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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 Our aim is to formulate and prove a weak form in equal characteristic \(p>0\) of the \(p\)-curvature conjecture. We also show the existence of a counterexample to a strong form of it. varieties in positive characteristic; stratified bundles; étale trivializable bundles; monodromy group; abelian varieties Esnault, H; Langer, A, On a positive equicharacteristic variant of the \(p\)-curvature conjecture, Doc. Math., 18, 23-50, (2013) Structure of families (Picard-Lefschetz, monodromy, etc.), Coverings in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Homotopy theory and fundamental groups in algebraic geometry, Positive characteristic ground fields in algebraic geometry, Abelian varieties of dimension \(> 1\) On a positive equicharacteristic variant of the \(p\)-curvature conjecture | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We establish the birational superrigidity of Fano double hypersurfaces of index one. Recall [\textit{A. V. Pukheikov}, Invent. Math. 134, 401-426 (1998; Zbl 0964.14011)] that a smooth Fano variety \(V\) is said to be birationally superrigid if, for each birational map \(\chi:V\to V'\) onto a uniruled variety of the same dimension and each linear system of divisors \(\Sigma'\) on \(V'\) without fixed components, the monotonicity condition \(c(D,V) \leq c(D',V')\) is satisfied, where \(D'\in \Sigma'\), \(D\in\Sigma =(\chi^{-1})_* \Sigma'\), i.e., \(\Sigma\) (the proper inverse transform of \(\Sigma'\), with respect to \(\chi)\) is a linear system of divisors on \(V\) without fixed components, and \(c(Y,X)\) stands for the threshold of canonical adjunction \(c(Y,X)= \sup\{b/a \mid b\in\mathbb{Z}_+, a\in\mathbb{Z}_+ \setminus\{0\}\), \(|aY+b K_X|\neq\emptyset\}\) for a variety \(X\) that is smooth in codimension one and for a Weil divisor \(Y\) on it. Let us choose integers \(M\geq 4\), \(m\geq 3\), and \(l \geq 1\) such that \(m+l=M+1\). Denote by the symbol \(\mathbb{P}\) the projective space \(\mathbb{P}^{M+1}\) over the complex field \(\mathbb{C}\). Let \(Q=Q_m \subset\mathbb{P}\) and \(W^*_{2l} \subset\mathbb{P}\) be hypersurfaces of degrees \(m\) and \(2l\), respectively, and let \(\sigma:V \to\mathbb{Q}\) be the double cover branched over \(W=W^*\cap Q\). Then \(V\) is a Fano variety of dimension \(M\) with Picard group \(\mathbb{Z} K_V\). Our principal result is the following assertion:
Theorem. A generic variety \(V\) is birationally superrigid.
The genericity is understood here as the genericity of the hypersurfaces \(Q\) and \(W^*\) in the sense of Zariski topology.
Corollary. (i) A generic variety \(V\) has no non-trivial structure of a Fano fibration. Moreover, it cannot be fibered into uniruled varieties by a rational mapping.
(ii) The groups of birational and biregular automorphisms of \(V\) coincide, \(\text{Bir} V =\Aut V\).
(iii) The variety \(V\) is non-rational. non-rational variety; birationally superrigid variety; smooth Fano variety; canonical adjunction; Fano fibration Birational automorphisms, Cremona group and generalizations, Fano varieties, Rational and unirational varieties, Coverings in algebraic geometry Birational automorphisms of Fano double covers | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The main objects of the paper under review are higher-dimensional analogues of two classical problems. The first one is the uniform boundedness conjecture (UB) stating that given a number field \(k\) and a positive integer \(g\), there exists a constant \(N=N(k,g)\) such that for any \(g\)-dimensional abelian \(k\)-variety the order of any its torsion \(k\)-point is at most \(N\) (this was proved by \textit{L.~Merel} [Invent. Math. 124, No. 1--3, 437--449 (1996; Zbl 0936.11037)] for \(g=1\)). A weaker variant, \(p\)-uniform boundedness conjecture (\(p\)UB), states that given \(k\), \(g\) as above and a fixed primed number \(p\), there exists a constant \(N=N(k,g,p)\) such that for any \(g\)-dimensional abelian \(k\)-variety \(A\) and any \(v\in A[p^{\infty}](k)\) the order of \(v\) is at most \(N\) (this was proved by \textit{Yu.~I.~Manin} [Izv. Akad. Nauk SSSR. Ser. Mat. 33, 459--465 (1969; Zbl 0191.19601)] for \(g=1\)). Both conjectures are widely open for \(g>1\).
The second one is the regular inverse Galois problem (RIGP): given a finite group \(G\) and a number field \(k\), does there exist a Galois extension \(L/k(T)\) with group \(G\) such that \(L\cap \bar k =k\)? An appropriate geometric reformulation of RIGP gives rise to the so-called modular tower conjecture (MT) of M.~Fried which, roughly, is a statement about the absence of rational points in certain Hurwitz spaces (moduli spaces of covers of curves).
The authors discuss higher-dimensional analogues of these two problems, (\(p\text{UB}_d\)) and (\(\text{MT}_d)\), respectively, where the field \(k\) is replaced with a \(d\)-dimensional \(k\)-scheme \(S\) of finite type. The main results of the paper state that both (\(p\text{UB}_1\)) and (\(\text{MT}_1\)) hold. Note that (\(p\text{UB}_1\)) in characteristic 0 as well as the implication \((p\text{UB}_1)\Longrightarrow (\text{MT}_1)\) in arbitrary characteristic were established in an earlier paper of the authors. abelian scheme; moduli space; fundamental group; inverse Galois problem; modular tower A. Cadoret and A. Tamagawa, ``Torsion of abelian schemes and rational points on moduli spaces'' in Algebraic Number Theory and Related Topics, 2007 , RIMS Kôkyûroku Bessatsu B12 , Res. Inst. Math. Sci. (RIMS), Kyoto, 2009, 7-29. Arithmetic ground fields for curves, Coverings of curves, fundamental group, Arithmetic ground fields for abelian varieties, Coverings in algebraic geometry, Inverse Galois theory Torsion of abelian schemes and rational points on moduli spaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Semi-algebraic geometry deals with subsets \(M\subset R^ n\) (R an arbitrary real closed field) which are boolean combinations of finitely many sets of the form \(\{x\in R^ n| \quad P(x)\geq 0\}\) where \(P\in R[X_ 1,...,X_ n]\). To study the geometry of such sets the authors introduced the category of semi-algebraic spaces over the real closed field R. However, certain questions cannot be treated satisfactorily in this setting. A case in point is the classification of coverings. This necessitates the introduction of a larger category, the category of locally semi-algebraic spaces. This paper contains the definition, examples and a few properties of these spaces.
A full account of the theory of locally semi-algebraic spaces may be found in the authors' book [''Locally semi-algebraic spaces'', Lect. Notes Math. 1173 (1985; Zbl 0582.14006)]. real closed field; coverings; locally semi-algebraic spaces Delfs, H.; Knebusch, M.: An introduction to locally semialgebraic spaces. Rocky mountain J. Math. 14, 945-963 (1984) Real algebraic and real-analytic geometry, Coverings in algebraic geometry An introduction to locally semialgebraic 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 quote the author's words: ``The aim of this paper is to study the torsion free cover of the \(k[x_ 1, x_ 2, \dots, x_ n]\)-module \(k\) (which \(x_ ik = 0)\) where \(n \geq 2\). This cover will be denoted by \(T(k)\). Our main tools will be the descriptions of a torsion free cover by Banaschewski and the injective envelope of \(k\) by Northcott''. cover of modules over polynomial rings Polynomial rings and ideals; rings of integer-valued polynomials, Coverings in algebraic geometry On the torsion free cover of the \(k[x_ 1,x_ 2,\dots,x_ n]\)-module \(k\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 book originates from a series of lectures and seminar talks for graduate students at various universities in Japan. It is a research monograph describing research by the author on finite branched coverings of projective complex manifolds in connection with the theory of algebraic functions of several complex variables. The author presents a theory generalizing earlier work in one complex variable, in particular work of \textit{A. Weil} [J. Math. Pures Appl., IX. Sér. 17, 47-87 (1938; Zbl 0018.06302)].
The book has three chapters: 1. ``Branched coverings of complex manifolds'', 2. ``Fields of algebraic functions'', 3. ``Weil-Tōyama theory''.
The main result in chapter 1 is a theorem providing a suitable sufficient condition for the existence of a finite Galois covering of a projective manifold, branched over a given divisor. In the case of compact Riemann surfaces, this originates in a problem posed by Fenchel. In addition, chapter 1 contains many well chosen examples of finite branched coverings. -- Chapter 2 studies finite abelian coverings using the theory of currents developed by \textit{G. de Rham} and \textit{K. Kodaira} in ``Harmonic integrals'', Lecture Notes Inst. Adv. Study (Princeton 1950). There are results providing necessary and sufficient conditions for the existence of a finite abelian covering onto a projective complex manifold branched over a given divisor. Furthermore, the set of all isomorphism classes of finite abelian branched coverings of a projective complex manifold is described using the notion of rational divisor classes. The results can be considered as higher dimensional generalizations of results of Iwasawa from 1952. -- Chapter 3 studies similar questions for finite Galois coverings, but according to the author the results obtained are not completely satisfactory. The book should be a good source for inspiration to further work. finite branched coverings of projective complex manifolds; algebraic functions; finite Galois coverings M. NAMBA, Branched coverings - algebraic functions, Pitman Research Notes in Mathematics Series 161. Zbl0706.14017 MR933557 Coverings of curves, fundamental group, Coverings in algebraic geometry, Algebraic functions and function fields in algebraic geometry, Transcendental methods of algebraic geometry (complex-analytic aspects), Low-dimensional topology of special (e.g., branched) coverings, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces, Research exposition (monographs, survey articles) pertaining to functions of a complex variable, Compact Riemann surfaces and uniformization Branched coverings and algebraic functions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper continues previous studies on the classification of first-order differential equations over \(C\left( z \right)\) and finding algebraic solutions of such equations see [\textit{Ngo, L.X. Chau, K. A. Nguyen} and \textit{M. van der Put}, ``Equivalence of differential equations of order one'', Preprint, Top, J., \url{arXiv:1303.4960}, (2013)].
Let \(C\) be an algebraically closed field of characteristic \(p>0\), and \(K\) be a finite separable extension of \(C\left( z \right)\). The authors consider equations of the form \(f\left( {y}',y \right)=0\), where \(f\in K\left[ S,T \right],\text{ }\,f\) is absolutely irreducible polynomial and \(\frac{df}{dS}\notin (f)\). In addition the natural continuation of the differentiation of \(K\) onto \(K\left[ S,\text{ }T \right]/(f)\) must satisfy the condition \({{D}^{p}}=0\) (admits a stratification). They give criteria for deciding whether the equation admits a stratification. It turns out that the autonomous (\(f\in C\left[ S,\text{ }T \right]\)) differential equation admitting a stratification has a separable algebraic solution. Also for such equations a Grothendieck-Katz conjecture is true, see \textit{M. van der Put} [Indag. Math., New Ser. 12, No. 1, 113--124 (2001; Zbl 1004.12005)].
All solutions of \(f\left( {y}',y \right)=0\) are algebraic if and only if for almost all primes \(p\) the reduced equation \(f \mod p\) is stratified. first order differential equation; positive characteristic; Grothendieck-Katz conjecture; stratification; algebraic curves Algebraic aspects (differential-algebraic, hypertranscendence, group-theoretical) of ordinary differential equations in the complex domain, Coverings in algebraic geometry, Ramification problems in algebraic geometry, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Abstract differential equations Stratified order one differential equations 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 Following an idea of \textit{C. Ciliberto} [Ricerche Mat. 29, 175--191 (1980; Zbl 0478.14001)] we show that double covers of projective \(r\)-space branched over an hypersurface of degree \(2d\) are unirational provided \(r\) is sufficiently big with respect to \(d\). Conte, A.; Marchisio, M.; Murre, J.: On unirationality of double covers of fixed degree and large dimension; a method of ciliberto. Algebraic geometry, 127-140 (2002) Rationality questions in algebraic geometry, Coverings in algebraic geometry, Rational and unirational varieties On unirationality of double covers of fixed degree and large dimension; a method of Ciliberto. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 0675.00006.]
The singular cohomology of a smooth complex projective variety, A, carries a Hodge structure which gives some information about subvarieties of A. Just now much information is not known, although an optimistic conjecture has been stated by \textit{A. Grothendieck} [Topology 8, 299-303 (1969; Zbl 0177.490)] in the process of modifying earlier speculations of Hodge.
This paper constitutes an attempt to test the conjecture when A is an abelian variety. Only certain special abelian varieties are considered. These are obtained from a compact Riemann surface C with an effective action of the group \({\mathbb{Z}}/m, m>2\) by taking \(A\subset Jac(C)\) to be the largest abelian subvariety such that \({\mathbb{Z}}/m\) acts by primitive characters on the tangent space at the origin. Such abelian varieties carry a Hodge structure, which as Weil pointed out, serves as a good testing ground for Hodge type conjectures. The paper associates two integers v and w to this action and then constructs a certain correspondence between A and the Fermat hypersurface of degree m and dimension v. This reduces `Grothendieck's generalized Hodge conjecture' for the Weil Hodge structure on A to the analogous question for a weight w Hodge structure on the Fermat hypersurface. It is often the case that w coincides with the level of the Weil Hodge structure. In such a situation the conjecture follows immediately. The case \(v=w=0\) was treated by the author in Compos. Math. 65, No.1, 3-32 (1988; Zbl 0663.14006). singular cohomology; abelian variety; Weil Hodge structure; Fermat hypersurface Transcendental methods, Hodge theory (algebro-geometric aspects), Algebraic theory of abelian varieties, Coverings in algebraic geometry, Cycles and subschemes Cyclic covers of \(P^ v\) branched along \(v+2\) hyperplanes and the generalized Hodge conjecture for certain abelian varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(f: X\to Y\) be a morphism of finite type of noetherian excellent \({\mathbb{Q}}\)-schemes. Then the main result asserts that the algebraic fundamental group of étale neighbourhoods of the fibres varies constructibly on Y, i.e. \(y\to \prod (X\times_ YSpec({\mathcal O}_{Y,y}))\quad is\) constructible on Y. In particular, the set where this group vanishes is constructible in Y. By an example it is shown that this set is not open in general. Similar results are obtained for the local fundamental group. An important tool in the proof of these results is the stratification theory of complex spaces. The detailed proof of the main result will appear in a forthcoming paper. algebraic fundamental group; local fundamental group; stratification theory of complex spaces J. Bingener,H. Flenner, Variation of the fundamental groups of schemes. Coverings in algebraic geometry, Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects), Homotopy theory and fundamental groups in algebraic geometry Variation of the fundamental group of schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We give an example of a family of 15 skew lines on a quintic such that its class is divisible by 3. We study properties of the codes given by arrangements of disjoint lines on quintics. quintic; cyclic cover; skew lines; codes; arrangements of disjoint lines on quintics S. Rams, Three-divisible families of skew lines on a smooth projective quintic, Trans. Amer. Math. Soc. 354 (2002), no. 6, 2359-2367 (electronic). Plane and space curves, Computational aspects of algebraic surfaces, Applications to coding theory and cryptography of arithmetic geometry, Arithmetic codes, Special varieties, Coverings in algebraic geometry Three-divisible families of skew lines on a smooth projective quintic | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 0624.00007.]
For rigid analytic varieties (analytic) covering maps are defined with respect to the Grothendieck topology of affinoid subspaces [see \textit{M. van der Put}, Ann. Inst. Fourier 33, No.1, 29-52 (1983; Zbl 0495.14017)]. This leads to corresponding notions of fundamental group and universal covering in rigid analysis. The author studies the basic properties of these concepts and gives a number of (mostly well-known) examples: The punctured affine line \(k^*\) is simply connected, as is any affinoid subspace of the projective line, and the Schottky uniformization of a Mumford curve is the universal covering. It is an open question whether the n-dimensional unit ball is simply connected for \(n\geq 2\). rigid analytic varieties; covering maps; Grothendieck topology; fundamental group; Schottky uniformization of a Mumford curve P. Ullrich, Rigid analytic covering maps , Proceedings of the conference on \(p\)-adic analysis (Houthalen, 1987), Vrije Univ. Brussel, Brussels, 1986, pp. 159-171. Local ground fields in algebraic geometry, Coverings in algebraic geometry, Non-Archimedean analysis Rigid analytic covering maps | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems ``In this paper we construct simply-connected algebraic surfaces of general type which have positive index. After the famous inequality \(c^ 2_ 1\leq 3c_ 2\) [Bogomolov-Miyaoka-Yau (1976)] was discovered, the empirical evidence for simply-connected surfaces of general type strongly indicated that in their case the inequality \(c^ 2_ 1\leq 2c_ 2\) must hold. The inequality \(c^ 2_ 1\leq 2c_ 2\) is equivalent of course to the fact that the index \(\tau (=(c^ 2_ 1- 2c_ 2)/3)\) is not positive. So the following conjecture was formulated [``watershed conjecture of Bogomolov'': cf. \textit{J.-M. Feustel} and \textit{R.-P. Holzapfel}, Math. Nachr. 111, 7-40 (1983; Zbl 0528.14015)]: If Y is a surface of general type with \(\tau (Y)>0\) then \(\pi_ 1(Y)\neq 0\) (or equivalently \(\pi_ 1(Y)\) is infinite).
Thus our results provide counter-examples to this conjecture. Take \(X={\mathbb{C}}P^ 1\times {\mathbb{C}}P^ 1\). Let \(\ell_ i\subseteq X\), \(\ell_ 1\subseteq {\mathbb{C}}P^ 1\times pt\), \(\ell_ 2=pt\otimes {\mathbb{C}}P^ 1\). Let \(E=a\ell_ 1+b\ell_ 2\), a,b\(\in {\mathbb{N}}\). Let \(X_{a,b}\) be the embedding of X into \({\mathbb{C}}P^ N\) with respect to the linear system \(| E|\). Take a canonical projection f of \(X_{a,b}\) to \({\mathbb{C}}P^ 2\), \(n=\deg (f)=2ab\). Let Y be its Galois cover that corresponds to the full symmetric group \(S_ n\). For \(a\gg 0\), \(b\gg 0\), a and b relatively prime, Y is the example we present. In our work we prove that the fundamental group \(\pi_ 1(Y)\) is a finite abelian group and that it is trivial when a,b are relatively prime.''
Irrelevant reviewer's remark: some names of authors are not correct, e.g. Bogamolov instead of Bogomolov, twice in the introduction, Holzupfel instead of Holzapfel in the references. algebraic surfaces of positive index; Chern numbers; Bogomolov-Miyaoka- Yau inequality; watershed conjecture; surface of general type; linear system; fundamental group B. Moishezon and M. Teicher, ''Simply-connected algebraic surfaces of positive index,'' Invent. Math., vol. 89, iss. 3, pp. 601-643, 1987. Topological properties in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry, Characteristic classes and numbers in differential topology, Coverings in algebraic geometry, Special surfaces Simply-connected algebraic surfaces of positive index | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Sei \(\pi\) : \(\tilde X\to X\) die universelle Überlagerung einer zusammenhängenden projektiven Mannigfaltigkeit X. Die Vermutung von Shafarevich besagt, daß dann \(\tilde X\) holomorph-konvex sein sollte. In vorliegender Arbeit wird diese Vermutung für projektive Flächen X unter gewissen Zusatzbedingungen bestätigt.
Der Beweis dazu basiert auf folgenden Tatsachen: (a). Eine vollständige Kähler-Mannigfaltigkeit mit beschränkter Geometrie, über der ein Geradenbündel \(L\to X\) gegeben ist, ist in geeigneten Situationen konvex bzgl. gewisser mittels L gegebener Bündel. Insbesondere gilt dann für eine projektive Mannigfaltigkeit X und ein positives Geradenbündel \(L\to X\), daß jede holomorphe Überlagerung \(\pi\) : \(\tilde X\to X\) ``Bündel''-konvex ist.
(b) Sei \(\pi\) : \(\tilde X\to X\) eine holomorphe Überlagerungsfläche einer komplexen Fläche X, und sei C eine eigentliche kompakte analytische Menge von X, so daß \(\tilde C:\)\(\pi\) \({}^{-1}(C)\) holomorph-konvex ist. Dann gibt es eine Umgebung \(V=V(C)\) und eine stetige plurisubharmonische Funktion auf \(\tilde V:\)\(=\pi^{-1}(V)\), die \(\tilde V\) relativ zu \(\tilde X\) ausschöpft. universal covering; smooth projective surface; holomorphically convex T. Napier, ''Convexity properties of coverings of smooth projective varieties,'' Math. Ann., vol. 286, iss. 1-3, pp. 433-479, 1990. Holomorphically convex complex spaces, reduction theory, Coverings in algebraic geometry Convexity properties of coverings of smooth projective varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author rewrites in a slightly different way the structure results for abelian covers of \textit{R. Pardini} [J. Reine Angew. Math. 417, 191--213 (1991; Zbl 0721.14009] and constructs a new example of minimal complex surface of general type as a \({\mathbb Z}_5^3\)-cover of the plane branched on a configuration of 6 lines, very similar to one of the famous examples given by \textit{F. Hirzebruch} [Contemp. Math. 9, 55--71 (1982; Zbl 0487.14009)]. abelian cover; ball quotient Gao, Y, A note on finite abelian covers, Sci China Math, 54, 1333-1342, (2010) Coverings in algebraic geometry, Ramification problems in algebraic geometry, Surfaces of general type A note on finite abelian covers | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper is devoted to prove that, given a quotient-cusp singularity \((V,p)\), its universal Abelian cover branched at \(p\) is a complete intersection cusp singularity. A quotient-cusp singularity \((V,p)\) is an isolated complex surface singularity which is double-covered by a cusp singularity. It is a rational singularity which is taut, i.e. the topology of its link determines the analytic type of \((V,p)\). In the paper, the universal Abelian cover \(({\widetilde V}, p)\) of \((V,p)\) branched at \(p\) is determined from the topology of the link \((V,p)\), and the proof is based on the description of the link of \((V,p)\) obtained by \textit{W. D. Neumann} [Trans. Am. Math. Soc. 268, 299-343 (1981; Zbl 0546.57002)], and some calculus. The authors also sketch a more geometric proof by giving an explicit group action on a different complete intersection representation of the cusp.
If the link of a normal complex surface singularity is a rational homology sphere, then the universal Abelian cover of the link is a finite covering, hence it gives rise to the universal Abelian cover of the singularity. The result in the paper, and the study by \textit{W. D. Neumann} [in: Singularities, Summer Inst., Arcata/Calif. 1981. Proc. Symp. Pure Math. 40, Part 2, 233-243 (1983; Zbl 0519.32010)] of universal Abelian covers of weighted homogeneous normal surface singularities is the motivation of the authors to state the following conjecture:
Let \((X,o)\) be a \(\mathbb Q\)-Gorenstein normal surface singularity whose link is a rational homology sphere, then the universal Abelian cover \(({\widetilde X}, 0)\) of \((X,o)\) is a complete intersection. quotient-cusp; universal Abelian cover; cusp singularity; group action; surface singularity; complete intersection Neumann, W.D., Wahl, J.: Universal abelian covers of quotient-cusps, Math. Ann. 326, 75--93 (2003) Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry, Coverings in algebraic geometry, Homogeneous spaces and generalizations, Group actions on varieties or schemes (quotients), Complex surface and hypersurface singularities Universal Abelian covers of quotient-cusps | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 topology of the complement of any nonsingular real algebraic curve of a given degree in \({\mathbb{C}}{\mathbb{P}}^ 2\) is standard. The situation varies when we consider the set A of complex points of a real curve and the complement of \({\mathbb{R}}{\mathbb{P}}^ 2\cup A\) in \({\mathbb{C}}{\mathbb{P}}^ 2\). One of the principal reasons for the research is to find rigid homotopic invariants of curves that would be finer than the complex scheme that can be restored by the topology of \(A\cup {\mathbb{R}}{\mathbb{P}}^ 2\). The author constructs a two-dimensional CW-complex which is homotopically equivalent to \({\mathbb{C}}{\mathbb{P}}^ 2\backslash(A\cup {\mathbb{R}}{\mathbb{P}}^ 2)\) for some class of curves. It allows him to compute the fundamental group of the complement. In particular the fundamental group is computed for all nonsingular real curves of a degree \(\leq 5\). The question whether the homotopic type of the complement of \(A\cup {\mathbb{R}}{\mathbb{P}}^ 2\) is finer invariant than the complex scheme remains undecided. - All necessary constructions are given in details; the proofs are absent. complement of nonsingular real algebraic curve; fundamental group; homotopic type Topological properties in algebraic geometry, Coverings in algebraic geometry, Special algebraic curves and curves of low genus, Homotopy theory and fundamental groups in algebraic geometry Topology of the complement of a real algebraic curve in \({\mathbb{C}}P^ 2\) | 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 The aim of the first part of the article is to construct an analog for triple coverings of Horikawa's resolution for double coverings.
Let \(W\) be a nonsingular complex analytic surface and \(\pi : \overline L \to W\) the \(\mathbb{P}^ 1\)-bundle associated to a line bundle \(L\) on \(W\). Let \(S\) be an irreducible reduced divisor on \(\overline L\) linearly equivalent to \(3T\), where \(T = {\mathcal O}_{\overline L} (1)\). Then we get a triple covering \(\varphi : S \to W\), we call \((S,W,L)\) a ``triple section surface'', and we want to construct a reduction process of singularities of the surface \(S\). There are two classes of singular points \(Q\) on \(S\): the ``inner double points'', for which \(\# \varphi^{ - 1} (\varphi (Q)) = 2\), and the ``target singularities'', for which \(\# \varphi^{ - 1} (\varphi (Q)) = 3\), and in this case we call \(P = \varphi (Q)\) a ``target point'' of \(W\). If \(\tau_ 1 : W_ 1 \to W\) is the blow-up of a target point \(P\) of \(W\), we can construct a new triple section surface \((S_ 1, W_ 1, L_ 1)\), where the line bundle \(L_ 1\) depends on \(L\) and on \(P\), such that we have a birational morphism \(\overline \tau_ 1 : \overline S_ 1 \to S\). We call this a ``triplet blow-up'' at \(P\), and we get the following result:
Theorem: Let \((S,W,L)\) be a triple section surface with \(S\) normal and
\[
(S,W,L) \gets (S_ 1, W_ 1, L_ 1) \gets \cdots \gets (S_ r, W_ r, L_ r) \gets \cdots
\]
be the reduction process by successive triplet blow-ups. Then this process terminates in finite steps, namely there exists \(r\) such that \((S_ r, W_ r, L_ r)\) has no target point.
\(\tau : (S_ r, W_ r, L_ r) \to (S,W,L)\) is called the ``canonical reduction'' and then the author studies the singularities of the (not necessarily normal) surface \(S_ r\) and the exceptional divisor of the resolution of \(S_ r\). He shows that the singularities are only relative cusps, relative nodes and isolated inner double points, all of multiplicity at most 2.
In the second part the author solves Durfee's conjecture for two- dimensional hypersurface singularities of multiplicity 3. If \((V,p)\) is a such singularity, there exists a triple section surface \((S,W,L)\) and a target singularity \(P\) on \(S\) such that \((V,p)\) and \((S,P)\) are analytically equivalent. We can use the results of the first part of the article and we get:
Theorem: Let \((V,p)\) be a normal two-dimensional hypersurface singularity of multiplicity 3. Then we have: \(\mu (V,p) \geq 6p_ g (V,p) + 2\), where \(\mu (V,p)\) is the Milnor number and \(p_ g (V,P)\) the geometric genus of the singularity. -- Especially the signature of the Milnor fiber of \((V,p)\) is negative.
Moreover, the equality holds if and only if \((V,p)\) is a simple elliptic singularity of type \(\widetilde E_ 6\) in the sense of Saito. resolution of surface singularities; triple section surface; target point; triple covering; inner double points; Durfee's conjecture; two- dimensional hypersurface singularities of multiplicity 3; Milnor number; simple elliptic singularity Némethi, A.: Dedekind sums and the signature of \(f(x,y)+z^N\). Selecta. Math. (N.S.) \textbf{4}(2), 361-376 (1998) Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry, Coverings in algebraic geometry, Complex surface and hypersurface singularities, Modifications; resolution of singularities (complex-analytic aspects) Normal two-dimensional hypersurface triple points and the Horikawa type resolution | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(\Gamma\)-reduction of a compact Kähler manifold, also known as Remmert reduction or the Shafarevich map, is a meromorphic fibration of the universal cover such that (among other technical requirements) any compact analytic subset passing through a very general point has to be contained in a fibre. In this version it was constructed by Campana. Here we see an orbifold version, valid for smooth orbifolds \((X/\Delta)\): that is, \(X\) is a compact Kähler manifold and \(\Delta\) is a \({\mathbb Q}\)-divisor with normal crossings of the form \(\Delta =\sum (1-{{1}\over{m_j}})\Delta_j\), where \(m_j>1\) are integers. There is a notion of orbifold universal cover \(\tilde X_\Delta\), and the result is that there is a \(\Gamma\)-reduction with the same properties as in the manifold case.
The methods are those of Campana. What one needs is a suitable Kähler metric, and the difference here is that one must allow singular metrics with suitable singularities along \(\Delta\). The main point, then, is to give a local description of this metric \(\omega_\Delta\) and check that it has the right properties: one must first modify \(\Delta\) slightly if the branching of the universal covering map is not exactly \(\Delta\). Then if \(\Delta_j=(s_j=0)\), where \(s_j\in H^0({\mathcal O}_X(\Delta_j))\) and \(\omega\) is a Kähler metric on \(X\), then for any choice of smooth metric on \({\mathcal O}_X(\Delta_j)\) and for \(C\in{\mathbb R}\) positive enough we may take
\[
\omega_\Delta =C\omega+\sum_j\sqrt{-1}\partial\bar\partial |s_j|^{2/m_j}.
\]
Shafarevich map; smooth orbifold; singular Kähler metric [14] Benoît Claudon, &\(\Gamma\)-reduction for smooth orbifolds&#xManuscripta Math.127 (2008) no. 4, p.~521Article | &MR~24 | &Zbl~1163. Families, moduli, classification: algebraic theory, Coverings in algebraic geometry, Compact Kähler manifolds: generalizations, classification \(\Gamma\)-reduction for smooth orbifolds | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 survey focuses on certain classical results on the bounded domains with projective algebraic or, equivalently, compact discrete quotients. The existence of arithmetic and uniform lattices \( \Gamma \) in semisimple real algebraic groups \(G\) of noncompact type justifies the presence of compact discrete quotients \(\Gamma \setminus G/K\) of the bounded symmetric domains \( G / K\). The Kodaira surfaces are exhibited as examples of projective algebraic manifolds, covered by bounded contractible non-symmetric domains. The basic ideas of Frankel-Nadel's uniformization of the compact complex manifolds with ample canonical bundle are briefly outlined. A strictly pseudoconvex domain with a smooth boundary, a Siegel domain of exponent \((c_{1}, \ldots,c_{k}) \in\) \textbf{R} and, in particular, a bounded circular domain is announced to admit a compact discrete quotient if and only if it is bounded symmetric. compact complex manifolds; uniformization; bounded domains A. Kasparian, Co-abelian toroidal compactifications of torsion free ball quotients , arXiv: Transcendental methods of algebraic geometry (complex-analytic aspects), Coverings in algebraic geometry, Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces, Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects) When does a bounded domain cover a projective manifold? (Survey) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 Cremona group \(\text{Bir}(\mathbb{P}^2)\) has been studied for a long time. Their modern treatment, using Mori theory, has been done in \textit{L. Bayle} and \textit{A. Beauville} [Asian J Math 4, 11--17 (2000; Zbl 1055.14012)] (order \(n = 2\)), \textit{D.-Q. Zhang} [J. Algebra 238, 560--589 (2001; Zbl 1057.14053)] (general \(n\)) and the current paper under review. The result in theorems A and B of the article is the classification of minimal pairs \((X, \sigma)\) where \(X\) is a smooth projective surface with non-nef \(K_X\) and \(\sigma \in \text{Aut}(X)\) is of prime order \(p\); see the article for the list (the equations of the surfaces and the actions on the coordinates are also given). As a consequence, the author gives the complete classification of Cremona transformations of prime order in \(\mathbb{P}^2\) (theorem E), where the un-resolved case \(E4\) has been proved by Bayle-Beauville [loc. cit.] to be conjugate to a linear automorphism of \(\mathbb{P}^2\). The author also describes (theorem F) the moduli spaces of conjugacy classes of prime order cyclic subgroups of \(\text{Bir}(\mathbb{P}^2)\). The reader will gain a lot more by reading the article. T. de Fernex. On planar Cremona maps of prime order. \textit{Nagoya Mathematical Journal}, \textbf{174} (2004), 1-28. Birational automorphisms, Cremona group and generalizations, Coverings in algebraic geometry On planar Cremona maps of prime order | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 0702.14023. galois automorphisms; fundamental group of the projective line minus three points 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 In this short paper the author calls attention to some problems in Schur indices, Hasse invariants and the Brauer equivalence for \(G\)-algebras (\(G\) is a finite group). He also gives some recent results of his without proofs. Brauer groups; Schur indices; Hasse invariants; Brauer equivalence; \(G\)- algebras Ordinary representations and characters, Galois cohomology, Brauer groups of schemes, Modular representations and characters Brauer and 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 Das Ziel der hier vorliegenden Arbeit ist es, eine torsionsfreie Riemann-Roch Formel für zyklische zwei- und dreiblättrige Überlagerungen mit einfachen Verzweigungen anzugeben. Dazu definieren wir Transfers für verzweigte Überlagerungen als Fortsetzung der Transfers, die Fulton and MacPherson für unverzweigte Überlagerungen definiert haben. branched cyclic covering Riemann-Roch theorems, Coverings in algebraic geometry Transfers and Riemann-Roch formulas for some cyclic coverings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems It is a classical result due to Zariski that a complex plane sextic \(C\) with 6 cusps is the branch locus of a non Galois triple cover of the plane if and only if it is defined by an equation of the form \(A^3+B^2=0\) where \(A\) and \(B\) are homogeneous polynomials of degree respectively \(2\) and \(3\). The paper under review contains an extension of this result to the case of a plane sextic with at most simple singularities. triple cover; cubic surface; plane sextic; torus curve H Ishida, H Tokunaga, Triple covers of algebraic surfaces and a generalization of Zariski's example, to appear in Adv. Stud. Pure Math., Math. Soc. Japan Coverings in algebraic geometry, Singularities of surfaces or higher-dimensional varieties Triple covers of algebraic surfaces and a generalization of Zariski's example | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this article the author gives an explicit method for constructing a finite dihedral Galois covering of a smooth projective variety \(Y\) with prescribed branch locus \(B\), i.e. a finite morphism \(\pi:X\to Y\) with \(X\) normal variety such that \(\mathbb{C}(X)\) is a Galois extension of \(\mathbb{C}(Y)\) having dihedral group \({\mathcal D}_{2n}\) as its Galois group. The branch locus \(\Delta=\Delta (X/Y)\) is defined by \(\Delta= \{y\in Y\mid \#^{-1}(y)< \deg\pi\}\).
If we choose generators \(\sigma\) and \(\tau\) of \({\mathcal D}_{2n}\) such that \(\sigma^2 = \tau^n= (\sigma \tau)^2 = \text{id}\), the invariant subfield \(\mathbb{C}(X)^\tau\) of \(\mathbb{C}(X)\) by \(\tau\) is a quadratic extension of \(\mathbb{C}(Y)\). Then if we note \(D(X/Y)=D\) the \(\mathbb{C}(X)^\tau\)-normalisation of \(Y\) we get a factorization of the covering \(\pi:X\to Y\) by \(D\), where \(\beta_2: X\to D\) and \(\beta_1: D\to Y\) are cyclic coverings respectively of degree \(n\) and 2. We can reduce the study of dihedral covering to that of these two coverings \(\beta_1\) and \(\beta_2\). We have to distinguish the cases \(n\) odd and \(n\) even, and we get the following:
Let \(f:Z\to Y\) be a smooth finite double covering of \(Y\) and \(\sigma\) be the involution on \(Z\) determined by \(f\). If we get three effective divisors \(D_1\), \(D_2\) and \(D_3\) in the \(n\)-odd case, or four effective divisors \(D_1\), \(D_2\), \(D_3\) and \(D_4\) in the \(n\)-even case, satisfying some convenient properties, then there exists a dihedral \({\mathcal D}_{2n}\) covering \(\pi: X\to Y\) such that: (i) \(D(X/Y)=Z\) and (ii) \(\Delta(X/Y) = \Delta(Z/Y)\cup f(\text{Supp} (D_1))\) in the \(n\)-odd case, and \(\Delta(X/Y) = \Delta(Z/Y) \cup f(\text{Supp} (D_1+D_2))\) in the \(n\)-even case.
This construction is universal and we get a converse of this result:
Let \(\pi: X\to Y\) be a dihedral \({\mathcal D}_{2n}\) covering such that \(D(X/Y)\) is smooth, and let \(\sigma\) be the involution on \(D(X/Y)\) determined by \(\beta_1\), then there exist three effective divisors \(D_1\), \(D_2\) and \(D_3\) in the \(n\)-odd case, or four effective divisors \(D_1\), \(D_2\), \(D_3\) and \(D_4\) in the \(n\)-even case, on \(D(X/Y)\) which satisfy the previous conditions and such that \(\text{Supp} (D_1+ \sigma^*D_1)\) in the \(n\)-odd case, or \(\text{Supp} (D_1+ \sigma^*D_1+D_2)\) in the \(n\)-even case, is the branch locus of \(\beta_2\).
In the last part of the article the author deduces from these propositions a result about dihedral covering of the projective plane \(\mathbb{P}^2\): Let \(\pi: S\to\mathbb{P}^2\) be a dihedral \({\mathcal D}_{2p}\) (with \(p\) odd prime) coverings of \(\mathbb{P}^2\) and let \(\Delta(S/ \mathbb{P}^2)\) be the branch locus of \(\pi\). Then \(\deg\Delta(S/ \mathbb{P}^2) \geq 3\). Furthermore if \(\deg\Delta(S/ \mathbb{P}^2)\leq 4\), the author gives the list of the all possibilities for the branch locus \(\Delta(S/ \mathbb{P}^2)\). dihedral group as Galois group; dihedral Galois covering of a smooth projective variety; prescribed branch locus Tokunaga H. (1994) On dihedral Galois coverings. Can. J. Math. 46: 1299--1317 Coverings in algebraic geometry, Ramification problems in algebraic geometry, Low codimension problems in algebraic geometry, Elliptic surfaces, elliptic or Calabi-Yau fibrations, Inverse Galois theory On dihedral Galois coverings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper under review mostly studies the existence of Ulrich line bundle on a double cover \(\pi: X \to \mathbb{P}^2\) of the projective plane \(\mathbb{P}^2\) branched along a smooth curve \(B\) of degree \(2s\), in the sense of \textit{M. Aprodu} and \textit{Y. Kim} [Ann. Univ. Ferrara, Sez. VII, Sci. Mat. 63, No. 1, 9--23 (2017; Zbl 1397.14054)], or equivalently, in the sense of \(\pi\)-Ulrich line bundles, cf. [\textit{R. Kulkarni}, \textit{Y. Mustopa}, and \textit{I. Shipman}, J. Algebra 474, 166--179 (2017; Zbl 1368.14028)]. It is well-known that such a double cover is a smooth quadric surface when \(s=1\), and is a del Pezzo surface of degree \(2\) if \(s=2\). In both cases, the structure of the Picard group is well-understood, and in particular, there are Ulrich line bundles. On the other hand, when \(s>2\) and \(B\) is sufficiently general, then \(\operatorname{Pic} (X) \simeq \mathbb{Z}\) and such a surface cannot have an Ulrich line bundle (see also Lemma 3.2 and Theorem 1.4).
To clarify the existence even when \(B\) is special, the authors provided equivalence conditions to have an Ulrich line bundle on \(X\) with respect to \(\pi\) in terms of ramified curves and of branch curves. In particular, the existence of Ulrich line bundle is obtained when there is a smooth curve \(C\) of degree \(s\) which is tangent to \(B\) of even order at every point of \(C \cap B\) (see Theorem 1.1). Hence, the non-existence of Ulrich line bundle on \(X\) when \(B\) is general implies that there is no such a curve \(C\) if the given branch curve \(B\) is general enough. One may ask its converse: fix a smooth curve \(C\) of degree \(s\) and ask whether there is a smooth curve \(B\) of degree \(2s\) which is tangent to \(C\) of even order at every point of intersection. The answer is positive: see Proposition 5.3.
The authors also studied the non-existence of Ulrich line bundles on a smooth \(d\)-fold cyclic cover of \(\mathbb{P}^n\) when \(d \le n\) and \(n \ge 3\): see Theorem 1.7. Ulrich bundles; double planes; cyclic coverings Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Coverings in algebraic geometry, Plane and space curves, Divisors, linear systems, invertible sheaves Ulrich line bundles on double planes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(C\) be a reduced projective plane curve of degree \(d\) having \(r\) irreducible components. The complement of \(C\) and a general line \(L\) has associated to it an infinite cyclic covering, and the Alexander polynomial of \(C\), denoted \(\Delta_ C\), is defined to be the Alexander polynomial of this cyclic covering. (This is independent of the choice of \(L\) as long as \(L\) is transverse to \(C\).) If one knows the fundamental group \(\pi_ 1(\mathbb{P}^ 2-C)\) then it is known how to compute \(\Delta_ C\). The main result of the present paper shows how to compute \(\Delta_ C\) without knowing \(\pi_ 1(\mathbb{P}^ 2-C)\). The main theorem (theorem 4.1) is a formula giving \(\Delta_ C\) as a product of polynomials associated to the singularity spectra of the various singular points of \(C\). The authors point out that a similar result (but formulated differently and making stronger hypotheses on the curve and its singularities) is obtained by \textit{A. Libgober} [Singularities, Summer Inst., Arcata/Calif. 1981, Proc. Symp. Pure Math. 40, Part 2, 135-143 (1983; Zbl 0536.14009)]. cyclic covering; Alexander polynomial; fundamental group Loeser, F.; Vaquié, M., Le polynôme d'Alexander d'une courbe plane projective, Topology, 29, 163-173, (1990) Coverings in algebraic geometry, Singularities of curves, local rings, Coverings of curves, fundamental group Le polynôme d'Alexander d'une courbe plane projective. (The Alexander polynomial of a projective plane curve) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(S\) be a smooth minimal surface of general type over the field of complex numbers and denote by \({\mathcal M} (S)\) the coarse moduli space of surface of general type homeomorphic to \(S\), \({\mathcal M} (S)\) is a quasi-projective variety by Gieseker's theorem. Since \(K^2_S>0\), the divisibility \(r(S)\) of the canonical class \(k_S= c_1(K_S) \in H^2 (S,\mathbb{Z})\) is well defined, i.e. \(r(S)= \max \{r\in\mathbb{N} \mid r^{-1} c_1(S) \in H^2 (S,\mathbb{Z})\}\). \(r(S)\) is a positive integer which is invariant under deformation and the set
\[
{\mathcal M}_d (S)=\bigl\{[S']\in {\mathcal M} (S) \mid r(S') =r(S)\bigr\}
\]
is a subvariety of \({\mathcal M} (S)\) and the number of connected components of \({\mathcal M}_d(S)\) is bounded by a function \(\delta\) of the numerical invariants \(K^2_S\), \(\chi ({\mathcal O}_S)\). It is known that \(\delta\) is not bounded [see \textit{M. Manetti}, Compos. Math. 92, N. 3, 285-297 (1994; Zbl 0849.14016)]. Here we prove that ``in general'' \(\delta\) takes quite large values, more precisely we have
Theorem A. For every real number \(4\leq \beta \leq 8\) there exists a sequence \(S_n\) of simply connected surfaces of general type such that:
(a) \(y_n= K^2_{S_n}\), \(x_n= \chi ({\mathcal O}_{S_n}) \to\infty\) as \(n\to \infty\).
(b) \(\lim_{n\to \infty} (y_n/x_n) =\beta\).
(c) \(\delta (S_n) \geq y_n^{(1/5) \log y_n}\) (here \(\delta (S_n)\) is the number of connected components of \({\mathcal M}_d (S_n))\).
Theorem A relies on the explicit description of the connected components in the moduli space of a wide class of surfaces of general type whose Chern numbers spread in all the region \({1\over 2} c_2 \leq c^2_1 \leq 2c_2 \).
Definition B. A finite map between normal algebraic surfaces \(p: X\to Y\) is called a simple iterated double cover associated to a sequence of line bundles \(L_1, \dots, L_n \in\text{Pic} (Y)\) if the following conditions hold:
(1) There exist \(n+1\) normal surfaces \(X=X_0, \dots, X_n=Y\) and \(n\) flat double covers \(\pi_i: X_{i-1} \to X_i\) such that \(p=\pi_n \circ \cdots \circ \pi_1\).
(2) If \(p_i: X_i\to Y\) is the composition of \(\pi_j\)'s \(j>i\) then we have for every \(i=1, \dots, n\) the eigensheaves decomposition \(\pi_{i*} {\mathcal O}_{X_{i-1}}= {\mathcal O}_{X_i} \oplus p^*_i (-L_i)\).
For any sequence \(L_1, \dots, L_n\in \text{Pic} (\mathbb{P}^1 \times \mathbb{P}^1)\) define \((N(L_1, \dots, L_n)\) as the image in the moduli space of the set of surfaces of general type whose canonical model is a simple iterated double cover of \(\mathbb{P}^1 \times \mathbb{P}^1\) associated to \(L_1,\dots,L_n\). The main theme of this paper is to determine sufficient conditions on the sequence \(L_1,\dots,L_n\) in such a way that the set \(N(L_1, \dots, L_n)\) has ``good'' properties; the conditions we find are summarized in the following definition:
Definition C. A sequence \(L_1, \dots, L_n\), \(L_i= {\mathcal O}_{\mathbb{P}^1 \times \mathbb{P}^1} (a_i,b_i)\), \(n\geq 2\) of line bundles on \(\mathbb{P}^1 \times\mathbb{P}^1\) is called a good sequence if it satisfies the following conditions.
(C1) \(a_i\), \(b_i\geq 3\) for every \(i=1, \dots, n\).
(C2) \(\max_{j<i} \min (2a_i-a_j\), \(2b_i-b_j) <0\).
(C3) \(a_n\geq b_n+2\), \(b_{n-1} \geq a_{n-1}+2\).
(C4) \(a_i\), \(b_i\) are even for \(i=2, \dots, n\).
(C5) For every \(i<n\), \(2a_i -a_{i+1} \geq 2\), \(2b_i- b_{i+1} \geq 2\).
The main result we prove is:
Theorem D. Let \(L_1,\dots,L_n\) be a good sequence in sense of definition C, then:
(a) \(N(L_1, \dots, L_n)\) is a nonempty connected component of the moduli space.
(b) \(N(L_1, \dots, L_n)\) is reduced, irreducible and unirational. (For (a) and (b) the condition C5 is not necessary.)
(c) The generic \([\text{S}] \in N(L_1, \dots, L_n)\) has \(\Aut (S)= \mathbb{Z}/2 \mathbb{Z}\).
(d) If \(M_1, \dots, M_m\) is another good sequence and \(N (L_1, \dots, L_n)= N(M_1, \dots, M_m)\) then \(n=m\) and \(L_i=M_i\) for every \(i=1, \dots, n\).
Theorem D gives us some new interesting examples of homeomorphic but not deformation equivalent surfaces of general type. divisibility of canonical class; Picard group; number of connected components of coarse moduli space; Chern class; surface of general type; simple iterated double cover Manetti, M, Iterated double covers and connected components of moduli spaces, Topology, 36, 745-764, (1996) Families, moduli, classification: algebraic theory, Enumerative problems (combinatorial problems) in algebraic geometry, Picard groups, Coverings in algebraic geometry, Moduli, classification: analytic theory; relations with modular forms, Complex-analytic moduli problems Iterated double covers and connected components of moduli spaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems See the preview in Zbl 0513.14008. abelian Galois covering surfaces; fundamental groups; Chern class Yamazaki, T., Yoshida, M.: On Hirzebruch's examples of surfaces withc 1 2 =3c 2. Math. Ann.266, 421-431 (1984) Homotopy theory and fundamental groups in algebraic geometry, Special surfaces, Characteristic classes and numbers in differential topology, Coverings in algebraic geometry On Hirzebruch's examples of surfaces with \(c^ 2_ 1=3c_ 2\) | 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 The authors describe new Zariski pairs, namely pairs of plane curves of the same degree and with the same list of singularities whose complements have non isomorphic fundamental groups. These new examples consist of pairs of real curves such that the associated ramified coverings of the complex projective plane have interesting resolutions. real curves; Zariski pairs; Zariski-van Kampen theorem Namba M., Tsuchihashi H.: On the fundamental groups of Galois covering spaces of the projective plane. Geometriae Dedicata 105, 85--105 (2004) Homotopy theory and fundamental groups in algebraic geometry, Coverings in algebraic geometry On the fundamental groups of Galois covering spaces of the projective plane | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The structure of the article is split into two parts.
In the first part the authors begin with a review of Mochizuki's theory of tame and purely imaginary harmonic bundles on quasi-projective varieties including the particular case of the smooth locus of a Kawamata log terminal (klt for short) variety. Also they review the notion of a Higgs sheaf on singular spaces, and discuss the stability of Higgs bundles defined on the smooth locus of a klt variety. Then they show a central existence result for harmonic structures.
In the second part, the authors show how existence of harmonic structures leads to nonabelian Hodge correspondences pertaining to local systems on the smooth locus of a klt space. Hereafter, they prove their main result on quasi-étale uniformisation (Theorem 1.5). harmonic metrics on Higgs bundles; klt varieties; quasiétale uniformisation problem; quotients of the unit ball; nonabelian Hodge correspondence Uniformization of complex manifolds, Coverings in algebraic geometry, Minimal model program (Mori theory, extremal rays), Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) Harmonic metrics on Higgs sheaves and uniformization of varieties 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 Let \(X,X'\) be complex connected projective smooth k-folds and let \(\pi: X'\to X\) be a cyclic covering branched along an ample divisor \(\Delta\). A Lefschetz type theorem due to \textit{M. Cornalba} [Boll. Unione Mat. Ital., V. Ser., A 18, 323-328 (1981; Zbl 0462.14007)] says that the Betti numbers of \(X,X'\) satisfy: \(b_ i(X)=b_ i(X')\) for \(i\leq k-1\) and \((*)\quad b_ k(X)\leq b_ k(X').\) For \(k=2\), (*) is always a strict inequality; the first part of the paper contains the classification of triplets \((X,X',\pi)\) for which \(b_ k(X')-b_ k(X)\leq 10\), in case \(k=2\). The last section of the paper is aimed at proving that equality in (*) is a rare phenomenon (obviously it happens for every odd k when \(\pi\) is the double cover of \({\mathbb{P}}^ k\) given by a smooth hyperquadric). Partial results are proved in case \(k\geq 3\) assuming that \(n=\deg(\pi)\geq k+2\) and that the line bundle \({\mathcal L}\in Pic(X)\), defined by the condition \(\Delta\in | {\mathcal L}^{\otimes n}|\), is ample and spanned by its global sections. The study is mainly based on the properties of the linear system \(| K_ X\otimes {\mathcal L}^{\otimes (n-1)}|\). Recently \textit{Wisniewski} [``On topological properties of some coverings'' (Preprint 1990)] has completed and extended the higher dimensional results, by proving that (*) is never an equality for \(n\geq k\) in a more general setting of triplets \((X,X',\pi)\) than cyclic covers considered in the paper. cyclic covering branched along an ample divisor; Lefschetz type theorem; Betti numbers Coverings in algebraic geometry, Topological properties in algebraic geometry, Ramification problems in algebraic geometry, Special surfaces Topological properties of cyclic coverings branched along an ample divisor | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 0686.00006.]
This paper records seminar talks explaining with examples, the author's method of exhibiting finite groups as Galois groups of regular extensions of the field \({\mathbb{Q}}(X)\) of rational functions over the rationals. The basic idea is to construct a (ramified) cover of the complex projective line with the desired group as group of deck transformations, and a rational structure on this cover, by taking the moduli space of such covers and finding a rational point on it. The number r of ramification points is a key parameter here: when \(r=3\), the method implies that the given finite group G can be generated in a very special way by two elements. For larger r, the situation becomes more complicated.
There are some quite general conditions on a group G, together with a faithful representation T of it as permutations of n letters and with an r-tuple \({\mathbb{C}}\) of conjugacy classes of G, that guarantee that G can be exhibited as the Galois group of a regular extension of \({\mathbb{Q}}(X)\) ramified at r rational points. The author constructs a parameter space \({\mathcal H}\), depending on the pair (\({\mathbb{C}},T)\), which parameterizes the covers associated to that pair. \({\mathcal H}\) is a cover of projective r- space with the discriminant locus deleted; for \(r>3\) it is required that there be a \({\mathbb{Q}}\) rational point on the pullback of \({\mathcal H}\) to the r-fold product of the projective line. One of the main results of the paper is the exhibition of a curve cover of the projective line, ramified over only zero, one and infinity (and identified with the projective normalization of the quotient of the upper by a certain finite index subgroup of \(PSL_ 2({\mathbb{Z}}))\) such that there is a rational point in the above pullback if and only if there is a rational point of the curve cover not in a fibre over a ramification point.
The author includes an extended example using \(G=A_ 5\) (the alternating group). number of ramification points; inverse Galois problem; finite groups; Galois groups; cover of the complex projective line; rational point M. D. Fried, ``Arithmetic of \(3\) and \(4\) branch point covers: A bridge provided by noncongruence subgroups of \({\mathrm SL}_ 2(Z)\)'' in Séminaire de theorie des nombres (Paris, 1987--88.) , Progr. Math. 81 , Birkhäuser, Boston, 1990, 77--117. Inverse Galois theory, Galois theory, Ramification problems in algebraic geometry, Coverings in algebraic geometry Arithmetic of 3 and 4 branch point covers. A bridge provided by noncongruence subgroups of \(SL_ 2({\mathbb{Z}})\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We show how to construct some old and new surfaces of general type with vanishing geometric genus from double planes, by computing explicit equations of their branch curves. surfaces of general type; double planes; branch curve Surfaces of general type, Coverings in algebraic geometry Surfaces of general type with vanishing geometric genus from double planes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The article contains two main results, both about branched coverings of Grassmannians.
The first asserts that if \(P: G(r,n) \to \mathbb{P}^{\binom{n}{r}-1}\) is the Plücker embedding of \(r\)-dimensional subspaces of \(\mathbb{C}^n\) and \(n\geq r+2\geq 1\), then there exists a projective manifold \(Y\) and branched covering \(f: Y\to G(r,n)\) of degree \(\dim G(r,n)\) such that there is no branched covering \(f': Y' \to \mathbb{P}^{\binom{n}{r}-1}\) of a projective manifold \(Y'\) with \(f\) the pullback of \(f'\) under \(P\).
The second result shows that an obvious and natural way to construct branched coverings only leads to trivial examples. More precisely, if \(G(r,n')\subset G(r,n)\) is the immersion defined by a field embedding \(\mathbb{C}^{n'} \subset \mathbb{C}^n\), and \(X\) is a subvariety of \(G(r,n)\) whose homology class is a positive multiple of the homology class of \(G(r,n')\). Then \(X\) is a translation of \(G(r,n')\) by an automorphism of \(G(r,n)\). Grassmannians; branched coverings; Plücker embedding Grassmannians, Schubert varieties, flag manifolds, Coverings in algebraic geometry Two results on branched coverings of Grassmannians | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(f: V\rightarrow W\) be a finite polynomial mapping of algebraic subsets \(V,W\) of \({\mathbf k}^n\) and~\({\mathbf k}^m,\) respectively, with \(n\leq m.\) \textit{M. Kwieciński} [J. Pure Appl. Algebra 76, No. 2, 151--153 (1991; Zbl 0753.14002)] proved that there exists a finite polynomial mapping \(F:{\mathbf k}^n\rightarrow {\mathbf k}^m\) such that \(F|_V=f.\) In this note we prove that, if \(V,W\subset {\mathbf k}^n\) are smooth of dimension \(k\) with \(3k+2\leq n,\) and \(f:V\rightarrow W\) is finite, dominated and dominated on every component, then there exists a finite polynomial mapping \(F: {\mathbf k}^n\rightarrow {\mathbf k}^n\) such that \(F|_V=f\) and \(\text{gdeg} F\leq (\text{gdeg} f)^{k+1}.\) This improves earlier results of the author. finite mapping; geometric degree; extension of mappings Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem), Coverings in algebraic geometry, Varieties and morphisms A note on geometric degree of finite extensions of mappings from a smooth variety | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 the definition of a flecnode on a surface in \(\mathbb{R}^3\) to a definition for a general immersed manifold in Euclidean space. Instead of considering a flecnode as a point on the manifold, we consider it as a pair of a normal vector and a tangent vector, called the flecnodal pair. The structure of this set is considered, as well as its connection to binomials and \(A_3\) singularities in the set of height functions. The specific case of a surface immersed in \(\mathbb{R}^4\) is studied in more detail, with the generic singularities of the flecnodal normals and the flecnodal tangents classified. Finally, the connection between the flecnodals and bitangencies are studied, especially in the case where the dimension of the manifold equals the codimension. Global theory of singularities, Coverings in algebraic geometry, Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces, Critical points of functions and mappings on manifolds The geometry of flecnodal pairs | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems One important invariant in the classification of algebraic surfaces is the fundamental group of the complement of the branch curve of a generic projection of the surface. To compute this fundamental group the surface is degenerated to a union of planes whose branch curve is a line arrangement.
The fundamental groups of such complements are quotients of Artin groups. In the paper certain quotients of Artin groups corresponding to different numerations of the 6-point singularity are identified. Isomorphisms between certain quotients of the groups corresponding to four particular 6-point ordering types are given. singularities; coverings; fundamental groups; surfaces; Coxeter and Artin groups Singularities in algebraic geometry, Coverings in algebraic geometry, Coverings of curves, fundamental group, Reflection and Coxeter groups (group-theoretic aspects) Algebraic invariants in classification of 6-points in degenerations of surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(U\) be an open neighborhood of \(0\in\mathbb{C}^n\) and \(\pi:X\to U\) be a finite Galois covering such that \(\pi^{-1}(0)=\{0\}\). Let \(B_\pi= \sum^s_{\nu=1} r_\nu B_\nu\) be brance locus \((B_i\) the irreducible components of the brance locus and \(r_i\) the ramification index of \(\pi\) along \(B_i)\).
Necessary, and sometimes sufficient, conditions for \((X,0)\) being a log-terminal or log-canonical singularity are given in terms of \(B_\pi\). Galois coverings \(\pi:(X,0)\to (\mathbb{C}^n,0)\) with \(B_\pi=D\) for a given divisor \(D\) are studied and characterised.
A method constructing explicitly resolutions of 2-dimensional Abel covering singularities is given. The self-intersection number, the genus of each irreducible component and the dual graphs of their exceptional sets are explicitly described in terms of \(B_\pi\) and the covering transformation group \(\text{Gal} (X\mid U)\). Finally, a necessary and sufficient condition for \((X,0)\) is given to be quasi-Gorenstein. log-canonical; Galois coverings; resolutions; singularities Local complex singularities, Coverings in algebraic geometry, Modifications; resolution of singularities (complex-analytic aspects) Galois covering singularities. I | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(x \in (X, \Delta)\) be a klt singularity defined over the complex number field \(\mathbf{C}\). The klt singularities form an important class of singularities in the minimal model prgoram, and they can be regarded as the local analog of Fano varieties. Recently, \textit{L. Braun} proved that the \textit{regional fundamental group} \(\pi_1^{\mathrm{reg}}(X, x)\) is finite [Invent. Math. 226, No. 3, 845--896 (2021; Zbl 1479.14029)]. This implies Kollár's conjecture: the \textit{local fundamental group} \(\pi_1^{\mathrm{loc}}(X, x)\) is finite. The main result of the paper under review (Theorem 2 and Corollary 1) is that the class of regional fundamental groups or local fundamental groups of klt singularities of dimension \(n\) satisfies the \textit{Jordan property}. This means that there exists a positive integer \(c=c(n)\) such that for any \(G:=\pi_1^{\mathrm{reg}}(X, x)\) or \(\pi_1^{\mathrm{loc}}(X, x)\) with \(\dim X = n\), there exists a normal abelian subgroup \(A \trianglelefteq G\) of rank at most \(n\) and index at most \(c\). This settles a conjecture of Shokurov. To prove the main theorem, the authors follow the global-to-local philosophy: they first estabilish the Jordan property for the orbifold fundamental groups of log Fano pairs using recent results of Prokhorov-Shramov (Jordan property for birational automorphism groups of Fano varieties) and Birkar (BAB conjecture), and then they study how the Jordan property can be lifted from the orbifold fundamental group of the Kollár component of a plt blowup to a suitable neighborhood of a klt singularity. Several interesting applications are given. Especially, they show the existence of effective simultaneous index 1 covers for klt singularities (Theorem 4). This gives a refinement of \textit{D. Greb} et al.'s result [Duke Math. J. 165, No. 10, 1965--2004 (2016; Zbl 1360.14094]. local fundamental group; klt singularities; Jordan property Singularities in algebraic geometry, Coverings in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry, Minimal model program (Mori theory, extremal rays) The Jordan property for local fundamental groups | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We describe all the factorial double covers of \(\mathbb{P}^3\) ramified along nodal quartic surfaces. double solid; factoriality; nodal surface Coverings in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, \(3\)-folds, Fano varieties, Divisors, linear systems, invertible sheaves Factorial 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 authors consider smooth complex varieties \(X\) such that the cotangent bundle \(\Omega_X^1\) is numerically effective. The main result of the paper under review is a bound for the degree of a finite surjective mapping \(f: X\to \mathbb{P}^n\), namely, \(\deg(f)\geq 2^{[\sqrt n]}\), where \([~]\) denotes the integer part. It appears somewhat unexpected that this bound is almost exponentially, but it comes out very natural from just two ingredients. First, the nef condition ensures the existence of points \(x\in X\) where \(df_x\) drops rank substantially, since the locus \(S_k=
\{x\in X\mid\operatorname{rank} df_x\leq n-k \}\) (whose expected dimension is \(n-k^2\)) can be shown to be non-empty for \(k=[\sqrt n]\). On the other hand, the local degree of \(f\) at such points must be larger than \(2^k\). Indeed, the authors indicate that this bound may be far from being optimal, and may be seen as a suggestion that the projective geometry associated to non-negative varieties grows rapidly with the dimension.
In the second part, this result is applied to subvarieties of abelian varieties, and it is shown that the top self-intersection of its canonical bundle is bounded from below by \(2^{[\sqrt n]}\). Furthermore, some variants of the estimate are given. numerically effective cotangent bundle; degree Embeddings in algebraic geometry, Coverings in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] A remark on projective embeddings of varieties with non-negative cotangent 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 In the paper under review, the authors study the problem of describing the irregularity of cyclic branched coverings of surfaces with abelian quotient singularities, and they use this description to find formulas for the particular case of the weighted projective plane \(\mathbb{P}^{2}_{w}\). Let us recall that an abelian quotient singular point in dimension two is necessarily a cyclic singularity. The main result of this paper describes the dimension of the equivariant spaces of the first cohomology of a \(d\)-cyclic cover \(\rho : X \rightarrow \mathbb{P}^{2}_{w}\) ramified along a curve \[C = \sum_{j}n_{j}C_{j}.\] The cover \(\rho\) naturally defines a divisor \(H\) such that \(dH\) is linearly equivalent to \(C\). If \(K_{\mathbb{P}^{2}_{w}}\) is the canonical divisor of \(\mathbb{P}^{2}_{w}\), and \[C^{(k)} = \sum_{j=1}^{r} \bigg\lfloor \frac{kn_{j}}{d}\bigg\rfloor C_{j}, \quad 0 \leq k < d,\] then these dimensions are given as the cokernel of the evaluation linear maps \[\pi^{(k)} : H^{0}(\mathbb{P}^{2}_{w}, \mathcal{O}_{\mathbb{P}^{2}_{w}}(kH + K_{\mathbb{P}^{2}_{w}} - C^{(k)})) \rightarrow \bigoplus_{P\in S}\frac{ \mathcal{O}_{\mathbb{P}^{2}_{w}}(kH + K_{\mathbb{P}^{2}_{w}} - C^{(k)})}{\mathcal{M}^{(k)}_{C,P}},\] where \(\mathcal{M}^{(k)}_{C,P}\) is defined as the quasi-adjunction-type \(\mathcal{O}_{\mathbb{P}^{2}_{w},P}\)-module, namely \[ \mathcal{M}^{(k)}_{C,P}:= \bigg\{ g \in \mathcal{O}_{\mathbb{P}^{2}_{w},P}(kH + K_{\mathbb{P}^{2}_{w}} - C^{(k)})\, : \, \mathrm{mult}_{E_{v}} \, \pi^{*}g > \sum_{j=1}^{r} \bigg[\frac{kn_{j}}{d}\bigg]m_{v_{j}}-v_{v}, \,\,\, \forall v \in \Gamma_{P} \bigg\}.\] Here the symbol \([\cdot ]\) denotes the decimal part of a rational number, and the multiplicities \(m_{v_{j}}\) and \(v_{v}\) are provided by \(\pi^{*}C_{j} = \hat{C}_{j} + \sum_{P \in S}\sum_{v \in \Gamma_{P}}m_{v_{j}}E_{v}\) and \(K_{\pi} = \sum_{P\in S}\sum_{v \in \Gamma_{P}}(v_{v}-1)E_{v}\) for an embedded \(\mathbb{Q}\)-resolution \(\pi\) of \(C \subset \mathbb{P}^{2}_{w}\), where \(S\) is the set of points of \(\mathbb{P}^{2}_{w}\) blown up in the resolution \(\pi\), and \(\Gamma_{P}\) is the dual graph associated with the resolution of \(P \in S\). As a consequence, one has \[h^{1}(X,\mathbb{C}) = 2\sum_{k=0}^{d-1}\dim\, \mathrm{coker} \, \pi^{(k)}.\]
As a non-trivial application of their studies, the authors present a Zariski pair of irreducible curves in a weighted projective plane, that is, two curves in the same plane \(\mathbb{P}^{2}_{w}\) with the same degree and local type of singularities, but whose embeddings are not homeomorphic. Zariski pairs; embedded Q-resolution; multiplier ideals; quotient singularity; Alexander polynomials; weighted projective plane; cyclic branched covering; normal surfaces; adjunction ideals; weighted blowup Coverings in algebraic geometry, Coverings of curves, fundamental group, Topological properties in algebraic geometry, Plane and space curves Cyclic branched coverings of surfaces with abelian quotient 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 The starting point of the paper is Segre's paper on the singularities of the branch curve \(B\) of a ramified cover \(\pi: X \to \mathbb{P}^2\), where \(X\) is a surface of any degree in \(\mathbb{P}^3\) (see \textit{B. Segre}, [Memorie Accad. d'Italia, Roma 1; Mat. Nr. 4, 31 p. (1930; JFM 56.0562.01)] and also \textit{O. Zariski} [Am. J. 51, 305--328 (1929; JFM 55.0806.01)]). Segre gave a necessary and sufficient condition for a curve to be the branch curve of a smooth surface in \(\mathbb{P}^3\).
The present paper generalizes Segre's result to any smooth surface \(X \subset \mathbb{P}^N\), giving a necessary and sufficient condition for a pair of plane curves \(B,E\) to be the branch curve of a surface \(X \subset \mathbb{P}^N\) and the image of the double curve of a \(\mathbb{P}^3\)-model of \(X\).
As a consequence the paper gives a new constructive proof of the \textit{O. Chisini} conjecture [Ist. Lombardo Sci. Lett., Rend., Cl. Sci. Mat. Natur., III. Ser. 8(77), 339--356 (1944; Zbl 0061.35305)], concerning the uniqueness of the generic ramified cover of the plane of degree \(\geq 5\) determined by its branch curve, so improving the results by \textit{Vik. S. Kulikov} [Izv. Math. 63, No. 6, 1139--1170 (1999); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 63, No. 6, 83--116 (1999; Zbl 0962.14005); 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 part I, cf. [\textit{M. Friedman, M. Leyenson} and \textit{E. Shustin}, Int. J. Math. 22, No. 5, 619--653 (2011; Zbl 1233.14022)]. branch curves; adjoint curves; ramified covers Coverings in algebraic geometry On ramified covers of the projective plane. II: Generalizing Segre's theory | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For the entire collection see Zbl 0633.00020.]
Let L be a reduced divisor of complex projective plane \({\mathbb{P}}^ 2 \)consisting of r distinct complex lines \(L_ 1,...,L_ r\) (r\(\geq 2)\). Let \(p:\quad X\to {\mathbb{P}}\quad 2\) be a Galois branched covering of \({\mathbb{P}}^ 2 \)associated with a \({\mathbb{Z}}/n{\mathbb{Z}}\) Hurewicz homomorphism. Then X admits a unique normal complex algebraic structure such that the unbranched covering \(p_ 0=p| X_ 0:\quad X_ 0\to {\mathbb{P}}\quad 2\) is locally biholomorphic, \(p:\quad X\to {\mathbb{P}}\quad 2\) is holomorphic and the Galois group G (\(\cong ({\mathbb{Z}}/n{\mathbb{Z}})^{r- 1})\) acts on X as a group of biholomorphisms of X onto itself where \(X_ 0:=p^{-1}({\mathbb{P}}\) 2). In this paper, the author refers to a pair (G,X) (or \(p:\quad X\to {\mathbb{P}}\quad 2\)) as the Kummer branched covering of \({\mathbb{P}}^ 2 \)with branch divisor nL and with \((G,X)=K({\mathbb{P}}^ 2:\)nL).
Theorem. Let L and L' be reduced divisors of \({\mathbb{P}}^ 2 \)consisting of r and r' complex lines, respectively, and let t(L) be the maximum of multiplicities of singular points of L. Suppose that: \((1)\quad each\) line \(L_ i\) contains at least 3 singular points of L; and \((2)\quad either\) t(L)\(\leq 3\) and \(n\geq 6\), or t(L)\(\geq 4\) and \(n\geq 5\cdot (t(L)- 1)\). If there is a biholomorphism \(h:\quad X\to X'\), it is an equivalence in the sense that \(G^{'h} (=\{h^{-1}\circ g'\circ h| g'\in G'\}) =G\). In particular, it induces a projective transformation \(h':\quad {\mathbb{P}}\quad 2\to {\mathbb{P}}\quad 2\) sending L onto L'. Galois covering; reduced divisor of complex projective plane; Galois branched covering; Hurewicz homomorphism; Kummer branched covering Coverings in algebraic geometry, Ramification problems in algebraic geometry, Holomorphic fiber spaces, Singularities of surfaces or higher-dimensional varieties On biholomorphisms between some Kummer branched covering spaces of complex projective plane | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author approaches the existence problem of a dihedral cover of Hirzebruch surfaces \(\Sigma_d\) with prescribed branch curves via the theory of elliptic surfaces and Mordell-Weil lattices. The elliptic surface \(S\) appears as a double cover of \(\Sigma_d\) and the potential branch curves on \(S\) are produced as sections of \(S/\mathbb{P}^1\) by the group operations on the Mordell-Weil groups.
The setting is as follows. Let \(\pi: X\rightarrow Y\) be a Galois cover with \(\mathrm{Gal}(\pi)\simeq D_{2p}\), the dihedral group of order \(2p\). The cover \(\pi\) is called \textit{elliptic} if (i) the canonical double cover \(f: D(X/Y)\rightarrow Y\) corresponding to the index two subgroup of \(\mathrm{Gal}(\pi)\) has a structure of a relatively minimal elliptic surface \(D(X/Y)=S\rightarrow \mathbb{P}^1\) with a section, and (ii) the covering transformation \(\sigma_f\) of \(f\) coincides with the fiberwise inversion of the elliptic surface \(S\). Note that then \(Y\) is ruled over \(\mathbb{P}^1\), hence is a blowup \(\widehat{\Sigma_d}\) of a Hirzebruch surface \(\Sigma_d\) for some even \(d\).
Under these notation, Theorem 3.3 is concisely as follows. Let \(p\) be an odd prime which does not divide the order of the torsion subgroup of \(\mathrm{MW}(S)\). Let \(s_1,s_2\in \mathrm{MW}(S)\) be sections such that \(s_i\) are not \(p\)-divisible in the Mordell-Weil group \(\mathrm{MW}(S)\). Then there exists a \(D_{2p}\)-cover \(X_{p}\rightarrow \widehat{\Sigma_d}\) such that the horizontal part of the branch of \(X_p/S\) is \(s_1+s_2+\sigma_f(s_1+s_2)\) if and only if the images \(\overline{s}_i\in \mathrm{MW}(S)\otimes \mathbb{Z}/p\mathbb{Z}\) are linearly dependent.
As an application, the author considers the case \(S\) is rational and constructs a Zariski pair of conic-line arrangement of degree \(7\). Dihedral cover; Mordell-Weil group; elliptic surface; Zariski pair Tokunaga, H.: Sections of elliptic surfaces and Zariski pairs for conic-line arrangements via dihedral covers. J. Math. Soc. Japan \textbf{66}(2), 613-640 (2014) Coverings in algebraic geometry, Elliptic surfaces, elliptic or Calabi-Yau fibrations, Rational and ruled surfaces Sections of elliptic surfaces and Zariski pairs for conic-line arrangements via dihedral 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 S. Iitaka conjectured that a compact Kähler manifold \(X^ n\) whose universal covering is \(\mathbb{C}^ n\) can be recovered by a complex \(n\)- torus. \(X^ n\) has to be assumed Kählerian in this conjecture, and Iitaka proved the conjecture for \(n =2\). The authors prove it in the category of projective manifolds for \(n = 3\): any projective threefold with universal covering \(\mathbb{C}^ 3\) can be covered by an abelian threefold. Furthermore, the following result is given: if \(X = \mathbb{C}^ n/\Gamma\) is compact Kählerian, and if, in terms of suitable coordinates in \(\mathbb{C}^ n\), the components of each element in \(\Gamma\) are rational functions, then \(X\) can be covered by a complex \(n\)-torus. covering by abelian threefold; Kähler manifold; universal covering; projective threefold Coverings in algebraic geometry, \(3\)-folds, Projective techniques in algebraic geometry, Compact Kähler manifolds: generalizations, classification On projective manifolds covered by space \(\mathbb{C}^ n\) | 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 We give a criterion of analytic irreducibility of plane curves valid in every characteristic, as well as an effective bound for possible truncation of the equations. This proceeding is a summary of our paper [Ann. Fac. Sci. Toulouse, VI. Sér., Math. 14, No. 3, 353--394 (2005; Zbl 1103.13013)] with a few proofs. We give some details and some proofs of ``very well known facts''. At the end, we give an example for the case of curves with one place at infinity and we propose some questions. V. Cossart and G. Moreno-Socías, Irreducibility criterion: a geometric point of view, Valuation theory and its applications, II , Saskatoon, SK, 1999, Fields Inst. Commun., 33 , Amer. Math. Soc., Providence, RI, 2003, pp.,27-42. Singularities in algebraic geometry, Valuations and their generalizations for commutative rings, Completion of commutative rings, Plane and space curves, Coverings in algebraic geometry Irreducibility criterion: A geometric point of view | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We show by toroidal methods that Igusa's desingularisation (and hence any desingularisation) of Satake's compactification of \(S_ 2/\Gamma_ 2(n)\) is simply-connected for \(n\geq 2\). Here \(S_ 2\) is Siegel's upper half plane of degree 2 and \(\Gamma_ 2(n)\) the congruence subgroup of level n in \(SP_ n{\mathbb{Z}}\). Siegel modular variety; Satake compactification; desingularisation; fundamental group Holger Heidrich and Friedrich W. Knöller, Über die Fundamentalgruppen Siegelscher Modulvarietäten vom Grade 2, Manuscripta Math. 57 (1987), no. 3, 249 -- 262 (German, with English summary). Special surfaces, Theta series; Weil representation; theta correspondences, Coverings in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Automorphic functions Über die Fundamentalgruppen Siegelscher Modulvarietäten vom Grade 2. (On the fundamental groups of Siegel modular varieties of degree 2.) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We calculate the first Betti number of an abelian covering of a \(CW\)-complex \(X\) as the number of points with cyclotomic coordinates (of orders determined by the Galois group) which belong to a certain subvariety of a torus constructed from the fundamental group of \(X\). This generalizes the classical formulas for the cyclic coverings due to Zariski and Fox. We also describe certain properties of these subvarieties of tori in the case when \(X\) is a complement to an algebraic curve in \(\mathbb{C}\mathbb{P}^ 2\) which are analogs of the Traldi-Torres relations from link theory and the divisibility theorem for Alexander polynomials of plane algebraic curves. Betti number of an abelian covering of a \(CW\)-complex; cyclotomic coordinates; fundamental group; complement to an algebraic curve; link; Alexander polynomials of plane algebraic curves Libgober A.: On the homology of finite abelian coverings. Topol. Appl. 43(2), 157--166 (1992) Coverings in algebraic geometry, Fundamental group, presentations, free differential calculus, Homotopy theory and fundamental groups in algebraic geometry, Coverings of curves, fundamental group, Covering spaces and low-dimensional topology On the homology of finite abelian 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 irreducible complex projective nonsingular algebraic surface endowed with an elliptic fibration \(p: S\to \Delta\). The authors prove the following theorem: (i) the universal covering space of S is holomorphically convex; (ii) any unramified covering of S is holomorphically convex provided that at least one singular fibre of p is not of type \(mI_ 0\) (i.e. not all singular fibres of p are smooth elliptic curves with some multiplicity). They also give an example of an abelian surface, which is the product of two elliptic curves, having an infinite cyclic covering with no nonconstant holomorphic functions. - As a corollary of (ii) the authors prove the following. Assume that p has at least one singular fibre not of type \(mI_ 0\); if \(C\subset S\) is an irreducible curve with \(C^ 2>0\), then the image of \(\pi_ 1(\bar C)\), (\(\bar C\) a nonsingular model of C) has finite index in \(\pi_ 1(S)\). In particular, if C is rational, it turns out that \(\pi_ 1(S)\) is finite. This is related to a question discussed by \textit{M. V. Nori} [Ann. Sci. Éc. Norm. Super., IV. Sér. 16, 305-344 (1983; Zbl 0527.14016)]. elliptic surfaces; holomorphic convexity; fundamental group; elliptic fibration; unramified covering R. V. Gurjar and A. R. Shastri, ''Covering spaces of an elliptic surface,'' Compositio Math., vol. 54, iss. 1, pp. 95-104, 1985. Special surfaces, Compact complex surfaces, Coverings in algebraic geometry Covering spaces of an elliptic 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 \(f:C\longrightarrow D\) be a nonconstant separable morphism between irreducible smooth projective curves defined over an algebraically closed field. We say that \(f\) is genuinely ramified if \(\mathcal{O}_D\) is the maximal semistable subbundle of \(f_*\mathcal{O}_C\) (equivalently, the induced homomorphism \(f_*: \pi_1^{\text{et}}(C)\longrightarrow\pi_1^{\text{et}}(D)\) of étale fundamental groups is surjective). We prove that the pullback \(f^*E\longrightarrow C\) is stable for every stable vector bundle \(E\) on \(D\) if and only if \(f\) is genuinely ramified. Coverings in algebraic geometry, Vector bundles on surfaces and higher-dimensional varieties, and their moduli Ramified covering maps and stability of pulled-back 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 To give a signature on a complex manifold one specifies an open cover of the manifold by simply-connected subsets \(U_{\alpha}\) together with surjective analytic maps \(\pi_{\alpha}: \tilde U_{\alpha}\to U_{\alpha '}\) where each \(\tilde U_{\alpha}\) is an open neighbourhood of zero in \({\mathbb{C}}^ n\) invariant under the action of a finite group \(\Gamma_{\alpha}\) of automorphisms of \({\mathbb{C}}^ n\), and the fibres of \(\pi_{\alpha}\) are orbits of \(\Gamma_{\alpha}\). This paper is concerned with the question whether, given a signature S on a compact connected algebraic manifold X, there exists a Galois covering \(\pi: Y\to X\) with Y projective space, affine space or the complex ball such that S is the signature induced by \(\pi\). Some necessary conditions and examples are given. Coxeter group of reflections; discrete group; automorphism groups of complex spaces; signature; Galois covering; projective space; affine space; complex ball V. P. Kostov, ''Versal deformations of differential forms of degree {\(\alpha\)} on the line,'' Funkts. Anal. Prilozhen.,18, No. 4, 81--82 (1984). Complex Lie groups, group actions on complex spaces, Reflection groups, reflection geometries, Other geometric groups, including crystallographic groups, Geometries with algebraic manifold structure, Coverings in algebraic geometry, Group actions on varieties or schemes (quotients) Discrete groups of reflections in the complex ball | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems It is known since \textit{A. Beauville} [Invent. Math. 55, 121--140 (1979; Zbl 0403.14006)] that the canonical map of a smooth complex algebraic surface of general type is, if finite, of degree \(d\) at most 36. So far only examples with degree up to 24 have been constructed. For the case of dimension 3, \textit{C. D. Hacon} [Proc. Japan Acad., Ser. A 80, No. 8, 166--167 (2004; Zbl 1068.14046)] has shown that if \(X\) is a Gorenstein minimal complex projective threefold of general type with at most locally factorial terminal singularities, then \(d\leq 576\). In this paper under review, this bound is improved to 360 (as in the case of surfaces, this theoretical limit can be achieved only for ball quotient varieties). The authors also classify the case where the canonical map is an abelian cover of \(\mathbb P^3\) (the possible canonical degrees are \(2^m\) with \(1\leq m\leq 5\)). Gorenstein threefold; canonical map; abelian cover --------, On the canonical degrees of Gorenstein threefolds of general type, Geom. Dedicata 185 (2016), no. 1, 123--130. \(3\)-folds, Coverings in algebraic geometry On the canonical degrees of Gorenstein threefolds of general type | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Two conjectures about the characters of the Hecke algebra \(H_ n(q)\) of type \(A_{n-1}\), evaluated at elements of its Kazhdan-Lusztig basis are announced and some supporting evidence for the two conjectures is provided. One of the two conjectures asserts that certain integral linear combinations of irreducible characters take values on the Kazhdan-Lusztig basis which are polynomials in \(q\) with nonnegative, symmetric, and unimodal integer coefficients. Certain permutations, called codominant, whose corresponding Schubert varieties are smooth and very simple to describe, can be picked. The second conjecture states that for each Kazhdan-Lusztig basis element \(C_ w'\) there is a sum \(C_{w_ 1}'+ \cdots + C_{w_ k}'\) of basis elements with \(w_ j\) codominant, such that \(\chi(C_ w') = \chi(C_{w_ 1}' + \cdots + C_{w_ k}')\) for every Hecke algebra character \(\chi\).
A conjectured immanant inequality for Jacobi-Trudi matrices defined in this paper is proved and it is shown how the two conjectures mentioned above would imply stronger inequalities of a similar kind. codominant permutations; Hecke algebra; Kazhdan-Lusztig basis; irreducible characters; Schubert varieties; immanant inequality for Jacobi-Trudi matrices M. Haiman. ''Hecke algebra characters and immanant conjectures''. J. Amer. Math. Soc. 6 (1993), pp. 569--595.DOI. Representation theory for linear algebraic groups, Linear algebraic groups over finite fields, Symmetric functions and generalizations, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Modular representations and characters, Grassmannians, Schubert varieties, flag manifolds, Other matrix groups over fields Hecke algebra characters and immanant conjectures | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(f: X\to Y\) be a finite galois covering (with group G) of projective varieties over a field k. Let \({\mathcal F}\) be a coherent sheaf on Y. We investigate the kG-module structure of the cohomology groups \(H^ i(X,f^*({\mathcal F})).\)
We have two general results. Theorem 1: These cohomology groups can be expressed as the homology of a finite complex of finitely generated free kG-modules; and its corollary, theorem 2: If all except one of these cohomology groups are trivial then the remaining one is a free kG-module.
We have two results where X and Y (of genus \(g)\) are curves. Theorem 3: (a corollary of theorem 2): If \({\mathcal L}\) is an invertible sheaf on Y with \(\deg({\mathcal L})>2(g-1)\) then \(H^ 0(X,f^*{\mathcal L}))\cong kG^{\deg ({\mathcal L})-g-1}\) (and \(H^ i=(0)\) for \(i>0)\); and theorem 4: Let \({\mathcal K}\) be the canonical sheaf of Y then, as a kG-module \(H^ 0(X,f^*({\mathcal K}))\) is determined by the fact it is the kernel of a projection from \(kG^ g\) to the augmentation ideal of kG. Finally, it is remarked (theorem 5) that the existence of the projection of theorem 4 places a restriction on the possible covering groups G which has not been previously known. Galois module structure; finite galois covering; cohomology groups Taylor, R.L.: On congruences between modular forms. Ph.D. Thesis, ProQuest LLC, Princeton University, Ann Arbor (1988) Étale and other Grothendieck topologies and (co)homologies, Coverings in algebraic geometry, Coverings of curves, fundamental group On Galois module structure of the cohomology groups of an algebraic variety | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 0665.00008.]
Let \(X'\) be a covering of \({\mathbb{P}}^ 2\) ramified along a union of lines with \(ramification\quad index\quad n\) and let X be its minimal desingularization, which admits a finite morphism \(X\to F\) where F is a suitable blow-up of \({\mathbb{P}}^ 2\). The Galois group A of X over F acts on X, hence on its cohomology. The main result of this paper is a formula which computes \(e(\alpha)=\sum_{i}\dim H^ i(\alpha) \), where \(\alpha\) is a character of A and \(H^ i(\alpha)\) is the corresponding eigenspace of \(H^ i(X)\). Then the notion of generic character is introduced: roughly speaking this is a character which is non trivial on the inertia subgroups of A at the divisors of F corresponding to the lines of the configuration and the exceptional divisors of the blowing up of \({\mathbb{P}}^ 2\). For such a character \(\alpha\) it is proved that \(H^ 1(\alpha)=H^ 3(\alpha)=0\) and the dimensions of \(H^ 2(\alpha)\) and of its Hodge components \(H^{2,0}(\alpha)\), \(H^{1,1}(\alpha)\), \(H^{0,2}(\alpha)\) are computed. It may be worth remarking that the interest on such coverings of \({\mathbb{P}}^ 2\) was called some years ago by \textit{F. Hirzebruch} [in Arithmetical and geometry Pap. dedic. I. R. Shafarevich, Vol. II: Geometry, Proc. Math. 36, 113-140 (1983; Zbl 0527.14033)] who proved that many surfaces \(X'\) of this kind are of general type and have Chern numbers in the non densely populated region \(2c_ 2(X)\leq c_ 1(X)^ 2\leq 3c_ 2(X)\). Hodge numbers; Kummer covering; Miyaoka-Yau-inequality; ramification; minimal desingularization; generic character; Chern numbers Coverings in algebraic geometry, Projective techniques in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Transcendental methods, Hodge theory (algebro-geometric aspects), Singularities in algebraic geometry, Characteristic classes and numbers in differential topology Hodge numbers of a Kummer covering of \({\mathbb{P}}^ 2\) ramified along a line configuration | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Using methods developed by Kollár, Voisin, ourselves and Totaro, we prove that a cyclic cover of \(\mathbb P_{\mathbb C}^n\), \(n\geq 3\), of prime degree \(p\), ramified along a very general hypersurface \(f(x_0,\dots, x_n)=0\) of degree \(mp\), is not stably rational if \(m(p-1) <n+1\leq mp\). In dimension 3 we recover double covers of \( \mathbb P^3_{\mathbb C}\) ramified along a very general surface of degree 4 (Voisin) and double covers of \( \mathbb P^3_{\mathbb C}\) ramified along a very general surface of degree 6 (Beauville). We also find double covers of \( \mathbb P^4_{\mathbb C}\) ramified along a very general hypersurface of degree 6. This method also enables us to produce examples over a number field. Коль\(###\)-Телэн, Ж.-Л.; Пирютко, Е. В., Циклические накрытия, которые не являются стабильно рациональными, Изв. РАН. Сер. матем., 80, 4, 35-48, (2016) Rationality questions in algebraic geometry, Rational and unirational varieties, (Equivariant) Chow groups and rings; motives, Algebraic cycles, Finite ground fields in algebraic geometry, Coverings in algebraic geometry Cyclic covers that are not stably rational | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A terminal \({\mathbb Q}\)-factorial Fano variety \(V\) with Picard group \({\mathbb Z}\) is birationally rigid if \(V\) is birationally equivalent neither to another Fano variety of the same type, nor to a fibration with a smooth general fiber of Kodaira dimension \(-\infty\); and if in addition \(\text{Bir}(V) = \text{Aut}(V)\) then \(V\) is birationally superrigid, cf. def.1 and 2. Theorem 3 states that any cyclic triple covering \(X\) of \({\mathbb P}^{2n}\), \(n \geq 2\) branched over a hypersurface \(S\) of degree \(3n\) with at most ordinary double points (such \(X\) is a terminal \({\mathbb Q}\)-factorial Fano variety \(V\) with Picard group \({\mathbb Z}\)) is birationally superrigid and the group \(\text{Bir}(X)\) is finite. In particular any such \(X\) is not rational.
The above theorem describes in particular all smooth birationally superrigid cyclic triple covers of projective spaces, see remark 5. By theorem 15, a triple cover \(X \rightarrow {\mathbb P}^{2n}\), \(n \geq 2\) as in theorem 3 (i.e. with a ramification hypersurface \(S\) with at most ordinary double points) can't be birationally equivalent to an elliptic fibration. But as seen in remark 17, if \(S\) has e.g. one ordinary triple point and \(n = 2\) then \(X\) is birationally equivalent to an elliptic fibration. Theorem 18, generalizing theorems 3 and 15, shows that if the ramification hypersurface \(S\) of the cyclic triple covering \(X \rightarrow {\mathbb P}^{2n}\) from theorem 3 admits besides ordinary double points also ordinary triple points then the conclusions of theorems 3 and 15 still hold, except in the case \(n = 2\) when \(S\) has a triple point and \(X\) is birational to an elliptic fibration by the construction given in remark 17. birationally rigid; cyclic triple cover Chel'tsov, I. A., Birationally super-rigid cyclic triple spaces, Izv. Ross. Akad. Nauk Ser. Mat., 68, 169-220, (2004) Rationality questions in algebraic geometry, Rational and birational maps, Coverings in algebraic geometry Birationally super-rigid cyclic triple 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 [For part I of this paper see \textit{A. Némethi}, same collection, Contemp. Math. 266, 89-128 (2000; see the preceding review Zbl 1017.14012).]
Summary: In this part II, we determine the resolution graph of the hypersurface singularity \((\{f+g^k=0\},0)\), where \((f,g): (\mathbb{C}^3,0) \to (\mathbb{C}^2,0)\) is an ICIS ``isolated complete intersection singularity'' and \(k\) is a sufficiently large integer. All these graphs are coordinated by a ``universal bi-colored graph'' \(\Gamma_{\mathcal C}\) associated with the ICIS \((f, g)\). Its definition is rather involved, and in concrete examples it is difficult to compute it. Nevertheless, we present a large number of examples. This is very helpful in the exemplification of its properties as well. Then we present our main construction which provides the dual resolution graph of the series \(\{f+g^k=0\}\) from the graph \(\Gamma_{\mathcal C}\) and the integer \(k\). This is formulated in a purely combinatorial algorithm. The result is a highly nontrivial generalization of the cyclic covering case of \((\mathbb{C}^2,0)\). resolution graph of hypersurface singularity; dual resolution graph; cyclic covering Némethi, A.; Szilárd, Á.: Resolution graphs of some surface singularities. II. generalized iomdin series, Contemp. math. 266, 129-164 (2000) Singularities of surfaces or higher-dimensional varieties, Complex surface and hypersurface singularities, Global theory and resolution of singularities (algebro-geometric aspects), Coverings in algebraic geometry, Applications of graph theory Resolution graphs of some surface singularities. II: Generalized Iomdin series | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 connected proper curve with marked points \(\{x_i\}\) over an algebraically closed field \(k\) of characteristic \(p\). Consider a Galois cover \(\phi : Y \to X\) of smooth connected curves branched only at \(\{x_i\}\). Abhyankar's conjecture (proved by \textit{M. Raynaud}, [Invent. Math. 116, 425--462 (1994; Zbl 0798.14013)] and \textit{D. Harbater} [Invent. Math. 117, 1--25 (1994; Zbl 0805.14014)]) determines exactly which groups \(G\) can be the Galois group of \(\phi\). An open problem is to determine which inertia groups and filtrations of higher ramification groups can be realized for such a cover \(\phi\).
In this article, the author deals with the case \(\phi : Y \to \mathbb{P}^1\) is branched at only one point. Such covers exist if and only if \(G\) is a quasi-\(p\) group, which means that \(G\) is generated by \(p\)-groups. \textit{D. Harbater} [Am. J. Math. 115, 487--508 (1993; Zbl 0790.14027)] proved that the Sylow \(p\)-subgroups of \(G\) can be realized as the inertia groups of such a cover \(\phi\). Under the assumption that the Sylow \(p\)-subgroups of \(G\) have order \(p\), the filtration of higher ramification groups is determined by one integer \(j\) for which \(p \nmid j\) namely by the lower jump or conductor. The author shows that all sufficiently large conductors occur and for \(p \neq 2\) gives an explicit bound for which conductors are sufficiently large.
Theorem. Let \(G\) be a finite quasi-\(p\) group whose Sylow \(p\)-subgroups have order \(p \neq 2\). There exists an integer \(J\) depending explicitly on \(p\), the \(p\)-weight of \(G\), and the exponent of the normalizer of a Sylow \(p\)-subgroup of \(G\) with the following property: if \(j \geq J\) and \(p \nmid j\) then there exists a \(G\)-Galois cover \(\phi : Y \to \mathbb{P}^1\) branched at only one point over which it has inertia group \(\mathbb{Z}/p\) and conductor \(j\).
The \(p\)-weight of \(G\) is defined in the following way. Let \(S\) be a chosen Sylow \(p\)-subgroup of \(G\). Let \(G(S)\) be the subgroup generated by all proper quasi-\(p\) subgroups \(G'\) such that \(G' \cap S\) is a Sylow \(p\)-subgroup of \(G'\). The group \(G\) is \(p\)-pure if \(G(S) \neq G\). Now consider all subgroups \(G' \subset G\) such that \(G'\) is quasi-\(p\) and \(p\)-pure and such that \(G' \cap S\) is a Sylow \(p\)-subgroup of \(G'\). The \(p\)-weight \(\omega(G)\) of \(G\) is the minimal number of such subgroups \(G'\) of \(G\) which are needed to generate \(G\) (it is independent of the choice of \(S\)).
The integer \(J\) is then given by the following definition: let \(m_e\) be the exponent of the normalizer \(N_G(S)\) of \(S\) in \(G\) divided by \(p\). One has \(J=m_e(2+1/(p-1))\omega(G)\) if \(p \nmid \omega(G)\) and \(J=m_e(2+1/(p-1)) \omega(G)+2\) otherwise. Galois cover; \(p\)-groups; sylow \(p\)-subgroups R. Pries, Conductors of wildly ramified covers, III, Preprint, 2001 Coverings of curves, fundamental group, Coverings in algebraic geometry, Ramification problems in algebraic geometry Conductors of wildly ramified covers. 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 From the Introduction: In this paper, we study the combinatorics of real double Hurwitz numbers with only real and positive branch points.
(Complex) Hurwitz numbers count genus \(g\) degree \(d\) covers of a curve of genus \(h\) with fixed ramification conditions at fixed points of the target. The \textit{ramification profile} of a point in the target, a partition of the degree \(d\), encodes how many sheets of the map come together above this point. We call a point a branch point if its ramification profile is not \((1^d)\). If the profile is \((2, 1^{d-2})\), then we say that the branch point (resp. the ramification) is \textit{simple}. The Riemann-Hurwitz formula implies how many branch points we need to fix in order to obtain a finite count. Such counts of covers date back to Hurwitz himself in the 19th century and have since then provided a fertile source for interesting problems connecting the geometry of covers, the moduli space of curves, the representation theory of the symmetric group, and matrix models in probability theory.
Double Hurwitz numbers are counts of covers of \(\mathbb{P}^1\), where we fix two special ramification profiles \(\mu\) and \(\nu\) about 0 and \(\infty\) and only simple ramification elsewhere. Double Hurwitz numbers feature a particularly rich structure investigated, for example, in [\textit{R. Cavalieri} et al., Adv. Math. 228, No. 4, 1894--1937 (2011; Zbl 1231.14023); \textit{I. P. Goulden} et al., Adv. Math. 198, No. 1, 43--92 (2005; Zbl 1086.14022); \textit{P. Johnson}, ``Double Hurwitz numbers via the infinite wedge'', \url{arXiv:1008.3266}; \textit{S. Shadrin} et al., Adv. Math. 217, No. 1, 79--96 (2008; Zbl 1138.14018)]. Tropical analogs of double Hurwitz numbers were introduced in [\textit{R. Cavalieri} et al., J. Algebr. Comb. 32, No. 2, 241--265 (2010; Zbl 1218.14058)] and have been a successful tool in obtaining structural results [Zbl 1231.14023].
In this paper, we study real covers of the projective line. In general, real algebraic geometry is much harder than complex algebraic geometry, that is, algebraic geometry over an algebraically closed field. This holds true also for real counts of covers. For example, counts of real covers may depend on the exact position of the branch points. We restrict our attention to real double Hurwitz numbers with only real and positive branch points. For this situation, the count does not depend on the exact position of the chosen branch points. There is an ambiguity in the definition of such Hurwitz numbers: we can either count them with their real structure (we call these numbers \(\tilde{H}_g(\mu,\nu)\)) or without (\(H_g(\mu,\nu)\)).
By matching a cover with a monodromy representation, the count of a Hurwitz number is equivalent to choices of n-tuples of elements of S d of fixed conjugacy class satisfying some conditions. This holds also true for the real double Hurwitz numbers in question. We exploit the symmetric group approach to Hurwitz numbers to investigate their combinatorial properties in two directions:
(a) We study tropical real double Hurwitz numbers.
(b) We investigate real double Hurwitz numbers in terms of paths in a subgraph of the Cayley graph of the symmetric group.
We construct tropical real double Hurwitz numbers as weighted counts of graphs mapping to a line (i.e., tropical covers) which are colored in a way reflecting the real structure. We obtain correspondence theorems for the numbers \(\tilde{H}_g(\mu,\nu)\) and \(H_g(\mu,\nu)\). The tropical interpretation of real double Hurwitz numbers uncovers the relation between these numbers: an easy corollary of our correspondence theorems (Corollary 3.24) implies that \(\tilde{H}_g(\mu,\nu) = H_g(\mu,\nu)\) if \(\mu\) and \(\nu\) are not both in \(\{d,(\frac{d}{2},\frac{d}{2})\}\). If \(\mu\) and \(\nu\) are both in \(\{d,(\frac{d}{2},\frac{d}{2})\}\), then their difference is also determined in Corollary 3.24.
The study of correspondence theorems for Hurwitz numbers relating them to their tropical counterparts is not limited to an approach in terms of the symmetric group. For complex Hurwitz numbers, there is a general version using topological methods. General correspondence theorems for real Hurwitz numbers are the topic of a forthcoming paper by Markwig and Rau.
By restricting to real double Hurwitz numbers with real positive branch points, of the symmetric group, which enables us to express the combinatorics both in terms of tropical covers as well as in terms of paths in the Cayley graph in a useful way. Hurwitz numbers; branch points; symmetric group; Cayley graph; tropical geometry M. Guay-Paquet, H. Markwig, and J. Rau, ''The combinatorics of real double Hurwitz numbers with real positive branch points,'' http://imrn.oxfordjournals.org/content/early/2015/05/14/imrn.rnv135.abstract (2015). Real algebraic and real-analytic geometry, Coverings of curves, fundamental group, Ramification problems in algebraic geometry, Coverings in algebraic geometry, , Symmetric groups, Enumerative problems (combinatorial problems) in algebraic geometry The combinatorics of real double Hurwitz numbers with real positive branch 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 See the following review. characteristic p; Zariski surface; divisor class group of the coordinate ring; fundamental group; Zariski open set purely inseparable descent Grant, A.; Lang, J., Applications of the fundamental group and Galois descent to the study of purely inseparable extensions, J. Algebra, 132, 2, 340-360, (1990) Special surfaces, Finite ground fields in algebraic geometry, Coverings in algebraic geometry, Special algebraic curves and curves of low genus Applications of the fundamental group and purely inseparable descent to the study of curves on Zariski surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This is a modest work particular about multiple coverings of ruled rational complex surfaces. The rationality of such coverings is obvious by construction. scrolls; multiple coverings of ruled rational complex surfaces Coverings in algebraic geometry, Rational and unirational varieties Multiple coverings of 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 The fibered surfaces are shown to be finite branched coverings of products of algebraic curves. As a consequence, the fundamental group of a finite surface turns to be commensurable with a product of the fundamental groups of Riemann surfaces. fibered surfaces; fundamental groups Compact Kähler manifolds: generalizations, classification, Coverings in algebraic geometry Fibered surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(K\) be a field, \(\mathcal{O}_K\) be a discrete valuation ring with field of quotients \(K\), and set \(S:={\text{Spec}}(\mathcal{O}_K)\). Let \(X\) be a geometrically connected smooth projective curve over \(K\). A normal scheme \(\mathcal{X}\), flat and projective over \(S\) with generic fibre isomorphic to \(X\), is called a model of \( X\) over \(\mathcal{O}_K\). It is easy to see that there exist always regular models of \(X\). \(\mathcal{X}\) is said to be a smooth model if the structural morphism \(\mathcal{X}\to S\) is smooth. Let \(\mathcal{X}'\) be a desingularization of \(\mathcal{X}\); now \(\mathcal{X}'\) is a regular model of \( X\). If the genus of \(X\) is \(\geq 1\), then there exists the minimal regular model of \( X\). The minimal regular model does not behave well functorially. Therefore, the author considers another type of model. A model \(\mathcal{X}\) of \(X\) is called ``semi-stable'' (resp. ``stable'') if every geometric fibre of the structural morphism \(\mathcal{X}\to S\) is a semi-stable curve (respectively stable curve). The stable reduction theorem of \textit{P. Deligne} and \textit{D. Mumford} [Publ. Math., Inst. Hautes Étud. Sci. 36, 75--109 (1969; Zbl 0181.48803)] states that, eventually after a finite separable base change, \(X\) admits a semi-stable model. There is no preferred semi-stable model of \(\mathbb{P}^1_K\). Therefore the author considers marked curves and models of marked curves. Let \(T\) be a scheme. A marked curve over \(T\) is a relative curve \(\mathcal{X}\) over \(T\) together with a finite set \(\mathcal{M}\) of sections \(T\to\mathcal{X}\) that have disjoint support and are contained in the smooth locus of \(\mathcal{X}\to T\). A marked model of a marked curve \((X,M)\) over \(S\) is a marked curve \((\mathcal{X},\mathcal{M})\) such that \(\mathcal{X}\) is a model of \(X\) and \(\mathcal{M}\) is the closure of \(M\) in \(\mathcal{X}\). Using the non-marked stable reduction theorem, the author get a stable reduction theorem for marked curves [cf.\ Th.\ 1]. In section 3 the author considers finite marked morphisms \(f\colon (X,M)\to (Y,N)\) of geometrically connected, smooth, projective curves over \(K\). A ``marked model'' of \(f\) over \(\mathcal{O}_K\) is a morphism \(\widetilde f\colon \mathcal{X}\to\mathcal{Y}\) of models \(\mathcal{X}\) and \(\mathcal{Y}\) of \(X\), respectively \(Y\) extending \(f\). In Prop.\ 3 the author proves that a morphism \(f\colon (X,M)\to (Y,N)\) admits, after a finite base change, a stable marked model \(\mathcal{X}\to \mathcal{Y}\). In section 4 let \(\mathcal{O}_K\) be a complete dicrete valuation ring with field of quotients \(K\). The author studies a finite marked morphism of geometrically connected, smooth, projective curves over \(K\): \(f\colon (X,M)\to (Y,N)\). A model of \(f\) over \(\mathcal{O}_K\) is a morphism \(\widetilde f\colon \mathcal{X}\to \mathcal{Y}\) between marked models \(\mathcal{X}\) and \(\mathcal{Y}\) of \(X\), respectively \(Y\) extending \(f: \widetilde f\vert_K=f\). algebraic curve; semi-stable model; marked curve; cover of curves; Belyi's theorem Coverings in algebraic geometry, Arithmetic ground fields for curves, Curves over finite and local fields, Local ground fields in algebraic geometry Properties of models of covers 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 A vector bundle \(E\) on an algebraic variety \(X\) is said to be ample if the tautological line bundle \({\mathcal O}(1)\) on the projective bundle \(\mathbb{P}(E)\) is ample. -- Surfaces with ample cotangent bundle are automatically of general type. \textit{F. Hirzebruch} [in Arithmetic and Geometry, Vol. 2: Geometry, Prog. Math. 36, 113-140 (1983; Zbl 0527.14033)] has constructed many interesting surfaces of general type as the desingularizations of certain Galois covers of the projective plane, branched on a union (``arrangement'') of lines, and \textit{A. Sommese} [Math. Ann. 268, 207-221 (1984; Zbl 0534.14020)] has characterized the arrangements which give rise to surfaces with ample cotangent bundle.
The main results of the paper under review is a criterion stating that if \(f:X \to Y\) is a finite map between smooth surfaces, branched over a normal crossing divisor with smooth components, and if \(Y\) has ample canonical bundle, then \(X\) has ample canonical bundle iff each component of the ramification divisor in \(X\) has strictly negative self intersection. The criterion is established by applying the Nakai criterion for the ampleness of a line bundle. -- Using this result, the author is able to produce new examples of surfaces with ample canonical bundles, that are constructed as double covers of some Hirzebruch surfaces. ample vector bundle; arrangements; Nakai criterion; surfaces with ample canonical bundles; Hirzebruch surfaces Spurr M.: Branched coverings of surfaces with ample cotangent bundle. Pac. J. Math. 164, 129--146 (1994) Surfaces of general type, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Coverings in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Branched coverings of surfaces with ample cotangent bundle | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper under review is concerned with the following central question: what geometric information does the geometric (resp. tame) fundamental group \(\pi_1(X)\) (resp. \(\pi^t_1(X)\)) of a curve \(X\) carry in positive characteristic? The author presents several conjectural answers and many new arguments to support them. First the author states some general conjectures comparing certain homomorphisms between connected \({\mathbb{F}}_p\)-schemes \(X\), \(Y\) with some classes of homomorphisms between \(\pi_1(X)\), \(\pi_1(Y)\) (resp. \(\pi^t_1(X)\), \(\pi^t_1(Y)\)). Then he considers many different open problems for a smooth \(k\)-curve \(X\): the determination of the genus of \(X\), the structure of the inertia subgroup of \(\pi_1(X)\), the transcendence degree of \(k\) over \(\overline{{\mathbb{F}}}_p\), Hopfian properties of \(\pi_1(X)\) (resp. \(\pi^t_1(X)\)), and the size of the maximal pro-free quotient of \(\pi_1(X)\) (resp. \(\pi^t_1(X)\)). Partial results are given, which illustrate the consequences of the general conjectures of the first section. geometric fundamental group; tame fundamental group Tamagawa, A.: Fundamental groups and geometry of curves in positive characteristic, Arithmetic fundamental groups and noncommutative algebra (Berkley, CA). Proc. Sympos. Pure Math. 70, Am. Math. Soc, Providence, RI, 2002, pp. 297--333 Coverings of curves, fundamental group, Coverings in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry, Finite ground fields in algebraic geometry, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory) Fundamental groups and geometry of curves 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 The author discusses the coverings of a connected and locally path- connected topological space X arising from the normal subgroups N of \(\pi_ 1(X)\) such that \(\pi_ 1(X)/N\) is abelian. No new results are given. fundamental group; coverings Coverings in algebraic geometry, Homotopy groups of special spaces On homology covering space and sheaf associated to the homology 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 Suppose that an ideal \(J\) of \(C^\infty\) functions on an open subset of \(\mathbf{R}^2\) is a Łojasiewicz ideal. We describe the set of \(C^\infty\) functions vanishing on the zeros of \(J\) explicitly using \(J\) in an open neighborhood of each point in zeros of \(J\), it can be obtained by taking real radical and closure starting from \(J\) repeatedly for a finite number of times. This gives an another affirmative answer to Bochnak's conjecture in dimension \(2\), which is first done by \textit{J.-J. Risler} [Ann. Inst. Fourier 26, No. 3, 73--107 (1976; Zbl 0324.46028)]. closed ideal; Łojasiewicz ideal; Nullstellensatz; real radical; zero property Real-analytic and semi-analytic sets, Real-analytic functions, \(C^\infty\)-functions, quasi-analytic functions, Rings and algebras of continuous, differentiable or analytic functions, Sums of squares and representations by other particular quadratic forms, Real-analytic manifolds, real-analytic spaces A nullstellensatz for ideals of \(C^\infty\) functions in dimension 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 A standard observation in algebraic geometry and number theory is that a ramified cover of an algebraic variety \(\tilde{X}\to X\) over a finite field \(\mathbb F_q\) furnishes the rational points \(x\in X(\mathbb F_q)\) with additional arithmetic structure: the Frobenius action on the fiber over \(x\).
For example, in the case of the Vieta cover of polynomials over \(\mathbb F_q\) this structure describes a polynomial's irreducible decomposition type.
Furthermore, the distribution of these Frobenius actions is encoded in the cohomology of \(\tilde{X}\) via the Grothendieck-Lefschetz trace formula.
This note presents a version of the trace formula that is suited for studying the distribution in the context of representation stability: for certain sequences of varieties (\(\tilde{X}_n\)) the cohomology, and therefore the distribution of the Frobenius actions, stabilizes in a precise sense.
We conclude by fully working out the example of the Vieta cover of the variety of polynomials. The calculation includes the distribution of cycle decompositions on cosets of Young subgroups of the symmetric group, which might be of independent interest. trace formula; representation stability; arithmetic statistics; polynomials over finite fields; symmetric group statistics Varieties over finite and local fields, Polynomials over finite fields, Arithmetic theory of polynomial rings over finite fields, Coverings in algebraic geometry, Finite ground fields in algebraic geometry, Exact enumeration problems, generating functions A trace formula for the distribution of rational \(G\)-orbits in ramified covers, adapted to representation stability | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Multivariable Alexander invariants of algebraic links are calculated in terms of algebro-geometric invariants (polytopes and ideals of quasiadjunction). The relations with log-canonical divisors, the multiplier ideals and a semicontinuity property of polytopes of quasiadjunction are discussed. algebraic links Libgober A.: Hodge decomposition of Alexander invariants. Manusc. Math. 107(2), 251--269 (2002) Knots and links in the 3-sphere, Covering spaces and low-dimensional topology, Singularities in algebraic geometry, Plane and space curves, Coverings in algebraic geometry, Coverings of curves, fundamental group, Milnor fibration; relations with knot theory Hodge decomposition of Alexander invariants | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 continues the study of finite flat covers of small degree started by the second author [see \textit{R. Miranda}, Am. J. Math. 107, 1123-1158 (1985; Zbl 0611.14011); see also \textit{R. Pardini}, Ark. Mat. 27, 319-341 (1989; Zbl 0707.14010), \textit{G. Casnati} and \textit{T. Ekedahl}, J. Algebr. Geom. 5, 439-460 (1996; Zbl 0866.14009), \textit{G. Casnati}, J. Algebr. Geom. 5, 461-477 (1996; Zbl 0921.14006)]. The approach to the problem is again ``from the bottom'', namely, given a variety \(Y\), one looks for data on \(Y\) that determine a degree 4 covering \(f: X\to Y\). If the characteristic of the ground field is different from 2, then there is a natural splitting \(f_*{\mathcal O}_X={\mathcal O}_Y\oplus E\), where \(E\) is a locally free sheaf of rank 4. Describing \(X\) is of course the same as describing the \({\mathcal O}_Y\)-algebra structure on \({\mathcal O}_Y\oplus E\). The multiplication is commutative, hence it determines a map \(\mu: S^2E\to E\oplus {\mathcal O}_Y\). The main result of the paper is the following:
(i) the second component \(\mu_2: S^2E\to {\mathcal O}_Y\) of \(\mu\) is determined by the first component \(\mu_1: S^2E\to E\);
(ii) there is a natural transformation \(\wedge^2S^2E^* \otimes \wedge^3 E\to \text{Hom} (S^2E, E)\) such that the maps \(\mu_1: S^2E\to E\) giving an associative multiplication are precisely the images of the totally decomposable elements of \(\wedge^2S^2E\otimes \wedge^3 E\). finite flat cover; quadruple cover D. W. Hahn - D. Miranda, Quadruple covers of algebraic varieties, J. Algebraic Geom. 8 (1999), 1-30. Zbl0982.14008 MR1658196 Coverings in algebraic geometry, Ramification problems in algebraic geometry, Finite ground fields in algebraic geometry Quadruple covers of algebraic varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We prove that every geometrically reduced projective variety of pure dimension \(n\) over a field of positive characteristic admits a morphism to projective \(n\)-space, étale away from the hyperplane \(H\) at infinity, which maps a chosen divisor into \(H\) and some chosen smooth points not on the divisor to points not in \(H\). This improves an earlier result of the author [C.R. Acad. Sci. Paris 335, 921--926 (2002; Zbl 1047.14051)] which was restricted to infinite perfect fields. We also prove a related result that controls the behavior of divisors through the chosen point. Kedlaya, More étale covers of affine spaces in positive characteristic, J. Algebraic Geometry 14 pp 187-- (2005) Coverings in algebraic geometry More étale covers of affine spaces 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 \(kE\) be the group algebra of an elementary abelian \(p\)-group \(E=\langle g_1,\ldots, g_r\rangle\) of finite rank \(r\) over an algebraically closed field \(k\) of characteristic \(p\). Let \(M\) be a finitely generated \(kE\)-module, \(X_i=g_i-1\), \(\alpha=(\alpha_1,\ldots,\alpha_r)\in A^r(k)\). The Jordan type of \(X_\alpha=\alpha_1g_1+\cdots+\alpha_rg_r\) on \(M\) is the partition \(\mathrm{JType}(X_{\alpha},M)=[p]^{a_p}[p- 1]^{a_{p-1}}\cdots[1]^{a_1}\), where \(X_{\alpha}\) acts on \(M\) via \(a_j\) Jordan blocks of length \(j\). The notion of a \(k\)-module of constant Jordan type was introduced by \textit{J. F. Carlson} et al. [J. Reine Angew. Math. 614, 191--234 (2008; Zbl 1144.20025)]. Investigation of the category \(\mathrm{cJt}(M)\) of \(kE\)-modules of constant Jordan type by means of functors \(\mathcal{F_i}\) to the category \(\mathrm{Vec}(\mathbb{P}^{r-1}(k))\) of vector bundles on the projective space \(\mathbb{P}^{r-1}(k)\) was initiated by \textit{D. Benson} and \textit{J. Pevtsova} [Trans. Am. Math. Soc. 364, No. 12, 6459--6478 (2012; Zbl 1286.20009)].
Two of the main results of the authors are as follows. A rank \(s\) shifted subgroup is a subalgebra of \(kE\) that is isomorphic to the group algebra \(kE'\), where \(E'\) is an elementary abelian \(p\)-group of rank \(s\). An embedding \(\varphi:kE'\to kE\) is called a homogeneously embedded \(s\)-shifted subgroup if for any generator system \(\{T_1,\ldots,T_s\}\) of \(\mathrm{Rad}(kE')\), the \(\varphi(T_j)\) are linear combinations of the \(X_i\). Let \(kE'\) be a homogeneously embedded \(s\)-shifted subgroup of \(kE\) and let \(f: \mathbb{P}^{s-1}(k)\to \mathbb{P}^{t-1}(k)\) be the corresponding closed immersion. If \(M\) is a \(kE\)-module of constant Jordan type, then for all \(1 \leq i \leq p\) we have \(f^{\ast}\mathcal{F_i}(M )\cong\mathcal{F_i}(M\downarrow_ {kE'})\). For \(M\) in rank two, its generic kernel \(\mathfrak{k}(M)\) is the largest submodule of \(M\) with the image of \(X_{\alpha}\) acting on \(M\) independently of the choice of \(\overline{\alpha}\in \mathbb{P}^{r-1}(k)\). If \(r=2\), \(M\) is a \(kE\)-module of constant Jordan type, \(J\) is the radical of \(kE\) then for each \(1 \leq i \leq p\), we have \(\mathcal{F_i}(M)\cong\mathcal{F_i}(J^{-i}\mathfrak{k}(M))/J^{i+1}\mathfrak{k}(M)\).
The paper is divided into 13 sections. After the introductory two, in the third section, homogeneously embedded subgroups are considered, and in the next two, the operator \(\theta_M\), the equal images property and vector bundles for W-modules. Section 7 deals with the generic kernel filtration, and the next one applies this to compute \(\mathcal{F_i}(M )\) in rank two. In Sections 9 and 10, the \(n\)th power generic kernel and the equal \(n\)-images property are studied, and in the next two this method is used to compute \(\mathcal{F_i}(M)\) along with further examples and applications. The last section discusses vector bundles in rank two. elementary abelian; group algebra; module; modular representation; constant Jordan type; vector bundle; projective space; coherent sheave Modular representations and characters, Group rings of finite groups and their modules (group-theoretic aspects), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Vector bundles on surfaces and higher-dimensional varieties, and their moduli Modules of constant Jordan type, pullbacks of bundles and generic kernel filtrations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Galois theory has such close analogies with the theory of coatings that algebraists use a geometric language to speak of body extensions, while topologists speak of ``Galois coatings''. This book endeavors to develop these theories in a parallel way, starting with that of coatings, which better allows the reader to make images. The authors chose a plan that emphasizes this parallelism. The intention is to allow to transfer to the algebraic framework of the Galois theory the geometric intuition that one can have in the context of the coatings.
This book is aimed at graduate students and mathematicians curious about a non-exclusively algebraic view of Galois theory.
For the French original see [Zbl 1076.12004]. Galois theory; Galois coatings Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to field theory, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to category theory, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic topology, Separable extensions, Galois theory, Coverings in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry, Coverings of curves, fundamental group Algebra and Galois theories. Translated from the French by Urmie Ray | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems It is known that the degree of the canonical map \(\varphi_X\) of a Gorenstein minimal projective general type surface or 3-fold with locally factorial terminal singularities is bounded. In fact precise bounds are known. For \(n\)-folds \(X\) with \(n\geq 4\) the situation is less clear. The case considered here is the special one where \(\varphi_X\) is an abelian Galois cover of \({\mathbb P}^n\), in which case \(X\) is called an abelian canonical \(n\)-fold. It is shown that the degree of the cover is then bounded (for fixed \(n\)): in fact the assumption that the base is \({\mathbb P}^n\) can be relaxed a little. Two examples are given of abelian canonical 4-folds of degrees 81 and 128. abelian canonical \(n\)-fold; canonical degree; canonical map; abelian cover \(n\)-folds (\(n>4\)), \(4\)-folds, Coverings in algebraic geometry On abelian canonical \(n\)-folds 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 We introduce the notion of Galois holomorphic foliation on the complex projective space as that of foliations whose Gauss map is a Galois covering when restricted to an appropriate Zariski open subset. First, we establish general criteria assuring that a rational map between projective manifolds of the same dimension defines a Galois covering. Then, these criteria are used to give a geometric characterization of Galois foliations in terms of their inflection divisor and their singularities. We also characterize Galois foliations on \(\mathbb P^2\) admitting continuous symmetries, obtaining a complete classification of Galois homogeneous foliations. Singularities of holomorphic vector fields and foliations, Birational automorphisms, Cremona group and generalizations, Coverings in algebraic geometry, Dynamical aspects of holomorphic foliations and vector fields Foliations and webs inducing Galois 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 One studies the properties of a complex surface of general type with a fibration \(f:\quad S\to C\) such that \(\omega^ 2_{S/C}<4\cdot \deg (f_*\omega_{S/C})\). For such a surface the image of the \(\pi_ 1\) of a fibre of f in \(\pi_ 1(S)\) is trivial, unless the fibres of f are hyperelliptic, and this image is \({\mathbb{Z}}_ 2\). One also shows a lower bound for \(\omega^ 2_{S/C}\), studies the stability of \(f_*\omega_{S/C}\), and gives several examples. fibered algebraic surfaces with low slope; image of fundamental groups Xiao, G., \textit{fibered algebraic surfaces with low slope}, Math. Ann., 276, 449-466, (1987) Special surfaces, Coverings in algebraic geometry Fibered algebraic surfaces with low slope | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(S\subset \mathbb{C}\mathbb{P}^r\) be a non-singular algebraic surface of degree \(\deg S=N\). It is well known that for almost all projections \(\text{pr}:\mathbb{C} \mathbb{P}^r\to \mathbb{C}\mathbb{P}^2\) the restrictions \(f:S \to \mathbb{C}\mathbb{P}^2\) of these projections to \(S\) satisfy the following conditions:
(i) \(f\) is a finite morphism of degree \(\deg f=\deg S\),
(ii) \(f\) is branched along an irreducible curve \(B\subset\mathbb{C} \mathbb{P}^2\) whose singularities are ordinary cusps and nodes only,
(iii) \(f^*(B)= 2R+C\), where the curve \(R\) is irreducible and non-singular and \(C\) is reduced,
(iv) \(f|_R: R\to B\) coincides with the normalization of \(B\).
We call such an \(f\) a generic morphism, and its branch curve \(B\) is called the discriminant curve of \(f\). Two generic morphisms \((S_1,f_1)\), \((S_2,f_2)\) with the same discriminant curve \(B\) are said to be equivalent if there is an isomorphism \(\varphi: S_1\to S_2\) such that \(f_1=f_2\circ \varphi\). The following assertion is known as ``Chisini's conjecture'':
Let \(B\) be the discriminant curve of a generic morphism \(f:S\to \mathbb{C}\mathbb{P}^2\) of degree \(\deg f\geq 5\). Then \(f\) is uniquely determined by the pair \((\mathbb{C}\mathbb{P}^2,B)\).
It is easy to see that the analogous conjecture for generic morphisms of projective curves to \(\mathbb{C} \mathbb{P}^1\) is not true. On the other hand, Chisini's conjecture holds for the discriminant curves of almost all generic morphisms of any projective surface. In particular, if \(S\) is any surface of general type with ample canonical class, then Chisini's conjecture holds for the discriminant curves of those generic morphisms \(f:S\to \mathbb{C}\mathbb{P}^2\) that are given by a three-dimensional linear subsystem of the \(m\)-canonical class of \(S\), where \(m\in\mathbb{N}\). The discriminant curves of such generic morphisms will be called \(m\)-canonical discriminant curves.
Let \(B\) be an algebraic curve of degree \(p\) in \(\mathbb{C}\mathbb{P}^2\). The topology of the embedding \(B\subset\mathbb{C} \mathbb{P}^2\) is determined by the braid monodromy of \(B\), which is described by a factorization of the ``full twist'' \(\Delta^2_p\) in the semi-group \(B_p^+\) of the braid group \(B_p\) on \(p\) strings. (In the standard generators, \(\Delta^2_p=(X_1\cdot\dots\cdot X_{p-1})^p.)\) If \(B\) is a cuspidal curve, then this factorization can be written as \(\Delta^2_p= \prod_iQ_i^{-1} X_1^{\rho_i}Q_i\), \(\rho_i\in(1,2,3)\), where \(X_1\) is the positive half-twist in \(B_p\).
Main problems:
Problem 1. Let \(B\subset\mathbb{C} \mathbb{P}^2\) be a cuspidal curve. Does the braid factorization type of the pair \((\mathbb{C}\mathbb{P}^2,B)\) uniquely determine the diffeomorphism type of this pair, and vice versa?
Problem 2. Let \(\Delta^2_p= {\mathcal E}_1\) and \(\Delta^2_p= {\mathcal E}_2\) be two braid monodromy factorizations. Does there exist a finite algorithm to recognize whether these two braid monodromy factorizations belong to the same braid factorization type?
One of the main results of this paper is the following theorem.
Theorem 1. Let \(B_1,B_2\subset\mathbb{C}\mathbb{P}^2\) be cuspidal algebraic curves. Suppose that the pairs \((\mathbb{C}\mathbb{P}^2,B_1)\) and \((\mathbb{C}\mathbb{P}^2, B_2)\) have the same braid factorization type. Then the pairs \((\mathbb{C} \mathbb{P}^2,B_1)\) and \((\mathbb{C} \mathbb{P}^2,B_2)\) are diffeomorphic.
It is well known that there exist four-dimensional smooth manifolds which are homeomorphic but not diffeomorphic. One of the most important problems of four-dimensional geometry is to find invariants that distinguish smooth structures on the same topological four-dimensional manifold.
Theorem 2. Let \(f_1:S_1\to\mathbb{C} \mathbb{P}^2\) and \(f_2:S_2\to \mathbb{C}\mathbb{P}^2\) be generic morphisms of non-singular projective surfaces, and let \(B_1,B_2\subset \mathbb{C}\mathbb{P}^2\) be their discriminant curves. Suppose that Chisini's conjecture holds for \((\mathbb{C}\mathbb{P}^2,B_1)\). If the pairs \((\mathbb{C}\mathbb{P}^2,B_1)\) and \((\mathbb{C}\mathbb{P}^2,B_2)\) have the same braid factorization type, then \(S_1\) and \(S_2\) are diffeomorphic.
Corollary. Let \(S_1,S_2\) be surfaces of general type with ample canonical class, and let \(B_1, B_2\) be \(m\)-canonical discriminant curves of generic morphisms \(f_1:S_1\to \mathbb{C} \mathbb{P}^2\) and \(f_2:S_2\to \mathbb{C}\mathbb{P}^2\) (respectively) given by three-dimensional linear subsystems of the \(m\)-canonical class of \(S_i\), where \(m\in \mathbb{N}\). If the pairs \((\mathbb{C}\mathbb{P}^2, B_1)\) and \((\mathbb{C}\mathbb{P}^2, B_2)\) have the same braid factorization type, then \(S_1\) and \(S_2\) are diffeomorphic. generic morphism; branch curve; discriminant curve; Chisini's conjecture; surface of general type; braid monodromy; cuspidal algebraic curves Kulikov S.\ V. and Teicher M., Braid monodromy factorizations and diffeomorphism types, Izv. Ross. Akad. Nauk Ser. Mat. 64 (2000), no. 2, 89-120. Plane and space curves, Differential topological aspects of diffeomorphisms, Coverings in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry, Braid groups; Artin groups Braid monodromy factorizations and diffeomorphism types | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Suppose given two complementary subspaces \({\mathbb{P}}^{m-1}\) of \({\mathbb{P}}^{g-1}\) with \(g=2m\geq 4\), let \(\Gamma \subset {\mathbb{P}}^{m-1}\) be a rational normal curve, and let S be the rational scroll (surface) consisting of the lines joining the corresponding points on the two copies of \(\Gamma\). Consider all canonically embedded curves \(C\subset {\mathbb{P}}^{g-1}\) of genus \(g\) lying on S. Then the linear system \(X=| C|\) on S has dimension \(2g+7\). Let I be the set of triples (C,z,H) with \(C\in X\), \(z\in S\), and H a hyperplane in \({\mathbb{P}}^{g-1}\) such that \(m_ z(C.H)\geq g\). The author proves there is exactly one component Y of I of dimension \(2g+7=\dim X\) mapping onto X. This can be phrased as saying that this system \(Y\to X\) of curves with a certain Weierstrass point has a transitive monodromy group, i.e. if \(L\supset K(X)\) is the normal closure of K(Y)\(\supset K(X)\), the group Gal(L/K(X)) acts transitively on the set of Weierstrass points \(z\in C\). For \(g=4\) this group turns out to be the full symmetric group. It may be that the same is true for any \(g=2m>4\). canonically embedded curve; algebraic monodromy; rational scroll; Weierstrass point Canuto C.,On the monodromy of Weierstrass points, Annali di Matematica pura ed appl.136 (1984), 49--63. Riemann surfaces; Weierstrass points; gap sequences, Curves in algebraic geometry, Coverings in algebraic geometry On the monodromy of Weierstrass 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 \textit{A. Tamagawa} [Compos. Math. 109, 135--194 (1997; Zbl 0899.14007)] proved an anabelian statement for affine hyperbolic curves defined over finite fields. Possibly, the statement extends to complete curves. Possibly also, the whole geometric fundamental group is not needed and one can expect that it is enough to consider some suitable metabelian quotient of the geometric fundamental group, one closely related with the theta divisor \(\theta\) of the sheaf of locally exact differentials on the curve.
This paper tentatively explores the geometry of \(\theta\) and its arithmetic. More precisely, if one considers smooth and proper curves of genus \(g\geq 2\), defined over \(k\), the algebraic closure of the prime field \(\mathbb{F}_p\), it looks plausible to expect that only finitely many of them have isomorphic fundamental groups. We prove such a result when \(g= 2\) or if we limit ourself to supersingular curves. affine hyperbolic curves; locally exact differentials Raynaud, M.: Sur le groupe fondamental d'une courbe complète en caractéristique p>0. Proc. sympos. Pure math. 70, 335-351 (2002) Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Coverings in algebraic geometry, Étale and other Grothendieck topologies and (co)homologies, Finite ground fields in algebraic geometry, Coverings of curves, fundamental group On the fundamental group of a complete curve in characteristic \(p>0\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(C\) be a smooth complex curve of genus \(2\). Let \(J^1\) be the space of degree \(1\) divisors on \(C\) and \({\mathcal S}{\mathcal U} _C(3)\) the moduli space of semi-stable rank 3 vector bundles on \(C\) with trivial determinant. Then \({\mathcal S}{\mathcal U} _C(3)\) is a projective variety of dimension \(8\), which is a double cover of \({\mathbb P}^8\), branched along a hypersurface of degree \(6\) (called ``Coble sextic'', denoted \({\mathcal C}_6\)), while \(J^1\) embedds naturally in the dual of \({\mathbb P}^8\) and there there is a unique cubic hypersurface singular along the \(J^1\), (called Coble cubic, denoted \({\mathcal C}_3\)).
The main result of the paper is the fact that these two Coble hypersurfaces are dual. As the author mentions, this result was conjectured by Dolgachev and proved by \textit{A. Ortega} [J. Algebr. Geom. 14, (2) 327--356 (2005; Zbl 1075.14031)], using ``some computer calculations''. In the paper under review no computer is used. The result induces a non-abelian Torelli result, namely: If, for two genus \(2\) curves \(C_1\) and \(C_2\) one has \({\mathcal S}{\mathcal U} _{C_1}(3) \cong {\mathcal S}{\mathcal U} _{C_2}(3)\), then \(C_1 \cong C_2\). Moreover, beautiful reinterpretations of certain classical results concerning the Segre-Igusa quartic, the Segre cubic, the Weddle quartic surface and the Kummer surface are given. vector bundles; moduli spaces; Coble hypersurfaces; classical geometry Quang Minh Nguyá» ... n, Vector bundles, dualities and classical geometry on a curve of genus two, Internat. J. Math. 18 (2007), no. 5, 535 -- 558. Vector bundles on curves and their moduli, Hypersurfaces and algebraic geometry, Coverings in algebraic geometry, Minimal model program (Mori theory, extremal rays), Torelli problem Vector bundles, dualities and classical geometry on a curve of genus two | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(K\) be a number field with ring of integers \(R\). For each integer \(g>1\) we consider the collection of abelian, étale \(R\)-coverings \(f:Y\to X\), where \(X\) and \(Y\) are connected proper curves over \(R\) and the genus of \(X\) is \(g\). We ask: Is there a positive integer \(B=B(K,g)\) which bounds the degree of such coverings? In this note we provide partial results towards such a bound and study the relationship with bounds on torsion in abelian varieties. covers of curves; degree of coverings; torsion of abelian varieties Coverings of curves, fundamental group, Arithmetic ground fields for abelian varieties, Coverings in algebraic geometry, Global ground fields in algebraic geometry On étale covers of curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper, the author observes that certain numbers occurring in Schubert calculus for \(\mathrm{SL}_n\) also occur as entries in intersection forms controlling decompositions of Soergel bimodules in higher rank. Applying the recent number theoretic results of \textit{J. Bourgain} and \textit{A. Kontorovich} [Ann. Math. (2) 180, No. 1, 137--196 (2014; Zbl 1370.11083)], he shows that these numbers grow exponentially. This observation gives many counter-examples to the expected bounds in Lusztig's conjecture on the characters of simple rational modules for \(\mathrm{SL}_n\) over fields of positive characteristic. The examples also give counter-examples to the James conjecture on decomposition numbers for symmetric groups. Soergel bimodules; Schubert calculus; Fibonacci numbers Williamson, Geordie, Schubert calculus and torsion explosion, J. Amer. Math. Soc., 30, 4, 1023-1046, (2017) Modular representations and characters, Representation theory for linear algebraic groups, Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds Schubert calculus and torsion explosion | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 Chern numbers \(c_1^2\) and \(c_2\) of a minimal complex surface of general type satisfy the famous Bogomolov-Miyaoka-Yau inequality \(c_1^2\leq 3c_2\).
It is quite hard to construct explicitly surfaces with \(c_1^2=3c_2\), and even surfaces with the ratio \(c_1^2/c_2\) close to 3 are not easily found. Some examples with \(c_1^2=3c_2\) have been given in \textit{Friedrich Hirzebruch} [Arithmetic and geometry, Pap. dedic. I. R. Shafarevich, Vol. II: Geometry, Prog. Math. 36, 113--140 (1983; Zbl 0527.14033)]. They are minimal desingularizations of Galois covers of the plane branched on certain arrangements of lines.
In the paper under review, the author generalizes Hirzebruch's construction by defining an arrangement of curves on a smooth surface \(S\) as a set of curves \(L_1,\dots L_t\) together with a set of divisors \(H_0,\dots H_k\) supported on \(L_1,\dots L_t\) such that some geometrical and combinatorial conditions are satisfied. It is shown that these conditions imply that for infinitely many integers \(n\) there exists a smooth Galois cover \(S_n\to S\) with Galois group \(Z_n^k\) branched on \(L_1,\dots L_t\) and the Chern numbers of \(S_n\) are computed.
The following example is worked out explicitly in the paper: \(S\) is the Fano surface of lines of the Fermat cubic hypersurface \(F\) defined by \(x_0^2+\cdots +x_4^2=0\) and the \(L_i\) are 30 smooth elliptic curves corresponding to the lines contained in certain special hyperplane sections of \(F\). In this case the construction gives a surface with \(c_1^2/c_2=3-1/27\) and an infinite sequence of surfaces with \(c_1^2/c_2\) converging to \(5/2\). surface arrangement; Chern numbers; covering; Fano surface Coverings in algebraic geometry, Fano varieties, Surfaces of general type Divisor arrangements and algebraic surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let k be a field and \(f: X\to Y\) a finite étale Galois covering of projective varieties over k. For a coherent sheaf \({\mathcal F}\) on Y, the group \(G=Gal(X/Y)\) naturally acts on the sheaf \(f^*({\mathcal F})\), and hence the cohomology group \(H^ i(X,f^*({\mathcal F}))\) is a k[G]-module \((i=0,1,2,...)\). We shall investigate the k[G]-module structure of \(H^ i(X,f^*({\mathcal F})).\)
Theorem 1. There exists a finite complex of k[G]-modules \((L): 0\to L^ 0\to L^ 1\to...\to L^ m\to 0\) with the following properties: (i) Each \(L^ i\) is a finitely generated free k[G]-module. (ii) The i-th cohomology group of the complex (L) is isomorphic to \(H^ i(X,f^*({\mathcal F}))\) as a k[G]-module \((i=0,1,2,...).\)
We can interpret theorem 1 as giving ''one relation'' among k[G]-modules \(H^ i(X,f^*({\mathcal F}))\) \((i=0,1,2,...)\). As a consequence we prove:
Theorem 2. Assume \(H^ i(X,f^*({\mathcal F}))=0\) for all indices i except for one value of i, say \(i=n\). Then \(H^ n(X,F^*({\mathcal F}))\) is a free k[G]-module.
Theorems 1 and 2 will be proved in {\S} 2. In {\S} 3, we shall apply our results when the dimension of X and Y equals 1. In this case we can determine the k[G]-module structure of \(H^ 0(X,f^*({\mathcal F}))\) when \({\mathcal F}\) is the canonical sheaf of Y (theorem 4). As an application of that, we get some information about the fundamental group of an algebraic curve (theorem 5). Galois module structure of the cohomology groups; finite étale Galois covering; fundamental group of an algebraic curve Nakajima, Shōichi: On Galois module structure of the cohomology groups of an algebraic variety. Invent. math. 75, No. 1, 1-8 (1984) Coverings in algebraic geometry, Classical real and complex (co)homology in algebraic geometry, Coverings of curves, fundamental group On Galois module structure of the cohomology groups of an algebraic variety | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Zariski's hyperplane section theorem says that for a hypersurface \(S\) in \(\mathbb{P}^n\) of dimension at least 2, we have \(\pi_1 (\mathbb{P}^2 \setminus (\mathbb{P}^2\cap S)) \cong \pi_1 (\mathbb{P}^n \setminus S)\), where \(\mathbb{P}^2 \subset \mathbb{P}^n\) is in general position with respect to \(S\). In the paper under review, the author gives a generalization of this result in the following case. Let \(f:X\to U\) be a morphism from a smooth connected variety \(X\) to a complex homogeneous variety \(U\); assume a connected affine algebraic group \(G\) acts transitively on \(U\). When
(1) \(U= \mathbb{P}^n\), \(G= \text{GL} (n+1)\),
(2) \(U= \mathbb{A}^n\), \(G\) is the group of affine automorphisms of \(\mathbb{A}^n\),
(3) \(U= \text{Grass} (r,v)\), \(G= \text{GL} (V)\),
the author calculates the fundamental group of \(\gamma f^{-1} (U\setminus D)\) in terms of the fundamental groups of \(X\) and \(U\setminus D\), where \(\gamma\) is a general element of \(G\) and \(D\) is a reduced effective divisor in \(U\). -- As a corollary we obtain the invariance of the fundamental group of the complement to an affine plane curve. Zariski hyperplane section theorem; fundamental group of the complement to an affine plane curve I. Shimada, Fundamental groups of complements to hypersurfaces. RIMS Kôkyûroku 1033, 27-33 (1998) Homotopy theory and fundamental groups in algebraic geometry, Plane and space curves, Coverings in algebraic geometry, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Fundamental groups of complements to 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 Let \(\mathcal{L}/\mathcal{K}\) be a finite Galois extension and let \(X\) be an affine algebraic variety defined over \(\mathcal{L}\). Weil's Galois descent theorem provides necessary and sufficient conditions for \(X\) to be definable over \(\mathcal{K}\), that is, for the existence of an algebraic variety \(Y\) defined over \(\mathcal{K}\) together with a birational isomorphism \(R:X \to Y\) defined over \(\mathcal{L}\). Weil's proof does not provide a method to construct the birational isomorphism \(R\). The aim of this paper is to give an explicit construction of \(R\). algebraic varieties; Galois extensions; fields of definition Compact Riemann surfaces and uniformization, Coverings in algebraic geometry, Families, moduli of curves (algebraic), Coverings of curves, fundamental group, Separable extensions, Galois theory, Computational aspects in algebraic geometry Weil's Galois descent theorem from a computational point of view | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The general structure of a triple cover of an algebraic variety is described by \textit{R. Miranda} in Am. J. Math. 107, 1123-1158 (1985; Zbl 0611.14011). A generalization of Miranda's result to the case of arbitrary schemes and the description of the characteristic 3 case are contained in the reviewer's paper in Ark. Mat. 27, No. 2, 319-341 (1989; Zbl 0707.14010).
In the present paper the author tries a new approach to the study of triple covers. Given a triple cover \(p:X\to Y\), he considers its Galoisization \(\hat p:\hat X\to Y\), namely the smallest Galois cover of \(Y\) such that there exists \(\alpha:\hat X\to X\) which is a morphism over \(Y\), and tries to deduce properties of \(X\) from the properties of \(\hat X\). --- This way, he proves that a totally ramified triple cover of a simply connected variety is cyclic and describes \(\hat X\) and \(X\) in detail in the case that \(X\) is a surface and the singularities of the branch curve of \(p\) are cusps. In addition, several examples are carefully computed. cyclic cover; structure of a triple cover; ramified triple cover Tokunaga, H.: Triple coverings of algebraic surfaces according to the cardano formula. J. math. Kyoto univ. 31, 359-375 (1991) Coverings in algebraic geometry, Surfaces and higher-dimensional varieties Triple coverings of algebraic surfaces according to the Cardano 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 gives some necessary and sufficiency conditions on the surjective of multiply maps \(H^{0}(R, L) \times H^{0}(R, K) \rightarrow H^{0}(R, L \otimes K),\) \((s, t) \rightarrow s \cdot t.\) Here \(R\) is a compact Riemann surface, \(L\) a holomorphic line bundle and \(K\) is the canonical line bundle on \(R.\) A line bundle \(L\) on \(R\) is called base point free, if sections in \(H^{0}(R, L)\) do not have a common zero point.
Theorem 1.1. For any compact Riemann surface \(R,\) the multiply maps \(H^{0}(R, L) \times H^{0}(R, K)\rightarrow H^{0}(R, L \otimes K)\) is surjective if and only if \(L\) is not a composite line bundle. Theorem 1.2. For a generic compact Riemann surface \(R,\) if the genus \(g\) of \(R\) is odd, then the map \(H^{0}(R, L) \times H^{0}(R, K) \rightarrow H^{0}(R, L \otimes K)\) is surjective if and only if \(L\) is base point free; if the genus \(g\) of \(R\) is even, and \(L\) is not the form \(L_{1} \otimes L_{2}\) with \(\deg(L_{1}) = \deg(L_{2}) = \frac{g}{2} + 1,\) and \(\dim H^{0}(R, L_{1}) = \dim H^{0}(R, L_{2}) = 2;\) \(\dim H^{0}(R, L_{1} \otimes L_{2}) = 3,\) then the map
\[
H^{0}(R, L) \times H^{0}(R, K) \rightarrow H^{0}(R, L \otimes K)
\]
is surjective if and only if \(L\) is base point free. compact Riemann surfaces; line bundles Coverings in algebraic geometry, Vector bundles on curves and their moduli, Riemann surfaces On multiply of sections of line bundles on compact Riemann 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 projective manifold, \(\rho :\tilde{X}\rightarrow X\) its universal covering and \({\rho}^*:\text{Vect}(X)\rightarrow \text{Vect}(\tilde{X})\) the pullback map for the isomorphism classes of vector bundles. It is still unknown whether the non-compact universal covers of projective varieties do have non-constant holomorphic functions. The Shafarevich conjecture claims that the universal cover \(\tilde{X}\) of a projective variety \(X\) is holomorphically convex (i.e. \(\tilde{X}\) has a lot of non-constant holomorphic functions).
In this paper, the authors prove the following interesting result: if a universal covering of a projective manifold \(X\) has no non-constant holomorphic functions, then the pullback map \({\rho}^*\) is almost an imbedding. This result can potentially be used to show that the universal covering of a projective manifold has a non-constant holomorphic function. holomorphic functions; vector bundles; universal coverings; projective manifolds F. Bogomolov and B. de Oliveira, ''Holomorphic functions and vector bundles on coverings of projective varieties,'' Asian J. Math., vol. 9, iss. 3, pp. 295-314, 2005. Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Coverings in algebraic geometry, Uniformization of complex manifolds, Holomorphically convex complex spaces, reduction theory Holomorphic functions and vector bundles on coverings 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 author studies the structure of \(\mathbb{Z}/3\)-covers of smooth algebraic surfaces. In particular he gives an explicit method of resolution of the singularities of such a cover, analogous to the standard resolution of singularities of double covers. Similar computations for a general abelian cover can be found in the reviewer's paper in J. Reine Angew. Math. 417, 191-213 (1991; Zbl 0721.14009). triple covers; resolution of singularities of a surface covering Tan S L. Galois triple covers of surfaces. Sci China Ser A, 1991, 34: 935--942 Global theory and resolution of singularities (algebro-geometric aspects), Coverings in algebraic geometry, Surfaces and higher-dimensional varieties Galois triple covers of surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let k be an algebraically closed field of characteristic p\(>0\). A purely inseparable covering of exponent 1 of the affine plane \({\mathbb{A}}^ 2:=Spec k[u,v]\), or affine Zariski surface, is a normal affine surface \(X:=Spec {\mathbb{A}}\) such that \(k[u,v]\subsetneqq A\subsetneqq k[x,y]\) with \(u=x^ p,\) \(v=y^ p.\)
\textit{J. Lang} [Mich. Math. J. 28, 375-380 (1981; Zbl 0495.14021)] has shown that if \(char(k)=2\), then \(A:=k[u,v(uv+1)x+vy]\) is a regular, factorial, irrational k-algebra, and we will show that this is true, in fact, for arbitrary \(p>0\). These examples suggest that affine Zariski surfaces have a rich geometry to be studied. One should contrast this with the situation when \(A\subset k[x,y]\) is an affine, regular k-algebra such that k[x,y] is generically separable over A. It is known that then A is a polynomial ring over k if A is factorial. characteristic p; purely inseparable covering of exponent 1; affine Zariski surfaces Miyanishi, M.; Russell, P.: Purely inseparable coverings of exponent one of the affine plane. J. pure appl. Algebra 28, 279-317 (1983) Coverings in algebraic geometry, Special surfaces Purely inseparable coverings of exponent one of the affine plane | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Using the ``spannedness'' and ``\(k\)-ampleness'' (in the sense of Sommese) of the vector bundle associated with a branched covering, \(f : X \to G\), of the Grassmannian \(G=\text{Gr}(r,n)\), we prove that \(f\) induces \(\mathbb{C}\)-cohomology isomorphisms between \(X\) and \(G\) in a certain range of cohomology dimensions, provided the degree of the covering is small enough. This generalizes Lazarsfeld's theorem on branched coverings of projective space. An analogous statement is proved true for the homotopy case. spannedness; ampleness; vector bundle; Grassmannian; branched covering Kim M. Barth-Lefschetz theorem for branched coverings of Grassmannians. J Reine Angew Math, 470: 109--122 (1996) Coverings in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Grassmannians, Schubert varieties, flag manifolds A Barth-Lefschetz type theorem for branched coverings of Grassmannians | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(S\) be a surface with at worst quotient singularities such that the anticanonical divisor \(-K_S\) is nef and big, i.e. \(- K_S \cdot C \geq 0\) for all curves \(C\) on \(S\) and \((- K_S)^2 > 0\). Such a surface is said to be quasi-log del Pezzo (it is log del Pezzo if \(- K_S\) is ample).
The following is shown: Let \(S\) be a quasi-log del Pezzo surface, \(S^0 : = S - \text{Sing} (S)\) the nonsingular part of \(S\). Then the fundamental group \(\pi_1 (S^0)\) is finite.
This is a generalization of a result of \textit{R. V. Gurjar} and \textit{D.-Q. Zhang}, J. Math. Sci. Tokyo 1, No. 1, 137-180 (1994) where it is shown for \(S\) log-del Pezzo. The author points out that it is impossible to relax the hypotheses by the condition \(\kappa (S,- K_X) = 2\), respectively to omit the condition ``\(- K_S\) is big''. singularities; finiteness of fundamental group; quasi-log del Pezzo surface Zhang, D. -Q.: Algebraic surfaces with nef and big anti-canonical divisor. Math. proc. Cambridge philos. Soc. 117, 161-163 (1995) Coverings in algebraic geometry, Divisors, linear systems, invertible sheaves, Special surfaces Algebraic surfaces with nef and big anti-canonical divisor | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 fundamental groups related with log-terminal singularities.
The main theorem shows that the fundamental group of a normal variety \(X\) is preserved by a resolution of its singularities if there exists an effective \(\mathbb{Q}\)-divisor \(\Delta\) such that \((X, \Delta)\) is a KLT (Kawamata log-terminal) pair.
\textit{J. Kollár} [Invent. Math. 113, No. 1, 177--215 (1993; Zbl 0819.14006)] proved the above statement in some special cases (as \(\dim X=3\)), and proved the analogous for algebraic fundamental groups.
The second theorem of this paper, a slight generalization of a previous result of the author [J. Algebr. Geom. 10, No. 4, 713--724 (2001; Zbl 1096.14009)], shows that fundamental groups are preserved also by proper surjective morphisms \(f:X \rightarrow S\) of normal varieties with connected fibres such that there exists a \(\Delta\) with \((X,\Delta)\) KLT and \(-(K_X+\Delta)\) \(f\)-nef and \(f\)-big.
As a corollary of its results the author can show that the fundamental group is invariant under various fundamental operations in the minimal model program, as contractions of extremal rays, flips and pluricanonical morphisms of minimal varieties of general type. fundamental group; \(L^2\)-index theorem; extremal rays; flips; pluricanonical morphisms Coverings in algebraic geometry, Singularities in algebraic geometry, Transcendental methods of algebraic geometry (complex-analytic aspects), Minimal model program (Mori theory, extremal rays) Local simple connectedness of resolutions of log-terminal 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 a natural number \(m\), let \(\mathcal{S}_m/\mathbb{F}_2\) be the \(m\) th Suzuki curve. We study the mod \(2\) Dieudonné module of \(\mathcal{S}_m\), which gives the equivalent information as the Ekedahl-Oort type or the structure of the \(2\)-torsion group scheme of its Jacobian. We accomplish this by studying the de Rham cohomology of \(\mathcal{S}_m\). For all \(m\), we determine the structure of the de Rham cohomology as a \(2\)-modular representation of the \(m\) th Suzuki group and the structure of a submodule of the mod \(2\) Dieudonné module. For \(m=1\) and \(2\), we determine the complete structure of the mod \(2\) Dieudonné module. Suzuki curve; Suzuki group; Ekedahl-Oort type; de Rham cohomology; Dieudonné module; modular representation Abelian varieties of dimension \(> 1\), Curves over finite and local fields, de Rham cohomology and algebraic geometry, Jacobians, Prym varieties, Modular representations and characters, Group schemes, Representations of finite groups of Lie type The de Rham cohomology of the Suzuki curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\), \(Y\) be smooth connected complex varieties of dimension \(n\), and let \(f\colon X\to Y\) be a finite morphism of degree \(d\), namely a branched covering of \(Y\); the sheaf \(f_*{\mathcal O}_X\) is locally free of rank \(d\) and the trace map \(f_*{\mathcal O}_X\to {\mathcal O}_Y\) is surjective. Denote by \({\mathcal E}^*\) the kernel of the trace map: \({\mathcal E}^*\) is a vector bundle of rank \(d-1\) on \(Y\). \textit{R. Lazarsfeld} [Math. Ann. 249, 153-162 (1980; Zbl 0434.32013)], proved that if \(Y={\mathbb P}^n\) then the dual bundle \({\mathcal E}\) of \({\mathcal E}^*\) is ample. As a consequence of this result, Lazarsfeld obtains the following Barth-Lefschetz type theorem:
The morphism \(f_*: H^i({\mathbb P}^n, {\mathbb C})\to H^i(X,{\mathbb C})\) is an isomorphism for \(i\leq n-d+1\).
Recently \textit{O. Debarre} [Manuscr. Math. 89, No. 4, 407-425 (1996; Zbl 0922.14033)] has conjectured that \({\mathcal E}\) is ample when \(Y\) is a homogeneous space with Picard number \(1\). In the paper under consideration the authors prove that \({\mathcal E}\) is ample when \(Y=LG_n\), where \(LG_n\) is the Lagrangian Grassmannian of maximal isotropic subspaces of a symplectic space of dimension \(2n\), and also when \(Y\) is a quadric of dimension \(3\leq n\leq 6\). In analogy with the case \(Y={\mathbb P}^n\), they obtain the following Barth-Lefschetz type statement:
If \(f: X\to LG_n\) is a branched covering of degree \(d\), then \(f_*: H^i(LG_n,{\mathbb C})\to H^i(X,{\mathbb C})\) is an isomorphism for \(i\leq n-d+1\).
In addition, they show that if \(Y\) is a homogeneous space not necessarily with Picard number \(1\), then \({\mathcal E}\) is generated by global sections. This gives evidence for a more general conjecture, stating that in this case \({\mathcal E}\) should be \(k\)-ample for a suitable \(k\). branched covering; homogeneous space; cohomology group; Lagrangian Grassmannian Kim M, Manivel L. On branched coverings of some homogeneous spaces. Topology, 38: 1141--1160 (1999) Coverings in algebraic geometry, Homogeneous spaces and generalizations, Topological properties in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds On branched coverings of some homogeneous spaces | 0 |
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