of basic definitions and notions of projective and Veronesean embeddings of point-line geometries, the author restricts himself to Grassmannians \(\mathcal B_{n,k}\) and \(\mathcal D_{n,k}\) of buildings of types \(B_n\) and \(D_n\) and seven associated embeddings. Some of the embeddings are known to be isomorphic in case the characteristic of the underlying field is not 2. Section 4 of the paper is mostly devoted to discussing the various embeddings in case of characteristic 2. Sketches of proofs are provided in order to convey the flavour of the arguments. The results of this section as well as of the following section on universality of embeddings are a summary of three papers by \textit{I. Cardinali} and \textit{A. Pasini} [J. Algebr. Comb. 38, No. 4, 863--888 (2013; Zbl 1297.14053); J. Comb. Theory, Ser. A 120, No. 6, 1328--1350 (2013; Zbl 1278.05052)] and [J. Group Theory 17, No. 4, 559--588 (2014; Zbl 1320.20041)].
The last section deals with universality of Weyl embeddings of \(\mathcal B_{n,k}\) and \(\mathcal D_{n,k}\). These are obtained from the fundamental dominant weights for the root system of types \(B_n\) and \(D_n\). \textit{A. Kasikova} and \textit{E. Shult} [J. Algebra 238, No. 1, 265--291 (2001; Zbl 0988.51001)] showed that most of these point-geometries admit universal projective embeddings. The author conjectures that the Weyl embeddings of \(\mathcal B_{n,k}\) and \(\mathcal D_{n,k}\) are universal for \(k = 2, \dots, n-1\); the ones when \(k = 1\) are known to be universal. He then considers the cases \(k = 2\) and 3 under additional assumptions and outlines a proof of universality in these situations. point-line geometry; orthogonal polar space; Grassmannian; Weyl module; Veronese variety; embedding; universal embedding Incidence structures embeddable into projective geometries, Buildings and the geometry of diagrams, Grassmannians, Schubert varieties, flag manifolds, Modular representations and characters, Linear algebraic groups over arbitrary fields, Lie algebras of linear algebraic groups, Modular Lie (super)algebras, Polar geometry, symplectic spaces, orthogonal spaces Embeddings of orthogonal Grassmannians | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A Gorenstein cover is a finite flat map \(f: X\to Y\) of algebraic varieties such that the fibre \(f^{-1}(y)\) is Gorenstein for every \(y\in Y\). In previous papers [\textit{G. Casnati} and \textit{T. Ekedahl}, J. Algebr. Geom. 5, 439-460 (1996; Zbl 0866.14009), \textit{G. Casnati}, J. Algebr. Geom. 5, 461-477 (1996; Zbl 0921.14006)], the author has given general structure theorems for Gorenstein covers of degree \(d\leq 5\) using the classification of 0-dimensional arithmetically Gorenstein subschemes of \(\mathbb{P} ^{d-2}\).
An analogous result for covers of degree 6 does not seem within reach, since there is at the moment no classification theorem of 0-dimensional arithmetically Gorenstein subschemes of \(\mathbb{P} ^4\).
In the paper under consideration, the author restricts his attention to a special class of degree 6 covers, the so-called Scandinavian covers. A Scandinavian cover of a variety \(Y\) is constructed as follows: Take locally free sheaves \({\mathcal E}\), \({\mathcal A}\) and \({\mathcal B}\) of ranks respectively 5,3, 3, consider the projective bundle \(\pi: \mathbb{P} ({\mathcal E})\to Y\) and a map \(\delta: \pi^*{\mathcal A}\to \pi^*{\mathcal B}(1)\), let \(X\subset \mathbb{P} ({\mathcal E})\) be the locus where \(\delta\) has rank \(\leq 1\) and denote by \(f: X\to Y\) the restriction of \(\pi\). If \(f\) has finite fibres, then it is a cover of degree 6.
The main result of the paper is an existence theorem for smooth Scandinavian covers under the following assumptions:
(i) \(Y\) is smooth of dimension \(\leq 2\);
(ii) \({\mathcal A}^{\vee}\otimes {\mathcal B}\otimes {\mathcal E}\) is globally generated;
(iii) \(\text{det}{\mathcal E}=(\text{det}{\mathcal B}^{-1}\otimes\text{det}{\mathcal A})^2\). covering; Scandinavian cover; Gorenstein covers Casnati G. (2001). Cover of algebraic varieties IV A Bertini theorem for scandinavian covers of degree 6. Forum Math. 13:21--36 Coverings in algebraic geometry, \(K3\) surfaces and Enriques surfaces, Ramification problems in algebraic geometry Covers of algebraic varieties. IV: A Bertini theorem for Scandinavian 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 In this article we show that a general octic surface in \({\mathbb{P}}^ 3\) has a finite number of everywhere tangent lines and using the Harris' technique [cf. \textit{J. Harris}, Duke Math. J. 46, 685-724 (1979; Zbl 0433.14040)], we study the monodromy group of these lines and prove that this group is the full symmetric group. monodromy group; everywhere tangent lines; octic surface Projective techniques in algebraic geometry, Special surfaces, Enumerative problems (combinatorial problems) in algebraic geometry, Questions of classical algebraic geometry, Coverings in algebraic geometry On the monodromy group of everywhere tangent lines to the octic surface in \({\mathbb{P}}^ 3\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The general motivation for this paper is ``Inverse Galois Theory'': the construction of Galois extensions of a field with a given finite group \(G\) as Galois group. For more on this topic, see the review by \textit{R. W. Odoni} in Bull. Lond. Math. Soc. 27, 90-92 (1995)] of the related books of \textit{J.-P. Serre} [Topics in Galois theory (Jones and Bartlett, 1992; Zbl 0746.12001)] and \textit{B. Matzat} [Konstruktive Galoistheorie, Lect. Notes Math. 1284 (Springer-Verlag, 1987; Zbl 0634.12011)].
The aim of this paper is the factorization of the polynomial \(Y^9 - XY^7 + e\), with \(e \neq 0\), \(e\) in a field with characteristic 7. This factorization shows that \(\text{PSL}(2,8)\) is the Galois group of a certain unramified covering of the affine line over an algebraically closed field of characteristic 7. To quote the paper ``\dots this paper may be regarded as a huge exercise in the high-school art of factoring polynomials. But the high-school is to be mixed with a goodly dose of things like valuations of algebraic function fields from college algebra, and resolution of singularities of plane curves from geometry, and so on.'' -- I liked very much the discussion and the use of plane curves and skipped the calculations.
The appendix by J.-P. Serre contains a proof of the fact that this extension of the affine line in characteristic 7 can be constructed from a similar extension in characteristic 0. The appendix contains also fascinating speculations of a ``modular'' interpretation of this construction. inverse Galois theory; algebraic fundamental group; plane curves; factorization of polynomials; resolution of plane curve singularities; hyperelliptic function fields; construction of Galois extensions; finite group; Galois group; PSL(2,8); unramified covering; affine line Shreeram S. Abhyankar, Square-root parametrization of plane curves, Algebraic geometry and its applications (West Lafayette, IN, 1990) Springer, New York, 1994, pp. 19 -- 84. Inverse Galois theory, Special algebraic curves and curves of low genus, Coverings of curves, fundamental group, Coverings in algebraic geometry Square-root parametrization of plane curves. Appendix by J.-P. Serre | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems It is shown that every Galois extension of the rationals with symmetric group \(S_ n\) can be obtained as a specialization of a suitable model of a Galois branched covering \({\mathcal C}\to P^ 1\) with the same Galois group, having at most \(2n-1\) branch points. (Here \({\mathcal C}\) is a connected complete and smooth curve over the complex field.) A similar result is obtained for finite abelian extensions of any number field. The proof is based on the work of \textit{D. Saltman} [Adv. Math. 43, 250-283 (1982; Zbl 0484.12004)]. branched covering; Galois group Beckmann, Sybilla, Is every extension of \(\mathbb{Q}\) the specialization of a branched covering?, J. algebra, 164, 2, 430-451, (1994) Separable extensions, Galois theory, Galois theory, Inverse Galois theory, Coverings in algebraic geometry Is every extension of \(\mathbb{Q}\) the specialization of a branched covering? | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Hypertoric varieties (or toric hyperkähler varieties) are hyperkähler analogues of toric varietes. They come with a symplectic form \(\omega\) on their regular locus. Such affine varieties \((Y,\omega)\) are called ``conical symplectic varieties''. Their fundamental geometric properties, deformation theory, birational geometry have been extensively studied. However, as the author suggests, notions of ``universal coverings'' have not been well-studied.
In [\textit{Y. Namikawa}, Kyoto J. Math. 53, No. 2, 483--514 (2013; Zbl 1277.32029)] the author asks whether for a concial symplectic variety \((Y,\omega)\) the fundamental group \(\pi_1\left(Y_{\operatorname{reg}}\right)\) is finite. Some partial results in this direction are later given by Namikawa.
Now, assuming this fundamental group is finite, Nagaoka defines in the present article the (singular) universal covering of a conical symplectic variety \((Y,\omega)\). In this way one obtains a new example of a conical symplectic variety.
One of the main motivations of the present article is the problem of describing these universal coverings and the fundamental groups \(\pi_1\left(Y_{\operatorname{reg}}\right)\).
Another reason to study universal coverings comes from an of analogue of the Bogomolov's decomposition, which asks if one can decompose the universal covering \(\left(\overline{Y}, \overline{\omega}\right)\) of \((Y, \omega)\) into a product \(\prod_i\left(Y_i, \omega_i\right)\) of irreducible conical symplectic varieties. Here, \(\omega_i\) is the unique conical symplectic structure on \(Y_i\) up to scalar. Such a decomposition result is conjectured by Namikawa for general conical symplectic varieties.
The present article gives a complete answer to the first problem for affine hypertoric varieties. It describes the universal covering of an affine hypertoric variety in terms of the combinatorics of the associated hyperplane arrangement. This is also an affine hypertoric variety. Also a description of \(\pi_1\left(Y_{\operatorname{reg}}\right)\) is given.
In the last section, the space of homogeneous \(2\)-forms on decomposable hypertoric varieties is determined. As an application, the author obtains the analogue of Bogomolov's decomposition for hypertoric varieties and refined classification results. hypertoric varieties; conical symplectic varieties; universal cover; fundamental group of regular locus; Bogomolov's decomposition; uniqueness of symplectic structure Coverings in algebraic geometry, Momentum maps; symplectic reduction, Toric varieties, Newton polyhedra, Okounkov bodies, Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.), Arrangements of points, flats, hyperplanes (aspects of discrete geometry) The universal covers of hypertoric varieties and Bogomolov's decomposition | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We study the endomorphism ring \(\text{End} (J(C))\) of the complex jacobian \(J(C)\) of a curve \(y^p=f(x)\) where \(p\) is an odd prime and \(f(x)\) is a polynomial with complex coefficients of degree \(n>4\) and without multiple roots. Assume that all the coefficients of \(f\) lie in a (sub)field \(K\) and the Galois group of \(f\) over \(K\) is either the full symmetric group \(S_n\) or the alternating group \(A_n\). Then we prove that \(\text{End} (J(C))\) is the ring of integers \(\mathbb{Z} [\zeta_p]\) in the \(p\)-th cyclotomic field \(\mathbb{Q} [\zeta_p]\) if \(p\) is a Fermat prime (e.g., \(p=3, 5, 17, 257)\). Notice that recently the author extended this result to the case of an arbitrary odd prime \(p\) [\textit{Yu. G. Zarkhin}, ``The endomorphism rings of jacobians of cyclic covers of the projective line'' (\url{http://xxx.lanl.gov/abs/math.AG/0103203})].
The case of positive characteristic \(\neq p\) (under additional assumptions that \(n>8\) and \(p\mid n)\) is discussed in another one of the author's papers [\textit{Yu. G. Zarkhin}, ``Endomorphism rings of certain jacobians'' (to appear)] there he proves that \(\mathbb{Z} [\zeta_p]\) coincides with its own centralizer in the endomorphism ring of the jacobian. cyclic covers of the projective line; endomorphism ring; jacobian Yu. G. Zarhin, ''Cyclic Covers of the Projective Line, Their Jacobians and Endomorphisms,'' J. Reine Angew. Math. 544, 91--110 (2002). Jacobians, Prym varieties, Coverings in algebraic geometry, Local theory in algebraic geometry Cyclic covers of the projective line, their jacobians and endomorphisms | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The theory of coverings of the two-dimensional torus is a standard part of algebraic topology and has applications in several topics in string theory, for example, in topological strings. This paper initiates applications of this theory to the counting of orbifolds of toric Calabi-Yau singularities, with particular attention to abelian orbifolds of \( {\mathbb{C}^D} \). By doing so, the work introduces a novel analytical method for counting abelian orbifolds, verifying previous algorithm results. One identifies a \(p\)-fold cover of the torus \( {\mathbb{T}^{D - 1}} \) with an abelian orbifold of the form \( {{{{\mathbb{C}^D}}} \left/ {{{\mathbb{Z}_p}}} \right.} \), for any dimension \(D\) and a prime number \(p\). The counting problem leads to polynomial equations modulo \(p\) for a given abelian subgroup of \(S_{D}\), the group of discrete symmetries of the toric diagram for \( {\mathbb{C}^D} \). The roots of the polynomial equations correspond to orbifolds of the form \( {{{{\mathbb{C}^D}}} \left/ {{{\mathbb{Z}_p}}} \right.} \), which are invariant under the corresponding subgroup of \(S_{D}\). In turn, invariance under this subgroup implies a discrete symmetry for the corresponding quiver gauge theory, as is clearly seen by its brane tiling formulation. D-branes; differential and algebraic geometry; conformal field models in string theory; superstring vacua Y.-H. He, P. Candelas, A. Hanany, A. Lukas and B. Ovrut eds. \textit{Computational Algebraic Geometry in String, Gauge Theory}, \textit{Advances in High Energy Physics}\textbf{2012} (2012) 431898. String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, Supersymmetric field theories in quantum mechanics, Calabi-Yau manifolds (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies, Coverings in algebraic geometry, Topology and geometry of orbifolds, Polynomials, Topological field theories in quantum mechanics, Representations of quivers and partially ordered sets Calabi-Yau orbifolds and torus coverings | 0 |
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