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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 It is well known that any triple cover can be constructed by a minimal cubic equation. Based on this fact, we compute the normalization, the rank two trace-free sheaf, the branch locus and the singular locus of a triple cover. We give also a canonical resolution of the singularities of a triple cover surface, which provides us the formulas to compute the global invariants. The classical criterion for cubic extensions to be Galois is presented for triple covers. Finally, we establish a relationship between Miranda's triple cover data and the minimal cubic equations. triple cover; minimal cubic equation; normalization; trace-free sheaf; branch locus; singular locus; canonical resolution Tan S L. Triple covers on smooth algebraic varieties. AMS/IP Stud Adv Math, 2002, 20: 143--164 Coverings in algebraic geometry Triple covers on smooth 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 In a famous paper of 1987 [in: Number theory, Semin. New York 1984/85, Lect. Notes Math. 1240, 165--195 (1987; Zbl 0627.12015)], \textit{D. Harbater} proved that any finite group \(G\) can be realized as the Galois group of a finite normal covering of the projective line over a given complete non-Archimedean field \(K\).
The present authors show that if \(\text{char}(K)= 0\), the covering curve can be chosen to be a Mumford curve. If, however, \(\text{char}(K)= p>0\), the authors find a necessary and sufficient condition for \(G\) to be realizable in the above sense. -- The first part of the proof is close to \textit{Q. Liu}'s proof of Harbater's theorem [in: Recent developments in the inverse Galois problem, Joint Summer Res. Conf., Univ. Washington 1993, Contemp. Math. 186, 261--265 (1995; Zbl 0834.12004)]. For the second part, the authors describe the uniformization of the Mumford curves involved. Note that the characteristic 0 result was independently obtained by \textit{P. E. Bradley} in his Ph.D. thesis [``\(p\)-adische Hurwitzräume'', Diss. (Karlsruhe 2002)] using the reviewer's characterization of discontinuous subgroups of \(\text{PGL}_2(K)\) [\textit{F. Herrlich}, Arch. Math. 39, 204--216 (1982; Zbl 0466.14007)]. finite group realized as Galois group; covering of the projective line; Mumford curve Van Der Put, M., Voskuil, H.: Mumford coverings of the projective line. Archiv der Mathematik 80, 98--105 (2003) Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Coverings of curves, fundamental group, Coverings in algebraic geometry Mumford coverings of the projective line | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A quasi-étale quotient of a product of two curves is the quotient of a product of two curves by the action of a finite group which acts freely out of a finite set of points. A quasi-étale surface is the minimal resolution of the singularities of a quasi-étale quotient. They have been successfully used in the last years by several authors to produce several interesting new examples of surfaces. In this paper we describe the principal results on this class of surfaces, and report the full list of the minimal quasi-étale surfaces of general type with geometric genus equal to the irregularity \(\leq 2\). Surfaces of general type, Coverings in algebraic geometry On quasi-étale quotients of a product of two curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For the entire collection see Zbl 0672.00003.]
We consider a ramified Galois cover \(\phi: \hat X\to {\mathbb{P}}^ 1_ X\) of the Riemann sphere \({\mathbb{P}}^ 1_ X\), with monodromy group G. The monodromy group over \({\mathbb{P}}^ 1_ X\) of the maximal unramified abelian exponent n cover of \(\hat X\) is an extension \(_ n\tilde G\) of G by the group \(({\mathbb{Z}}/n{\mathbb{Z}})^{2g}\), where g is the genus of \(\hat X\). Denote the set of linear equivalence classes of divisors of degree \(k\) on \(\hat X\) by \(Pic^ k(\hat X)=Pic^ k\). This is equipped with a natural G-action. We show that the equivalence class of the extension \(_ n\tilde G\to G\) is determined by the cocycle of \(H^ 1(G,Pic^ 0)\) representing \(Pic^ 1\) ({\S} 2.2). From this we give an effective criterion (involving the Schur multiplier of G) to decide when this group extension splits for all n ({\S} 4.2). In particular we easily produce examples from this of cases where \(\hat X\) has \(G\) invariant divisor classes of degree 1, but no \(G\) invariant divisor of degree 1 ({\S} 5.1).
The extension \(_ n\tilde G\to G\) naturally factors into a sequence \(_ n\tilde G\to H\to G\) where H is the smallest quotient of \(_ n\tilde G\) giving a Frattini cover ({\S} 1.1) that fits between \(_ n\tilde G\) and G. Extension of the main result of {\S} 4.2 would consider all maximal quotients M of \(_ n\tilde G\) such that \(M\to G\) splits. We note that the sequence including such an M factors through H, and by example we demonstrate that such maximal quotients M may not be unique ({\S} 5.2). abelian extensions of Galois covers; ramified Galois cover; divisor; Frattini cover; maximal quotients Fried, M.; Völklein, H.: Unramified abelian extensions of Galois covers. Proceedings of symposia in pure mathematics 49, 675-693 (1989) Coverings of curves, fundamental group, Ramification problems in algebraic geometry, Theta functions and abelian varieties, Coverings in algebraic geometry, Group actions on varieties or schemes (quotients), Homogeneous spaces and generalizations, Extensions, wreath products, and other compositions of groups Unramified abelian extensions of Galois covers | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper is a report on work in progress on the irregularity of cyclic coverings of surfaces. It has been motivated by the attempt to understand and clarify some classical results by \textit{M. de Franchis} [Atti R. Accad. Lincei, Rend., Cl. Sci. Fis. Math. Natural. 13, 688-695 (1904; JFM 35.0423.01) and Rend. Circ. Mat. Palermo 20, 331-334 (1905; JFM 36.0491.01)], \textit{G. Bagnera} and \textit{M. de Franchis} [Mem. Mat. Fis., Soc. Ital. Sci. (3) 15, 253-343 (1908; JFM 39.0698.03)] and \textit{A. Comessatti} [Rend. Circ. Mat. Palermo 31, 369-386 (1911; JFM 42.0448.01)] on the subject. De Franchis' theorem says that every double covering of a regular surface with geometric genus zero has Albanese dimension 1. Comessatti extended this result to triple coverings, proving that any such a cyclic covering of a regular surface with geometric genus zero has Albanese dimension 1, unless it belongs to one family of exceptions.
De Franchis' approach, essentially followed by Comessatti, consists in the application of the classical trick of producing independent holomorphic 1-forms whose wedge product is zero. We exploit this idea in \(\S 4\). In \(\S 5\) we review and slightly simplify Bagnera-De Franchis' classification. Using these results and those from \(\S 4\), one is able to considerably extend De Franchis' and Comessatti's theorems mentioned above to the case of cyclic coverings of order \(n\leq 12\), \(n\neq 7,9,11\). However for sake of brevity we stated the result only in the case \(n=4\). Albanese varieties; irregularity of cyclic coverings of surfaces; JFM 35.0423.01; JFM 36.0491.01; JFM 39.0698.03; JFM 42.0448.01; Albanese dimension F. Catanese and C. Ciliberto, ''On the irregularity of cyclic coverings of algebraic surfaces,'' in Geometry of Complex Projective Varieties, Rende: Mediterranean, 1993, vol. 9, pp. 89-115. Coverings in algebraic geometry, Special surfaces On the irregularity of cyclic coverings of algebraic surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be an integral scheme over a connected Dedekind scheme \(S\) with \(f: X \to S\) a faithfully flat morphism of finite type. Let \(x \in X(S)\). \textit{C. Gasbarri} defined in his paper [Duke Math. J. 117, No. 2, 287--311 (2003; Zbl 1026.11057)] the abelianalized fundamental scheme \(\pi_1(X,x)\). This is a generalization of Nori's definition of the fundamental group scheme of a reduced connected and proper scheme \(X\) over a prefect field \(k\), which is a generalization of the classical étale fundamental group.
Assume that the existence of the Picard scheme \(\mathrm{\mathbf Pic}_{X/S}\), \(\mathrm{\mathbf Pic}^0_{X/S}\) as an open \(S\)-group subscheme of \(\mathrm{\mathbf Pic}_{X/S}\) is defined such that for any point \(s \in S\), \(\mathrm{\mathbf Pic}^0_{X/S} \times_S k(s) \simeq \mathrm{\mathbf Pic}^0_{X_s/k(s)}\). It exists with assumptions that \(\mathrm{\mathbf Pic}_{X/S}\) is separated over \(S\) and \(\mathrm{\mathbf Pic}^0_{X_s/k(s)}\) are smooth and have the same dimensions for all \(s \in S\). When \(\mathrm{\mathbf Pic}^0_{X/S}\) is a projective abelian scheme, set \(\mathrm{\mathbf Alb}_{X/S} := (\mathrm{\mathbf Pic}^0_{X/S} )^*\) and call it the Albanese scheme of \(X \to S\). Let \(\mathrm{\mathbf Pic}^{\tau}_{X/S}\) be an open scheme of \(\mathrm{\mathbf Pic}_{X/S}\) defined by \(\cup_n n^{-1} (\mathrm{Pic}^0_{X/S})\) where \(n:\mathrm{\mathbf Pic}_{X/S} \to \mathrm{\mathbf Pic}_{X/S}\) is the multiplication by \(n\) homomorphism. Let \(\mathrm{NS}^{\tau}_{X/S}\) be the \(S\)-scheme representing the quotient sheaf associated with respect to the fpqc topology to the functor \(T \mapsto \mathrm{\mathbf Pic}_{X/S}^{\tau}(T)/\mathrm{\mathbf Pic}^0_{X/S}(T)\).
The main result of this paper is that there is an exact sequence of commutative group schemes:
\[
0 \to (\mathrm{\mathbf NS}^{\tau}_{X/S})^{\wedge} \to \pi_1(X,x)^{\mathrm{ab}} \to \pi_1(\mathrm{\mathbf Alb}_{X/S}, 0_{\mathrm{\mathbf Alb}_{X/S}}) \to 0.
\]
As a corollary, when \(f: C \to S\) is a smooth and projective curve with integral geometric fibres, the natural morphism \(\pi_1(C, x)^{\mathrm{ab}} \to \pi_1(J,0_J)\) is an isomorphism where \(J\) is the Jacobian of \(C\). fundamental group scheme; Albanese scheme Antei, M., \textit{on the abelian fundamental group scheme of a family of varieties}, Israel J. Math., 186, 427-446, (2011) Group schemes, Picard groups, Coverings in algebraic geometry On the abelian fundamental group scheme of a family 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 Questions about modular representation theory of finite groups can often be reduced to elementary abelian subgroups. This is the first book to offer a detailed study of the representation theory of elementary abelian groups, bringing together information from many papers and journals, as well as unpublished research. Special attention is given to recent work on modules of constant Jordan type, and the methods involve producing and examining vector bundles on projective space and their Chern classes. Extensive background material is provided, which will help the reader to understand vector bundles and their Chern classes from an algebraic point of view, and to apply this to modular representation theory of elementary abelian groups. The final section, addressing problems and directions for future research, will also help to stimulate further developments in the subject. With no similar books on the market, this will be an invaluable resource for graduate students and researchers working in representation theory. Research exposition (monographs, survey articles) pertaining to group theory, Modular representations and characters, \(p\)-adic representations of finite groups, Finite nilpotent groups, \(p\)-groups, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Representations of elementary abelian \(p\)-groups and vector bundles | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Given an Enriques surface \(T\), its universal \({K3}\) cover \(f : S \to T\), and a genus \(g\) linear system \(|C|\) on \(T\), we construct the relative Prym variety \(P_H = \text{Prym}_{v, H} (\mathcal{D/C})\), where \(\mathcal{C} \to |C|\) and \(\mathcal{D} \to |f^{\ast}C|\) are the universal families, \(v\) is the Mukai vector \((0, [D], 2-2g)\), and \(H\) is a polarization on \(S\). The relative Prym variety is a \((2g-2)\)-dimensional possibly singular variety, whose smooth locus is endowed with a hyperkähler structure. This variety is constructed as the closure of the fixed locus of a symplectic birational involution defined on the moduli space \(M_{v,H}(S)\). There is a natural Lagrangian fibration \(\eta \colon P_H \to |C|\) that makes the regular locus of \(P_H\) into an integrable system whose general fiber is a \((g-1)\)-dimensional (principally polarized) Prym variety, which in most cases is not the Jacobian of a curve. We prove that if \(|C|\) is a hyperelliptic linear system, then \(P_H\) admits a symplectic resolution which is birational to a hyperkähler manifold of \({K3}^{[g-1]}\)-type, while if \(|C|\) is not hyperelliptic, then \(P_H\) admits no symplectic resolution. We also prove that any resolution of \(P_H\) is simply connected and, when \(g\) is odd, any resolution of \(P_H\) has \(h^{2,0}\)-Hodge number equal to one. Arbarello, E.; Saccà, G.; Ferretti, A., The relative Prym variety associated to a double cover of an Enriques surface, J. Differential Geom., 100, 2, 191-250, (2015) \(K3\) surfaces and Enriques surfaces, Families, moduli of curves (algebraic), Jacobians, Prym varieties, Coverings in algebraic geometry, Relationships between algebraic curves and integrable systems, Vector bundles on surfaces and higher-dimensional varieties, and their moduli Relative Prym varieties associated to the double cover of an Enriques surface | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For any \(n> 1\), the author constructs examples of branched Galois coverings \(M\to\mathbb{P}^n\) where \(M\) is one of \((\mathbb{P}^1)^n\), \(\mathbb{C}^n\) and \((\mathbb{B}_1)^n\), where \(\mathbb{B}_1\) is the 1-ball. In terms of orbifolds, this amounts to giving examples of orbifolds over \(\mathbb{P}^n\) uniformized by \(M\). The author also discusses the related ``orbifold braid groups''. branched Galois covering; orbifold braid groups Coverings in algebraic geometry, Braid groups; Artin groups, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Projective techniques in algebraic geometry, Ramification problems in algebraic geometry On branched coverings of \(\mathbb P^n\) by products of discs | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 provides a series of examples of groups which cannot be isomorphic to \(\pi_1 (\mathbb{C}^2 - C)\) where \(C\) is an algebraic curve. The examples stem from the fact that the Alexander polynomial of \(\pi_1 (\mathbb{C}^2 - C)\) is cyclotomic. Using this property one can show that some groups (e.g. the fundamental group of a figure eight knot) are not isomorphic to \(\pi_1 (\mathbb{C}^2 - C)\) and are note ruled out by previously considered restrictions [see \textit{J. W. Morgan}, Publ. Math., Inst. Hautes Étud. Sci. 48, 137-204 (1978; Zbl 0401.14003) and 64, 185 (1986; Zbl 0617.14013)]. It is shown that the cyclotomic property is particularly elementary in the case when \(C\) is transversal to the line at infinity. In this case it is a consequence of the fact that the fundamental group of a projective curve is the quotient of the fundamental group of the affine portion by the cyclic subgroup of the center. complements to hypersurfaces; Alexander polynomial; fundamental group A. S. Libgober, ``Groups which cannot be realized as fundamental groups of the complements to hypersurfaces in \(C^ N\)'' in Algebraic geometry and Its Applications (West Lafayette, Ind., 1990) , Springer, New York, 1994, 203--207. Homotopy theory and fundamental groups in algebraic geometry, Coverings in algebraic geometry, Fundamental groups and their automorphisms (group-theoretic aspects) Groups which cannot be realized as fundamental groups of the complements to hypersurfaces in \({\mathbb{C}}^ N\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems See the preview in Zbl 0539.14009. cyclic covering; local Torelli problem; periods of holomorphic; n-forms; Kuranishi space; Hirzebruch surfaces [Kon] Konno, K.: On deformations and the local Torelli problem of cyclic branched coverings. Math. Ann.271, 601-617 (1985) Coverings in algebraic geometry, Period matrices, variation of Hodge structure; degenerations, Transcendental methods, Hodge theory (algebro-geometric aspects), Ramification problems in algebraic geometry, Jacobians, Prym varieties, Formal methods and deformations in algebraic geometry On deformations and the local Torelli problem of cyclic 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 \(S\) be an hyperelliptic surface of general type, \(f:S\to C\) a genus \(g\) hyperelliptic fibration. In this paper, we prove that if the maximal \(\mathbb Z_2\)-quotient rank of the vertical part of the fundamental group of \(S\) is \(r\), then its slope
\[
\lambda(f) \geq \begin{cases} 4+ \frac{4r-8}{g(r+1)-r^2},\quad &\text{if \(r\) is even},\\ 4+\frac{4r-8}{g(r+3)-(r+1)^2},\quad &\text{if \(r\) is odd},\end{cases}
\]
with equality only if the branch locus \(R\) of the double cover induced by the hyperelliptic involution on \(S\) has \((r+1\to r+1)\) type singularities (if \(r\) is even), or \((r+2\to r+2)\) type singularities (if \(r\) is odd). double cover; hyperelliptic surface of general type; hyperelliptic fibration; fundamental group; slope; singularities Surfaces of general type, Coverings in algebraic geometry, Fibrations, degenerations in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry, Singularities of surfaces or higher-dimensional varieties The slopes of hyperelliptic surfaces with \(Z_2\)-quotient ranks | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems O. Zariski, building-up on a theorem by Severi asserting that the moduli space of irreducible nodal curves with given degree and number of double points is irreducible, proved 1929 that the complement of a nodal curve C in \({\mathbb{P}}^ 2\) has an abelian fundamental group. However, Severi's proof was not correct. (J. Harris announced some weeks ago a new proof.) \textit{W. Fulton} [Ann. Math., II. Ser. 111, 407-409 (1980; Zbl 0406.14008)] proved an algebraic version of ''Zariski's theorem'': the algebraic fundamental group is abelian. His proof relies on his connectivity theorems and therefore is completely algebraic. \textit{P. Deligne} ''translated'' topologically Fulton's argument and proved the ''full theorem of Zariski'' [cf. Sémin. Bourbaki, 32e année, Vol. 1979/80, Exposé 543, Lect. Notes Math. 842, 1-10 (1981; Zbl 0478.14008)]. - Applied to the vertex of the cone over a nodal curve, Fulton-Deligne's theorem says that the local fundamental group of the complement of the cone in \(({\mathbb{C}}^ 3,0)\) is abelian. The authors generalize this last statement to the non homogeneous case: let X be a hypersurface with normal crossings up to codimension 2 in a point P; then the local fundamental group at P of the complement of X in the ambient space is abelian. Their method is topological and strongly inspired by Deligne's argument. The higher singularities in codimension 2 appearing in the statement are without influence on the fundamental group by some Lefschetz-type theorems due to D. Cheniot, H. Hamm and Lê D. T. and P. Deligne. - Finally the authors consider a question raised by K. Saito on the relation between poles of the residues of holomorphic forms with logarithmic behavior along X and the shape of the singularities of X. Their result gives a partial answer, relating it to the fundamental group. Zariski theorem on fundamental group; complement of a normal crossing divisor Lê D\tilde{}ung Tráng and Kyoji Saito. The local \({\pi}\)1of the complement of a hypersurface with normal crossings in codimension 1 is abelian. Ark. Mat., 22(1):1--24, 1984. Homotopy theory and fundamental groups in algebraic geometry, Low codimension problems in algebraic geometry, Coverings in algebraic geometry The local \(\pi _ 1\) of the complement of a hypersurface with normal crossings in codimension 1 is abelian | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 develop a theory of group actions and coverings on Brauer graphs that parallels the theory of group actions and coverings of algebras. In particular, we show that any Brauer graph can be covered by a tower of coverings of Brauer graphs such that the topmost covering has multiplicity function identically one, no loops, and no multiple edges. Furthermore, we classify the coverings of Brauer graph algebras that are again Brauer graph algebras. classification of coverings of Brauer graph algebras Green, Edward L.; Schroll, Sibylle; Snashall, Nicole, Group actions and coverings of Brauer graph algebras, Glasg. Math. J., 56, 2, 439-464, (2014) Group actions on combinatorial structures, Representations of quivers and partially ordered sets, Coverings in algebraic geometry, Graded rings and modules (associative rings and algebras), Variational aspects of group actions in infinite-dimensional spaces Group actions and coverings of Brauer graph algebras | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems When an abelian variety has a principal polarization, it is self-dual. However, when the normalization is not principal, the problem of giving an explicit formula for the ``inversion'' (by which the author means taking the opposite with respect to the group law), on the Heisenberg group associated to the polarizing bundle, becomes subtle and important.
The author generalizes results, both classical [cf. \textit{C. Birkenhake, H. Lange}, Complex abelian varieties. 2nd augmented ed. Grundlehren der Mathematischen Wissenschaften 302. (Berlin: Springer) (2004; Zbl 1056.14063)] and proved by \textit{D. Mumford} [Invent. Math. 1, 287--354 (1966); ibid. 3, 75--135, 215--244 (1967; Zbl 0219.14024)] and \textit{G. R. Kempf} [Am. J. Math. 111, No.1, 65--94 (1989; Zbl 0673.14023)] in any characteristic (different from 2), and is able to give very precise results, even for (ample) line bundles that satisfy fewer technical conditions than the previously considered ones (he also compares definitions, and can show that certain conditions follow, in certain cases). These formulas should be very useful, potentially in mathematical physics where the Heisenberg group and the sections of various theta divisors, viewed as `duals' of conformal blocks [cf. e.g. \textit{T. Abe}, On \(\text{SL}(2)\)--\(\text{GL}(n)\) strange duality, preprint] are of great interest.
To give an oversimplified statement of the main result, given a separable ample line bundle \(L\) over an abelian variety \(X\), with the only requirement of being symmetric with respect to the `minus' involution \(\iota\), \(\iota^\ast L\cong L\), the author considers generalized Heisenberg groups \({\mathcal G}(\delta )\) where \(\delta =(\delta_1,\ldots ,\delta_g)\) is the polarization and a theta structure \({\mathcal G}(\delta ) {\cong\atop{{\rightarrow\atop{\;\;}}}} {\mathcal G}(L)\). He gives the appropriate notion of even/odd line bundle algebraically equivalent to \(L\), counts the number of symmetric sections in \(H(L)\) that are invariant under the natural Heisenberg action, using characters of the Heisenberg group, and shows that for such an even \({\mathcal L}\cong T^\ast_a(L)\), taking the opposite results in adding \(2u_0\) to \(u\) (where \(u\) is the point in \(X\) corresponding to the projection of the Heisenberg group and \(u_0\) corresponds to \(a\) and dividing by the character corresponding to \((u+u_0)\). He compares this with the classical and with Kempf's formula.
The methods are clever uses of linear-algebraic properties for the representation of the Heisenberg group (on spaces of sections), in the presence of suitably chosen symplectic forms and attendant Arf invariants. abelian varieties; polarization; theta structure Algebraic theory of abelian varieties, Theta functions and abelian varieties, Quadratic and bilinear forms, inner products, Modular representations and characters, Singularities in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] On the inverse formula | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author studies the problem of existence of finite Galois covers of complex projective manifolds with assigned Galois group. In particular he proves that, given a smooth projective manifold M and a Galois cover \(X\to M\) with group G, there exist a projective space \({\mathbb{P}}\), a finite subgroup \(G'\) of Aut(\({\mathbb{P}})\) isomorphic to G, and a meromorphic map \(M\to {\mathbb{P}}/G\), such that \(X\to M\) is obtained from \({\mathbb{P}}\to {\mathbb{P}}/G\), by base changing with \(M\to {\mathbb{P}}/G\), and normalizing.
As an application, he shows that for any given \(manifold\quad M\) and \(group\quad G\) there exists a Galois cover of M with group G.
The main ingredient in all the proofs is a result by Deligne and Kita on the existence of generalized Fuchsian equations with assigned monodromy. finite Galois covers of complex projective manifolds with assigned Galois group; generalized Fuchsian equations; monodromy DOI: 10.2969/jmsj/04130391 Coverings in algebraic geometry, Projective techniques in algebraic geometry, Structure of families (Picard-Lefschetz, monodromy, etc.) On finite Galois coverings of projective manifolds | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We introduce the notion of \(\pi \)-cosupport as a new tool for the stable module category of a finite group scheme. In the case of a finite group, we use this to give a new proof of the classification of tensor ideal localising subcategories. In a sequel to this paper, we carry out the corresponding classification for finite group schemes. cosupport; stable module category; finite group scheme; localising subcategory; support; thick subcategory Cohomology of groups, Representations of associative Artinian rings, Modular representations and characters, Cohomology theory for linear algebraic groups, Group schemes Stratification and \(\pi\)-cosupport: 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 The Chevalley-Weil theorem is a powerful tool in Diophantine Geometry. It roughly asserts that given an unramified morphism of finite degree \(\pi \colon W \to V\) of projective algebraic varieties defined over a number field \(k\), one can lift the rational points \(p \in V(k)\) to points of \(W\) defined over a fixed number field.
One key point of this result is the assumption that the morphism is ``unramified'', since otherwise, if \(\operatorname{deg}(\pi) >1\) and if \(V\) has enough rational points, then one would expect that the field of definitions of the fibers \(\pi^{-1}(p)\) vary with \(p\), so as to generate a field of infinite degree.
A proof of this result was first given by \textit{C. Chevalley} and \textit{A. Weil} [C. R. Acad. Sci., Paris 195, 570--572 (1932; Zbl 0005.21611; JFM 58.0182.04)] in the case of curves. Since then, many proofs have appeared in the literature, also for arbitrary dimension (e.g. in the works of Bombieri and Gubler, Lang, Serre, etc.).
The main goal of the present note is to illustrate a self-contained proof of this theorem with a rather different presentation and assumptions of purely topological content (see Theorem 1.1). In order to this, the authors discuss and compare various concepts of ``ramification''. They finally adopt a purely topological notion of ramification via the notion of ``topological cover'', with which they are able to exploit the absence of it and deducing it for the number fields after specialization. In previous proofs, the notion ``unramified'' was always formulated in an algebraic way (e.g.~ ``étale'' or ``\(\Omega^1_{W/V} = 0\)'').
Finally, the authors apply Theorem 1.1 to the study of solutions of generalized Fermat equations.
Even though this note does not contribute to substantially new results, the notion of topological cover is new in this context, and it provides a more accessible statement of the Chevalley-Weil theorem which is easier to use for applications. Chevalley-Weil theorem; covers; ramification; Diophantine equations Rational points, Ramification and extension theory, Coverings in algebraic geometry Around the Chevalley-Weil theorem | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper collects properties and constructions related to the Hirzebruch surfaces \(F_k\). Let \(F_k'\) be the surface \(F_k\) embedded into a projective space via a very ample divisor. After presenting braid monodromy in general, the author computes the braid monodromy of the branch curve \(S\) of a generic projection of \(F_k'\) to the projective plane. This helps to characterize the fundamental group of the complement of \(S\), which is in turn linked to the fundamental group of the Galois cover of \(F'\). The paper ends with a formula expressing the Chern numbers of the Galois cover of \(F_k'\) in terms of suitable invariants. Hirzebruch surfaces; braid monodromy; fundamental group; Galois cover Special surfaces, Coverings in algebraic geometry Hirzebruch surfaces: Degenerations, related braid monodromy, Galois covers | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(G\) be a finite group over a field \(k\) of positive characteristic. A full triangulated subcategory \(\mathcal{C}\) of the stable module category \(StMod \, G\) of possibly infinite-dimensional \(G\)-modules is called a \textit{colocalising subcategory} if it is closed under set-indexed products. It is \(\mathrm{Hom}\) closed if whenever \(M\) is in \(\mathcal{C}\), so is \(\mathrm{Hom}_k(L,M)\) for any \(G\)-module \(L\). The main result of the paper gives a bijection between the Hom closed colocalising subcategories of \(StMod \, G\) and the subsets of \(\mathrm{Proj} \, H^*(G,k)\) where the latter is the set of homogeneous prime ideals not containing \(H^{\geq 1}(G,k)\). This bijection is given by sending \(\mathcal{C}\) to its \(\pi\)-cosupport.
In earlier work [the first author et al., J. Am. Math. Soc. 31, No. 1, 265--302 (2018; Zbl 1486.16011)] the authors had classified the tensor closed localising subcategories of \(StMod \;G\). Combined with the present work the assignment \(\mathcal{C} \mapsto \mathcal{C}^{\bot}\) gives a bijection between the tensor closed localising subcategories of \(StMod \, G\) and the Hom closed colocalising subcategories of \(StMod \, G\). cosupport; stable module category; finite group scheme; colocalising subcategory Benson, Dave; Iyengar, Srikanth B.; Krause, Henning; Pevtsova, Julia, Colocalising subcategories of modules over finite group schemes, Ann. K-Theory, 2379-1683, 2, 3, 387\textendash408 pp., (2017) Modular representations and characters, Derived categories, triangulated categories, Group schemes, Cohomology theory for linear algebraic groups, Cohomology of groups Colocalising subcategories of modules over 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 We construct inductively an equivariant compactification of the algebraic group \(\mathbb{W}_n\) of Witt vectors of finite length over a field of characteristic \(p>0\). We obtain smooth projective rational varieties \(\overline{\mathbb{W}}_n\), defined over \(F_p\); the boundary is a divisor whose reduced subscheme has normal crossings.
The Artin-Schreier-Witt isogeny \(F-1: \mathbb{W}_n \rightarrow \mathbb{W}_n\) extends to a finite cyclic cover \(\Psi_n: \overline{\mathbb{W}}_n \rightarrow \overline{\mathbb{W}}_n\) of degree \(p^n\) ramified at the boundary. This is used to give an extrinsic description of the local behavior of a separable cover of curves in characteristic \(p\) at a wildly ramified point whose inertia group is cyclic. M. A. Garuti, ''Linear systems attached to cyclic inertia,'' in Arithmetic Fundamental Groups and Noncommutative Algebra, Providence, RI: Amer. Math. Soc., 2002, vol. 70, pp. 377-386. Coverings in algebraic geometry, Ramification and extension theory, Witt vectors and related rings Linear systems attached to cyclic inertia. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The aim of this lecture is to first state the following conjecture of Abhyankar: Let \(k\) be a field of positive characteristic \(p>0\), and let \(G\) be a finite group. Denote by \(p(G)\) the normal subgroup generated by all Sylow \(p\)-subgroups of \(G\). Let \(X^*\) be a smooth projective curve of genus \(g\) over \(k\), and let \(X\) denote the complement in \(X^*\) of \(r\) points of \(X^*\) (\(r\geq 1\)). Then there exists a connected étale Galois cover of \(X\) of group \(G\) if and only if the quotient group \(G/p(G)\) is generated by \(2g+r-1\) elements. The necessity of this condition on \(G/p(G)\) was shown by Grothendieck in SGA1. In 1994 this conjecture has been solved in the affirmative by \textit{M. Raynaud} if \(X\) is the affine line [Invent. Math. 116, 425-462 (1994; Zbl 0798.14013)], and by \textit{D. Harbater} in the general case [Invent. Math. 117, 1-25 (1994; Zbl 0805.14014)].
The aim of this lecture of the proceedings is to prove a result of Raynaud (loc. cit.) which implies Abhyankar's conjecture for the affine line. The proof presented here involves describing the necessary constructions of covers in rigid geometry. In the last part one presents the proof of Pop of Abhyankar's conjecture [see \textit{F. Pop}, Invent. Math. 120, 555-578 (1995; Zbl 0842.14017)]. étale cover of curve; characteristic \(p\); Galois groups; Abhyankar's conjecture Rigid analytic geometry, Coverings in algebraic geometry, Coverings of curves, fundamental group, Galois theory Abhyankar's conjecture. I: Construction of coverings in rigid geometry | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We survey some applications of parity sheaves and Soergel calculus to representation theory. Modular representations and characters, Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects), Hecke algebras and their representations, Representations of finite symmetric groups, Sheaves in algebraic geometry Parity sheaves and the Hecke category | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(R_{5,5}\) be the elliptic modular surface of level \(5\). In this paper elliptic fibrations on various double covers \(X\) of \(R_{5,5}\) are classified, each of which is a \(K\)3 surface.
The authors discuss first double covers \(X\) of \(S\) in some generality and then classify all fibrations on \(X\) in the case that \(X\to R_{5,5}\) is ramified along the two \(I_5\)-fibers and in the case that \(X\to R_{5,5}\) is ramified along two smooth fibres. elliptic fibrations; \(K3\) surfaces \(K3\) surfaces and Enriques surfaces, Coverings in algebraic geometry Elliptic fibrations on covers of the elliptic modular surface of level 5 | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 give a geometric construction of the Jacquet-Langlands transfer for automorphic forms of higher weights. Our method is by studying the geometry of the mod \(p\) fibres of Hodge type Shimura varieties which satisfy a mild assumption and the cohomological correspondences between them. geometric Satake equivalence; Hodge-type Shimura varieties; Jacquet-Langlands transfer; moduli of local Shtukas Modular and Shimura varieties, Arithmetic aspects of modular and Shimura varieties, Geometric Langlands program (algebro-geometric aspects), Modular representations and characters A geometric Jacquet-Langlands transfer for automorphic forms of higher weights | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [This article was published in the book announced in Zbl 0342.00010].
From the author's introduction: ``The rational double points of surfaces in characteristic zero are related to the finite subgroups \(G\) of \(SL_2\). Namely, if \(V\) denotes the affine plane with its linear \(G\)-action, then the variety \(X=V/G\) has a singularity at the origin, which is the one corresponding to \(G\).
Let \(p\) be a prime integer. If \(p\) divides the order of \(G\), this subgroup will degenerate when reduced modulo \(p\), and the smooth
reduction of \(V\) will usually not be compatible with an equisingular reduction of \(X\). Nevertheless, it turns out that every rational double point in characteristic \(p\) has a finite (possibly ramified) covering by a smooth scheme. In this paper we prove the existence of such a covering by direct calculation, and we compute the local fundamental groups of the singularities.''
The paper has an excellent brief list of references from which the reader can
obtain a perspective of the subject. Bibliography Artin, M., Coverings of the rational double points in characteristic \textit{p}, (Complex Analysis and Algebraic Geometry, (1977), Iwanami Shoten Tokyo), 11-22, MR 0450263 Coverings in algebraic geometry, Rational points, Local ground fields in algebraic geometry, Singularities in algebraic geometry Coverings of the rational double points 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 concepts of almost-algebraic and essentially transcendental sets are introduced and the following results are proved:
A. Let \(S\subset {\mathbb{C}}^ k\) be a bounded holomorphically convex domain and let \(f: S\to {\mathbb{C}}^ n\) be a proper holomorphic map. Then f(S) is an essentially transcendental set.
B. Let X be an irreducible almost-algebraic set in \({\mathbb{C}}^ n\) and f be a bounded holomorphic function on X. Then f is constant. essentially transcendental sets; almost-algebraic set Analytic subsets of affine space, Holomorphically convex complex spaces, reduction theory, Proper holomorphic mappings, finiteness theorems, Transcendental methods of algebraic geometry (complex-analytic aspects), Stein spaces, Coverings in algebraic geometry, Generalizations (algebraic spaces, stacks), Transcendental methods, Hodge theory (algebro-geometric aspects) Almost-algebraic sets | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Finite covers of strictly minimal countable sets appear in the investigation of totally categorical theories. The author approaches the description of such covers by analysing their arities. If covers can be obtained by adding binary relations to principal finite covers, then these covers split and in certain situations they can be characterized completely.
In the first section, the main definitions and some technical results are given. The facts of the second section are needed for the proof of the main theorems. Using Ziegler's description of finite covers of disintegrated strictly minimal sets (Ziegler, 1922) simple covers are introduced and their arities are completely described in this case. In Sections 3 and 4 the following results are presented:
Every covering expansion of a finite principal cover \(M_ 0 = N \cup (N \times F)\) of a strictly minimal structure \(N\) is a reduct of some minimal cover.
Every binary covering expansion \(M\) of \(M_ 0\) is splitting; by this a binary covering expansion is a covering expansion obtained by adding binary relations only.
The kernels \(K\) of the binary covering expansions \(M\) are characterized by their groups \(L(K)\) and \(H(K)\).
The arity of a finite principal cover \(M_ 0\) of a strictly minimal set \(N\) (disintegrated or projective, there is no structure on fibres) is less or equal to the arity of a cover \(M\) of \(N\) which is an expansion of \(M_ 0\).
Let \(N\), \(M_ 0\), and \(M\) be as above and assume \(G_ 0 \subseteq \text{Aut}(M)\), where \(G_ 0\) is a certain subgroup of \(\text{Aut}(M_ 0)\). Then the action of \(\text{Aut}(M)\) on \(N \times F\) can be characterized in two special forms as a wreath product or a direct product of certain permutation groups.
Let \(M\) be a binary covering expansion of a cover \(M_ 0\) of a 2- transitive structure \(N\). Further let the automorphism group of a fibre be 2-transitive as well. Let \(G_ 0 \subset \text{Aut}(M)\) be the group of those automorphisms of \(M_ 0\) for which for every \(\alpha \in \text{Aut}(F)\) the permutation \((a,f) \mapsto (a,\alpha(f))\) is the identity on \(N\). Then the action of \(\text{Aut}(M)\) on \(N \times F\) can be described by \((\text{Aut}(F),F) \text{Wr }(\text{Aut}(N),N)\) or \((\text{Aut}(N),N) \times (\text{Aut}(F),F)\).
In the last section, the author investigates ternary expansions of a finite principal cover of a 2-transitive structure. Some results of the other sections can be applied to such an expansion. kernel fibre; envelope; two-graph; switching class; totally categorical theories; arities; finite covers; strictly minimal sets; covering expansion; strictly minimal structure; 2-transitive structure Ivanov, A. A.: Some combinatorial aspects of the cover problem for totally categorical theories. Automorphisms of first-order structures, 215-232 (1994) Categoricity and completeness of theories, Graphs and abstract algebra (groups, rings, fields, etc.), Group actions on posets and homology groups of posets [See also 06A09], Coverings in algebraic geometry, Infinite automorphism groups Some combinatorial aspects of the cover problem for totally categorical theories | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 a new interpretation of an example of Zariski pairs introduced by the reviewer [\textit{E. Artal Bartolo}, J. Algebr. Geom. 3, No. 2, 223-247 (1994; Zbl 0823.14013)]. The example consists of curves with four irreducible components: a smooth cubic and three lines in general position which are tangent to inflection points of the cubic. In the reviewer's paper it was proven that the topological type of the embedding of these curves in the projective plane depends on the position of the inflection points on a line. The author proves this fact with the help of his theory about dihedral coverings. One main point of this paper is the new relationship of Zariski pairs and Mordell-Weil groups of elliptic \(K3\) surfaces. Zariski pairs; dihedral coverings; elliptic fibrations; topological type of the embedding; Mordell-Weil groups of elliptic \(K3\) surfaces Hiro-o Tokunaga, A remark on E. Artal-Bartolo's paper: ''On Zariski pairs'' [J. Algebraic Geom. 3 (1994), no. 2, 223 -- 247; MR1257321 (94m:14033)], Kodai Math. J. 19 (1996), no. 2, 207 -- 217. Coverings in algebraic geometry, Rational points, Elliptic surfaces, elliptic or Calabi-Yau fibrations, \(K3\) surfaces and Enriques surfaces, Special algebraic curves and curves of low genus A remark on Artal's paper | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 dedicated to descent varieties meaning the following. Given a cover \(f:X\to B\) over a field, produce a parameter space \(V\) for families of models of \(f\) satisfying the following versal property: each model of \(f\) is a fiber of the family, so it corresponds to points of \(V\). The parameter space is an algebraic variety which is smooth, geometrically irreducible and defined over the field of moduli of \(f\).
One application of the result is the problem of covers whose field of moduli is a number field such as \(\mathbb{Q}\). It is known that although \(\mathbb{Q}\) may not be a field of definition, the cover has to be defined over \(\mathbb{Q}_p\) for almost all prime numbers \(p\), avoiding the bad cases where \(p\) divides the order of the group of the cover or where branch points coalesce modulo \(p\) [\textit{P. Dèbes} and \textit{D. Harbater}, J. Reine Angew. Math. 498, 223--236 (1998; Zbl 0905.14015); \textit{M. Emsalem}, in: Application of covers over finite fields, Contemp. Math. 245, 117--132 (1999; Zbl 0978.14012)]. As a consequence of the main result a local-global principle on varieties follows. More precisely, let \(V\) be a geometrically irreducible smooth variety defined over the field \(\mathbb{Q}^{tp}\) of totally \(p\)-adic numbers. If \(V(\mathbb{Q}_p)\neq\emptyset\) for each embedding \(\mathbb{Q}^{tp}\hookrightarrow\mathbb{Q}_p\), then \(V(\mathbb{Q}^{tp})\neq\emptyset\). This is a special case of a result of \textit{G. Laumon} and \textit{L. Moret-Bailly} [Champs Algébriques, Ergebn. Math. Grenzgeb. 39, Springer-Verlag (2000; Zbl 0945.14005)]; other proofs were also obtained by \textit{B. Green, F. Pop} and \textit{P. Roquette} [Jahresber. Dtsch. Math.-Ver. 97, 43--74 (1995; Zbl 0857.11033)] and \textit{F. Pop} [Ann. Math. (2)144, 1--34 (1996; Zbl 0862.12003)].
In order to state the main result, let \(B\) be a regular projective geometrically irreducible variety defined over a field \(K\), and \(f\) is a mere cover of \(B\), i.e., a finite, surjective, geometrically unramified morphism \(f:X\to B\) defined over \(K\) with \(X\) a normal and geometrically irreducible \(K\)-variety. This cover is said to be a \((G)\)-cover if its group of automorphism is \(G\). Assume that the field of moduli of \(f\) is \(K\) itself. The main theorem A states that there exists an affine variety \(V\) defined over \(K\), a \((G)\)-cover \(\mathcal{F}:\mathcal{X}\to V\times B\) such that
(i) for each \(v\in V\), the fiber \(\mathcal{F}_v:\mathcal{X}_v\to B_{K(v)}\) is a \(K(v)\)-model of \(f\).
(ii) If \(k\) is an extension of \(K\) and \(\tilde{f}:\tilde{X}\to B(k)\) is a \(k\)-model of \(f\), then there exists \(v\in V(k)\) such that \(\tilde{f}\) is isomorphic to the fiber cover \(\mathcal{F}_v:\mathcal{X}_v\to B_k\) (as \((G)\)-covers of \(B_k\)).
Moreover, \(V\) is smooth and for every extension \(k\) of \(K\) for which \(V(k)\neq\emptyset\), \(V\) is unirational over \(k\).
There are various corollaries to this main result A. The first concerns existentially closed fields. A field \(k\) is existentially closed in a regular extension \(\Omega\) of \(k\) if for each smooth geometrically irreducible \(k\)-variety \(V\), \(V(\Omega)\neq\emptyset\) implies \(V(k)\neq\emptyset\). Under the hypothesis that \(K\) is existentially closed in some field of definition \(k_0\) of \(f\), they show that \(f\) itself is already defined over \(K\).
For the second consequence, let \(K\) be a global field, \(\Sigma\) a non-empty finite set of places of \(K\), \(K^{\Sigma}\) the maximal extension of \(K\) in a separable closure \(K^s\) of \(K\) which is totally split over every \(v\in\Sigma\). If \(f:X\to B_{K^s}\) is a \((G)\)-cover defined over \(K^s\), the field of moduli of \(f\) is contained in \(K^{\Sigma}\) and for each \(v\in\Sigma\) there exists a \(K_v\)-model \(f_v\) for \(f\), then \(f\) itself is defined over \(K^{\Sigma}\).
For the third application, it is assumed that \(B\) is a curve. A place \(v\) of \(K\) is said to be good with respect to \(f\) if the residue characteristic \(p\) does not divide the order of the group \(G\) of the cover, the base space \(B\) has good reduction at \(v\) and the branch locus of \(f\) is smooth at \(v\). It can be shown that if \(K\) is the field of moduli of \(f\), then \(K_v\) is a field of definition for all good places \(v\) of \(K\). This result was first proved for \(G\)-covers of \(\mathbb{P}^1\) [Dèbes and Harbater, loc. cit.] and then generalized [Emsalem, loc. cit.]. As a consequence they prove that if \(f:X\to B_{K^s}\) is a \((G)\)-cover of curves, the field of moduli of \(f\) is \(K\), \(\Sigma\) is a finite non-empty set of good places of \(K\), then \(f\) is defined over \(K^{\Sigma}\).
The main theorem A has a global counterpart, main theorem B in which instead of dealing with one specific cover \(f\) one considers the whole moduli of covers and the descent variety will then be over Hurwitz spaces. More specifically, for each integer \(d>0\), finite subgroup \(G\) of the group \(S_d\) and integer \(r\geq3\), one considers covers \(f:X\to\mathbb{P}^1\) over fields of characteristic 0 with the following geometric invariants: the monodromy group (i.e., the Galois group of the Galois closure of \(f\otimes\overline{k}\)); the monodromy action is then (up to equivalence) given by \(G\subset S_d\), where \(d=\deg(f)\); the number of branch points is \(r\). These invariants are associated to moduli spaces of covers, Hurwitz spaces.
The second main result is described as follows. There exists a smooth quasi-projective \(\mathbb{Q}\)-scheme \(\mathcal{V}\) and a Hurwitz family \(f:X\to\mathbb{P}^1_{\mathcal{V}}\) over \(\mathcal{V}\) with the following properties
(i) the classifying moduli map \(\gamma_f:\mathcal{V}\to\mathcal{H}\) is smooth with geometrically irreducible fibers;
(ii) for every field \(k\) of characteristic 0, every cover of \(\mathbb{P}^1_k\) of the type corresponding to \(\mathcal{H}\) appears as the fiber of the Hurwitz family \(f\) at some \(k\)-point of \(\mathcal{V}\);
(iii) (local description of \(\mathcal{V}\) over \(\mathcal{H}\)) There exists an integer \(n\geq0\), a right action of \(\text{GL}_{n,\mathbb{Q}}\) on \(\mathcal{V}\) with finite stabilizers such there there is an étale surjective morphism \(\rho:U\to\mathcal{H}\) such that \(U\times_{\mathcal{H}}\mathcal{V}\) is \(U\)-isomorphic to \(\Gamma\backslash\text{GL}_{n,U}\) (with the natural projection on \(U\) and the natural right action of \(\text{GL}_{n,U}\)), where \(\Gamma\) is a subgroup of \(\text{GL}_{n,U}\).
This result has various applications to the field of definition of covers. One long-standing question in descent of covers is whether a \((G)\)-cover is ``often'' defined over its field of moduli. More precisely, for any field \(k\) of characteristic 0, let \(\mathcal{H}(k)^{\text{no ob}}\subset\mathcal{H}(k)\) be the set of points \(h\in\mathcal{H}(k)\) such that the corresponding cover \(f_h:X_h\to\mathbb{P}^1_{\overline{k}}\) is defined over \(k\). The first corollary of theorem B then states that the set \(\mathcal{H}(k)^{\text{no ob}}\) is the image of \(\mathcal{V}(k)\) into \(\mathcal{H}(k)\). In particular, if \(k\) is a large field, then for every connected component \(\mathcal{Z}\) of \(\mathcal{H}_k:=\mathcal{H}\times_{\text{Spec}(\mathbb{Q})}\text{Spec}(k)\), the set \(\mathcal{H}(k)^{\text{no ob}}\cap\mathcal{Z}\) is either empty or Zariski-dense in \(\mathcal{Z}\).
The second corollary slightly differs from the first one. Instead of taking only \(k\)-rational points of \(\mathcal{H}\) one considers all closed points. More specifically, let \(\mathcal{H}^{\text{no ob}}\) be the subset of \(\mathcal{H}\) of points \(h\in\mathcal{H}\) whose corresponding cover \(f_h:X_h\to\mathbb{P}^1_{\overline{k}}\) can be defined over the residue field \(\kappa(h)\) of \(h\). The second corollary says that \(\mathcal{H}^{\text{no ob}}\) is Zariski-dense in \(\mathcal{H}\).
Finally it also implies the following corollary about covers of \(\mathbb{P}^1\) over ``totally \(\sigma\)-adic'' fields. Let \(K\) be a number field, \(\Sigma\) a finite non-empty set of places of \(K\), \(K^{\Sigma}\) the maximal extension of \(K\) in a fixed separable closure \(K^s\) of \(K\) which is totally split over every \(v\in\Sigma\), \(\mathcal{H}_0\) a connected component of \(\mathcal{H}_K=\mathcal{H}\times_{\text{Spec}(\mathbb{Q})}\text{Spec}(K)\), geometrically connected over \(K\). Assume that for each \(v\in\Sigma\) there exists a \(K_v\)-\((G)\)-cover \(f_v:X_v\to\mathbb{P}^1_{K_v}\) corresponding to the point \(h\in\mathcal{H}_0(K_v)\). Then there exists a \(K^{\Sigma}\)-\((G)\)-cover \(f:X\to\mathbb{P}^1_{K^{\Sigma}}\) corresponding to a point of \(\mathcal{H}_0(K^{\Sigma})\). fields of moduli; fields of definition Dèbes, P., Douai, J.-C., Moret-Bailly, L.: Descent varieties for algebraic covers. J. fur die reine und angew. Math. 574, 51--78 (2004) Coverings in algebraic geometry, Arithmetic theory of algebraic function fields, Generalizations (algebraic spaces, stacks), Algebraic functions and function fields in algebraic geometry, Coverings of curves, fundamental group, Other nonalgebraically closed ground fields in algebraic geometry Descent varieties for algebraic covers | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A Galois covering of a complex manifold \(Y\) is a finite flat map \(f: X\to Y\), with \(X\) normal and irreducible, such that the corresponding field extension \({\mathbb C}(X)/{\mathbb C}(Y)\) is Galois. The Galois group of the covering \(G:= \text{Gal}({\mathbb C}(X)/{\mathbb C}(Y))\) acts faithfully on \(X\) in such a way that \(f: X\to Y=X/G\) \ is the projection onto the quotient. The locally free sheaf \(f_*{\mathcal O}_X\) has a natural \(G\)-action compatible with the \({\mathcal O}_Y\)-algebra structure. At the generic point of \(Y\) this action corresponds to the regular representation of \(G\) on \({\mathbb C}(Y)\). The case in which \(G\) is abelian was studied by \textit{R. Pardini} [J. Reine Angew. Math. 417, 191-213 (1991; Zbl 0721.14009)]. Here the sheaf \(f_*{\mathcal O}_X\) is a direct sum of line bundles \(\bigoplus L_{\chi}^{-1}\), where \(\chi\) runs over all the irreducible representations (``characters'') of \(G\), and \(G\) acts on the summand \(L_{\chi}^{-1}\) via the character \(\chi\). In this case, if \(Y\) is compact then the \(G\)-cover \(f: X\to Y\) is determined by the so-called building data, namely the \(L_{\chi}\) and the irreducible components of the branch divisor \(D\) of \(f\), together with some combinatorial data. The building data are subject to some compatibility relations and, conversely, given a set of building data satisfying these relations one can explicitly construct a \(G\)-cover with those data.
The situation is far more complicated when the group \(G\) is not abelian, since \(f_*{\mathcal O}_X\) still decomposes according to the irreducible representations of \(G\) but the summands have higher rank. The simplest non abelian groups one can consider are perhaps the dihedral groups \(D_{2n}\) (where \(2n\) is the order of the group). This situation was studied by \textit{H. Tokunaga} [Can. J. Math. 46, No. 6, 1299-1317 (1994; Zbl 0857.14009)]. In this case \(f\) is a composition \(X\to Z\to Y\), where \(Z\to Y\) is a double cover and \(X\to Z\) is a \({\mathbb Z}_n\)-cover. Assuming that \(Z\) is smooth, Tokunaga gives building data on the variety \(Z\) that encode the geometry of \(f\) in a way similar to the abelian case. The aim of this thesis under review is to obtain a theory of Galois coverings with dihedral Galois groups closer to the one existing for abelian covers, namely to describe the coverings in terms of data on \(Y\), without resorting to the auxiliary variety \(Z\). In order to keep calculations simpler, the author concentrates on the case \(G=D_8\), although her methods work in principle for any dihedral groups. She proves the existence of a Zariski open set \(U\subset Y\) that meets all the components of the branch divisor \(D\) of \(f\) and such that \(f_*{\mathcal O}_X|_U\) splits as a direct sum of line bundles and she gives an explicit algorithm for this decomposition. She then writes down multiplication tables for the restriction of \(f_*{\mathcal O}_X\) both to \(U\) and to \(Y\setminus D\), and she uses these results to give a structure theorem for \(D_8\)-coverings. More precisely, she gives data (effective divisors) on \(Y\) and an explicit way to construct a \(D_8\)-covering of \(Y\) from these data. The coverings \(f\colon X\to Y\) thus obtained have the property that \(f_*{\mathcal O}_X\) splits as a direct sum of line bundles, although of course the decomposition is not compatible with the group action. The author also gives examples where such a decomposition does not exist, thus showing that not every \(D_8\)-cover can be obtained from the above mentioned construction. The first part of the thesis contains a nice survey of Galois coverings of smooth manifolds and a careful analysis of the relations between results obtained via different approaches to the problem. branched covering; Galois covering; dihedral covering; branch divisor Coverings in algebraic geometry, Ramification problems in algebraic geometry On dihedral coverings in 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 We provide a homological construction of unitary simple modules of Cherednik and Hecke algebras of type \(A\) via BGG resolutions, solving a conjecture of Berkesch-Griffeth-Sam. We vastly generalize the conjecture and its solution to cyclotomic Cherednik and Hecke algebras over arbitrary ground fields, and calculate the Betti numbers and Castelnuovo-Mumford regularity of certain symmetric linear subspace arrangements. Cherednik algebras; quiver Hecke algebras; Castelnuovo-Mumford regularity Associative rings and algebras arising under various constructions, Modular representations and characters, Hecke algebras and their representations, Configurations and arrangements of linear subspaces On BGG resolutions of unitary modules for quiver Hecke and Cherednik algebras | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This survey is based on the lecture given by the author at the International Conference on Representations of Algebras in Bielefeld in 2012. \textit{J. F. Carlson} gave a talk on this subject at ICRA XII in Toruń in 2007 with the survey [in: Trends in representation theory of algebras and related topics. Proceedings of the 12th international conference on representations of algebras and workshop (ICRA XII), Toruń, Poland, August 15--24, 2007. Zürich: European Mathematical Society (EMS). 167--200 (2008; Zbl 1210.20013)] published in the same series. In the current article we try to pick up where Carlson left off although some overlaps to set the stage were unavoidable. We also focus on the general case of a finite group scheme as opposed to a finite group highlighted in [loc. cit.]. cohomology of finite group schemes; \(\pi\)-points; modules of constant Jordan type; coherent sheaves Modular representations and characters, Group schemes, Cohomology of groups Representations and cohomology of finite group schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper under review is mostly a survey of known results on the structure of finite surjective holomorphic maps \(f \colon X \rightarrow Y\) between projective or compact Kähler manifolds. To any such map one associates the vector bundle \(\mathcal{E}_f=(f_{\ast}(\mathcal{O}_X)/\mathcal{O}_Y)^{\ast}\). The paper is divided in three parts.
The first part studies the problem of classifying all \(Y\) such that \(\mathcal{E}_f\) is ample or nef for all \(f\) and a summary of most known results is given. It is also shown, this is a new result, that if \(Y\) is a projective manifold such that if \(\mathcal{E}_f\) is ample for all finite surjective maps \(f \colon X \rightarrow Y\), then either \(Y\) is Fano with \(b_2(Y)=1\) or \(K_Y\) is nef. An example is given of a Fano \(Y\) with \(b_2(Y)=1\) and a map \(f \colon X \rightarrow Y\) such that \(\mathcal{E}_f\) is not ample showing that the converse is not true. The author goes on to conjecture that if \(Y\) is a rational homogeneous manifold with \(b_2(Y)=1\), then \(\mathcal{E}_f\) is ample for all \(f\).
The second part of the paper studies the deformation theory of finite surjective holomorphic maps \(f \colon X \rightarrow Y\) between projective manifolds. Let \(\mathrm{Hom}_f(X,Y)\) be the connected component of \(\mathrm{Hom}(X,Y)\) containing \(f\). A review of some known results on the structure of \(\mathrm{Hom}_f(X,Y)\) is given as well as a summary of most known results on the problem when a deformation of \(f\) is induced by \(\mathrm{Aut}^0(Y)\). Two new results are presented. The first one is that if \(f \colon X \rightarrow Y\) is a finite map between projective manifolds with \(Y \not= \mathbb{P}^n\) and \(\mathcal{E}_f\) is ample, then all deformations of \(f\) come from \(\mathrm{Aut}^0(Y)\). The second is that if \(Y\) is a Del Pezzo surface which is the blow up of \(\mathbb{P}^2\) at \(d \geq 3\) points and \(f \colon X \rightarrow Y\) is a finite map from a smooth projective surface, then any deformation of \(f\) comes from \(\mathrm{Aut}^0(Y)\).
The third part contains known results on the structure of projective manifolds \(X\) admitting a surjective endomorphism \( f \colon X \rightarrow X\). It is also conjectured that if \(X\) is a Fano manifold with \(b_2(X)=1\) admitting a surjective endomorphism \(f \colon X \rightarrow X\), then \(X \cong \mathbb{P}^n\). finite morphism; deformation; Galois covering; ample vector bundle; Fano manifold; surjective endomorphism Nie C X, Wu C X. Space-like hyperspaces with parallel conformal second fundamental forms in the conformal space (in Chinese). Acta Math Sinica Chin Ser, 2008, 51: 685--692 Coverings in algebraic geometry, Families, fibrations in algebraic geometry, Fano varieties Finite morphisms of projective and Kähler manifolds | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A celebrated result of Mumford states that a complex normal surface germ \((X,x)\) is smooth if and only if its local fundamental groups is trivial. The same conclusion holds if the topological fundamental group is replaced by the étale fundamental group of the punctured neighbourhood \(U\) of the singularity. This result breaks down in positive characteristic. \textit{M. Artin} [Complex Anal. algebr. Geom., Collect. Pap. dedic. K. Kodaira, 11--22 (1977; Zbl 0358.14008)] asked whether, if \(\pi^{\text{ét}}(U)\) is finite, there always is a finite morphism from a smooth scheme.
This note answers the question with the étale fundamental group replaced by the local Nori fundamental group scheme \(\pi_{\text{loc}}^N(U,X,x)\), which the authors define here by adapting Nori's original construction. The main result is that a surface singularity over an algebraically closed field is a rational singularity, if \(\pi_{\text{loc}}^N(U,X,x)\) is a finite group scheme, and if \(\pi_{\text{loc}}^N(U,X,x)=0\), then it is a rational double point. Using Artin's list it follows that in the last case \(X\) admits a finite morphism from a smooth scheme, and in fact \((X,x)\) is smooth except possibly in the cases \(E_8^1\) (\(p=2,3\)) and \(E_8^3\) (\(p=2\)). covering, local fundamental group; Nori group scheme; rational singularity Esnault, H.; Viehweg, E.: Surface singularities dominated by smooth varieties, J. reine angew. Math. 649, 1-9 (2010) Singularities in algebraic geometry, Coverings in algebraic geometry Surface singularities dominated by smooth varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The articles of this volume will be reviewed individually. Proceedings, conferences, collections, etc. pertaining to associative rings and algebras, Representation theory of associative rings and algebras, Modular representations and characters, Noncommutative algebraic geometry, Proceedings of conferences of miscellaneous specific interest Expository lectures on representation theory. Maurice Auslander distinguished lectures and international conference, Woods Hole Oceanographic Institute, Quisset Campus, Falmouth, MA, USA, April 25--30, 2012. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper under review deals with the classification of surfaces of general type and of their automorphisms. A numerical Godeaux surface is a minimal complex surface of general type \(S\) with \(p_g(S)=0\), \(K_S^2=1\), \(\chi(\mathcal O_S)=1\).
\textit{J. Keum} and \textit{Y. Lee} [Math. Proc. Camb. Philos. Soc. 129, No. 2, 205--216 (2000; Zbl 1024.14019)] and \textit{A. Calabri, C. Ciliberto} and \textit{M. Mendes Lopes} [Trans. Am. Math. Soc. 359, No. 4, 1605--1632 (2007; Zbl 1124.14036)] classified the numerical Godeaux surfaces possessing an involution. From a result of \textit{E. Stagnaro} [J. Ann. Univ. Ferrara, Nuova Ser., Sez. VII 43, 1--26 (1997; Zbl 0927.14017)] can be derived an example of a numerical Godeaux surface with an automorphism of order 5.
In this paper it is proved that a numerical Godeaux surface cannot have an automorphism of order 3. The result is part of the author's Ph.D. thesis which can also be found at \url{http://ricerca.mat.uniroma3.it/dottorato/Tesi/tesipalmieri.pdf}. Godeaux surfaces; automorphisms of surfaces of general type Surfaces of general type, Coverings in algebraic geometry Automorphisms of order three on numerical Godeaux surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A finite extension \(R[\alpha]\) of a UFD \(R\) is called a Bring-Jerrard extension (a B-J extension) if \(\alpha\) is the root of an irreducible polynomial of the form \(z^n+sz+t\in R[z]\). The aim of this paper is to calculate explicitely the integral closure of a B-J extension and of some extensions of degree 4 and 5 (an explicit expression is known only for extensions of degree \(\leq 3\)).
In this paper, \(R\) is assumed to be a noetherian UFD containing a field whose characteristic is coprime to \(n\) and \(n-1\) and \(A=R[\alpha]\) is an extension of \(R\) of degree \(n\). When \(A\) is a B-J extension, the discriminant, the ramification divisor and the integral closure of \(A\) are computed.
For extensions \(A\) of degree \(4\) or \(5\) and some conditions on the characteristic, the authors calculate the integral closure of \(A\) by reducing \(A\) to a type B-J extension, using Tschirnhaus transformation.
In the last section, the computation of the integral closure is applied to algebraic geometry. The structure sheaf of every Bring-Jerrard covering space is determined explicitely in terms of coefficients of the equation defining the covering. A geometric criterion for a finite morphism of degree 3 to be Galois is also gotten and a partial result for the case of degree 5 is proved. ramification divisor; Galoisness; Bring-Jerrard extension; Tschirnhaus transformation Tan S L, Zhang D Q. The determination of integral closures and geometric applications. Adv Math, 2004, 185: 215--245 Integral closure of commutative rings and ideals, Coverings in algebraic geometry, Ramification and extension theory The determination of integral closures and geometric 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 This paper deals with the problem of conjugacy of Cartan subalgebras for affine Kac-Moody Lie algebras. Unlike the methods used by Peterson and Kac, our approach is entirely cohomological and geometric. It is deeply rooted on the theory of reductive group schemes developed by Demazure and Grothendieck, and on the work of J. Tits on buildings. affine Kac-Moody Lie algebra; conjugacy; reductive group scheme; torsor; Laurent polynomials; non-abelian cohomology Chernousov, V.; Gille, P.; Pianzola, A.; Yahorau, U.: A cohomological proof of Peterson-Kac's theorem on conjugacy of Cartan subalgebras for affine Kac-Moody Lie algebras. J. algebra 399, 55-78 (2014) Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Galois cohomology of linear algebraic groups, Group schemes, Coverings in algebraic geometry A cohomological proof of Peterson-Kac's theorem on conjugacy of Cartan subalgebras for affine Kac-Moody Lie algebras | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) and \(Y\) be normal projective algebraic varieties defined over the field of complex numbers. Recall that a \(G\)-cover is a finite surjective morphism \(X\rightarrow Y\) such that the induced field extension \({\mathbb C}(X)/{\mathbb C}(Y)\) is Galois, with Galois group equal to \(G\). Furthermore, a \(G\)-cover \(X\rightarrow Y\) is called versal if for any other \(G\)-cover \(W\rightarrow Z\) there exists a \(G\) equivariant rational map \(W\dasharrow X\) whose image is not fixed pointwisely by \(G\).
The essential dimension \(\mathrm{ed}_{\mathbb{C}}(G)\) [introduced by \textit{J. Buhler; Z. Reichstein}, Compos. Math. 106, No. 2, 159--179 (1997; Zbl 0905.12003)] of a finite group \(G\) is equivalent to the minimum of the dimension of \(X\) where \(X\rightarrow Y\) is a versal \(G\)-cover. It is known that finite groups with essential dimension \(1\) are diedral and cyclic groups.
In this article, the authors studies groups with essential dimension \(2\). More precisely, they study examples of versal \(G\)-covers where \(G\) is either the symmetric group on \(4\) letters or the alternative group on \(5\) letters. In both cases, using classical tools of birational geometry of surfaces (like Noether's inequality), they provide distinct \(G\)-covers and prove that these are not birationally equivalent. This yields explicit distinct embeddings of \(G\) into the \(2\)-dimensional Cremona group. versal Galois covers; Cremona group Coverings in algebraic geometry, Group actions on varieties or schemes (quotients), Birational automorphisms, Cremona group and generalizations A note on embeddings of \(S^4\) and \(A_5\) into the two-dimensional Cremona group and versal Galois covers | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We determine the fundamental group of the complement of the three-cuspidal quartic minus the tangent lines through the cusps. The existence of a rigid covering of this complement whose monodromy group is isomorphic to the simple group \(\text{PSL}_{2}(\mathbb{F}_{7})\) is proved. Coverings of curves, fundamental group, Coverings in algebraic geometry On rigid covers associated to the three-cuspidal quartic | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(V\subseteq {\mathbb P}^n(\overline{k})\) be a projective algebraic variety over a zero characteristic field \(k\) which is the set of common zeros of a family of homogeneous polynomials of degree less than \(d\) in \(n+1\) variables with coefficients from \(k\). A smooth cover of \(V\) is a finite family \(\{ V_\alpha\}\), \(\alpha\in A\), of quasiprojective smooth algebraic varieties \(V_\alpha\subseteq{\mathbb P}^n(\overline{k})\) such that \(V=\bigcup_{\alpha\in A}V_\alpha\) and all the irreducible components of \(V_\alpha\) have the same dimension (depending on \(\alpha\)). A smooth cover \(\{ V_\alpha\}\) of \(V\) is called a smooth stratification of \(V\), if for any \(\alpha_1,\alpha_2\in A, \alpha_1 \neq \alpha_2\) we have \(V_{\alpha_1}\cap V_{\alpha_2}=\emptyset\). In this paper a new representation of quasiprojective algebraic varieties is introduced. Using this representation, the author proves the existence of a smooth cover and a smooth stratification \(V=\bigcup_{\alpha\in A}V_\alpha\) such that for every \(\alpha\in A\) the degree \(\overline{V}_\alpha\) of the Zariski closure \(\overline{V}_\alpha\) is bounded above by \(2^{2^{n^C}}d^n\) where \(C>0\) is an absolute constant, and the order of \(A\) is bounded above by \(2^{2^{n^C}}d^n\) in the case of smooth cover and by \(2^{2^{n^C}}d^{n(n+1)/2}\) in the case of smooth stratification. In the algorithmic part of this paper the field \(k\) is assumed to be finitely generated over \(\mathbb Q\). A method is presented to construct sequences of local parameters for components of algebraic varieties at a time that is polynomial in the size of the input and in the value \(2^{2^{n^C}}d^n\). Using the results of his previous work [\textit{A. L. Chistov}, J. Math. Sci., New York 108, 897-933 (2002); translation from Zap. Nauch. Semin. POMI 258, 7-59 (1999; Zbl 1081.14527) and in: Proc. St. Petersburg Math. Soc. vii. Transl., Ser. 2, Am. Math. Soc. 203, 201-231 (2001; Zbl 1086.14511)], the author gives an algorithm for constructing regular sequences and sequences of local parameters in the local ring of a component of dimension \(n-s\) of \(V\) at a time that is polynomial in the size of the input and in \(n^{2^{s^C}}d^n\). In all these results the constants \(C\) can be computed explicitly. representation of quasiprojective algebraic varieties; smooth stratification; smooth cover A. L. Chistov, ''Efficient smooth stratification of an algebraic variety in zero-characteristic and its applications,'' J. Math. Sci., 113, No. 5, 689--717 (2003). Coverings in algebraic geometry, Stratifications in topological manifolds, Computational aspects of higher-dimensional varieties Efficient smooth stratification of an algebraic variety in zero characteristic 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 We prove the following result that answers a question of M. Chen: Let \(X\) be a Gorenstein minimal complex projective 3-fold of general type with locally factorial terminal singularities. If \(|K_X|\) defines a generically finite map \(\varphi:X \to\mathbb{P}^{p_g-1}\), then \(\deg (\varphi)\leq 576\). For any positive integer \(m>0\), we give infinitely many examples of (non-Gorenstein) 3-folds of general type with canonical map of degree \(m\). C. D. Hacon, On the degree of the canonical maps of \(3\)-folds, Proc. Japan Acad. Ser. A Math. Sci. 80 (2004), no. 8, 166--167. \(3\)-folds, Coverings in algebraic geometry, Divisors, linear systems, invertible sheaves On the degree of the canonical maps of 3-folds | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 motivic tensor-triangulated geometry, more explicitly, the study of the tt-geometry of Voevodsky's (tensor) triangulated category \(\mathrm{DM}^{\mathrm{gm}}(F; k)\) of geometric motives over a field \(F\), with coefficients \(k\).
The authors focus on the tt-subcategory of geometric mixed Artin-Tate motives \(\mathrm{DATM}^{\mathrm{gm}} (F; k)\) (generated by the Tate objects \(k(i)\) and the Artin motives \(M (E)\), for \(E/F\) a finite separable extension). Moreover, they take \(F\) to be a real closed field (\(\mathbb{R}\) for this review) and concentrate largely upon the most interesting coefficients, \(k = \mathbb{Z}/2\). This extends work of the second author [\textit{M. Gallauer}, Compos. Math. 155, No. 10, 1888--1923 (2019; Zbl 1498.14049)] for the case \(F\) algebraically closed.
Their main result shows that the spectrum (in the sense of [\textit{P. Balmer}, J. Reine Angew. Math. 588, 149--168 (2005; Zbl 1080.18007)]) of \(\mathrm{DATM}^{\mathrm{gm}} (\mathbb{R}; \mathbb{Z}/2)\) has six points, has Krull dimension two and has six irreducible closed subsets. They show that the lattice of closed subsets has fourteen points, hence \(\mathrm{DATM}^{\mathrm{gm}} (\mathbb{R}; \mathbb{Z}/2)\) has precisely fourteen tt-ideals. The spectrum has three `maximal' irreducible closed subsets; the authors represent these as the support of explicit Artin-Tate motives, corresponding to the three motives of the title.
Using these results, they determine the spectrum of \(\mathrm{DATM}^{\mathrm{gm}} (\mathbb{R}; \mathbb{Z})\) (up to that of \(\mathrm{DATM}^{\mathrm{gm}} (\mathbb{R}; \mathbb{Q})\), which is conjectured to be a point); the most interesting part of the structure arises from \(\mathbb{Z}/2\)-coefficients. These results hint at the richness of motivic tensor-triangulated geometry.
The main result is proved using analogous results for the modular representation theory of the cyclic group \(C_2\). The relationship is provided by the profound result of Positselski [\textit{L. Positselski}, Mosc. Math. J. 11, No. 2, 317--402 (2011; Zbl 1273.12004)], which exploits the Milnor conjecture (as proved by Voevodsky).
Namely, for \(\mathcal{A}\) the category of finite-dimensional \(k C_2\)-modules, the authors consider the category \(\mathcal{A}^{\mathrm{fil}}_{\mathrm{ex}}\) of filtered objects in \(\mathcal{A}\), equipped with its `minimal' exact structure; this is a Frobenius exact category. Positselski established the equivalence \[ \mathrm{D}_b (\mathcal{A}^{\mathrm{fil}}_{\mathrm{ex}}) \stackrel{\simeq} {\rightarrow} \mathrm{DATM}^{\mathrm{gm}} (\mathbb{R}; \mathbb{Z}/2), \] where the domain is the bounded derived category of the exact category \( \mathcal{A}^{\mathrm{fil}}_{\mathrm{ex}}\). (The authors revisit some elements of the proof using alternative methods.) Although the equivalence is not known to preserve the tensor structure, it has sufficient structure to ensure that these tt-categories have the same spectrum, thus reducing to the study of the spectrum of \( \mathrm{D}_b (\mathcal{A}^{\mathrm{fil}}_{\mathrm{ex}}) \).
The authors first identify the spectrum of the homotopy category \(\mathrm{K}_b (A)\) of bounded complexes in \(A\), which is a three point space (the more familiar spectrum of \(\mathrm{D}_b (\mathcal{A})\) identifies as a two-point open subset). They then analyse \(\mathrm{D}_b (\mathcal{A}^{\mathrm{fil}}_{\mathrm{ex}})\) by using two tt-functors, \(\mathrm{gr}\) and \(\widetilde{\mathrm{fgt}}\), from \(\mathrm{D}_b (\mathcal{A}^{\mathrm{fil}}_{\mathrm{ex}})\) to \(\mathrm{K}_b (A)\). The functor \(\mathrm{gr}\) is induced by passage to the associated graded whereas \(\widetilde{\mathrm{fgt}}\) is a `twisted' version of the functor induced by forgetting the filtration. An intricate and interesting analysis shows that these detect the six points of the spectrum of \(\mathrm{D}_b (\mathcal{A}^{\mathrm{fil}}_{\mathrm{ex}})\) and determines the topology. Artin-Tate motives; tensor-triangular geometry; modular representation theory; classification Motivic cohomology; motivic homotopy theory, Homological algebra in category theory, derived categories and functors, Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects), Modular representations and characters Three real Artin-Tate motives | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(\mathbf{G}\) be a connected reductive \(\mathbb{F}_q\)-group endowed with the Frobenius isogeny \(F\), where \(q\) is a power of a prime number \(p\). For an element \(w\) in the Weyl group of \(\mathbf{G}\), Deligne and Lusztig constructed a finite étale morphism \(\pi: X(w) \to Y(w)\) that is \(\mathbf{G}^F\)-equivariant, as well as a compactification \(j: X(w) \hookrightarrow \bar{X}(w) \) à la Demazure. When \(w\) is a Coxeter element, \(\bar{X}(w)\) admits a stratification indexed by the \(F\)-stable parabolic subgroups: let \(\mathbf{P}\) be an \(F\)-stable parabolic with unipotent radical \(\mathbf{U}\), the corresponding stratum is
\[
i_{\mathbf{P}}: \bar{X}_{\mathbf{P}}(w) := \bar{X}(w)^{\mathbf{U}^F} \to \bar{X}(w).
\]
Put \(\Lambda = \mathbb{Z}/\ell^m\) for some prime number \(\ell \neq p\) and \(m\). The pull-back
\[
R\Gamma(X(w), \pi_* \Lambda) = R\Gamma( \bar{X}(w), Rj_* (\pi_*\Lambda) ) \to R\Gamma( \bar{X}_{\mathbf{P}}(w), i^*_{\mathbf{P}} Rj_* (\pi_*\Lambda) )
\]
then induces a natural morphism
\[
R\Gamma(X(w), \pi_* \Lambda)^{\mathbf{U}^F} \to R\Gamma( \bar{X}_{\mathbf{P}}(w), i^*_{\mathbf{P}} Rj_* (\pi_*\Lambda) ).
\]
In this paper, it is proved that the arrow above is an isomorphism when \(\mathbf{G} = \mathrm{GL}_d\). This result is motivated by the study of the cohomology of Drinfeld's symmetric space over a \(p\)-adic field \(K\) with residual field \(\mathbb{F}_q\), for which \(Rj_*(\pi_* \Lambda)\) appears as certain nearby cycles. Such issues are pertinent in the geometric realization of the local Langlands conjecture and the Jacquet-Langlands correspondence over \(K\). Deligne-Lusztig varieties; Coxeter element; compactification Wang, H, Sur la cohomologie des compactifications de variétés de Deligne-Lusztig, Ann. Inst. Fourier (Grenoble), 64, 2087-2126, (2014) Classical groups (algebro-geometric aspects), Étale and other Grothendieck topologies and (co)homologies, Representation theory for linear algebraic groups, Cohomology theory for linear algebraic groups, Linear algebraic groups over finite fields, Modular representations and characters, Representations of finite groups of Lie type, Langlands-Weil conjectures, nonabelian class field theory On the cohomology of the compactification of the Deligne-Lusztig varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors present new examples of simply-connected algebraic surfaces of general type and non-negative signature. These examples include three spin manifolds with zero signature and a countably infinite number of spin manifolds with positive signature. The examples are Galois covers of generic projections of Hirzebruch (rational ruled) surfaces with respect to the symmetric group. Hirzebruch surfaces; simply-connected algebraic surfaces of general type; non-negative signatue; spin manifolds; Galois covers Moishezon B, Robb A, Teicher M. On Galois covers of Hirzebruch surfaces. Math Ann, 305: 493--539 (1996) Coverings in algebraic geometry, Surfaces of general type, Families, moduli, classification: algebraic theory, Braid groups; Artin groups On Galois covers of Hirzebruch 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 authors classify (up to étale coverings) all non-projective threedimensional compact Kähler manifolds \(X\) which admit a surjective holomorphic map \(f:X\rightarrow X\) whose general fiber has \(d\geq 2\) elements. The map \(f\) is necessarily finite and can only be ramified when \(X\) is uniruled. General results of \textit{F. Campana} [Invent. Math. 63, 187--223 (1981; Zbl 0436.32024)] and \textit{N. Nakayama} and \textit{D.-Q. Zhang} [Math. Ann. 346, No. 4, 991--1018 (2010; Zbl 1189.14043)] imply that the general fiber of the Iitaka fibration of a non-projective compact Kähler manifold \(X\) is not an elliptic curve, hence \(\kappa(X)\leq 1\). For \(X\) non-uniruled \(\kappa(X)\) equals zero or one and is used as classification parameter. If \(\kappa(X)=0\), then \(X\) is either a complex torus or an étale covering of a product \(E\times Y\), \(E\) an elliptic curve, \(Y\) bimeromorphic to a K3 surface. If \(\kappa(X)=1\), then \(X\) is an étale covering of a product \(C\times S\), either \(C\) a surface of general type and \(S\) a complex torus or \(C\) an elliptic curve and \(S\) an elliptic surface. The case \(X\) uniruled is more involved since \(f\) may be ramified. The algebraic dimension of \(X\) serves here as a classification parameter. \(X\) is an étale covering of a projectivized line bundle \(\mathbb P(E)\) over a complex torus \(A\) or a product \(S\times \mathbb P^1\), further details depending on \(a(X)\).
The proofs are based on Mori theory for compact Kähler threefolds, see also [the second author, Math. Ann. 311, No. 4, 729--764 (1998; Zbl 0919.32016)]. The authors achieve new results in this area, in particular they establish a minimal model program for compact Kähler threefolds admitting non-trivial finite étale coverings. \newline For endomorphisms of smooth projective threefolds see \textit{Y. Fujimoto} and \textit{N. Nakayama} [J. Math. Kyoto Univ. 47, No. 1, 79--114 (2007; Zbl 1138.14023)]. compact Kähler manifold; endomorphism; torus fibrations; minimal model program; non-projective manifold Andreas, H.; Peternell, T., Non-algebraic compact Kähler threefolds admitting endomorphisms, Sci. Chin. Math., 54, 1635-1664, (2011) Compact complex \(3\)-folds, Compact Kähler manifolds: generalizations, classification, Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables, Coverings in algebraic geometry, Minimal model program (Mori theory, extremal rays) Non-algebraic compact Kähler threefolds admitting 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 For part I of this paper see: \textit{R. F. Pries}, ibid. 481-484 (2002; see the preceding review Zbl 1030.14012). Consider a wildly ramified \(G\)-Galois cover of curves \(\varphi:Y \to\mathbb{P}^1_k\) branched at only one point over an algebraically closed field \(k\) of characteristic \(p\). In this note, the author proves using formal patching that all sufficiently large conductors occur for such covers \(\varphi\) when the Sylow \(p\)-subgroups of \(G\) have order \(p\). Galois cover of curves; characteristic \(p\); large conductors Pries, R.: Conductors of wildly ramified covers II. C. R. Acad. sci. Paris sér. I math. 335, No. 1, 485-487 (2002) Coverings of curves, fundamental group, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Local ground fields in algebraic geometry, Coverings in algebraic geometry Conductors of wildly ramified covers. II | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The fundamental group of the complement of a singular plane curve introduced by Zariski is an interesting object to study in different areas in mathematics. One of the ways to study this object is to study the ramification curves of generic projections of surfaces. More precisely:
Let \(S\) be a non-singular algebraic surface of degree \(\nu \) in \(\mathbb P^{3}\) (the field is algebraically closed of characteristic 0). and \(\pi : S \rightarrow \mathbb P^{2}\) be a projection with center \(0\in \mathbb P^{3}-S\). Let \(B\) be the branch curve of \(\pi \) having as singularities nodes and cusps.
According to a conjecture of \textit{O. Chisini} [``Sulla identita birazionale di due funzioni algebriche di piu variabili, dotate di un a medesima varieta di diramazione'', (Italian) Ist. Lombardo Sci. Lett., Rend., Cl. Sci. Mat. Natur., III. Ser. 11(80), 3--6 (1949; Zbl 0041.28002)] the projection \(\pi \) is uniquely determined by \(B\) if \(\nu \geq 5\). This is proved by \textit{V. S. Kulikov} [``On Chisini's conjecture. II.'', Izv. Math. 72, No. 5, 901--913 (2008); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 72, No. 5, 63--76 (2008; Zbl 1153.14012)] for \(\nu \geq 11\).
This leads to study the problem whether a plane curve is obtained as the branch curve of a projection \(\pi \). For example, a plane sextic with six cusps is a branch curve of a generic projection if and only if its six cusps form a special 0-cycle on the plane.
\textit{B. Segre}'s classical result [``Sulla caratterizzazione delle curve di diramazione dei piani multipli generali'', Memorie Accad. d'Italia, Roma 1; Mat. Nr. 4, 31 p. (1930; JFM 56.0562.01)] gives a necessary and sufficient condition for a plane curve to be a branch curve of a generic projection of a smooth surface in \( \mathbb P^{3}\). Segre proves that a nodal-cuspidal plane curve \(B\) of degree \(\nu (\nu - 1)\) with \(n(\nu )\) nodes and \(c(\nu )\) cusps is a branch curve of a generic projection of a smooth surface of degree \(\nu \geq 3\) in \( \mathbb P^{3}\) if and only if there are two adjoint curves of degrees \(a(\nu )\) and \(a(\nu ) + 1\) passing through the \(0\)-cycle of singularities of \(B\) and having separated tangents and these singularities.
The article under review gives a different proof for Segre's result in terms of Picard and Chow groups of \(0\)-cycles on a singular plane curve \(B\). And, the Appendix, written by Shustin, presents a new pair \((B,C)\) of \textit{O. Zariski} [``On the problem of existence of algebraic functions of two variables possessing a given branch curve'', Amer. J. 51, 305--328 (1929; JFM 55.0806.01)] where \(B\) denotes a branch curve of a smooth surface \(S\) in \( \mathbb P^{3}\) and \(C\) denotes a nodal cuspidal curve which is not a branch curve. branch curve; adjoint curves; \(0\)-cycles; Zariski pairs Friedman, M., Leyenson, M.: On ramified covers of the projective plane I: Segre's theory and classification in small degrees (with an appendix by Eugenii Shustin). Int. J. Math. 22, 619-653 (2011) Special algebraic curves and curves of low genus, Special divisors on curves (gonality, Brill-Noether theory), Coverings in algebraic geometry On ramified covers of the projective plane. I: Interpreting Segre's theory (with an appendix by Eugenii Shustin) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper the authors study dihedral covers of smooth varieties, in particular they provide a structure theorem (Section 5) for the Galois coverings \(\pi: X\to Y\), with \(Y\) smooth and Galois group the dihedral group \(D_n\) of order \(2n\). Given a smooth variety \(Y\), they describe the algebraic ``building data'' on \(Y\) which are equivalent to the existence of such covers \(\pi:X\to Y\).\newline The underlying idea is to factor \(\pi: X\to Y\) as the composition of a cyclic covering of order \(n\), \(p: X\to Z:=X/H\), (\(H \subset D_n\) is the group of rotations) and a (singular) double covering \(q:Z\to Y\). A technical novelty here consists in describing Weil divisors and the Picard group of divisorial sheaves on a normal double cover \(Z\) (Section 3). They also describe in detail the special case of Picard group of hyperelliptic curves (Section 4).
In the second part of the paper (Section 6 and 7) the authors concentrate on two very explicit classes of dihedral covers of algebraic varieties: the \textit{simple} and the \textit{almost simple} dihedral covers. A simple dihedral covering \(X\) is contained in a rank two split vector bundle of the form \(\mathbb L\oplus \mathbb L\) over \(Y\), and is defined by equations
\[
\begin{cases} u^n+v^n= 2a \,, & a\in H^0(\mathcal{O}_Y(nL)),\\ uv= F \,, & F\in H^0(\mathcal{O}_Y(2L)). \end{cases}
\]
The dihedral action is generated by an element \(\sigma\) of order \(n\) such that \(u\mapsto \xi u\,, v \mapsto\xi^{-1}\), where \(\xi\) is a primitive \(n\)-th root of \(1\), and and by an involution \(\tau\) exchanging \(u\) with \(v\). The authors prove that the branch locus is the divisor \(B = \{a^2 - F^n = 0\}\); \(X\) is smooth if \(B\) is smooth outside of \(F = 0\) and the two divisors \(\{a = 0\}\) and \(\{F = 0\}\) intersect transversally.\newline The almost simple dihedral covers are defined on the fibre product of two \(\mathbb P^1\)-bundles: \(X\) is the subset of \(\mathbb P(\underline {\mathbb C} \oplus \mathbb L)\times _Y\mathbb P(\underline {\mathbb C} \oplus \mathbb L)\) defined by the equations:
\[
\begin{cases} u_1v_1-u_0v_0F=0\,,& F\in H^0(\mathcal{O}_Y(2L))\,,\\ a_\infty v_1^nu_0^n-2a_0v_0^n u_0^n+a_\infty v_0^n u_1^n=0 \,, & a_0\in H^0(\mathcal{O}_Y(A_0)), a_\infty\in H^0(\mathcal{O}_Y(A_\infty)) \end{cases}
\]
where, \(v_1, u_1\) are fibre coordinates on the geometric line bundle \(\mathbb L\to Y\), \(v_0, u_0\) are fibre coordinates on the trivial geometric line bundle \(\underline {\mathbb C} =Y\times \mathbb C \to Y\), and \(A_0, A_\infty\) are effective divisors on \(Y\) such that \(A_0\equiv nL+A_\infty\), \(A_\infty=\{a_\infty=0\}\) and \(A_0=\{a_0=0\}\). The dihedral group \(D_n\) acts on \(X\) in the following way:
\[
\begin{aligned} \sigma([u_0:u_1],[v_0:v_1]) =([u_0:\xi u_1],[v_0:\xi^{-1}v_1])\,\\ \tau([u_0:u_1],[v_0:v_1])= ([v_0,v_1],[u_0:u_1])\,. \end{aligned}
\]
The authors prove that the branch locus is the divisor \(B = \{a_\infty(a_0^2-a_\infty^2 F^n )= 0\}\); \(X\) is smooth if the locus \(\{a_0^2-a_\infty^2 F^n = 0\}\) is smooth outside of \(F = 0\), \(A_0\) intersects transversally \(\{F = 0\}\) , \(A_0\cap A_\infty = \emptyset\) and \(A_\infty\) is smooth. Galois covers; dihedral covers; direct image sheaves; algebraic varieties; classification of algebraic varieties Coverings in algebraic geometry, Surfaces and higher-dimensional varieties Dihedral Galois covers of algebraic varieties and the simple cases | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let S be a projective normal surface defined over the field \({\mathbb{C}}\) of complex numbers. S is said to be a log del Pezzo surface if S has only quotient singularities and the anticanonical bundle \(-K_ S\) is ample. The rank of S is the Picard rank \(\rho (S)=\dim_{{\mathbb{Q}}}Pic(S)\otimes {\mathbb{Q}}\). Moreover S is Gorenstein, i.e. \(K_ S\) is a Cartier divisor, if and only if S has only rational double points provided S is a log del Pezzo surface. The main result of the paper is the following theorem:
Let S be a Gorenstein log del Pezzo surface. Suppose that S is either minimal or singular. Then S is a Gorenstein, algebraic compactification of the affine plane if and only if the smooth locus \(S^ 0\) is simply connected.
In the case when S is a Gorenstein log del Pezzo surface of rank 1, the authors compute explicitly the fundamental group \(\pi_ 1(S^ 0)\), \(S^ 0=S-Sing(S)\), as well as the group \(H^ 2(S^ 0,{\mathbb{Z}})/H^ 2(S,{\mathbb{Z}})\). The computation of \(\pi_ 1(S^ 0)\) leads to a number of interesting consequences. quotient singularities; Gorenstein singularities; canonical bundle; Gorenstein log del Pezzo surface; algebraic compactification of the affine plane; fundamental group M. Miyanishi and D.-Q. Zhang, ''Gorenstein log del Pezzo surfaces of rank one,'' J. Algebra 118(1), 63--84 (1988). Special surfaces, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Coverings in algebraic geometry, Families, moduli, classification: algebraic theory Gorenstein log del Pezzo surfaces of rank 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 A duality theorem for the stable module category of representations of a finite group scheme is proved. One of its consequences is an analogue of Serre duality, and the existence of Auslander-Reiten triangles for the \(\mathfrak{p}\)-local and \(\mathfrak{p}\)-torsion subcategories of the stable category, for each homogeneous prime ideal \(\mathfrak{p}\) in the cohomology ring of the group scheme. Serre duality; local duality; finite group scheme; stable module category; Auslander-Reiten triangle Group schemes, Representations of associative Artinian rings, Modular representations and characters, Cohomology theory for linear algebraic groups, Cohomology of groups, Derived categories, triangulated categories Local duality for representations of finite group schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The present paper is devoted to the study of general properties of the moduli spaces of surfaces of general type (i.e., given a minimal model S of a surface of general type, one looks at the moduli space M(S) parametrizing all the isomorphism classes of complex structures on the 4- dimensional oriented compact topological manifold underlying S: M(S) is a quasi projective variety by a theorem of \textit{D. Gieseker} [Invent. Math. 43, 233-282 (1977; Zbl 0389.14006)].
Theorem A proves that for each natural number n there are positive integers \(d_ 1<d_ 2<...<d_ n\) and a surface of general type S such that M(S) has irreducible components \(Y_ 1,...,Y_ n\) of respective dimensions \(d_ 1,...,d_ n\) (this is in sharp contrast with the case of a curve C of genus \(g\geq 2\) where M(C) is irreducible of dimension 3g- 3). In view of this result, in the paper is given an upper bound for the dimension of an irreducible component of M(S), to wit, if d(S) is the dimension of M(S) at the point corresponding to S, theorem B proves that d(S) is at most \(10\chi +3c^ 2_ 1+108,\) and (theorem C) at most \(10\chi +q+1\) if S contains a smooth canonical curve (where \(\chi =1- q+p_ g,\) one should notice that \(\chi\), q, \(c^ 2_ 1\) are topological invariants, in fact, and that a lower bound for d(S) is given by the Kodaira-Spencer-Kuranishi theory of deformations, giving: d(S) is at least \(10\chi -2c^ 2_ 1).\)
Finally, the last chapter studies irregular surfaces without irrational pencils, since in the literature there are scattered correct and incorrect results: using some of these results, \textit{G. Castelnuovo} claimed [Atti. Accad. Naz. Lincei, VIII. Ser., Rend., CE. Sci. Fiz. Mat. Nat. 7, 3-11 (1949; Zbl 0035.372)] that for those surfaces d(S) is at most \(p_ g+2q\). In this paper a counterexample is given (showing that at least, asymptotically on \(p_ g\), the coefficient 4 is needed in front of \(p_ g)\), but theorem D proves that if q is at least 3, and there exist 2 holomorphic 1-forms whose wedge product defines a reduced irreducible curve, then d(S) is at most \(p_ g+3q-3\), and moreover \(c^ 2_ 1\) is at least 6\(\chi\).
The first chapter deals with smooth Galois covers with abelian group, studying their fundamental groups [the method used is to prove that the fundamental group of the complement of the branch locus is abelian: theorem 1.6 is, for instance, a generalization of a previous result of \textit{R. Mandelbaum} and \textit{B. Moishezon}, Trans. Am. Math. Soc. 260, 195-222 (1980; Zbl 0465.57014), but has, in the meantime, been further generalized by Nori].
The main ingredients to prove theorem A are a careful study of the deformations of Galois covers in the special case where the group is \(({\mathbb{Z}}/2)^ 2\), plus an application of Freedman's result on diffeomorphisms of 4-manifolds [\textit{M. H. Freedman}, J. Differ. Geom. 17, 357-453 (1982; Zbl 0528.57011)], and a number-theoretic lemma proved by \textit{E. Bombieri} in the appendix to this paper: in fact, the surfaces considered for theorem A, are just \(({\mathbb{Z}}/2)^ 2\) Galois covers of \({\mathbb{P}}^ 1\times {\mathbb{P}}^ 1\). upper bound for the dimension of an irreducible component of moduli space; abelian fundamental group of the complement of the branch locus; moduli spaces of surfaces of general type; deformations of Galois covers Catanese, F., On the moduli spaces of surfaces of general type, \textit{J. Differential Geom.}, 19, 2, 483-515, (1984) Families, moduli, classification: algebraic theory, Fine and coarse moduli spaces, Coverings in algebraic geometry, Moduli, classification: analytic theory; relations with modular forms, Special surfaces, Complex-analytic moduli problems, Formal methods and deformations in algebraic geometry On the moduli spaces of surfaces of general type. Appendix: Letter of E. Bombieri written to the author | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems By a classical result of \textit{A. Beauville} [Invent. Math. 55, 121--140 (1979; Zbl 0403.14006)] if the canonical image of a smooth surface of general type \(X\) is a surface \(\Sigma\), denoting by \(S\) a minimal desingularization of \(\Sigma\), either \(p_g(S)=0\) or the resolution map \(S \to \Sigma\) is the canonical map of \(S\).
It is very easy to construct examples with \(p_g(S)=0\), but it is much more difficult to give examples of the latter case with degree \(d \geq 2\), being \(d\) the degree of the canonical map of \(X\): in this case the dominant rational map \(X \rightarrow S\) is called a {good canonical cover} of degree \(d\). Beauville in the above mentioned paper could give only one example, and its paper motivated many authors in constructing other examples of good canonical covers.
In this paper the authors construct three sequences of good canonical covers of degree \(2\) with \(X\) regular and unbounded invariants \(p_g\) and \(K^2\). Only sporadic examples of surfaces with these properties were previously known.
Sequences of examples were already known, but only of irregular surfaces. For example, the same authors [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 29, No. 4, 905--938 (2000; Zbl 1016.14020)] gave a method for constructing infinite sequences of good canonical covers with \(d=2\), but all examples constructed by this method have \(q(X)\geq 2\).
Roughly speaking, the authors' idea is to find actions of a group \(G\) on some of those good canonical covers \(X \rightarrow S\), {killing the irregularity} of \(X\) and letting the quotient \(X/G \rightarrow S/G\) remain a good canonical cover.
They could find three sequences of examples, always using the group \(G=\mathbb{Z}/3\mathbb{Z}\). In the last section they could show that in one of these three cases the surface \(X\) is simply connected. canonical maps; double covers Surfaces of general type, Coverings in algebraic geometry Regular canonical 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 \(\Sigma\) be a smooth projective surface, let \(f': S'\to\Sigma\) be a double cover of \(\Sigma\) and let \(\mu:S\to S'\) be the canonical resolution of \(S'\). Put \(f= f'\circ\mu\). An irreducible curve \(D\) on \(\Sigma\) is said to be a splitting curve with respect to \(f\) if \(f^*D\) is of the form \(D^+ +D^-+ E\), where \(D^+\neq D^-\), \(D^-= \sigma^*_f D^+\), \(\sigma_f\) being the covering transformation of \(f\) and all irreducible components of \(E\) are contained in the exceptional set of \(\mu\).
In this article, we consider ``reciprocity'' concerning splitting curves when \(\Sigma\) is a rational ruled surface. quadratic residue curve; Mordell-Weil group Coverings in algebraic geometry, Coverings of curves, fundamental group, Plane and space curves, Rational and ruled surfaces A note on quadratic residue curves on rational ruled surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a smooth complex projective variety. A classical question asks for a description of \(X\) admitting surjective ramified endomorphisms \(f: X \to X\) of degree \(d\geq 2\). In the paper under review the authors prove that \(X\) must be uniruled and investigate the case of Fano manifolds of Picard number \(\rho(X)=1\). It is conjectured that, if \(X\) is a Fano manifold with \(\rho(X)=1\) and \(f: X \to X\) is an endomorphism of degree \(>1\), then \(X={\mathbb P}^n\). This conjecture has been proved (i) in case of surfaces, (ii) in case of threefolds by \textit{E. Amerik, M. Rovinsky} and \textit{A. Van de Ven} [Ann. Inst. Fourier 49, No. 2, 405--415 (1999; Zbl 0923.14008)], \textit{J.-M. Hwang} and \textit{N. Mok} [J. Algebr. Geom. 12, 627--651 (2003; Zbl 1038.14018)], \textit{C. Schuhmann} [J. Algebr. Geom. 8, 221--244 (1999; Zbl 0970.14022)], (iii) in case of rational homogeneous manifolds by \textit{J.-M. Hwang} and \textit{N. Mok} [Invent. Math. 136, 209--231 (1999; Zbl 0963.32007)], \textit{K. H. Paranjape} and \textit{V. Srinivas} [Invent. Math. 98, 425--444 (1989; Zbl 0697.14037)], (iv) in case of toric varieties by \textit{G. Occhetta} and \textit{J. A. Wiśniewski} [Math. Z. 241, 35--44 (2002; Zbl 1022.14016)], (v) in case of varieties containing a rational curve with trivial normal bundle by Hwang and Mok [Zbl 1038.14018].
Let \({\mathcal E}_f\) be the vector bundle canonically associated to \(f\). In the paper under review the authors prove the conjecture if one of the following holds: (i) \({\mathcal E}_{f_k}\) is ample for some iterate \(f_k\) of \(f\) and \(h^0(X,f_k^*(T_X))>h^0(X,T_X)\); (ii) \(f\) is Galois and \(h^0(X,f_k^*(T_X))>h^0(X,T_X)\) for some \(k\); (iii) \(X\) is almost-homogeneous and \(h^0(X,T_X)>\dim X\) and \({\mathcal E}_{f_k}\) is ample for sufficiently large \(k\).
Moreover, they prove that, if \(X\) is a Fano manifold with \(\rho(X)=1\) and \(f: X \to X\) is an endomorphism of degree \(d\), then \(d=1\) if one of the following holds: (i) the index of \(X\) is \(\leq 2\) and there exists a line in \(X\) not contained in the branch locus of \(f\); (ii) \(X\) satisfies the Cartan--Fubini condition, \(X\) is almost homogeneous and \(h^0(X,T_X)>\dim X\); (iii) \(X\) satisfies the Cartan--Fubini condition, \(X\) is almost homogeneous and either branch or the ramification divisor of \(f\) meets the open orbit of \(\text{Aut}^0(X)\); (iv) \(X\) is a del Pezzo manifold of degree \(5\).
The basic idea is to study a ramified finite covering \(f\) of degree \(d\) through the properties of \({\mathcal E}_f\), which tends to inherit positivity properties from the ramification divisor. In particular, when \(f: X \to Y\) is a Galois covering of projective manifolds which does not factor through an étale covering (this is the case, for instance, for Fano manifolds with \(\rho(X)=1\)) and all irreducible components of the ramification divisor are ample, they prove that \({\mathcal E}_f\) is ample.
Moreover, if \(X\) is a projective manifold with \(\rho(X)=1\) and if \({\mathcal F}\subset T_X\), a coherent subsheaf of positive rank, is ample, they prove that \(X\) is isomorphic to the projective space, generalizing a theorem of \textit{M. Andreatta} and \textit{J. A. Wiśniewski} [Invent. Math. 146, 209--217 (2001; Zbl 1081.14060)]. Coverings; Fano varieties Kollár, J.: Rational Curves on Algebraic Varieties, vol.~32 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer, Berlin (1996) Coverings in algebraic geometry, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Fano varieties Galois coverings and endomorphisms of projective varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The Severi inequality is a theorem claimed first by \textit{F. Severi} in [Comment. Math. Helv. 4, 268--326 (1932; Zbl 0005.17602)], stating that any minimal surface of general type \(S\) of maximal Albanese dimension, i.e., such that the image of the Albanese morphism is a surface, has canonical degree \(K^2_S\) not smaller than \(4\) times the Euler characteristic \(\chi({\mathcal O}_S)\).
Severi's original proof was wrong [\textit{F. Catanese}, Lect. Notes Math. 997, 90--112 (1983; Zbl 0517.14011); \textit{M. Reid}, Lect. Notes Math. 732, 534--544 (1979; Zbl 0423.14021)] and therefore the Severi inequality became a conjecture, and remained such for almost 30 years, up to \textit{R. Pardini}'s proof in [Invent. Math. 159, No. 3, 669--672 (2005; Zbl 1082.14041)].
It is a general principle in the geography of surface of general type that, when an inequality is true, the surfaces for which the inequality is an equality should be ``simpler'', and therefore a classification of them is (obviously) interesting and (maybe) possible. So, once the Severi inequality is proved, it is natural to try to attack a classification of the surfaces on the Severi line, that is of the minimal surfaces of general type \(S\) with maximal Albanese dimension \(2\) and \(K^2_S=4\chi({\mathcal O}_S)\). Unfortunately, since the argument of Pardini's is based on a limit procedure, is not easily usable to get informations on the surfaces on the Severi line.
In this paper, the authors classify these surfaces, showing that the canonical models of surfaces on the Severi line are exactly the surfaces with irregularity \(2\) arising as double covers of an abelian surface with ample branch divisor having only simple singularities. The main tools are a covering trick already used in Pardini's proof and the properties of the continuous rank ([\textit{M. A. Barja}, Duke Math. J. 164, No. 3, 541--568 (2015; Zbl 1409.14013)]).
The same result has been obtained independently at the same time by \textit{X. Lu} and \textit{K. Zuo} [``On the Severi type inequalities for irregular surfaces'', Preprint, \url{arXiv:1504.06569}] with a different proof. It is worth mentioning that \textit{M. Manetti} had proved in [Math. Nachr. 261--262, 105--122 (2003; Zbl 1044.14017)] the same result under the additional hypothesis that \(K_S\) ist ample. Manetti's paper, that appeared few years before Pardini's proof, contains in fact a proof of the Severi inequality under this further assumption. surfaces of general type; Severi inequality; étale coverings; irregular varieties Barja, M. Á.; Pardini, R.; Stoppino, L., Surfaces on the Severi line, J. Math. Pures Appl. (9), 105, 5, 734-743, (2016) Surfaces of general type, Coverings in algebraic geometry, Divisors, linear systems, invertible sheaves Surfaces on the Severi 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 Here we consider bihyperelliptic curves, i.e., double covers of hyperelliptic curves. By applying the theory of quadruple covers, among
other things we prove that the bihyperelliptic locus in the moduli space of smooth curves is irreducible and unirational for \(g \geq 4\gamma + 2 \geq 10.\) Edoardo Ballico, Gianfranco Casnati, and Claudio Fontanari, On the geometry of bihyperelliptic curves, J. Korean Math. Soc. 44 (2007), no. 6, 1339 -- 1350. Coverings of curves, fundamental group, Families, moduli of curves (algebraic), Special algebraic curves and curves of low genus, Coverings in algebraic geometry On the geometry of bihyperelliptic 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 Concerns the author's paper reviewed in Zbl 0860.11022. Hershel M. Farkas, Yaacov Kopeliovich, and Irwin Kra, Uniformizations of modular curves, Comm. Anal. Geom. 4 (1996), no. 1-2, 207 -- 259. Hershel M. Farkas, Yaacov Kopeliovich, and Irwin Kra, Corrigendum to: ''Uniformizations of modular curves'', Comm. Anal. Geom. 4 (1996), no. 4, 681. 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 Corrigendum to ``Uniformization 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 In this paper the authors study certain Galois covers \(X_{\mathcal L}(n)\) of the projective plane with group \((\mathbb Z/n)^r\) branched in a configuration \(\mathcal L\) of \(r+1\) lines. The minimal resolution of \(X_{\mathcal L}(n)\) is known as the Hirzebruch-Kummer cover \(HK(n,\mathcal L)\). The first interesting example occurs for the complete quadrangle \(\mathcal C \mathcal Q\) and \(n=5\). The surface \(HK(5,\mathcal C \mathcal Q)\) as shown by Hirzebruch, is a ball quotient, in particular a rigid surface admitting a Hermitian metric with
strongly negative curvature. In a previous work, the authors could prove that the surface \(HK(n,\mathcal C \mathcal Q)\) is rigid if and only if \(n \geq 4\). Since the rigidity of the cover implies the rigidity of the line configuration, it is natural to ask, if \(HK(n,\mathcal L)\) is a rigid surface for any rigid configuration \(\mathcal L\) and \(n\) sufficiently large.
In the first part of the paper the rigidity of the equisingular deformations of \(X_{\mathcal L}(n)\) is established under the assumption that the configuration is singularly saturated, meaning that it has at least four singular points, each line in the configuration contains at least two singular points and the line defined by any two singular points belongs to the configuration. About the question of the existence of a metric with negative curvature very few is known. In fact it is only settled for \(\mathcal C \mathcal Q\) in the special case where \(5\) divides \(n\), thanks to the work of \textit{F. Zheng} [Commun. Anal. Geom. 7, No. 4, 755--786 (1999; Zbl 0937.53036)]. Motivated by this question, the authors establish in the second part of the paper explicit equations for \(HK(n,\mathcal C \mathcal Q)\) as a submanifold of the product of four Fermat curves of degree \(n\). Their idea is to consider \(HK(n,\mathcal C \mathcal Q)\) as a cover of the blowup of the plane in the four points of the complete quadrangle, which is the Del Pezzo surface of degree \(5\). They first derive determinantal equations for the Del Pezzo surfaces of degree \(9-k \leq 6\) as submanifolds of \((\mathbb P^1)^k\). Then they use the equations for the Del Pezzo of degree \(5\), together with the natural \((\mathbb Z/n)^2\) cover from the Fermat curve of degree \(n\) to the projective line, to derive also equations for \(HK(n,\mathcal C \mathcal Q)\). rigid complex manifolds and varieties; branched coverings; Hirzebruch Kummer coverings; deformation theory; configurations of lines; del Pezzo surfaces Local deformation theory, Artin approximation, etc., Moduli, classification: analytic theory; relations with modular forms, Surfaces of general type, Fano varieties, Coverings in algebraic geometry, Vanishing theorems in algebraic geometry, Deformations of complex structures, Equisingularity (topological and analytic), Compact complex surfaces, Arrangements of points, flats, hyperplanes (aspects of discrete geometry) Del Pezzo surfaces, rigid line configurations and Hirzebruch-Kummer coverings. Del Pezzo surfaces, rigid line configurations and Hirzebruch-Kummer covering. In beloved memory of Paolo (de Bartolomeis) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems An isolated singularity x of a complex surface (X,\({\mathcal O}_ x)\) is \textit{perfect}, or \textit{homological trivial}, if the local fundamental group \(\pi_ 1(\partial U_ x)\) is a perfect group, where \(U_ x\) is a contractible neighborhood of x in X. A graph \(\Gamma\) is called \textit{perfect} if there exist integer weights \(n_ i\) on the vertices of \(\Gamma\) for which \(\Gamma (n_ 1,...,n_ k)\) is the weighted dual intersection graph of the minimal resolution of a perfect surface singularity whose minimal resolution is normal.
In this paper we use techniques for graphical evaluation of determinants to characterize most kinds of perfect graphs and to relate this problem to Diophantine questions involving partial fraction representations of integers. These questions, in turn, have independent interest in number theory, involving techniques of continued fractions and Egyptian fractions. weighted dual intersection graph of the minimal resolution of a perfect surface singularity; Diophantine questions; partial fraction representations; continued fractions; Egyptian fractions Brenton, D.; Drucker, D., Perfect graphs and complex surface singularities with perfect local fundamental group, Tohoku Math. J., 41, 507-525, (1989) Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects), Local complex singularities, Continued fractions, Coverings in algebraic geometry, Graph theory Perfect graphs and complex surface singularities with perfect local fundamental group | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let G be a finite group and K an algebraically closed field of characteristic \(p>0\). To any finitely generated KG-module M a homogeneous affine variety V(M) is associated via cohomology. In [J. Algebra 85, 104- 143 (1983; Zbl 0526.20040), Thm. 8.2] the author had shown that if G is an elementary abelian group of order \(p^ n\) and M an indecomposable periodic KG-module then V(M) is a line in \(K^ n\). Here he obtains the best generalization of this result and shows that if \=V(M) is the corresponding projective variety, then Theorem: If M is an indecomposable KG-module then \=V(M) is connected. connectedness of variety of indecomposable module; homogeneous affine variety; cohomology; indecomposable periodic KG-module; projective variety Carlson, J.F.: The variety of an indecomposable module is connected. Invent. Math. 77, 291--299 (1984) Homological methods in group theory, Modular representations and characters, Homogeneous spaces and generalizations, Group rings of finite groups and their modules (group-theoretic aspects), (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.), Topological properties in algebraic geometry The variety of an indecomposable module is connected | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(f : X \rightarrow Y\) be a separable finite surjective map between irreducible normal projective varieties defined over an algebraically closed field \(k\). The map \(f\) is called genuinely ramified if induced homomorphism between étale fundamental groups
\[
f_{*} : \pi_1^{\text{et}}(X) \rightarrow \pi_1^{\text{et}}(Y)
\]
is surjective. In this paper, the authors provide equivalent conditions for the map f to be genuinely ramified. By using these equivalent conditions, the authors prove that that the pullback of stable vector bundles via \(f\) are stable.
Given a stable vector bundle \(E\) on \(X\), the authors also prove that there is a vector bundle \(W\) on \(Y\) with \(f^*(W)\) isomorphic to \(E\) if and only if \(f_{*}(E)\) contains a stable subbundle \(F\) such that
\[
\frac{\deg(F)}{\text{rk}(F)} = \frac{1}{\deg(f)} \frac{\deg(E)}{\text{rk}(E)}.
\] genuinely ramified map; maximal pseudostable bundle; stable bundle Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Homological functors on modules of commutative rings (Tor, Ext, etc.), Coverings in algebraic geometry Genuinely ramified maps and stable vector bundles | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper concerns fields of definition and fields of moduli of \(G\)-Galois covers of the line over \(p\)-adic fields, and more generally over henselian discrete valuation fields. We show that the field of moduli of a \(p\)-adic cover will be a field of definition provided that the residue characteristic \(p\) does not divide \(| G|\) and that the branch points do not coalesce modulo \(p\) (or in the more general case, that the branch locus is smooth on the special fibre). Hence if \(p\) does not divide \(| G|\), then a \(G\)-Galois cover of the \(\overline \mathbb{Q}\)-line with field of moduli \(\mathbb{Q}\) will be defined over a number field contained in \(\mathbb{Q}_p\) if the branch points do not coalesce modulo \(p\). This provides an explicit global-to-local principle for \(p\)-adic covers. Galois covers; \(p\)-adic fields; characteristic \(p\); \(p\)-adic covers; fields of definition; valuation fields; global-to-local principle Pierre Dèbes and David Harbater, Fields of definition of \?-adic covers, J. Reine Angew. Math. 498 (1998), 223 -- 236. Coverings of curves, fundamental group, Local ground fields in algebraic geometry, Valued fields, Coverings in algebraic geometry, Arithmetic ground fields for curves, Henselian rings Fields of definition of \(p\)-adic covers | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For the entire collection see Zbl 0644.00012.]
The author presents certain results and examples concerning the fundamental group \(\pi_ 1(X-D)\), where X is a smooth complex space and \(D\subset X\) an analytic hypersurface.
The central example in the global case is the discriminant D(P) of an irreducible polynomial P in \({\mathbb{C}}^ n\), and in the local case the result of \textit{Lê Dũng Tráng} and \textit{K. Saito} [Ark. Math. 22, 1- 24 (1984; Zbl 0553.14006)]. local singularities; fundamental group Local complex singularities, Complex singularities, Fundamental group, presentations, free differential calculus, Coverings in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry On the fundamental group of the complement of an analytic hypersurface | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems As it is well-known, special real-valued solutions \(u(x,t)\) of the famous Korteweg-de Vries equation
\[
u_t = 6uu_x - u_{xxx}
\]
can be constructed from hyperelliptic Riemann surfaces of genus \(g\) and their Riemann theta functions (with respect to a period matrix \(B)\) by the formula
\[
u(x,t) = - 2 {\partial^2 \over \partial x^2} \ln \theta (Ux + Vt + W,B) + C,
\]
where \(U,V\) and \(W\) are certain complex \(g\)- vectors and \(C\) is a constant. These solutions appear as quasi-periodic potentials of the Schrödinger operators
\[
H(t) = - {\partial^2 \over \partial x^2} + u(x,t)
\]
which, in turn, define the KdV equation as an operator equation in Lax form. The basic question of reduction theory is the following: Under what conditions, for the chosen hyperelliptic Riemann surface \(\Gamma_g\) and the data \(U,V,W\) and \(B\), is the so- called \(g\)-gap potential \(u(x,t)\) defined above reducible to such ones described in terms of elliptic functions?
Generally, this is the problem of reducing Riemann theta functions and Abelian integrals to those in lower genus, especially in genus one, and in this setting it is a classic topic developed by Weierstrass, Poincaré, and others afterwards.
In a whole series of previous papers, published between 1982 and 1989, the authors have investigated this problem for \(g\)-gap potentials under various aspects. A rather comprehensive account of their work can be found in their (and other co-authors') recent monograph, there mainly in Chapter 7 [\textit{E. D. Belokolos}, \textit{A. I. Bobenko}, \textit{V. Z. Enol'skij}, \textit{A. R. Its} and \textit{V. B. Matveev}, Algebro-geometric approach to nonlinear integrable equations (1994; Zbl 0809.35001)].
The present paper may be regarded as a (nearly self-contained) continuation of their earlier work, with special emphasis on the reduction of \(g\)-gap potentials to elliptic ones, including the classical Lamé potential and the more recent Treibich-Verdier potentials [\textit{A. Treibich} and \textit{J.-L. Verdier}, Solitons elliptiques (1990; Zbl 0726.14024)]. The exposition is very lucid and detailed, partly a sort of systematic survey on the topic as a whole, however enhanced by several new results in reduction theory and numerous concrete examples. theta functions; elliptic functions; reduction theory; algebraic curves; Schrödinger operator; KdV-like equations; Lamé equation E. D. Belokolos and V. Z. Ènol\(^{\prime}\)skiĭ, Reduction of theta functions and elliptic finite-gap potentials, Acta Appl. Math. 36 (1994), no. 1-2, 87 -- 117. KdV equations (Korteweg-de Vries equations), Research exposition (monographs, survey articles) pertaining to partial differential equations, Special algebraic curves and curves of low genus, Theta functions and curves; Schottky problem, Coverings in algebraic geometry Reduction of theta functions and elliptic finite-gap potentials | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 subject of this paper are Calabi-Yau manifolds which are obtained by resolving the singularities of double coverings of \(\mathbb P^3\) branched along an octic surface. These surfaces are allowed to be reducible and to have isolated and non-isolated singularities of certain specified types, more general than those considered in the paper of \textit{S. Cynk} and \textit{T. Szemberg} [in: Singularities Symposium-Łojasiewicz 70, Cracow 1996, and Seminar Singularities and Geometry, Warsaw 1996, Banach Cent. Publ. 44, 93-101 (1998; Zbl 0915.14025)].
The main result of this paper states (similar to the paper of Cynk and Szemberg) the existence of a resolution of such a double solid which is Calabi--Yau. The topological Euler number is computed for such manifolds and a table is presented, which shows the possible values. Using the methods and results of the paper of Cynk and Szemberg mentioned above and by allowing isolated singularities on the branch octic, the author is able to fill most of the gaps in the table of possible Euler numbers which was presented in his joint paper with T. Szemberg (loc. cit.). octic arrangements; Calabi-Yau resolution; topological Euler number; small resolution of double solid; isolated singularities S. Cynk, Double coverings of octic arrangements with isolated singularities , Adv. Theor. Math. Phys. 3 (1999), 217-225. Calabi-Yau manifolds (algebro-geometric aspects), Coverings in algebraic geometry, Topological properties in algebraic geometry, Singularities of surfaces or higher-dimensional varieties Double coverings of octic arrangements with isolated 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 For X a scheme of finite type over some base ring R, the group \(\pi_ 1^{ab}(X)\) classifies the abelian unramified coverings of X. As in the class field theory of number fields, the primary goal for schemes is to obtain a description of \(\pi_ 1^{ab}(X)\) solely in terms of X itself, via some kind of ''reciprocity'' map. When R is a finite field, Lang in 1956, defined a reciprocity map from \(Z_ 0(X)\), the free abelian group on the closed points, and showed that for normal X the map had a dense image. Subsequent work led to the study of \(CH_ 0(X)\), the 0- dimensional Chow group of X, which for smooth projective varieties over a field is \(Z_ 0(X)\) modulo rational equivalence, and which in general can be defined using algebraic K-theory. It turns out that for X proper over \({\mathbb{Z}}\), there is a reciprocity map from \(CH_ 0(X)\) to the quotient \({\tilde \pi}{}_ 1^{ab}(X)\) of \(\pi_ 1^{ab}(X)\) which classifies the unramified abelian covers that split completely over any real valued point of X. (For schemes over finite fields, this quotient is just \(\pi_ 1^{ab}(X).)\)
The main part of the current paper studies arithmetical surfaces, that is, proper smooth surfaces over finite fields, or connected regular surfaces which are proper and flat over \({\mathbb{Z}}\). For surfaces over finite fields, the degree map exhibits \(CH_ 0(X)\) as an extension of \({\mathbb{Z}}\) by \(CH_ 0(X)^ 0\), the subgroup of degree 0 cycle classes. Correspondingly, there is a map from \(\pi_ 1^{ab}(X)\) onto \({\hat {\mathbb{Z}}}\), the Galois group of the algebraic closure of the ground field. Call \(\pi_ 1^{ab}(X)^ 0\) the kernel of this map. In 1981 \textit{N. Katz} and \textit{S. Lang} [Enseign. Math., II. Sér. 27, 285-319 (1981; Zbl 0495.14011)] showed that \(\pi_ 1^{ab}(X)^ 0\) is finite, while work of Bloch and Milne established the finiteness of \(CH_ 0(X)^ 0\). Milne further established the p-primary injectivity of the reciprocity map \(CH_ 0(X)\to \pi_ 1^{ab}(X),\) assuming a certain condition on X. In this paper the authors show that for any smooth projective geometrically irreducible scheme over a finite field, the reciprocity map is always injective, and induces an isomorphism of the finite groups \(CH_ 0(X)^ 0\) and \(\pi_ 1^{ab}(X)^ 0\) so that, roughly speaking, \(\pi_ 1^{ab}(X)\) is obtained from \(CH_ 0(X)\) by replacing \({\mathbb{Z}}\) by \({\hat {\mathbb{Z}}}\). According to the authors, they had originally obtained these results for surfaces, and an elegant induction argument of Colliot-Thélène established the general case.
For X a regular connected surface proper and flat over \({\mathbb{Z}}\), the main result established in this paper is that the reciprocity map \(CH_ 0(X)\to {\tilde \pi}_ 1^{ab}(X)\) is an isomorphism of finite abelian groups. Earlier, Bloch had proved this result in the case X is smooth over the ring of integers in a number field. To establish the unramified class field theory of arithmetical surfaces X in general, the authors concentrate on a fibration \(X\to C\), where C is a one dimensional scheme, and where the fibration may have singular fibres. In essence, the authors are able to improve on the earlier results of Bloch and Milne because they have a technique for treating the class field theory of a singular curve over a local field. Abelian fundamental group; abelian unramified coverings; reciprocity; 0- dimensional Chow group; arithmetical surfaces; unramified class field theory of arithmetical surfaces Kazuya Kato & Shuji Saito, ``Unramified class field theory of arithmetical surfaces'', Ann. Math.118 (1983) no. 2, p. 241-275 Arithmetic ground fields for surfaces or higher-dimensional varieties, Coverings in algebraic geometry, Parametrization (Chow and Hilbert schemes), Coverings of curves, fundamental group, Applications of methods of algebraic \(K\)-theory in algebraic geometry Unramified class field theory of arithmetical 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 \(S\) be an algebraic surface over an algebraically closed field \(k\), and \(P\) a point on a reduced curve \(D\) on \(S\). Let \(\mu:S'\to S\) be an embedded resolution of singularities for the germ at \(P\) of the pair \((S,D)\). Denote by \(K_S\) and \(K_{S'}\) the canonical divisors of \(S\) and \(S'\). If the numerical pull-back \(\mu^*(K_S+D)\) is written in the form \(\mu^*(K_S+D) = K_{S'} +D'+E\), \(D'\) the strict transform of \(D\), then the pair \((S,D)\) is said to be log terminal (respectively canonical) at \(P\) if and only if all coefficients of \(E\) are strictly less than 1 (respectively non-positive). Assume that \((S,D)\) is log terminal at \(P\) and let \(r\) be the minimal positive integer such that \({\mathcal O}_S (r(K_S+D))\) is Cartier. For each nowhere vanishing section \(\theta\) of \({\mathcal O}_S (r(K_S+D))\) there is an index 1 cover associated to \(\theta\), \(\pi:T\to S\). The paper proves that if \(\text{char} k\neq 2,3\) and \(\theta\) is general enough, then the pair \((T,\pi^*D)\) is canonical at \(\pi^{-1}(P)\). Counterexamples for the cases \(\text{char} k=2\) or 3 are given. curve on surface; embedded resolution of singularities Y. Kawamata, Index one covers of log terminal surface singularities, J. Alg. Geom. 8 (1999), 519--527. Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects), Coverings in algebraic geometry Index 1 covers of log terminal surface singularities | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We consider the class of singular double coverings \(X \to \mathbb P^3\) ramified in the degeneration locus \(D\) of a family of 2-dimensional quadrics. These are precisely the quartic double solids constructed by Artin and Mumford as examples of unirational but nonrational conic bundles. With such a quartic surface \(D\), one can associate an Enriques surface \(S\) which is the factor of the blowup of \(D\) by a natural involution acting without fixed points (such Enriques surfaces are known as nodal Enriques surfaces or Reye congruences). We show that the nontrivial part of the derived category of coherent sheaves on this Enriques surface \(S\) is equivalent to the nontrivial part of the derived category of a minimal resolution of singularities of \(X\). Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], \(K3\) surfaces and Enriques surfaces, Coverings in algebraic geometry, \(3\)-folds On nodal Enriques surfaces and quartic 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 The paper is devoted to the Jacobian conjecture: A polynomial mapping \(f:\mathbb{C}^2 \to\mathbb{C}^2\) with a constant nonzero Jacobian is polynomially invertible. The main result of the paper is as follows. There is no four-sheeted polynomial mapping whose Jacobian is a non-zero constant such that after the resolution of the indeterminacy points at infinity there is only one added curve whose image is not a point and does not belong to infinity. ramified covering; degree of a mapping; Jacobian conjecture; polynomial mapping A. V. Domrina and S. Yu. Orevkov, ''On Four-Sheeted Polynomial Mappings of \(\mathbb{C}\)2. I: The Case of an Irreducible Ramification Curve,'' Mat. Zametki 64(6), 847--862 (1998) [Math. Notes 64, 732--744 (1998)]. Jacobian problem, Ramification problems in algebraic geometry, Coverings in algebraic geometry On four-sheeted polynomial mappings of \(\mathbb{C}^2\). I: The case of an irreducible ramification 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 Examples of simply connected algebraic surfaces of general type with \(\tau =c_ 1^ 2-2c_ 2\geq 0\) are constructed, which disproves two previous conjectures. These surfaces are Galois coverings of \(X={\mathbb{P}}_ 1({\mathbb{C}})\times {\mathbb{P}}_ 1({\mathbb{C}})\) with respect to \(f=\pi \circ i\), where \(\pi: {\mathbb{P}}_ N({\mathbb{C}})\to {\mathbb{P}}_ 2({\mathbb{C}})\) is a generic projection, and \(i: X\to {\mathbb{P}}_ N({\mathbb{C}})\) is the embedding given by a divisor of type (a,b) with \(a\geq 3\), \(b\geq 2\), and (a,b) relatively prime. The simple connectedness is proved by a very delicate argument involving in particular a degeneration of f, and the braid group of \({\mathbb{C}}\) with respect to a finite subset. Chern numbers; simply connected algebraic surfaces of general type; embedding Boris Moishezon and Mina Teicher, Existence of simply connected algebraic surfaces of general type with positive and zero indices, Proc. Nat. Acad. Sci. U.S.A. 83 (1986), no. 18, 6665 -- 6666. Special surfaces, Coverings in algebraic geometry, Characteristic classes and numbers in differential topology Existence of simply connected algebraic surfaces of general type with positive and zero indices. (Galois covering/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 author considers a 3-sheeted covering of \({\mathbb{P}}^ 2\), branched along the union \(S_ 1\cup S_ 2\) of two non singular curves with normal crossings, and its minimal model \(\tilde X.\) In a previous paper [''Covering spaces of \(P^ 2\) branched along two non singular curves with normal crossings, Comment. Math. Univ. St. Pauli 33, 163-190 (1984)] the author has shown that X is a normal surface whose singularities are either rational double points or rational triple points. Moreover the irregularity of \(\tilde X\) vanishes [see \textit{S. Kawai}, Comment. Math. Univ. St. Pauli 30, 87-103 (1981; Zbl 0496.14012)]. - In this paper \(\tilde X\) is classified according to the degrees of \(S_ 1\) and \(S_ 2\). The proof uses in an essential way the results of the author's paper cited above. branch locus; 3-sheeted covering; minimal model Yamamoto, S.: The relatively minimal models of certain 3-sheeted branched covering spacesof P2. Indag. math. 46, 453-459 (1984) Minimal model program (Mori theory, extremal rays), Families, moduli, classification: algebraic theory, Coverings in algebraic geometry The relatively minimal models of certain 3-sheeted branched covering spaces of \(P^ 2\) | 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 variety over an algebraically closed field \( k\) of positive characteristic. We define and study a general notion of regular singularities for stratified bundles (i.e. \(\mathcal {O}_X\)-coherent \( \mathfrak{D}_{X/k}\)-modules) on \( X\) without relying on resolution of singularities. The main result is that the category of regular singular stratified bundles with finite monodromy is equivalent to the category of continuous representations of the tame fundamental group on finite dimensional \( k\)-vector spaces. As a corollary we obtain that a stratified bundle with finite monodromy is regular singular if and only if it is regular singular along all curves mapping to \( X\). Singularities in algebraic geometry, Coverings in algebraic geometry, Ramification problems in algebraic geometry, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials Regular singular stratified bundles and tame ramification | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 authors consider the following situation: let \(X\) be a smooth varieties of general type of dimension \(m\geq 3\) for which the canonical map induces a triple cover onto \(Y\), where \(Y\) is either a projective bundle over \(\mathbb P^1\) or a projective space or a quadric hypersurface, embedded by a complete linear series (except \(\mathbb Q_3\) embedded in \(\mathbb P^4\)).
They prove that the general member of a deformation of the canonical morphism of \(X\) with base a smooth curve is again canonical and induces a triple cover.
Giving explicit examples, they also find components of the moduli of threefolds \(X\) of general type with \( K^3_X= 3p_g- 9\), \(K^3_X \neq 6\), whose general members correspond to canonical triple covers.
In the paper under review the authors also prove that for an algebraic surface of general type, there are no canonical triple covers of rational scrolls \(Y\) or a quadric surface \(\mathbb Q_2\) or non-linearly embedded \(\mathbb P^2\), whether they are embedded as a variety of minimal degree or not.
They give special emphasis on those \(Y\) which are varieties of minimal degree because this case does not occur in either curves or surfaces. Indeed, the degree of the canonical morphism for curves is bounded by 2, and there are no odd degree canonical covers of smooth surfaces of minimal degree other than linear \(\mathbb P^2\) as shown in [\textit{F. J. Gallego} and \textit{B. P. Purnaprajna}, in: A tribute to C. S. Seshadri. A collection of articles on geometry and representation theory. Basel: Birkhäuser. 241--270 (2003; Zbl 1067.14015)]. varieties of general type; deformations; covering space Coverings in algebraic geometry, Families, moduli, classification: algebraic theory, Surfaces of general type Deformations of canonical triple covers | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let X, Y be algebraic varieties. A finite flat morphism \(f: X\to Y\) of degree 3 is called a triple cover. The author obtains a criterion for X to be a triple section of an ample line bundle over Y. This criterion is effective in dimension \(\geq 4.\)
As a corollary, he shows that if X is smooth (\(\dim(X)\geq 4)\) and if Y is a complete intersection in \(P^ N\), then X is a triple section of a line bundle \({\mathcal O}_ Y(k)\) for some positive integer k. He also gives an example (of dimension 3) of a triple cover which is not of triple section type.
For related topics, we refer to the following articles: \textit{R. Miranda} [Am. J. Math. 107, 1123-1158 (1985; Zbl 0611.14011)] and \textit{R. Lazarsfeld} [Math. Ann. 249, 153-162 (1980; Zbl 0434.32013)]. triple cover; triple section of an ample line bundle Takao Fujita, Triple covers by smooth manifolds, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 35 (1988), no. 1, 169 -- 175. Coverings in algebraic geometry Triple covers by smooth 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 Given a Riemann surface \(X\) and an orientation-preserving involution \(\sigma\) of \(X\), one considers the double cover \(\pi: \tilde{X} \rightarrow X\) with a lifted involution \(\tilde{\sigma}\) of \(\tilde{X}\). The surface \(\tilde{X}/ \tilde{\sigma}\) is called a twist of the surface \((X, \sigma)\). This paper is devoted to study torus twists, that is to say, twists \(\tilde{X} / \tilde{\sigma}\) of genus \(1\).
The surface \(X\) is presented by edge-identified polygons in the complex plane. If all identifications are translations, the surface is called a translation surface, and otherwise a half-translation surface. If one selects a one-form on \(X\), it twists into a quadratic differential. When a quadratic differential is not the square of a one-form, it is called strict.
The main result of the paper is that torus twists of a hyperelliptic translation surface are strict quadratic differentials. The work ends by listing all covers and quotients of surfaces of genus \(2\). The moduli space of translation surfaces of genus 2 is stratified in two strata, corresponding to surfaces with a single cone point of order \(2\), and to surfaces with two cone points of order 1. Surfaces in the first stratum are represented by a Swiss cross, and those in the second one by a regular decagon. All the possible covers in either case are listed in an Appendix. translation surfaces; quadratic differential; hyperelliptic surfaces; Riemann surfaces; branched covers; torus-twists Coverings in algebraic geometry, Differentials on Riemann surfaces Hyperelliptic translation surfaces and folded tori | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 zero, let \(X\) be a geometrically integral \(k\)-variety of dimension \(n\) and let \(K\) be its field of functions. We prove that, given an element \(\alpha\in H^m(K,\mu^{\otimes m}_r)\), there exist \(n^2\) functions \(\{f_i\}_{,i=1,\ldots,n^2}\) such that \(\alpha\) becomes unramified in \(L=K(f^{1/r}_1,\ldots,f^{1/r}_{n^2})\). Brauer group; Galois cohomology; unramified cohomology; period-index problem Brauer groups of schemes, Valuation rings, Coverings in algebraic geometry A bound to kill the ramification over function fields | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper is a continuation of two others by the authors [in The Grothendieck Festschrift, Vol. III, Prog. Math. 88, 437-480 (1990; Zbl 0726.14024); ``Variétés de Kritchever de solitons elliptiques'', in Proc. Indo-French Conf. Geometry, Bombay 1989)], which were devoted to the study of the structure of the solutions of the KdV equation called elliptic solitons. To a hyperelliptic curve \(C\) of genus \(g\), equipped with a double point \(p\) of the hyperelliptic involution and a divisor \(D\) of degree \(g-1\), is associated a meromorphic function \(u\) on the complex plane, which is a so-called potential with a finite number of instability zones, and from which the triple \((C,p,D)\) can be recovered (the Schrödinger equation with potential \(u\) on the plane has a family of eigenfunctions parametrized by \(C\) with poles at \(D)\). The function associated with the divisor \((g-1)p\) is called the source potential. This paper is concerned with the case where \(C\) is a ``tangential at \(p\)'' covering of an elliptic curve \(E\). Here ``tangential at \(p\)'' means that the images of \(C\) and \(E\) in Jac\(C\) by the morphisms defined by \(p\) are tangent at the origin. The potentials associated to such a covering ``live'' on \(E\) and are combinations of Weierstrass functions there. To each such tangential covering of an elliptic curve by a hyperelliptic one is associated naturally a four-tuple of (intersection) numbers. And some of these four-tuples determine (for each \(E)\) a unique such tangential hyperelliptic covering. These are the so-called exceptional coverings.
The paper studies the above correspondence between ``exceptional'' four- tuples and combinations of Weierstrass functions, characterizing in particular those combinations which are reached by source potentials. The formulas describing this picture are full of triangular numbers, whence the title. KdV equation; elliptic solitons; tangential covering of an elliptic curve; hyperelliptic covering; Weierstrass functions Treibich, A. and Verdier, J.-L., Revétements exceptionnels et sommes de 4 nombres triangulaires, Duke Math. J., 1992, vol. 68, no. 2, pp. 217--236. Coverings in algebraic geometry, KdV equations (Korteweg-de Vries equations), Relations of PDEs on manifolds with hyperfunctions, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials Exceptional coverings and sums of 4 triangular numbers | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper uses the theory of gerbes developed by Giraud. We first outline some of the definitions needed. Let \(K\) be a field and \(B\) be a regular projective geometrically irreducible \(K\)-variety. A gerbe \({\mathcal G}(f)\) is assigned to each finite branched cover \(f: X\to B\) defined over the separable closure \(K_{s}\) of \(K\). Let \(G(K)\) be the group of this cover \(f\) and \(N\) the normalizer of \(G(K)\) in \(S_{d}\), where \(d\) is the degree of \(f\). If the usual map \(\lambda : G(K)\to N/CG(K)\), where \(CG(K)\) is the centralizer of \(G(K)\) in \(N\), lifts to a map \(\Lambda : G(K)\to N/G(K)\), then one says that the (\(\lambda\)/Lift)-condition holds. Let \(\phi :\Pi_{K_{s}}\to G\) be the morphism associated to \(f\). Here \(\Pi_{K_{s}}\) is the arithmetic fundamental group. The field \(K\) is the field of moduli of \(f\) if the ramification locus is \(G(K)\)-invariant and, for each \(u\in\Pi_{K}\), there is a \(\psi_{u}\in N\) such that \(\phi (x^{u}) = \phi(x)^{\psi_{u}}\) for all \(x\in \Pi_{K_{s}}\). Suppose now that the (\(\lambda\)/Lift)-condition holds and \(K\) is the field of moduli of \(f\). The main theorem of the paper, given the context by the above statements, shows that the following are equivalent:
(i) the gerbe \({\mathcal G}(f)\) is neutral,
(ii) the cover \(f\) can be defined over \(K\) with \(\Lambda\) as constant extension map,
(iii) the class \([{\mathcal G}(f)_{\Lambda}]\), viewed as an element of \(H^{2}(K,Z(G))\), is trivial. gerbes; covers; Galois cohomology; variety; moduli field Dèbes, P.; Douai, J. -C.: Gerbes and covers. Comm. algebra 27, 577-594 (1999) Nonabelian homological algebra (category-theoretic aspects), Coverings in algebraic geometry Gerbes and 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 paper we study the Hodge numbers of a branched double covering of a smooth, complete algebraic threefold. The involution on the double covering gives a splitting of the Hodge groups into symmetric and skew-symmetric parts. Since the symmetric part is naturally isomorphic to the corresponding Hodge group of the base we study only the skew-symmetric parts and prove that in many cases it can be computed explicitly. threefold; Hodge groups; double covering Cynk, S.: Cohomologies of a double covering of a non-singular algebraic 3-fold, Math. Z. 240, 731-743 (2002) \(3\)-folds, Coverings in algebraic geometry, Transcendental methods, Hodge theory (algebro-geometric aspects) Cohomologies of a double covering of a non-singular algebraic 3-fold | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 famous Severi inequality states that a minimal projective surface of general type \(S\) with maximal Albanese dimension satisfies \(K_S^2\geq 4\chi (\mathcal O_S)\). This inequality has a long history (see the review of [\textit{R. Pardini}, Invent. Math. 159, No. 3, 669--672 (2005; Zbl 1082.14041)] and it was finally proven by \textit{R. Pardini} [Invent. Math. 159, No. 3, 669--672 (2005; Zbl 1082.14041)] in characteristic zero. Later \textit{X. Yuan} and \textit{T. Zhang} [Adv. Math. 259, 89--115 (2014; Zbl 1297.14045)] proved that the result holds in every characteristic.
\textit{M. Á. Barja} et al. [J. Math. Pures Appl. (9) 105, No. 5, 734--743 (2016; Zbl 1346.14102)] and \textit{X. Lu} and \textit{K. Zuo} [Int. Math. Res. Not. 2019, No. 1, 231--248 (2019; Zbl 1430.14079)] showed (with completely different proofs) that in characteristic zero a minimal smooth projective surface of maximal Albanese dimension satisfies \(K_S^2= 4\chi (\mathcal O_S)\) if and only if \(q(X)=2\) and the canonical model of \(X\) is a double cover of \(\mathrm{Alb}(X)\) branched on an ample divisor with at most negligible singularities.
The present paper extends to every characteristic of the ground field this result. The proof uses both ideas of Pardini and of Lu and Zuo and the authors need to establish several difficult technical results with special focus in characteristic 2 where as usual many difficulties arise. algebraic surface of general type; Severi inequality; Severi line; double covers; irregular varieties; maximal Albanese dimension Surfaces of general type, Special surfaces, Coverings in algebraic geometry Surfaces on the Severi line 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 \(U={\mathbb{P}}_ k^ 1-\{0,1,\infty\}\) and assume \(k\) is a number field of finite degree over \({\mathbb{Q}}\). For a fixed odd prime \(\ell\), let \(\pi'_ 1(U)\) denote the profinite fundamental group of \(U\) divided by the kernel of the geometric fundamental group to its pro-\(\ell\) completion \(\pi_ 1(U\otimes \bar k)^{pro-\ell}\). In this paper, it is shown that every group automorphism of \(\pi'_ 1(U)\) respecting the canonical augmentation to \(Gal(\bar k/k)\) must come from a \(k\)-automorphism of \(U\) up to inner automorphisms of \(\pi_ 1(U\otimes \bar k)^{pro-\ell}\). galois automorphisms; fundamental group of the projective line minus three points Hiroaki Nakamura, On Galois automorphisms of the fundamental group of the projective line minus three points, Math. Z. 206 (1991), no. 4, 617 -- 622. Coverings of curves, fundamental group, Galois theory, Global ground fields in algebraic geometry, Coverings in algebraic geometry On galois automorphisms of the fundamental group of the projective line minus three points | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors (a) explain a relationship between rank-2 vector bundles (and Tschirnhausen modules) and algebraic-geometric codes, (b) give a detailed explicit example using the Klein quartic, (c) describe how algebraic-geometric codes can be formulated using adeles and pseudo-differentials in the sense of A. Weil. A very interesting and well-written paper. Coles, D.; Previato, E.: Goppa codes and tschirnhausen modules, Series on coding theory and cryptography (Advances in coding theory and cryptography) 3, 81-100 (2007) Geometric methods (including applications of algebraic geometry) applied to coding theory, Coverings in algebraic geometry, Ramification problems in algebraic geometry Goppa codes and Tschirnhausen modules | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 classify the analytic coverings of a weighted projective space \(P=P(Q)\) whose branching sets are unions of the form \(P_{\sin g}\cup H\), where \(P_{\sin g}\) denotes the singular part of P and H is a normal hypersurface in P. It turns out that all such coverings are cyclic and their total spaces are hypersurfaces in suitable weighted projective spaces. The computation of the fundamental groups involved is made using the simply-connectedness of some Milnor fibers. - Similar results hold for coverings of quasi-smooth weighted complete intersections and will be given elsewhere. analytic coverings; branching sets; fundamental groups; Milnor fibers A. Dimca and S. Dimiev. On analytic coverings of weighted projective spaces , Bull. London Math. Soc. 17 (1985) pp. 234-238. 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 We generalize an argument of a previous paper [\textit{M. Manetti}, Topology 36, No. 3, 745-764 (1997; Zbl 0889.14014)] for proving a result about automorphisms of generic simple cyclic covers of smooth algebraic varieties. A finite map \(\pi:S\to X\) is called a simple cyclic cover if there exists an invertible sheaf \({\mathcal L}\) on \(X\) such that \(\pi_*{\mathcal O}_S= \bigoplus_{i=0}^{n-1} {\mathcal L}^{-i}\) [cf. \textit{W. Barth}, \textit{C. Peters} and \textit{A. Van de Ven}, ``Compact complex surfaces'' (1984; Zbl 0718.14023); I. 17]. Here we prove under some ``mild'' assumptions on the triple \(X,{\mathcal L},n\) that for the generic cyclic cover \(S\) the group \(\Aut(S)\) of biregular automorphisms equals the group \(\mu_n\) of automorphisms of the branched cover \(\pi\). automorphisms of generic simple cyclic covers; biregular automorphisms Coverings in algebraic geometry, Automorphisms of curves Automorphisms of generic 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 very short paper (translated from Russian) studies the situation of a two-sheeted covering \(X\) of a non-singular real algebraic surface \(Y\) ramified along a non-singular real algebraic curve \(Z\). Several results are given, precising conditions for \(X\) to be an \(M\)-surface in terms of the topology of \(Y\) and \(Z\). The last two propositions consist of congruences modulo 8 relating some Euler characteristic with the number of self-intersections of \(Z\) in \(Y\) and based of the Gudkov-Krakhnov-Kharlamov congruences. The paper is too short to give fully detailed proofs.
Notations are not always very clear. \(M\)-surface; two-sheeted covering; real algebraic surface; Euler characteristic Topology of real algebraic varieties, Coverings in algebraic geometry, Arithmetic ground fields for surfaces or higher-dimensional varieties Two-sheeted coverings of real algebraic surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The main result of the paper is the following. Let \(K\) be field such that all Sylow subgroups of its absolute Galois group are infinite. Let \(X\) be a smooth and irreducible variety over \(K\) with function field \(F\). Let \(E/F\) be a finite separable extension and let \(Y\) be the normalization of \(X\) in \(E/F\). If there exists a closed point of \(X\) which does not split completely in \(Y\to X\), then the set of these points in Zariski dense in \(X\). This result generalizes Theorem 7.1 (1) of \textit{S. Saito} [J. Number Theory 21, 44--80 (1985; Zbl 0599.14008)], where \(K\) is a local field and \(\dim X = 1\). The hypotheses of the theorem are satisfied, in particular, when \(K\) is an Hilbertian field. geometric class field theory; coverings G. Wiesend, Covers of varieties with completely split points, Israel J. Math., in press Geometric class field theory, Coverings in algebraic geometry, Class field theory Covers of varieties with completely split points | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A real algebraic curve in projective \(n\)-space is called unramified if the degree of any divisor cut out by a real hyperplane overcomes the number of points in the divisor by at most \(n-1\) (here a pair of imaginary conjugate points is thought of as one point). Over the complex field, the unramified curves are known to be rational normal. Real plane unramified curves are conics [\textit{J. Huisman}, An unramified real plane curve is a conic, Matematiche 55, 459--467 (2000); \url{http:// fraise.univ-brest.fr/~huisman/recherche/publications.html}].
In the present note, the author shows that a real plane unramified nonspecial curve in an even-dimensional space is rational normal, and in an odd-dimensional space he shows that it is an \(M\)-curve consisting of pseudo-lines, provided that the curve under consideration has many real branches and, possibly, few ovals. The proof is based on the study of coverings of the real part of the Picard variety by unions of real branches of a curve. rational normal curve; \(M\)-curve; unramified curve; Picard variety Real algebraic sets, Plane and space curves, Coverings in algebraic geometry Unramified nonspecial real space curves having many real branches and few ovals. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Over the past thirty years, cohomological support varieties have seen extensive study and played a key role in the development of modular representation theory. The notion was first introduced in the context of finite groups. Let \(G\) be a finite group and \(k\) be an algebraically closed field of prime characteristic \(p\). The cohomological variety \(|G|\) of \(G\) is defined to be the maximal ideal spectrum of the even-dimensional portion of the cohomology ring \(H^*(G,k)\). For a \(kG\)-module \(M\), the cohomological support variety \(|G| _M\) is the closed conical subset of those maximal ideals which contain the annihilator of \(\text{Ext}_G^*(M,M)\).
These varieties have a number of nice properties involving sums, tensor products, complexity, and projectivity of modules. One of the most interesting properties is a cohomology-free description of these varieties. For an arbitrary group, a variety can be described in terms of those for elementary Abelian \(p\)-subgroups, and then, for an elementary Abelian \(p\)-group, these varieties can be identified with certain ``rank varieties'' which are defined purely representation-theoretically. More recently, the notion of support varieties has been extended to arbitrary (i.e., possibly infinite dimensional) modules as well. The work for finite groups spawned great interest in other areas as the idea has been extended to Lie algebras, infinitesimal group schemes, and certain general types of Hopf algebras where analogous properties and non-cohomological descriptions have been shown to hold.
The goal of the authors is to construct a framework for obtaining non-cohomological identifications of support varieties which generalize both the finite group and infinitesimal group scheme settings. Let \(G\) be a finite group scheme over \(k\). That is, an affine group scheme whose coordinate algebra is finite dimensional. An ordinary finite group as well as the restricted enveloping algebra of a restricted Lie algebra can be considered in this context. Indeed, \(G\) can be identified as a semi-direct product of an infinitesimal group scheme (one whose coordinate algebra is local) and an ordinary finite group. The authors introduce the set \(P(G)\) of Abelian \(p\)-points of \(G\) which is a set of algebra homomorphisms from the group algebra \(k\mathbb{Z}_p\) to the ``group algebra'' \(kG\) modulo an equivalence relation. Here \(kG\) denotes the \(k\)-linear dual of the coordinate algebra of \(G\). Given a finite dimensional \(kG\)-module \(M\), they also associate a subset \(P(G)_M\subset P(G)\). The set \(P(G)\) is given the Noetherian topology with \(\{P(G)_M\}\) being the closed subsets. The main (and beautiful) result of the paper is the construction of a homeomorphism from \(P(G)\) to a ``projectivization'' of \(|G|\) which restricts to a homeomorphism from \(P(G)_M\) to the projectivization of \(|G|_M\). They also show that the \(P(G)_M\) (and hence the projectivization of \(|G|_M\)) satisfies all of the ``usual'' properties.
It should be noted that in more recent work [\(\Pi\)-supports for modules for finite group schemes over a field, (preprint)], the authors refine these ideas even further to allow one to work over an arbitrary field of prime characteristic and with arbitrary modules. finite group schemes; cohomology rings; cohomological support varieties; projectivizations; elementary Abelian \(p\)-groups E. M. Friedlander and J. Pevtsova, Representation-theoretic support spaces for finite group schemes, Amer. J. Math. 127 (2005), no. 2, 379-420. Modular representations and characters, Group schemes, Representation theory for linear algebraic groups, Cohomology theory for linear algebraic groups, Cohomology of groups Representation-theoretic support spaces for finite group schemes. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We use the gluing construction introduced by Jia Huang to explore the rings of invariants for a range of modular representations. We construct generating sets for the rings of invariants of the maximal parabolic subgroups of a finite symplectic group and their common Sylow \(p\)-subgroup. We also investigate the invariants of singular finite classical groups. We introduce parabolic gluing and use this construction to compute the invariant field of fractions for a range of representations. We use thin gluing to construct faithful representations of semidirect products and to determine the minimum dimension of a faithful representation of the semidirect product of a cyclic \(p\)-group acting on an elementary abelian \(p\)-group. modular invariants; gluing groups Actions of groups on commutative rings; invariant theory, Modular representations and characters, Geometric invariant theory, Linear algebraic groups over finite fields Modular invariants of finite gluing 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 the third part of my thesis [at the Mathematics Institute of Fudan Univ., P.R. China (1987)]. Roughly speaking, the topics discussed in my thesis are so-called surfaces of general type with \(\chi\) (\({\mathcal O}_ S)=1\) and fibrations of genus two. For this kind of surfaces, we may find a double covering over a ruled surface, say P. Therefore, one can classify them (as we did in part I and II) by giving the invariants of P and classify the branch locus of the corresponding double covering.
Once we know this basic method, we may use certain results of \textit{Xiao}, \textit{Persson} and \textit{Beauville} for double covering. For example, one can easily know that basically, we only need to classify the surfaces with \(q(S)=p_ g(S)=0, 1\) and 2. For example for surfaces with \(q=0\) in our case, P should be a product of two projective lines. In this situation, using a result of \textit{Horikawa}, one can show that there are only a few choices for the branch curves. - Now we may study the singular points on the branch locus. Suppose its non-fiber part are the summation of curves \(C_ j\). We at first determine the types of \(C_ j's\). Then we give the singularities on \(C_ j's\). Finally, we also show the situation for the intersection among \(C_ i\) and \(C_ j\) for \(i\neq j\). In this sense, we classify all the surfaces of general type with \(\chi\) (\({\mathcal O}_ S)=1\) and fibration of genus 2. As a by-product, we also can give the 2-torsion of the algebraic fundamental group of our surface.
Having classified those surfaces, we try to construct them. We pay our most attention to the surfaces with \(p_ g=0\), as they are very interesting. - By our classification, now the self-intersection of the classical sheaf is 1 or 2. Such surfaces do exist by the constructions of \textit{Oort-Peters} and \textit{Xiao} respectively. Now in this part III (under review), we construct another one. We give the exact bipolynomials, which define the curves in the branch locus. By the way, in a paper of \textit{Reid}, there is also constructed another example by our classification. Now by additional work of mine, there are only four types which we do not know if they exist. For this, please see a forthcoming book written by \textit{Xiao} about surfaces with fibrations. classification of surfaces of general type; construction of surfaces of general type; double covering; branch locus Families, moduli, classification: algebraic theory, Coverings in algebraic geometry, Surfaces of general type, Moduli, classification: analytic theory; relations with modular forms, Families, fibrations in algebraic geometry Surfaces of general type with \(\chi({\mathcal O}_ S)=1\) and fibrations of genus two. III | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(S\) be a smooth simply-connected complex projective surface, and let \(A\) be a finite abelian group. We define invariants \(T_A\), \(F_A\) and \(\sigma_A\) for curves \(B\) on \(S\) by means of étale Galois coverings of the complement of \(B\) with the Galois group \(A\), and show that they are useful in finding examples of Zariski pairs of curves on \(S\). We also investigate the relation between these invariants and the fundamental group of the complement of \(B\). Zariski pair; Galois covering; lattice; discriminant group; fundamental group Plane and space curves, Coverings in algebraic geometry Topology of curves on a surface and lattice-theoretic invariants of coverings of the 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 an algebraically closed field of characteristic \(p>0\). In the paper ``Coverings of algebraic curves'', Am. J. Math. 79, 825-856 (1957; Zbl 0087.036), \textit{S. Abhyankar} proved that the automorphism group of an étale Galois cover of the affine line over K is generated by its p- Sylow subgroups and conjectures that the converse should be true, namely that if a finite group G is generated by its p-Sylow subgroups then there exists an étale Galois cover of the affine line over K with Galois group G.
The author proves a weaker form of this conjecture, i.e. the following theorem: Let \(\tilde G\) be a finite group that is generated by its p- Sylow subgroups, let N be a normal subgroup of \(\tilde G\) and let G be the quotient group G/N. Assume that N is solvable and that G is the Galois group of an étale cover of the affine line. - Under these assumptions there exists an étale Galois cover of the affine line over K with Galois group \(\tilde G.\)
In particular, this result implies Abhyankar's conjecture for a solvable group. existence of étale Galois cover Serre, J.-P., \textit{construction de revêtements étales de la droite affine en caractéristique \textit{p}}, C. R. Acad. Sci. Paris Sér. I Math., 311, 341-346, (1990) Coverings of curves, fundamental group, Local ground fields in algebraic geometry, Coverings in algebraic geometry Construction de revêtements étales de la droite affine en caractéristique p. (Construction of étale coverings of the affine line 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 We study the global monodromy on the middle homology group of the universal coverings of the complements to non-singular affine hypersurfaces which intersect the hyperplane at infinity transversely. This monodromy can be regarded as a deformation of the monodromy on the middle homology group of the affine hypersurfaces. We show that this representation becomes irreducible when the deformation parameter is generic. Picard-Lefschetz theory; universal coverings; complements to non-singular affine hypersurfaces; deformation of the monodromy Shimada, I., Picard-Lefschetz theory for the universal covering of complements to affine hypersurfaces, Publ RIMS, Kyoto Univ., 32 (1996), 835-928. Structure of families (Picard-Lefschetz, monodromy, etc.), Coverings in algebraic geometry, Milnor fibration; relations with knot theory, Hypersurfaces and algebraic geometry Picard-Lefschetz theory for the universal coverings of complements to affine hypersurfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The famous Jacobian conjecture states that a polynomial map \(f: {\mathbb C}^n\to {\mathbb C}^n\) with non-zero constant Jacobian is one-to-one. This conjecture, which is still completely open, is the motivation for the main result of this paper, which is the following:
Theorem. Let \(f: {\mathbb C}^2\to {\mathbb C}^2\) be a polynomial map with non-zero constant Jacobian. If there exists a curve \(\Gamma\) homeomorphic to \({\mathbb C}\) such that the restricted map \(f: {\mathbb C}^2-\Gamma\to {\mathbb C}^2-\Gamma\) is a topological covering, then \(f\) is bijective. Jacobian conjecture; exceptional value set; polynomial automorphism Coverings in algebraic geometry, Jacobian problem A remark on Vitushkin's 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 It is proved that every algebraic Kummer surface can be an unramified double cover of some Enriques surface. The main tools used in the proof are Nikulin's results on lattices, the Torelli theorem for \(K3\)-surfaces, and the surjectivity of the period map for marked \(K3\)-surfaces and Enriques surfaces. cover of Enriques surface; algebraic Kummer surface; Torelli; theorem; period map Jong Hae Keum, Every algebraic Kummer surface is the \?3-cover of an Enriques surface, Nagoya Math. J. 118 (1990), 99 -- 110. \(K3\) surfaces and Enriques surfaces, Coverings in algebraic geometry, Families, moduli, classification: algebraic theory Every algebraic Kummer surface is the \(K3\)-cover of an Enriques surface | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Grothendieck's anabelian conjecture on the pro-\(l\) fundamental groups of configuration spaces of hyperbolic curves is reduced to the conjecture on those of single hyperbolic curves. This is done by estimating effectively the Galois equivariant automorphism group of the pro-\(l\) braid group on the curve. The process of the proof involves the complete determination of the groups of graded automorphisms of the graded Lie algebras associated to the weighted filtration of the braid groups on Riemann surfaces. Galois representation; anabelian geometry; braid group; pro-\(l\) fundamental groups; groups of graded automorphisms; graded Lie algebras DOI: 10.1090/S0002-9947-98-02038-8 Coverings in algebraic geometry, Braid groups; Artin groups, Separable extensions, Galois theory, Fundamental groups and their automorphisms (group-theoretic aspects) Galois rigidity of pro-\(l\) pure braid groups of algebraic curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let S be a complex projective nonsingular minimal surface of general type and let \({\mathcal M}(S)\) be the coarse moduli space of complex structures on the oriented topological 4-manifold underlying S. It is known that \({\mathcal M}(S)\) is a quasi-projective variety. The paper under review is the third of a series [(1) J. Differ. Geom. 19, 483-515 (1984; Zbl 0549.14012); (2) Algebraic Geometry, Open Problems, Proc. Conf., Ravello/Italy 1982, Lect. Notes Math. 997, 90-112 (1983; Zbl 0517.14011); (4) J. Differ. Geom. 24, 395-399 (1986)] devoted to the study of general properties of \({\mathcal M}(S)\). This study was carried out by using a test class of simply connected surfaces obtained by deforming bidouble covers of \({\mathbb{P}}^ 1\times {\mathbb{P}}^ 1\), i.e. Galois covers with group \(({\mathbb{Z}}/2)^ 2\). The interest in these bidouble covers is evident in view of the analogy with hyperelliptic curves in dimension \(1\).
Here the author studies in the large the deformations of these surfaces. A bidouble cover as above looks like the subvariety of the total space of the bundle \({\mathcal O}_{{\mathbb{P}}^ 1\times {\mathbb{P}}^ 1}(a,b)\oplus {\mathcal O}_{{\mathbb{P}}^ 1\times {\mathbb{P}}^ 1}(n,m)\) defined by equations \(z^ 2=f(x,y)\), \(w^ 2=g(x,y)\), where f and g are bihomogeneous forms of bidegrees (2a,2b) and (2n,2m) respectively. Let \({\mathcal N}_{(a,b),(n,m)}\) be the subset of the moduli space corresponding to smooth natural deformations; the author supplies a complete description of the closure of \({\mathcal N}_{(a,b),(n,m)}\), for \(a>2n\), \(m>2b\). This is achieved by exploiting the relations between deformations of bidouble covers of \({\mathbb{P}}^ 1\times {\mathbb{P}}^ 1\) and degenerations of \({\mathbb{P}}^ 1\times {\mathbb{P}}^ 1\) to normal surfaces with certain singularities, which the author calls ''1/2 rational double points'' and studies in great detail. minimal surface of general type; coarse moduli space; bidouble covers; deformations; 1/2 rational double points Catanese F.: Automorphisms of rational double points and moduli spaces of surfaces of general type. Compos. Math. 61(1), 81--102 (1987) Families, moduli, classification: algebraic theory, Coverings in algebraic geometry, Formal methods and deformations in algebraic geometry, Special surfaces, Singularities of surfaces or higher-dimensional varieties Automorphisms of rational double points and moduli spaces of surfaces of general type | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper is dedicated to developing a positive characteristic version of the Kazhdan-Lusztig cells. In [Contemp. Math. 683, 333--361 (2017; Zbl 1390.20001)], the author and \textit{G. Williamson} defined the \(p\)-canonical basis for the Hecke algebra of a crystallographic Coxeter system. It can be thought of as a positive characteristic analogue of the Kazhdan-Lusztig basis. The \(p\)-canonical basis shares strong positivity properties with the Kazhdan-Lusztig basis (similar to the ones described by the Kazhdan-Lusztig positivity conjectures), but it loses many of its combinatorial properties. Replacing the Kazhdan-Lusztig basis by the \(p\)-canonical basis in the definition of the left (resp. right or two-sided) cells leads to the notion of left (resp. right or two-sided) p-cells. These \(p\)-cells are the main subject of the present paper.
The first properties of \(p\)-cells are proved in Section 3. Left and right \(p\)-cells are related by taking inverses (see Lemma 3.6), just like for Kazhdan-Lusztig cells. The set of elements with a fixed left descent set decomposes into right \(p\)-cells (see Lemma 3.4). The most important result of this section is a certain compatibility of \(p\)-cells with parabolic subgroups that shows that any right \(p\)-cell preorder relation in a finite standard parabolic subgroup \(W_I\) induces right \(p\)-cell preorder relations in each right \(W_I\)-coset (see Theorem 3.9). It is also shown that unfortunately Kazhdan-Lusztig cells do not always decompose into \(p\)-cells, but the author expects that this may still be the case when the prime \(p\) is good for the corresponding algebraic group. Indeed, in Section 4 it is proved that Kazhdan-Lusztig left cells decompose into left \(p\)-cells in finite types B and C for \(p > 2\). This is done by studying the consequences of the Kazhdan-Lusztig star-operations for the \(p\)-canonical basis, these operations appear as the main technical tool of the present work. Kazhdan-Lusztig cells; Hecke algebra; \(p\)-canonical basis Other geometric groups, including crystallographic groups, Representation theory for linear algebraic groups, Combinatorial aspects of representation theory, Hecke algebras and their representations, Representations of finite symmetric groups, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Grassmannians, Schubert varieties, flag manifolds, Modular representations and characters, Reflection and Coxeter groups (group-theoretic aspects) The ABC of \(p\)-cells | 0 |
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