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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 [For the entire collection see Zbl 0516.00012.]
Let K be a two-dimensional arithmetic field. The authors show how to describe \(Gal(K^{ab}/K)\) in terms of algebraic K-theory \((K{}_ 2\) in fact). If X is a scheme (regular and connected), with function field K, which is proper over \({\mathbb{Z}}\), they construct ''idèle class groups'' \(\bar C_ m(X)\) and \(C_ m(X)\) associated with a modulus, m, on X. These groups are defined in terms of algebraic K-theory. Their class field theorem asserts, for example,
\[
(*)\quad Gal(K^{ab}/K)\quad \cong \quad \lim_{\overset \leftarrow m}\bar C_ m(X)
\]
as m ranges over admissible moduli on X.
To prove (*), the authors develop their two dimensional local and global class field theories. In addition, they describe how their results are related to the class field theory of S. Lang [cf., e.g., \textit{N. M. Katz} and \textit{S. Lang}, Enseign. Math., II. Sér. 27, 285-314; 315-319 (1981; Zbl 0495.14011)]. proper scheme over \({\mathbb{Z}}\); two dimensional class field theories; two- dimensional arithmetic field; \(K_ 2\); idèle class groups; algebraic K-theory Kazuya Kato and Shuji Saito, Two-dimensional class field theory, Galois groups and their representations (Nagoya, 1981) Adv. Stud. Pure Math., vol. 2, North-Holland, Amsterdam, 1983, pp. 103 -- 152. Class field theory; \(p\)-adic formal groups, Class field theory, Arithmetic ground fields for abelian varieties, \(K\)-theory of global fields, \(K\)-theory of local fields, Transcendental field extensions, Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects), Applications of methods of algebraic \(K\)-theory in algebraic geometry, Coverings in algebraic geometry Two dimensional class field 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 We give a motivic proof of finiteness of S-integral points on punctured projective line. We do this by studying torsors over different notions of unipotent fundamental groups attached to an open curve defined over a number field and the algebraic spaces which parametrize these torsors. This reduces the finiteness of integral points of such curves to a strict inequality between some global and local Galois cohomology groups. When the curve is a punctured projective line, we use abelian categories of mixed Tate motives over the base number field and localizations of its ring of integers to replace the global cohomology groups by algebraic \(K\)-groups of the base number field. Finally for totally real number fields, we use Borel's explicit calculations to conclude finiteness of S-integral points. M. HADIAN-JAZI, Motivic fundamental groups and integral points, Ph.D. thesis, UniversitaÈt Bonn, 2010. Research exposition (monographs, survey articles) pertaining to algebraic geometry, Motivic cohomology; motivic homotopy theory, Coverings in algebraic geometry, Rational points Motivic fundamental groups and integral points | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \({\mathcal C}\) be a quasi-analytic ring of smooth functions in \({\mathbb R}^n\) enjoying the property of resolution of singularities as described by Bierstone and Milman in [Sel. Math., New Ser. 10, No. 1, 1-28 (2004; Zbl 1078.14087)]. The Łojasiewicz radical of an ideal \(I\) in \({\mathcal C}\) is defined as the ideal \(\root{L}\of{I}\) of all functions \(g\in {\mathcal C}\) for which that are a function \(f\in I\) and an integer \(m\geq 1\) such that \(f\geq g^{2m}\) on \({\mathbb{R}}^n\). The saturation of an ideal \(J\) in \({\mathcal C}\) is defined as the ideal \(\widetilde{J}\) of all functions \(g\in {\mathcal C}\) such that for any \(x\in {\mathbb{R}}^n\), we have \(g_x\in J{\mathcal C}_x\), where the subscript \(x\) denotes the germs at \(x\). It is shown that if \(I\) is finitely generated, then the ideal of functions vanishing on the zero variety of \(I\) coincides with \(\widetilde{\root{L}\of{I}}\). When \(I\) is not finitely generated, a similar characterization is given by a variant of the Łojasiewicz radical, computed on all compact subsets of \({\mathbb R}^n\). quasianalytic classes of functions; real Nullstellensatz; Lojasiewicz radical Real-analytic and semi-analytic sets, \(C^\infty\)-functions, quasi-analytic functions, Real algebraic and real-analytic geometry, Rings and algebras of continuous, differentiable or analytic functions A global Nullstellensatz for ideals of Denjoy-Carleman 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 We generalize to the ``secant'' case the methods previously developed [cf. \textit{A. Treibich}, Duke Math. J. 59, No. 3, 611--627 (1989; Zbl 0698.14029) and \textit{A. Treibich} and \textit{J.-L. Verdier}, in: The Grothendieck Festschrift, Collect. Artic. in Honor of the 60th Birthday of A. Grothendieck. Vol. III, Prog. Math. 88, 437--480 (1990; Zbl 0726.14024)] for studying the tangential covers. We construct, in particular, all complete and integral curves over an algebraically closed field of characteristic 0, which are secant to an elliptic curve in their generalized Jacobians. secant curve to an elliptic curve; secant to an ellipuc curve; generalized Jacobians Jacobians, Prym varieties, Coverings in algebraic geometry Discrete analogs of polynomials and tangential covers | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(G\) be a linear algebraic group over an algebraically closed field \(k\) of characteristic \(p>0\). We assume \(G\) is of exponential type, which means roughly that for every \(p\)-nilpotent element \(x\) in the Lie algebra \(\mathfrak g\) there is a 1-parameter subgroup whose Lie algebra is spanned by \(x\).
The aim of the paper is to develop a theory of support varieties for rational representations of \(G\). It will not be based on cohomology but on 1-parameter subgroups. Let \(V_r(G)\) be the affine scheme of homomorphisms of group schemes from the \(r\)-th Frobenius kernel \(\mathbb G_{a(r)}\) of \(\mathbb G_a\) to \(G\). Let \(V(G)\) be the set of 1-parameter subgroups \(\sigma\colon\mathbb G_a\to G\), viewed as a topological subspace of \(\varprojlim_rV_r(G)(k)\). The support varieties will be subsets of \(V(G)\). First one studies the case \(G=\mathbb G_a\) extensively. Recall that \(kG\), known as the group algebra or the hyperalgebra or the algebra of distributions, is the direct limit of the duals of the coordinate rings of the Frobenius kernels of \(G\). Given a representation \(M\) and a 1-parameter subgroup \(\sigma\colon\mathbb G_a\to G\) the author constructs a \(p\)-nilpotent element of \(kG\) and calls its action on \(M\) ``the action of \(G\) on \(M\) at \(\sigma\)''. The construction is more complicated than the terminology suggests.
A \(p\)-nilpotent operator on a vector space \(V\) defines an action of \(k[t]/t^p\) on \(V\). If this makes \(V\) into a free \(k[t]/t^p\)-module, then we say the operator acts freely. So if \(V\) is finite dimensional then a \(p\)-nilpotent operator acts freely if and only if all its Jordan blocks have size \(p\). Now the support variety \(V(G)_M\) of a rational \(G\)-module \(M\) is defined to be the subset of \(V(G)\) consisting of the \(\sigma\) at which the ``action of \(G\) on \(M\) at \(\sigma\)'' does \textit{not} act freely on \(M\). It is shown that this \(V(G)_M\) has properties that are familiar from the theory of support varieties of finite groups. ``Non-maximal \(j\)-rank varieties'' are introduced that detect finer information about Jordan types. linear algebraic groups; support varieties; 1-parameter subgroups; rational representations; group schemes; Frobenius kernels; \(p\)-nilpotent elements; Jordan blocks Friedlander, E., Support varieties for rational representations, Compos. math., 151, 765-792, (2015) Representation theory for linear algebraic groups, Modular representations and characters, Cohomology theory for linear algebraic groups, Group schemes Support varieties for rational representations. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\mathbb K\) be an algebraically closed field of characteristic zero. For any \(n>0\), let \(\mathbb T\) be the torus \((\mathbb K^*)^n\). A pair of nonsingular \(n\times n\) matrices \(M,N\) with integer coefficients defines two homomorphisms \(\phi(M),\phi(N):\mathbb T\rightarrow\mathbb T\). Hence, the data \(f=(\mathbb T,\phi(M),\phi(N))\) defines a dominant rational self-correspondence called monomial correspondence.
The main result of the paper is the computation of the \(k\)-th dynamical degree of \(f\), denoted by \(\lambda_k(f)\).
Other results concern the growth of the degree of \(f^p\), and, if \(\mathbb K=\overline{\mathbb Q}\), a computation of the arithmetic degree of a rational point of \(\mathbb T\).
Finally, as a direct application of the results by \textit{J.-L. Lin} and \textit{E. Wulcan} [Ann. Inst. Fourier 64, No. 5, 2127--2146 (2014; Zbl 1322.14076)], the authors prove that there exists a projective toric variety with at worst quotient singularities on which \(f\) is algebraically \(k\)-stable.
These results are obtained using the fact that even if the iterated correspondence \(f^p\) may not be monomial, it is closely related to the maps \(\mathbb T\rightarrow\mathbb T\) induced by the matrix \((\det(M)NM^{-1})^p\) and by the diagonal matrix of order \(n\) with eigenvalue \(\det(M)\). arithmetic dynamics; algebraic dynamics; monomial correspondences; dynamical degrees Height functions; Green functions; invariant measures in arithmetic and non-Archimedean dynamical systems, Families, moduli of curves (algebraic), Rational and birational maps, Coverings in algebraic geometry Dynamical invariants of monomial correspondences | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 natural symplectic structure on the space of representations \(\pi\) \(\to G\), where \(\pi\) is the fundamental group of a Riemann surface and G is a semisimple Lie group, is introduced. The author considers this result as a common cause for the existence of a symplectic structure on different moduli spaces associated with Riemann surfaces. For example if \(G=PSL_ 2({\mathbb{R}})\) then the symplectic structure coincides with the Weil- Petersson's one on Teichmüller space. Weil-Petersson structure; free differential calculus; symplectic structure; fundamental group of a Riemann surface; moduli spaces; Teichmüller space 16. Goldman, William M. The symplectic nature of fundamental groups of surfaces \textit{Adv. in Math.}54 (1984) 200--225 Math Reviews MR762512 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Coverings of curves, fundamental group, General geometric structures on manifolds (almost complex, almost product structures, etc.), Riemann surfaces, Coverings in algebraic geometry, Complex-analytic moduli problems, Semisimple Lie groups and their representations, Families, moduli of curves (analytic) The symplectic nature of fundamental groups of surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In the first section the notions of asymptotic vectors, binormal vectors, parabolic points, and inflection points are extended to an immersion \(s:M^n\rightarrow \mathbb{R}^{2n}\). The second section is focused on the case \(n=3\), namely the case of \(3\)-manifolds immersed in \(\mathbb{R}^6\). The possible generic algebraic structures of asymptotic vectors at a parabolic point or an inflection point, and the generic topological structures of the parabolic surface are classified. This results are directly connected to the author's papers [Adv. Geom. 3, No. 4, 453--468 (2003; Zbl 1038.53006)] and [Contemp. Math. 459, 1--12 (2008; Zbl 1156.53301)]. asymptotic vector; binormal vector; parabolic point; inflection point; immersed manifold; parabolic surface Dreibelbis, Daniel, Self-conjugate vectors of immersed 3-manifolds in \(\mathbb {R}^6\), Topology Appl., 0166-8641, 159, 2, 450-456, (2012) 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, Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) Self-conjugate vectors of immersed 3-manifolds in \(\mathbb{R}^6\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(D\) be a curve of genus \(g\) over an algebraically closed field of positive characteristic \(p\). We write \(\pi_A(D)\) for the set of isomorphism classes of of finite groups occurring as Galois groups of unramified covers of \(D\). This paper is concerned with giving necessary and sufficient conditions for a family of groups to lie in \(\pi_A(D)\). The main tool is a criterion of Nakajima concerning generators of group rings. The author first reviews the known results and gives a comparison between some necessary conditions. In particular, she shows that for many groups, Nakajima's criterion does not follow from previously known criteria. Then the author goes on to consider a specific class of groups for which she shows that Nakajima's condition is not only necessary but also sufficient. In the last section more specific results about tame covers of genus two curves are derived. fundamental group; positive characteristic; projective curves; tame covers; Galois groups Stevenson, K. F.: Conditions related to \({\pi}\)1. J. number theory 68, 62-79 (1998) Coverings of curves, fundamental group, Coverings in algebraic geometry, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Homotopy theory and fundamental groups in algebraic geometry Conditions related to \(\pi_1\) of projective 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 It is known that the Jacobian variety of the Klein curve is isomorphic to \(E\times E\times E\), where \(E\) is an elliptic curve. In this paper we compute the defining equation of \(E\) in Weierstrass normal form and show that the Klein curve covers \(E\) doubly and triply. orbit space; theta matrix; coverings; Klein curve; Jacobian variety Jacobians, Prym varieties, Elliptic curves, Coverings in algebraic geometry, Coverings of curves, fundamental group A note on Klein 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 An algebraic surface M of general type with \(K_{M^ 2}=9\) and \(p_ g=q=0\) has been studied by Mumford. This surface is called Mumford's fake projective plane, because it has the same Betti numbers as the complex plane. By its construction M has an unramified Galois covering \(V\to M\) of order 8. Moreover, a simple group G of order 168 acts on V and M is the quotient of V by a 2-Sylow subgroup of G. The quotient surface \(Y=V/G\) and its minimal desingularization \(\tilde Y\) is studied. (Thus M is a branched covering over Y of order 21.) It is shown that Y has only 4 isolated singularities as singularites, 3 of them are rational double points of type \(A_ 2\) and the 4-th singularity is the quotient singularity of type (7,3) by the cyclic group of order 7. Moreover, it is shown that \(\tilde Y\) is an elliptic surface with the following data: the Kodaira dimension 1, the Euler number 12, \(p_ g=q=0\), the second pluri- genus 1, the third pluri-genus 1, the combination of singular fibers \(_ 2I_ 0+_ 3I_ 0+I_ 3+I_ 3+I_ 3+I_ 3\). Since Mumford's paper is based on the theory of the 2-adic unit ball due to Mustafin and Kurihara, the theory of schemes over the ring \({\mathbb{Z}}_ 2\) of 2-adic integers is the main tool in this paper. Mumford's fake projective plane M N Ishida, An elliptic surface covered by Mumford's fake projective plane, Tohoku Math. J. \((2)\) 40 (1988) 367 Elliptic surfaces, elliptic or Calabi-Yau fibrations, Families, moduli, classification: algebraic theory, Special surfaces, Coverings in algebraic geometry An elliptic surface covered by Mumford's fake projective plane | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let X be a hypersurface of degree n in \({\mathbb{P}}^ n_{{\mathbb{C}}}\) for \(n\geq 4\). Then the author shows that X contains a family of plane conic curves which cover X. The result for \(n=4\) is given by \textit{B. R. Tennison} [Proc. Lond. Math. Soc., III. Ser. 29, 714-734 (1974; Zbl 0308.14005)], and that for \(n=5\) is due to \textit{A. Conte} and \textit{J. P. Murre} [Math. Ann. 238, 79-88 (1978; Zbl 0373.14006)]. The author proves the result by reducing it to the argument for a generic hypersurface X. family of plane conic curves Families, moduli of curves (algebraic), Coverings in algebraic geometry, \(4\)-folds On hypersurfaces admitting a covering by rational 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 \(G\) be a finite group, and let \(k\) be a field of characteristic~\(p\). A \(kG\)-module is said to be generic if it is indecomposable, of infinite length over \(kG\), but of finite length as a module over its endomorphism ring. Let \(V_G\) denote the maximal ideal spectrum of the cohomology ring \(H^*(G,k)\). Let \(U\) be the collection of closed homogeneous irreducible subvarieties of \(V_G\) that are not irreducible components, and let \(F_U\) be the corresponding Rickard idempotent module. It is shown that \(F_U\) is a finite sum of generic modules corresponding to the irreducible components of \(V_G\). An explicit construction of \(F_U\) is given. generic modules; idempotent modules; cohomological varieties; finite groups; cohomology rings Auslander, M.: A functorial approach to representation theory. In: Representations of Algebras (Puebla, 1980) of Lecture Notes in Math., vol. 944, pp 105-179. Springer, Berlin (1982) Modular representations and characters, Special varieties, Group rings of finite groups and their modules (group-theoretic aspects), Homological methods in group theory Generic idempotent modules for a finite group | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(A\) be a smooth complex projective \(n\)-fold, \(n \geq 2\), and let \(\pi : A \to \mathbb{P}^ 2\) be a double cover. The authors' aim is to classify all the smooth \((n + 1)\)-folds \(X\) containing \(A\) as a very ample divisor. Let \(L = {\mathcal O}_ X (A)\). Then for \(n \geq 3\) the pair \((X,L)\) can only be either \((\mathbb{P}^{n + 1}, {\mathcal O}_{\mathbb{P}^{n + 1}} (2))\) or \((\mathbb{Q}^{n + 1}, {\mathcal O}_{\mathbb{Q}^{n + 1}} (1))\), while for \(n = 2\) the situation is far richer. Actually in this case \(K_ A = \pi^* {\mathcal O}_{\mathbb{P}^ 2} (a)\) with \(a \geq - 2\). Equality gives the same pairs as above, but if \(a = 0\), then \(X\) is a Fano 3-fold of the principal series while, for \(a \geq 1\), \((X,L)\) is a conic bundle over a smooth surface. Case \(a=-1\) is the more interesting one and leads to 6 different classes of pairs related to special varieties arising in adjunction theory, like quadric fibrations over \(\mathbb{P}^ 1\), Veronese bundles over \(\mathbb{P}^ 1\) and scrolls over a smooth surface \(S\). In particular this last class gives rise to 4 special pairs which are thoroughly studied by combining the scroll structure over \(S\) with a structure of the conic bundle \(\Pi : X \to \mathbb{P}^ 2\) given by a morphism extending \(\pi\). double solids; hyperplane sections; double cover as a very ample divisor; Fano 3-fold; adjunction theory; quadric fibrations; Veronese bundles; scrolls over a smooth surface A. Lanteri, M. Palleschi and A. J. Sommese, Double covers of \(\bm{P}^{n}\) as very ample divisors, Nagoya Math. J., 137 (1995), 1-32. Divisors, linear systems, invertible sheaves, Coverings in algebraic geometry, \(3\)-folds, Projective techniques in algebraic geometry Double covers of \(\mathbb{P}^ n\) as very ample divisors | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper studies a relation between fundamental group of the complement to a plane singular curve and the orbifold pencils containing it. The main tool is the use of Albanese varieties of cyclic covers ramified along such curves. Our results give sufficient conditions for a plane singular curve to belong to an orbifold pencil, that is, a pencil of plane curves with multiple fibers inducing a map onto an orbifold curve whose orbifold fundamental group is nontrivial. We construct an example of a cyclic cover of the projective plane which is an abelian surface isomorphic to the Jacobian of a curve of genus 2 illustrating the extent to which these conditions are necessary. Singularities of curves, local rings, Pencils, nets, webs in algebraic geometry, Coverings in algebraic geometry, Coverings of curves, fundamental group, Singularities in algebraic geometry Albanese varieties of cyclic covers of the projective plane and orbifold pencils | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Consider a 2-dimensional singularity \((X,x)\) in \(\mathbb{C}^n\). If we intersect \(X\) with a small ball around the singularity and remove the singular point itself, we get a space \(V\). The paper studies the universal cover of \(V\), namely it is shown that the universal cover is a Stein manifold if and only if the fundamental group \(G\) of \(V\) is infinite. The \(G\) finite case follows from results of Mumford and Prill. The infinite case is easier if the first homology group is infinite also. In this case the authors study the exceptional divisor at a resolution of the singularity. If its graph is a tree the result follows from a work of Napier; if there is a cycle the authors use it to construct an infinite covering with suitable properties. The other case when the homology is finite but the fundamental group is infinite is reduced to the first case using results of Luecke and Heil. complement of singularity; universal covering; Stein manifold M. Colt\?oiu and M. Tib\? ar, Steinness of the universal covering of the complement of a 2-dimensional complex singularity, Math. Ann. 326 (2003), 95-104. Modifications; resolution of singularities (complex-analytic aspects), Stein manifolds, , Coverings in algebraic geometry, Stein spaces, Steinness of the universal covering of the complement of a 2-dimensional complex singularity | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(A\) be a complex abelian varitey of dimension \(n\), and let \(\pi:\mathbb{C}^n\to A\) be its covering map. It follows from the work of \textit{J. Ax} [Am. J. Math. 94, 1205--1213 (1972; Zbl 0266.14018)] that given an algebraic variety \(X\) in \(\mathbb{C}^n\) the Zariski closure of \(\pi(X)\) is a union of finitely many cosets of abelian subavrieties of \(A\). In [J. Reine Angew. Math. 741, 47--66 (2018; Zbl 1408.14140); Q. J. Math. 68, No. 2, 359--367 (2017; Zbl 1386.14165)] \textit{E. Ullmo} and \textit{A. Yafaev} considered the topological closure of \(\pi(X)\) in the above setting and also in the case that \(X\) is a set definable in an o-minimal expansion of the real field and showed in this case a result similar to the one of Ax in the case that \(X\) is an algebraic curve.
In the article under review the authors give a full description of the topological closure \(\mathrm{cl}(\pi(X))\) when \(X\) is an algebraic subvariety of \(\mathbb{C}^n\) of arbitrary dimension and \(A\) is more general a compact complex torus and also when \(X\subset \mathbb{R}^n\) is definable in an o-minimal structure over the reals and \(\pi:\mathbb{R}^n\to \mathbb{T}\) is the covering map of a compact real torus.
Here are the main results:
Theorem. Let \(\pi:\mathbb{C}^n\to \mathbb{T}\) be the covering map of a compact complex torus and let \(X\) be an algebraic subvariety of \(\mathbb{C}^n\). Then there are finitely many algebraic subvarieties \(C_1,\ldots,C_m\subset \mathbb{C}^n\) and finitely many real subtori (i.e. real Lie subgroups) \(\mathbb{T}_1,\ldots,\mathbb{T}_m\subset \mathbb{T}\) of positive dimension such that
\[\mathrm{cl}(\pi(X))=\pi(X)\cup\bigcup_{i=1}^m(\pi(C_i)+\mathbb{T}_i).\]
In addition,
(i) For every \(i=1,\ldots,m,\) we have \(\dim_\mathbb{C}C_i<\dim_\mathbb{C}X\).
(ii) If \(\mathbb{T}_i\) is maximal with respect to inclusion among the subtori then \(C_i\) is finite.
Theorem. Let \(\pi:\mathbb{R}^n\to \mathbb{T}\) be the covering map of a compact real torus and let \(X\) be a closed set definable in an o-minimal expansion of the real field. Then there are finitely many definable closed sets \(C_1,\ldots,C_m\subset \mathbb{R}^n\) and finitely many real subtori \(\mathbb{T}_1,\ldots,\mathbb{T}_m\subset \mathbb{T}\) of positive dimension such that
\[\mathrm{cl}(\pi(X))=\pi(X)\cup\bigcup_{i=1}^m(\pi(C_i)+\mathbb{T}_i).\]
In addition,
(i) For every \(i=1,\ldots,m,\) we have \(\dim_\mathbb{R}C_i<\dim_\mathbb{R}X\) (where \(\dim_\mathbb{R}\) is the o-minimal dimension).
(ii) If \(\mathbb{T}_i\) is maximal with respect to inclusion among \(\mathbb{T}_1,\ldots,\mathbb{T}_m\) then \(C_i\) is bounded in \(\mathbb{R}^n\) and in particular \(\pi(C_i)\) is closed.
It is also shown that the conjectured generalization of the above result by Ullmo and Yafaev to the case of an arbitrary dimension needs modifications and stronger assumptions. algebraic flows; o-minimal flows; orbit closure; o-minimal; algebraically closed valued fields Coverings in algebraic geometry, Model theory of ordered structures; o-minimality Algebraic and o-minimal flows on complex and real tori | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The aim of this short note is to give an explicit description of birationally equivalent models of quartic double solids with at least one node. A quartic double solid is a double covering of \(\mathbb{P}^3\) branched along a quartic surface. The nodes of the double solid are in bijection with the nodes of the quartic surface \(B\subset\mathbb{P}^3\). Using the projection \(\mathbb{P}^3\to \mathbb{P}^2\), the center of which is a node of \(B\), we obtain a birational map to a conic bundle over \(\mathbb{P}^2\). We describe explicitly a birational equivalence between such a conic bundle and a cubic hypersurface in \(\mathbb{P}^4\). We obtain this by projecting the cubic from a smooth point. birationally equivalent models; quartic double solids with at least one node; conic bundle \(3\)-folds, Twistor theory, double fibrations (complex-analytic aspects), Compact complex \(3\)-folds, Rational and birational maps, Coverings in algebraic geometry Another description of certain 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 study pluriregular varieties \(X\) of general type with base-point-free canonical bundle whose canonical morphism has degree \(3\) and maps \(X\) onto a variety of minimal degree. The degrees of minimal generators of the canonical ring of \(X\) are described. This is applied to studying the projective normality of the images of the pluricanonical morphisms of \(X\). It is shown that if \(\text{dim} X\geq 3\), then there does not exist a converse to a theorem of \textit{M. Green} [Duke Math. J. 49, 1087--1113 (1982; Zbl 0607.14005)] that bounds the degree of the generators of the canonical ring of \(X\). The nonexistence of some a priori plausible examples of triple canonical covers of varieties of minimal degree is proved. The targets of flat canonical covers of varieties of minimal degree are characterized. canonical bundle; canonical ring Gallego, F. J.; Purnaprajna, B. P., Triple canonical covers of varieties of minimal degree, (A Tribute to C.S. Seshadri. A Tribute to C.S. Seshadri, Chennai, 2002. A Tribute to C.S. Seshadri. A Tribute to C.S. Seshadri, Chennai, 2002, Trends Math., (2003), Birkhäuser: Birkhäuser Basel), 241-270 Coverings in algebraic geometry, Divisors, linear systems, invertible sheaves, \(n\)-folds (\(n>4\)) Triple canonical covers of varieties of minimal degree | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author is interested by methods to compute algebraic models for coverings of the projective line (over \(\mathbb{C})\) with a prescribed ramification data. Such coverings are described by moduli spaces, the Hurwitz spaces. The specification of the ramification data implies a lot of calculations, to compute the coefficients of the equations defining the coverings, and, unless for small cases, it is not possible to make the calculations, to solve the nonlinear equations giving the coefficients. \textit{J.-M. Couveignes} and \textit{L. Granboulan} [in: The Grothendieck theory of dessins d'enfants, Lond. Math. Soc. Lect. Notes Ser. 200, 79-113 (1994; Zbl 0835.14010)] have given a today famous example of computation of a covering of \(\mathbb{P}^1_\mathbb{C}\) with 4 branch points and with a big monodromy group (Mathieu group of degree 24). The arguments and the different steps of this coverings are explained, and the paper contains also a list of other effective methods.
Note that the pages of the paper are not well numbered, three of them have to be changed following the cycle (51, 52, 53): page 51 is in fact page 52, this one being page 53 and this last one being page 51. coverings of the projective line; Hurwitz spaces; ramification Jean-Marc Couveignes, Tools for the computation of families of coverings, Aspects of Galois theory (Gainesville, FL, 1996) London Math. Soc. Lecture Note Ser., vol. 256, Cambridge Univ. Press, Cambridge, 1999, pp. 38 -- 65. Coverings of curves, fundamental group, Ramification problems in algebraic geometry, Coverings in algebraic geometry, Inverse Galois theory Tools for the computation of families of coverings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\), \(Y\) be algebraic varieties defined over the complex number field \(\mathbb{C}\), and \(\text{Hom}(X,Y)\) the set of \((\mathbb{C})\)-morphisms of \(X\) into \(Y\). For any \(f\in \text{Hom}(X,Y)\), denote the induced mapping on the associated analytic spaces by \(f^{an} : X^{an} \to Y^{an}\). We say that \(f \in \text{Hom}(X,Y)\) is strongly rigid (in the algebraic sense) if every morphism \(g \in \text{Hom}(X,Y)\) with \(g^{an}\) homotopic to \(f^{an}\) must coincide with \(f\) itself. We also call a morphism \(f : X \to Y\) \(\pi_ 1\)-dominant, if the induced homomorphism \(\pi_ 1(X) \to \pi_ 1(Y)\) on the algebraic fundamental groups is open. Notice that any dominant morphism of smooth varieties is necessarily \(\pi_ 1\)-dominant, and that the Lefschetz hyperplane section theorem produces many \(\pi_ 1\)-dominant morphisms which are not dominant. We will obtain, for example, the following result (corollary 4.10) without the use of transcendental calculus.
Let \(Y\) be a connected finite etale cover of an Artin good neighborhood with hyperbolic successive fibres. Then every \(\pi_ 1\)-dominant morphism into \(Y\) is strongly rigid. More generally, the following statement holds even when the base field \(\mathbb{C}\) is replaced by any algebraically closed field of characteristic 0: if two \(\pi_ 1\)- dominant morphisms \(f,g: X\to Y\) induce the same homomorphism of \(\pi_ 1 (X)\) into \(\pi_ 1(Y)\) up to inner automorphisms of \(\pi_ 1(Y)\), then \(f = g\).
Our method is affected by A. Grothendieck's letter to G. Faltings in 1983. strongly rigid morphism; Galois rigidity of algebraic mappings; hyperbolic varieties; Galois action; algebraic fundamental groups; \(\pi_ 1\)-dominant morphisms , Galois rigidity of algebraic mappings into some hyperbolic varieties, Internat. J. Math. 4 (1993), no. 3, 421-438. Mohamed Saïdi and Akio Tamagawa Coverings in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry, Hyperbolic and Kobayashi hyperbolic manifolds, Birational geometry Galois rigidity of algebraic mappings into some hyperbolic varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper under review is devoted to study the equivariant \(K\)-stability of log Fano varieties. To be more precise, the authors prove the following theorem.
{Theorem.} Let \((X,D)\) be a log Fano pair. Let \(G<\text{Aut}(X,D)\) be a finite subgroup. Then the following statements hold.
\begin{enumerate}
\item If \((X,D)\) is \(G\)-equivariantly \(K\)-semistable, then it is \(K\)-semistable.
\item If \((X,D)\) is \(G\)-eqvivariantly \(K\)-polystable, then it is \(K\)-polystable.
\end{enumerate}
Moreove, the authors also study the behaviour of \(K\)-semistablity and \(K\)-polystability under crepant finite surjective morphism and they show that the \(K\)-semistablity (resp. \(K\)-polystability) is preserved under crepant finite surjective morphisms (resp. Galois morphisms) of log Fano pairs. As an application, the authors show that smooth cubic fourfolds of cyclic cover type are \(K\)-stable.
We remark that the theorem above in full generality was proved by \textit{Z. Zhuang} [Invent. Math. 226, No. 1, 195--223 (2021; Zbl 1483.14076)] for any algebraic group actions and the \(K\)-stability of arbitrary smooth cubic fourfolds was established by \textit{Y. Liu} [J. Reine Angew. Math. 786, 55--77 (2022; Zbl 1491.14061)]. finite group actions on log Fano pairs; equivariant \(K\)-stability; \(K\)-moduli of Fano varieties Fano varieties, Group actions on varieties or schemes (quotients), Coverings in algebraic geometry Equivariant \(K\)-stability under finite group action | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We show the finiteness of étale coverings of a variety over a finite field with given degree whose ramification bounded along an effective Cartier divisor. The proof is an application of P. Deligne's theorem [\textit{H. Esnault} and \textit{M. Kerz}, Acta Math. Vietnam. 37, No. 4, 531--562 (2012; Zbl 1395.11103)] on a finiteness of \(l\)-adic sheaves with restricted ramification. By applying our result to a smooth curve over a finite field, we obtain a function field analogue of the classical Hermite-Minkowski theorem. algebraic fundamental groups; ramification; \(l\)-adic sheaves Coverings in algebraic geometry, Curves over finite and local fields, Coverings of curves, fundamental group, Finite ground fields in algebraic geometry, Varieties over finite and local fields A Hermite-Minkowski type theorem of varieties over finite fields | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For the entire collection see Zbl 0549.00004.]
This lecture notes were written by a student attending the lecture, and contain results now appeared in J. Differ. Geom. 19, 483-515 (1984; Zbl 0549.14012). upper bound for the dimension of an irreducible component of moduli space; abelian fundamental group of the complement of the branch locus; moduli spaces of surfaces of general type; deformations of Galois covers CATANESE, F.: Moduli of surfaces of general type (Proc. Conf. Alg. Geom. Ravello, 1982) Lecture Notes in Math. Vol. 997, Springer, Berline, 90-111, (1983) Families, moduli, classification: algebraic theory, Complex-analytic moduli problems, Special surfaces, Moduli, classification: analytic theory; relations with modular forms, Formal methods and deformations in algebraic geometry, Fine and coarse moduli spaces, Coverings in algebraic geometry Moduli of surfaces of general type | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We prove some consequences of an old unpublished connectivity result of Mumford. These mostly deal with the fundamental group of some (ramified) coverings of homogeneous spaces; for example, it is shown that the fundamental group of a complex projective irreducible normal variety which is a covering of a simple (in a sense defined in the paper) homogeneous space \(V\), of degree \(\leq\dim(V)\), is isomorphic to \(\pi_1(V)\). connectivity; fundamental group; coverings of homogeneous spaces Debarre O., Manuscr. Math 89 pp 407-- (1996) Homogeneous spaces and generalizations, Homotopy theory and fundamental groups in algebraic geometry, Coverings in algebraic geometry, Topological properties in algebraic geometry On a connectivity theorem of Mumford for homogeneous spaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The main result of this paper is a general structure theorem for Gorenstein coverings \(\rho:X\to Y\), namely finite flat maps between schemes (with \(Y\) integral) such that the scheme theoretic fibre \(\rho^{-1}(y)\) is Gorenstein for every \(y\) in \(Y\). More precisely, given a Gorenstein cover \(\rho:X\to Y\) of degree \(d\), denote by \(\mathcal E\) the cokernel of the natural inclusion \({\mathcal O}_Y\to\rho_*{\mathcal O}_X\): Then \(\mathcal E\) is a locally free rank \(d-1\) sheaf, \(X\) can be embedded in the \({\mathbb{P}}^{d-2}\)-bundle \({\mathbb{P}}(\mathcal E)\) in such a way that \({\mathcal O}_X(1)=\omega_{X|Y}\), and there exists a locally free resolution
\[
0\to N_{d-2}(-2-d)\to\ldots\to N_1(-2)\to {\mathcal O}_{\mathbb{P}}\to {\mathcal O}_X\to 0.
\]
Moreover, the ranks of the bundles \(N_i\) are computed and it is shown that the resolution is unique up to unique isomorphism. For \(d=3,4\), the result is made more explicit and sufficient conditions for the existence of smooth covers with given \(\mathcal E\) are given. These results are used to establish the existence of a coarse moduli space for Enriques surfaces with a polarization of degree \(4\) and to show that this moduli space is an irreducible unirational quasi-projective variety of dimension \(10\).
[For part II of this paper see \textit{G. Casnati}, J. Algebr. Geom. 5, No. 3, 461-477 (1996)]. Gorenstein coverings; coarse moduli space for Enriques surfaces Casnati, G.; Ekedahl, T., Covers of algebraic varieties. I. A general structure theorem, covers of degree \(3,4\) and Enriques surfaces, J. Algebraic Geom., 5, 3, 439-460, (1996) Coverings in algebraic geometry, \(K3\) surfaces and Enriques surfaces Covers of algebraic varieties. I: A general structure theorem, covers of degree 3, 4 and Enriques surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author [\textit{P. Balmer}, J. Reine Angew. Math. 588, 149--168 (2005; Zbl 1080.18007)] has introduced the notion of a triangular spectrum \(\text{Spc}(\mathcal{K})\) for a tensor triangulated category \(\mathcal{K}\). From the previous work of the author and others it is clear that \(\text{Spc}(\mathcal{K})\) is a natural home for triangular geometry. However, computing this spectrum is a very hard problem. It is essentially equivalent to the problem of classifying the tensor thick ideals of \(\mathcal{K}\). The latter is a very rich topic and has been studied in the fields of stable homotopy theory, homological algebra, algebraic geometry and modular representation theory. Therefore \(\text{Spc}(\mathcal{K})\) is a natural and interesting object of study.
In the paper under review the author constructs a natural and continuous map
\[
\rho_{\mathcal{K}} : \text{Spc}(\mathcal{K}) \rightarrow \text{Spec} (\text{End}_{\mathcal{K}}(\mathbb{I}))
\]
from the triangular spectrum \(\text{Spc}(\mathcal{K})\) to the Zariski spectrum \(\text{Spec} (\text{End}_{\mathcal{K}}(\mathbb{I}))\) of the endomorphism ring of the unit object \(\mathbb{I}\) in the tensor triangulated category \(\mathcal{K}\). It is shown that this map is often surjective but far from injective in general. For instance, it is shown that when \(\mathcal{K}\) is connective, i.e., \(\text{Hom}_{\mathcal{K}}(\Sigma^{i}(\mathbb{I}), \mathbb{I}) = 0\) for \(i < 0\), then \(\rho_{\mathcal{K}}\) is surjective. This applies to derived categories of rings and also to the stable homotopy category of finite spectra. Graded version of this result is also proved in Theorem 7.3 and Corollary 7.4. Using these results the author is able to shed some light on \(\text{Spc}(\mathcal{K})\) in examples coming from \(\mathbb{A}^{1}\)-homotopy theory and non-commutative topology where \(\text{Spc}(\mathcal{K})\) has not been known; see Corollary 10.1 and Corollary 8.8. tensor triangular geometry; spectra P. Balmer, Spectra, spectra, spectra--tensor triangular spectra versus Zariski spectra of endomorphism rings, Algebr. Geom. Topol. 10 (2010), no. 3, 1521-1563. Derived categories, triangulated categories, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Kasparov theory (\(KK\)-theory), Modular representations and characters, Stable homotopy theory, spectra, Abstract and axiomatic homotopy theory in algebraic topology, Nonabelian homotopical algebra, Simplicial sets, simplicial objects (in a category) [See also 55U10] Spectra, spectra, spectra -- tensor triangular spectra versus Zariski spectra of endomorphism rings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We describe a series of Calabi-Yau manifolds which are cyclic coverings of a Fano 3-fold branched along a smooth divisor. For all the examples we compute the Euler characteristic and the Hodge numbers. All examples have small Picard number \(\varrho =h^{1,1}\). Calabi-Yau manifolds; cyclic coverings; singularities; Fano 3-folds; Hodge numbers; Picard number Cynk S. (2003). Cyclic coverings of Fano threefolds. Ann. Polon. Math. 80:117--124 Coverings in algebraic geometry, \(3\)-folds, Germs of analytic sets, local parametrization, Singularities in algebraic geometry, Calabi-Yau manifolds (algebro-geometric aspects) Cyclic coverings of Fano threefolds | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors use the result that for any complex algebraic variety \(X\) and a closed subvariety \(Y\) of \(X\) there is a triangulation of \(X\) for which \(Y\) is a subcomplex. Two generalizations of Zariski's result on irregularity of cyclic multiple planes are offered. In both the conclusion is that the first Betti number of a surface \(S\) is \(0\). One result assumes that \(f(x,y)\in\mathbb{C}[x,y]\) is reduced, \(n=p^l\) for some prime \(p\) and \(l\geq 1\) an integer, where \(f=0\) is connected and \(S=\{ z^n-f=0\}\). In the second \(f(x,y)\) is irreducible, \(n=p^l\) and \(S\) is a resoltution of singularities of \(\{ z^n-f=0\}\). These results are valid in higher dimensions e.g., for \(x_{m+1}^n-f(x_1,\ldots,x_m)=0\), \(m\geq 2\). Finally, the authors show how to derive a know theoretic result from a theorem of Goldsmith (which they prove by correcting the original proof which contains an error). Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem), Coverings in algebraic geometry, Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants), Low-dimensional topology of special (e.g., branched) coverings, Knots and links (in high dimensions) [For the low-dimensional case, see 57M25] Cyclic multiple planes, branched covers of \(S{^n}\) and a result of D. L. Goldsmith | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 0511.00009.]
An arithmetic field, K, is one which is finitely generated over its prime field. The author gives a survey of the local and global class field theory for fields of dimension \(>1\). The one-dimensional case is classical class field theory. The object of the class field theory is to give explicit descriptions of \(Gal(K^{ab}/K)\), the Galois group of the maximal Abelian extension, and to describe \(\pi_ 1^{ab}(X)\), the Abelianised fundamental group of an arithmetic scheme.
Developing ideas of \textit{S. Bloch} [Ann. Math., II. Ser. 114, 229-265 (1981; Zbl 0512.14009)] the author accomplishes this aim in an impressive series of papers (often joint with S. Saito) which cover the global and (several types of) local fields of higher dimension. The descriptions are in terms of Milnor K-theory, \(K^ M_ i(K)\). The results are numerous and complicated so I will content myself with one of the most important ones as an illustration.
Theorem: Let K be an n-dimensional, non-Archimedean local field. Then there exists an isomorphism \(K^ M_ n(K)/N_{L/K}(K^ M_ n(L))\cong Gal(L/K)\) for each finite Abelian extension, L/K. Here \(N_{L/K}\) is the norm map for Milnor K-theory. arithmetic field; class field theory; Galois group of the maximal Abelian extension; Abelianised fundamental group of an arithmetic scheme; fields of higher dimension; Milnor K-theory; non-Archimedean local field; norm map Class field theory; \(p\)-adic formal groups, Class field theory, Arithmetic ground fields for abelian varieties, \(K\)-theory of global fields, \(K\)-theory of local fields, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Transcendental field extensions, Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects), Coverings in algebraic geometry Class field theory and algebraic K-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 We present a new technique to handle the study of homogeneous rings of a projective variety endowed with a finite or a generically finite morphism to another variety \(Y\) whose geometry is easier to handle. Under these circumstances it is possible to use the information given by the algebra structure of \({\mathcal O}_X\) over \({\mathcal O}_Y\) to describe the homogeneous ring associated to line bundles which are the pullbacks of line bundles on \(Y\).
In this article we illustrate our technique to study the canonical ring of curves (a well-known ring that we revisit with this new technique) equipped with a suitable finite morphism and homogeneous rings of a certain class of Calabi-Yau threefolds. very ampleness; canonical curve; finite cover; homogeneous rings of a projective variety; finite morphism; Calabi-Yau threefolds F. J. Gallego and B. P. Purnaprajna, Some homogeneous rings associated to finite morphisms, Preprint. To appear in ``Advances in Algebra and Geometry'' (Hyderabad Conference 2001), Hindustan Book Agency (India) Ltd. Rational and birational maps, Coverings in algebraic geometry, Arithmetic ground fields for surfaces or higher-dimensional varieties, Coverings of curves, fundamental group Some homogeneous rings associated to finite morphisms | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors apply the theory of covers of degree \(d\) to prove the following result.
Theorem. There exists an explicit construction for smooth, minimal, pluriregular (i.e. \(h^1(X,{\mathcal O}_X)=h^2(X,{\mathcal O}_X)=0)\) threefolds of general type \(X\) with \(p_g (X)=5\), \(K^3_X=2d=8,10,12\), whose canonical map is a cover \(\rho:X \to Q\) of degree \(d\) onto a smooth quadric \(Q\subseteq \mathbb{P}^4_\mathbb{C} \).
In the cases \(K^3_X=8,10\) (respectively, \(K^3_X=12)\) this is (respectively, is not) the only possible way to obtain every threefold with those invariants and with such a canonical map. Moreover, in the case \(d= 4\), we give a rough description of the locus \({\mathcal M}^{\text{quadric}}_{8,5}\) of such threefolds inside the moduli space \({\mathfrak M}_{8,5}\) of smooth, pluriregular threefolds of general type \(X\) with \(p_g(X)=5\), \(K^3_X=8\). The authors prove that \({\mathfrak M}^{\text{quadric}}_{8,5}\) is unirational of dimension 126 and that the unique component of \({\mathfrak M}_{8,5}\) containing \({\mathcal M}^{\text{quadric}}_{8,5}\) has dimension 128. Furthermore, the general deformation of each threefold corresponding to a point of \({\mathcal M}^{\text{quadric}}_{8,5}\) has birational canonical morphism onto a hypersurface of degree 8 of \(\mathbb{P}^4_\mathbb{C}\) with ordinary singularities.
The authors obtain some partial results about covers of degee \(d=5,6\) of some quadrics. They also give some examples of threefolds of general type \(X\) with \(h^1(X,{\mathcal O}_X)=h^2(X, {\mathcal O}_X)=0\), \(K^3_X= d(p_g(X)-3)\) and odd \(p_g(X)\geq 7\), as covers of threefolds of minimal degree in \(\mathbb{P}^{p_g(X)-1}_\mathbb{C}\). Casnati G., Supino P. (2002). Construction of threefold with finite canonical map. Glasgow Math. J. 44:65--79 \(3\)-folds, Coverings in algebraic geometry, Moduli, classification: analytic theory; relations with modular forms, Rational and unirational varieties Construction of threefolds with finite canonical map. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 an important research in Galois theory of schemes. As in his paper in Trans. Am. Math. Soc. 262, 399-415 (1980; Zbl 0461.14001), but in more general cases, the author uses the notion of mock cover to construct Galois coverings of curves. The main results are formulated in terms of Galois extensions of domains. More precisely, in various cases the existence of regular Galois extensions of the given domain T with the given (arbitrary finite) Galois group G is proved. Among these cases are: \((i)\quad T=k[x][[t]],\) where k is a field containing the primitive n-th roots of unity for all n (with \(char k=0);\) \((iii)\quad T=k[x][[t]]\) or \(T=k[x],\) where k is an algebraically closed field with \(char k\neq 0;\) \((iii)\quad T={\mathfrak O}[[t]],\) where \({\mathfrak O}\) is the ring of integers of a number field; \((iv)\quad T={\mathbb{Z}}_{r+}[[t]],\) the subring of \({\mathbb{Z}}[[t]]\) consisting of power series with radius of convergence greater than r, where \(0<r<1\). The notion of Galois extension of domains is defined as follows. Let \(S\supset T\) be domains and G be a finite group which acts on S; S is a Galois extension of T with group G, if the field of fractions of S is a Galois extension of the field of fractions of T with group G, where S is finite over T. The Galois extension \(S\supset T\) is regular, if T is considered as A-algebra such that A is algebraically closed in T, and if A is algebraically closed in S as well. ramification; Galois theory of schemes; mock cover; existence of regular Galois extensions; power series D. Harbater,Mock covers and Galois extensions, Journal of Algebra91 (1984), 281--293. Coverings of curves, fundamental group, Ramification problems in algebraic geometry, Galois theory and commutative ring extensions, Coverings in algebraic geometry, Extension theory of commutative rings Mock covers and Galois extensions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper, we introduce the terminology of matroids into the study of Zariski-pairs related to rational elliptic surfaces, aiming to simplify the presentation and arguments involved. As an application, we provide new examples of Zariski \(N\)-ples of relatively low degree. Namely we show that a Zariski 102-ple of degree 18 exists. elliptic surfaces; Mordell-Weil lattice; matroids; Zariski-pairs Elliptic surfaces, elliptic or Calabi-Yau fibrations, Coverings in algebraic geometry, Combinatorial aspects of matroids and geometric lattices The matroid structure of vectors of the Mordell-Weil lattice and the topology of plane quartics and bitangent lines | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author applies the theory of abelian covers [\textit{R. Pardini}, J. Reine Angew. Math. 417, 191--213 (1991; Zbl 0721.14009)] to construct some examples of surfaces of general type, the most interesting of which is a surface with \(p_g=4\), \(K^2=31\) and birational canonical map. The examples are constructed as desingularizations of singular abelian covers of \({\mathbb P}^1\times{\mathbb P}^1\). abelian cover; surface of general type; canonical map Liedtke C. (2003) Singular abelian covers of algebraic surfaces. Manuscr. Math. 112(3): 375--390 Surfaces of general type, Singularities of surfaces or higher-dimensional varieties, Special surfaces, Coverings in algebraic geometry Singular abelian covers of algebraic surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems From a viewpoint of algebraic function theory, abelian integrals and Fuchsian differential equations, this expository paper introduces finite ramified coverings of complex manifolds, in particular, the basic theory of finite Galois coverings, presents examples and results now available, describes unsolved problems and the present status in this direction. The paper is divided into two parts dealing with the case of closed Riemann surfaces and that of complex manifolds, in particular, projective manifolds, respectively. finite Galois coverings; closed Riemann surfaces; complex manifolds Riemann surfaces; Weierstrass points; gap sequences, Compact Riemann surfaces and uniformization, Coverings in algebraic geometry, Coverings of curves, fundamental group Finite ramification covering of complex manifolds | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \({\mathfrak C}\) be a class of finite groups closed under the formation of subgroups, quotients, and group extensions. For an algebraic variety \(X\) over a number field \(k\), let \(\pi_1^{\mathfrak C} (X)\) denote the (\(\mathbb{C}\)-modified) profinite fundamental group of \(X\) having the absolute Galois group \(\text{Gal} (k/k)\) as a quotient with kernel \(\pi_1^{\mathfrak C}(X_{\overline{k}})\), the maximal pro-\({\mathfrak C}\) quotient of the geometric fundamental group of \(X\). The purpose of this paper is to show certain rigidity properties of \(\pi_1^{\mathfrak C}(X)\) for \(X\) of hyperbolic type by the study of the outer automorphism group \(\text{Out } \pi_1^{\mathfrak C}(X)\) of \(\pi_1^{\mathfrak C}(X)\). In particular, we show finiteness of \(\text{Out } \pi_1^{\mathfrak C}(X)\) when \(X\) is a certain typical hyperbolic variety and \({\mathfrak k}\) is the class of finite \(l\)-groups (\(l\): odd prime).
Indeed, we have a criterion of Gottlieb type for center-triviality of \(\pi_1^{\mathfrak C}(X_{\overline{k}})\) under certain good hyperbolicity condition on \(X\). Then our question on finiteness of \(\text{Out } \pi_1^{\mathfrak C}(X)\) for such \(X\) is reduced to the study of the exterior Galois representation \(\varphi_X^{\mathfrak C}: \text{Gal} (\overline{k}/k)\to \text{Out } \pi_1^{\mathfrak C}(X_{\overline{k}})\), especially to the estimation of the centralizer of the Galois image of \(\varphi_X^{\mathfrak C}\). In \S 2, we study the case where \(X\) is an algebraic curve of hyperbolic type, and give fundamental tools and basic results. We devote \S 3, \S 4 and the appendix to detailed studies of the special case \(X= M_{0,n}\), the moduli space of the \(n\)-point punctured projective lines \((n\geq 3)\), which are closely related with topological work of \textit{N. V. Ivanov}, arithmetic work of \textit{P. Deligne} and {Y. Ihara}, and categorical work of \textit{V. G. Drinfeld}. Section 4 deals with a Lie variant suggested by P. Deligne. profinite fundamental group; rigidity; outer automorphism group; exterior Galois representation Nakamura H., Galois rigidity of pure sphere braid groups and profinite calculus, J. Math. Sci. Univ. Tokyo 1 (1994), no. 1, 71-136. Homotopy theory and fundamental groups in algebraic geometry, Braid groups; Artin groups, Inverse Galois theory, Fundamental groups and their automorphisms (group-theoretic aspects), Coverings in algebraic geometry, Galois theory, Limits, profinite groups Galois rigidity of pure sphere braid groups and profinite calculus | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors study the formal moduli of wildly ramified Galois coverings. They determine the dimensions of the global and local versal deformation rings. dimensions of deformation rings; formal moduli; wildly ramified Galois coverings Bertin, Déformations formelles des revêtements sauvagement ramifiés de courbes algébriques, Invent. Math. 141 (1) pp 195-- (2000) Coverings of curves, fundamental group, Coverings in algebraic geometry, Formal methods and deformations in algebraic geometry Formal deformations of wildly ramified coverings of algebraic curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The semisimplicity of the Alexander automorphism (the monodromy operator) is proved on the cohomology \(H^1(X_\infty)_{\neq 1}\) of the infinite cyclic covering of the complement to a plane non-reduced algebraic curve, and, in particular, the semi-simplicity of \(H^1 (X_\infty)\) in the case of an irreducible curve. A natural mixed Hodge structure on \(H^1 (X_\infty)\) is introduced and the irregularity of cyclic coverings of \(\mathbb{P}^2\) is calculated in terms of the number of roots of the Alexander polynomial of the branch curve. monodromy operator; cyclic covering; complement to a plane non-reduced algebraic curve; Hodge structure; Alexander polynomial; branch curve V. S. Kulikov and Vik. S. Kulikov, On the monodromy and mixed Hodge structure on the cohomology of the infinite cyclic covering of the complement to a plane algebraic curve, Izv. Math. 59 (1995), 367--386. Coverings in algebraic geometry, Special algebraic curves and curves of low genus, Transcendental methods, Hodge theory (algebro-geometric aspects) On the monodromy and mixed Hodge structure on cohomology of the cyclic infinite covering of the complement to a plane algebraic curve | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(k\) be an algebraically closed field of characteristic \(p\). Let \(S\) be a normal surface over \(k\). A divisor \(\Delta=\sum d_i D_i\) on \(S\) is called a standard \(\mathbb Q\)-boundary if \(d_i=1-\frac{1}{b_i}, b_i\in \mathbb N\cup \{\infty\}\). Let \(f:S'\to S\) be a proper birational morphism, \(S'\) normal, and write
\[
K_{S'}+\Delta'=f^\ast(K_S+\Delta)+\sum_E a(E, \Delta) E,
\]
\(\Delta'\) the strict transform of \(\Delta, E\) the exceptional divisor. Let \(\text{discrep}(S, \Delta) = \inf_E\{a(E, \Delta) \mid E \text{ exceptional divisor}\}.\) \((S, \Delta)\) is called purely \(\log\) terminal (resp. canonical) if discrep \((S, \Delta) > -1\) (resp. \(\geq 0\)).
The index \(r\) of \((S, \Delta)\) is the smallest positive integer such that \(r(K_S+\Delta)\) is a Cartier divisor. Let \(\varphi\) be from the function field of \(S\) such that \(\text{div} (\varphi)=r(K_S+\Delta)\). The normalization \(\widetilde{S}\to S\) in the field extension defined by \(\sqrt[r]{\varphi}\) is called an index 1 cover associated to \(\varphi\). Index 1 covers \(\widetilde{S}\) of the purely \(\log\) terminal pair \((S, \Delta)\) are considered. Under the condition that \(S\) is smooth, char \((k)\geq 3\) and some other conditions it is proved that \(\widetilde{S}\) is canonical. In characteristic 2 this result is wrong. The counterexamples are given. normal surface; cover; log terminal singularity; positive characteristic Coverings in algebraic geometry, Coverings of curves, fundamental group On index one covers of two-dimensional purely log terminal singularities 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 recalls C. Chevalley's and E. Warning's theorem on the number of rational points of an affine variety defined over a finite field and the related results of \textit{J. Ax} [Am. J. Math. 86, 255--261 (1964; Zbl 0121.02003)] and \textit{N. Katz} [Am. J. Math. 93, 485--499 (1971; Zbl 0237.12012)]; he then briefly describes the Weil conjectures and mentions the different cohomology theories inspired by those conjectures. The author pays particular attention to the rigid cohomology defined by \textit{P. Berthelot} [Mém. Soc. Math. Fr., Nouv. Sér. 23, 7--32 (1986; Zbl 0606.14017)] and describes several applications of that cohomology theory: a theorem of Chevalley-Warning type, a recent theorem of \textit{H. Esnault} [Invent. Math. 151, No. 1, 187--191 (2003, Zbl 1092.14010)] asserting in particular that \(|X({\mathbb F}_q)|\equiv 1\pmod q\) for a Fano variety \(X\) defined over the finite field \({\mathbb F}_q\) of \(q\) elements, a rather technical theorem, attributed by the author to T. Ekedahl, which implies in particular that
\[
|X({\mathbb F}_q)|\equiv|Y({\mathbb F}_q)|\pmod q
\]
for any two smooth proper birationally equivalent to each other geometrically connected varieties \(X\) and \(Y\) over \({\mathbb F}_q\), and a theorem asserting that the fundamental group of a smooth proper rationally chain connected variety over an algebraically closed field of positive characteristic \(p\) is finite, of order not divisible by \(p\). Fano varieties; chain rationally connected varieties; rigid cohomology; Weil cohomologies Chambert-Loir, A., Points rationnels et groupes fondamentaux: applications de la cohomologie \textit{p}-adique, Astérisque, 294, 125-146, (2004), (d'après P. Berthelot, T. Ekedahl, H. Esnault, etc.) \(p\)-adic cohomology, crystalline cohomology, Fano varieties, Finite ground fields in algebraic geometry, Rational points, Coverings of curves, fundamental group, Rigid analytic geometry, Varieties over finite and local fields, Coverings in algebraic geometry, Rational and unirational varieties Rational points and fundamental groups: applications of the \(p\)-adic cohomology (following P. Berthelot, T. Ekedahl, H. Esnault, etc.). | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The purpose of this paper is to study the geometry of the Harris-Mumford compactification of the Hurwitz scheme. The Hurwitz scheme parametrizes certain ramified coverings \(f:C \to\mathbb{P}^1\) of the projective line by smooth curves. Thus, from the very outset, one sees that there are essentially two ways to approach the Hurwitz scheme:
(1) We start with \(\mathbb{P}^1\) and regard the objects of interest as coverings of \(\mathbb{P}^1\).
(2) We start with \(C\) and regard the objects of interest as morphisms from \(C\) to \(\mathbb{P}^1\).
One finds that one can obtain the most information about the Hurwitz scheme and its compactification by exploiting interchangeably these two points of view. Our first main result is the following theorem.
Let \(b,d\), and \(g\) be integers such that \(b=2d+ 2g-2\), \(g\geq 5\) and \(d>2g+4\). Let \({\mathcal H}\) be the Hurwitz scheme over \(\mathbb{Z} [{1\over b!}]\) parametrizing coverings of the projective line of degree \(d\) with \(b\) points of ramification. Then \(\text{Pic} ({\mathcal H})\) is finite.
The number \(g\) is the genus of the ``curve \(C\) upstairs'' of the coverings in question. Note, however, that the Hurwitz scheme \({\mathcal H}\), and hence also the genus \(g\), are completely determined by \(b\) and \(d\). -- Note that although in the statement of the theorem we spoke of ``the'' Hurwitz ``scheme,'' there are in fact several different Hurwitz schemes used in the literature, some of which are, in fact, not schemes, but stacks. The main idea of the proof is that by combinatorially analyzing the boundary of the compactification of the Hurwitz scheme, one realizes that there are essentially three kinds of divisors in the boundary, which we call excess divisors, which are ``more important'' than the other divisors in the boundary in the sense that the other divisors map to sets of codimension \(\geq 2\) under various natural morphisms. On the other hand, we can also consider the moduli stack \({\mathcal G}\) of pairs consisting of a smooth curve of genus \(g\), together with a linear system of degree \(d\) and dimension 1. The subset of \({\mathcal G}\) consisting of those pairs that arise from Hurwitz coverings is open in \({\mathcal G}\), and its complement consists of three divisors, which correspond precisely to the excess divisors. Using results of Harer on the Picard group of \({\mathcal M}_g\), we show that these three divisors on \({\mathcal G}\) form a basis of \(\text{Pic} ({\mathcal G}) \otimes_\mathbb{Z} \mathbb{Q}\), and in fact, we even compute explicitly the matrix relating these three divisors on \({\mathcal G}\) to a certain standard basis of \(\text{Pic} ({\mathcal G}) \otimes_\mathbb{Z} \mathbb{Q}\). The above theorem then follows formally.
Crucial to our study of the Hurwitz scheme is its compactification by means of admissible coverings and we prove a rather general theorem concerning the existence of a canonical logarithmic algebraic stack \(({\mathcal A}, M)\) parametrizing such coverings:
Fix non-negative integers \(g,r,q,s,d\) such that \(2g-2+r =d(2q-2+s) \geq 1\). Let \({\mathcal A}\) be the stack over \(\mathbb{Z}\) defined as follows: For a scheme \(S\), the objects of \({\mathcal A}(S)\) are admissible coverings \(\pi:C\to D\) of degree \(d\) from a symmetrically \(r\)-pointed stable curve \((f:C\to S\); \(\mu_f \subseteq C)\) of genus \(g\) to a symmetrically \(s\)-pointed stable curve \((h:D \to S\); \(\mu_h \subseteq D)\) of genus \(q\); and the morphisms of \({\mathcal A} (S)\) are pairs of \(S\)-isomorphisms \(\alpha: C\to C\) and \(\beta: D\to D\) that stabilize the divisors of marked points such that \(\pi\circ \alpha= \beta\circ \pi\). Then \({\mathcal A}\) is a separated algebraic stack of finite type over \(\mathbb{Z}\). Moreover, \({\mathcal A}\) is equipped with a canonical log structure \(M_{\mathcal A} \to {\mathcal O}_{\mathcal A}\), together with a logarithmic morphism \(({\mathcal A}, M_{\mathcal A}) \to \overline {{\mathcal M} {\mathcal S}}^{\log}_{q,s}\) (obtained by mapping \((C;D;\pi) \mapsto D)\) which is log étale (always) and proper over \(\mathbb{Z} [{1\over d!}]\). finite Picard group; Hurwitz scheme; Hurwitz coverings; admissible coverings S. Mochizuki, ''The geometry of the compactification of the Hurwitz scheme,'' Publ. Res. Inst. Math. Sci., vol. 31, iss. 3, pp. 355-441, 1995. Families, moduli of curves (algebraic), Picard groups, Coverings in algebraic geometry The geometry of the compactification of the Hurwitz scheme | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A toric variety \(X_k(\Sigma)\) defined over a field \(k\) is determined by a complex \(\Sigma\) of rational cones in \(\mathbb{R}^d\) and will have an algebraic embedding into projective space if there is an integral convex polytope \(P\) which is dual to \(\Sigma\) in an appropriate sense. Having chosen such a \(P\), there is a natural map \(\mu_p\) from \(X_k (\Sigma)\) to a certain projective space \(\mathbb{P}^r_k\), and in the event that \(P\) is large enough, this map is an algebraic embedding. In particular, if \(nP\) denotes the \(n\)-fold scaling of \(P\), then for \(n\) sufficiently large, \(\mu_{nP}\) is an algebraic embedding.
In this paper, we consider the weaker question of when \(\mu_P\) is injective, giving necessary and sufficient conditions on \(P\) which depend only on a certain arithmetic property of the field \(k\). When the field is \(\mathbb{R}\) or \(\mathbb{C}\), injectivity implies that the map will be a topological embedding (in the metric topology).
We conclude by giving an example \(\mu_P: X_\mathbb{C}(\Sigma) \to\mathbb{P}^r_\mathbb{C}\) which is a topological embedding but not an algebraic embedding and an example \(\mu_P:X_\mathbb{C} (\Sigma)\to \mathbb{P}^r_\mathbb{C}\) which is not a topological embedding, but whose restriction \(\mu_P: (\Sigma)\to \mathbb{P}^r_\mathbb{R}\) is a topological embedding. projective embeddings; toric variety Toric varieties, Newton polyhedra, Okounkov bodies, Coverings in algebraic geometry Projective embeddings of toric varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let K be a compact Riemann surface which is an n-sheeted covering of a sphere C. Assume that the branching points \(t_ 1\), \(t_ 2=0\) and \(t_ 3\), as well as the generators of the monodromy group of K, \(\sigma_ 1=(2,1,...,1)\), \(\sigma_ 2=(n)\), \(\sigma_ 3=(n_ 1)(n_ 2)\), \(n_ 1+n_ 2=n\), \(\sigma_ 1\sigma_ 2\sigma_ 3=e\), are given. Then the problem to find a construction of an algebraic equation of K arises. The author gives the solution to this problem, namely, he gives the construction of a normal basis of algebraic functions and finds the canonical matrix of solutions of the corresponding vector-matrix Riemann problem, and calculates the particular indices. algebraic equation of compact Riemann surface; monodromy Riemann surfaces; Weierstrass points; gap sequences, Coverings of curves, fundamental group, Compact Riemann surfaces and uniformization, Coverings in algebraic geometry Effective construction of a field of algebraic functions corresponding to an n-sheeted covering of a sphere and applications | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(F_N: X^N+ Y^N= 1\) be the Fermat curve of \(N\)th degree, with \(N\geq 4\). In this paper the author obtains a basis for the singular homology group \(H_1(F_N,\mathbb{Z})\) and specifies the intersection product in \(H_1(F_N,\mathbb{Z})\). His method is based on the construction of a fundamental domain for \(F_N\) using basic facts from hyperbolic geometry. Furthermore, the same computations are developed for the quotient curves of the Fermat curves of prime exponent. uniformization; Fermat curve; singular homology group; intersection product; fundamental domain; quotient curves Guàrdia, J.: A fundamental domain for the Fermat curves and their quotients. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales 94, 391--396 (2000) Coverings of curves, fundamental group, Curves of arbitrary genus or genus \(\ne 1\) over global fields, Coverings in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry A fundamental domain for the Fermat curves and their quotients | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 Zariski \(k\)-plet is a set of \(k\) reduced (complex projective) plane curves having all homeomorphic tubular neighborhoods but pairwise not exchanged by an homeomorphism of \(\mathbb{P}^2\). The existence of Zariski \(k-\)plets is related with the number of connected components of equisingular families of plane curves: the curves in a \(k-\)plet are equisingular but lie in different connected components of the corresponding family. In this paper the authors find new examples of Zariski \(k\)-plets for arbitrary high \(k\). More precisely, for every \(m\geq 4\), they construct a Zariski \(\left[\frac{m}{2}\right]\)-plet of curves of degree \(m+2\), all union of a smooth conic and a rational nodal curve of degree \(m\) tangent to the conic in \(m\) points. The idea is the following. Taking a smooth conic \(C_0\), the double cover of \(\mathbb{P}^2\) branched on it, say \(f\), is \(\mathbb{P}^1 \times \mathbb{P}^1\). Then the Zariski \(\left[\frac{m}{2}\right]\)-plet is obtained by taking curves that are the union of \(C_0\) with \(f(D)\), with \(D\) general rational nodal curve of bidegree \((a,b)\) with \(a+b=m\) (varying \((a,b)\) with \(b \leq a\)). They prove that two such a curve obtained taking different \((a,b)\) are not exchanged by an homeomorphism of \(\mathbb{P}^2\) since they can show (in slightly more general hypotheses) that, for \(n\geq 3\), there exists a \(D_{2n}\) Galois cover of \(\mathbb{P}^2\) branched on \(2C_0+nf(D)\) if and only if \(n\) divides \((a-b)\). Artal, E.; Tokunaga, H., Zariski \textit{k}-plets of rational curve arrangements and dihedral covers, Topol. Appl., 142, 227-233, (2004) Coverings in algebraic geometry Zariski \(k\)-plets of rational curve arrangements and 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 Let \(W\) be a smooth complex projective variety of dimension \(n\), endowed with an ample line bundle \(h\). A line of \((W, h)\) is a rational curve in \(W\) of \(h\)-degree \(1\). Let \(F(W)\) be the space of lines of \((W, h)\). If non-empty, its dimension is \(\dim(F(W)) \leq\exp\dim (F(W))= (-K_W) \cdot \ell + n - 3\), where \(\ell\) is a line of \((W, h)\), and it is of interest to know when this is an equality. This happens e.g., for a general hypersurface \(W \subset \mathbb P^{n+1}\) of degree \(d > 2n-1\), with \(h\) being the hyperplane bundle. In the paper under review, the authors consider the space of lines in cyclic covers of projective spaces and prove the following result. Let \(m, n, d\) be positive integers such that \(md > 2n-3\) and \(k := 2(n-1)-d(m-1) \geq 0\), let \(w : Y \to \mathbb P^n\) be a cyclic cover of degree \(m\), branched along a general hypersurface \(X \subset \mathbb P^n\) of degree \(md\) and \(h = w^*\mathcal O_{\mathbb P^n}(1)\). Then \(F(Y)\) is smooth of dimension \(k\) and irreducible if \(k \geq 1\); in particular, \(F(Y)\) has the expected dimension. For \(n = 3, m = d = 2\), this was already proven by [\textit{A. S. Tikhomirov} [Izv. Akad. Nauk SSSR, Ser. Mat. 44, 415--442 (1980; Zbl 0434.14023)]. The result is proven by connecting the lines of \((Y, h)\) to the \(m\)-contact order lines in \(X\), i.e., lines \(\ell \subset \mathbb P^n\) whose local intersection number with \(X\) at each point of \(\ell \cap X\) is a multiple of \(m\). Actually, the authors study the subvariety \(S_m(X)\) of the Grassmannian \(G\) of lines of \(\mathbb P^n\) parameterizing such lines, proving that it is smooth of dimension \(k\) and show that \(F(Y)\) is an unramified cover of \(S_m(X)\) of degree \(m\). Furthermore, when \(k = 0\), they obtain an explicit enumerative formula expressing the number of \(m\)-contact order lines of \(X\) in terms of intersection numbers of Schubert cycles on \(G\). cyclic cover; Fano scheme; bitangents Projective techniques in algebraic geometry, Coverings in algebraic geometry, Fano varieties The space of lines in cyclic covers of projective space | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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's paper [ibid. 2, No. 4, 508--513 (2015; Zbl 1409.14031)] contains a crucial error (which was pointed out to me by T. Graber) in Proposition 3: the quadric fibration \(Q\to \Sigma\) is not locally trivial for the Zariski topology (it becomes so only by passing to the double étale covering of \(\Sigma\) parametrizing the planes contained in a rank 2 quadric of \(L\)). Unfortunately, this invalidates the main result of the paper. Beauville, A.: A very general quartic double fourfold or fivefold is not stably rational. Algebraic Geom. (to appear) Coverings in algebraic geometry, Rationality questions in algebraic geometry Erratum: ``A very general quartic double fourfold or fivefold is 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 For closed, oriented topological manifolds, the signature is multiplicative under finite covers. This property is no longer true for general Poincaré duality spaces, as shown by examples of \textit{C. T. C. Wall} [Ann. Math. (2) 86, 213--245 (1967; Zbl 0153.25401)].
In order to take into account the existence of singularities, one replaces the usual homology by the intersection homology of \textit{M. Goresky} and \textit{R. MacPherson} [Topology 19, 135--165 (1980; Zbl 0448.55004)]. The latter depends on a parameter, called perversity, and satisfies a Poincaré duality for pseudomanifolds, involving different perversities. A first possibility for an autoduality in the presence of singularities is represented by Witt spaces for which the intersection homologies of the two middle perversities coincide. For them, \textit{P. H. Siegel} has defined a bordism-invariant signature [Am. J. Math. 105, 1067--1105 (1983; Zbl 0547.57019)].
\textit{M. Banagl} [Extending intersection homology type invariants to non-Witt spaces. Providence, RI: American Mathematical Society (AMS) (2002; Zbl 1030.55005)] introduced a more general situation than Witt conditions. Under some hypotheses involving the existence of Lagrangian sheaves, he constructs a self-dual sheaf ``between'' the two middle perversities and satisfying Poincaré duality. Pseudomanifolds having such a structure are called L-pseudomanifolds; they include Witt spaces. \textit{M. Banagl} [Ann. Math. (2) 163, No. 3, 743--766 (2006; Zbl 1101.57013)] proved that the signature of the self-dual sheaf of an L-pseudomanifold is a bordism-invariant, independent of the choice of the self dual sheaf. He also introduced homological characteristic L-classes, that are also independent of this choice. Recall that these classes agree with the Goresky-MacPherson-Siegel L-classes in the case of a Witt space.
The first main result of the paper under review is the multiplicativity of the signature and, more generally, the topological L-class of closed, oriented L-pseudomanifolds. The second theorem is the proof of the Brasselet-Schürmann-Yokura conjecture for normal complex projective 3-folds with at most canonical singularities, trivial canonical class and positive irregularity. This conjecture is the equality of topological and Hodge L-class for compact complex algebraic rational homology manifolds. signature; characteristic classes; pseudomanifolds; stratified spaces; intersection homology; perverse sheaves; Hodge theory; canonical singularities; varieties of Kodeira dimension zero; Calabi-Yau varieties Characteristic classes and numbers in differential topology, Intersection homology and cohomology in algebraic topology, Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects), Singularities of surfaces or higher-dimensional varieties, \(3\)-folds, Coverings in algebraic geometry Topological and Hodge L-classes of singular covering spaces and varieties with trivial canonical class | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(V(a,b)\) be the irreducible \(G:=SL_3({\mathbb C})\)-module with highest weight with numerical labels \(a\), \(b\), where \(a\), \(b\) are nonnegative integers. The paper gives a simple method to obtain a basis for \(\text{Hom}_G (V(a,b)\otimes V(c,d), V(e,f))\). In fact Theorem 3.1 gives explicit formulas for decomposing such maps into certain blocks.
As an application, one proves the rationality of \({\mathbb P}(V(4,4))/G\), using an extension (Prop. 5.2) of the \textit{double bundle method} [cf. \textit{F. Bogomolov, P. Katsylo}, Math. USSR, Sb. 54, 571--576 (1986); translation from Mat. Sb., Nov. Ser. 126(168), No.4, 584--589 (1985; Zbl 0591.14040)].
Another application, which cannot be obtained by direct application of Prop. 5.2, is the rationality of the moduli space of plane curves of degree \(34\). representation theory; rationality; moduli spaces; plane curves; group quotients Böhning Chr., Graf v. Bothmer H.-Chr.: A Clebsch--Gordan formula for \$\$\{\(\backslash\)mathrm\{SL\}\_3 (\(\backslash\)mathbb\{C\})\}\$\$ and applications to rationality. Adv. Math. 224, 246--259 (2010) Families, moduli of curves (algebraic), Representation theory for linear algebraic groups, Algebraic moduli problems, moduli of vector bundles, Coverings in algebraic geometry, Group actions on varieties or schemes (quotients) A Clebsch-Gordan formula for \(\text{SL}_3(\mathbb C)\) and applications to rationality | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 maximizing sextic. We study a \({\mathcal D}_{2p}\) (\(p\): odd prime) covering of \(\mathbb{P}^2\) branched along \(C\). Suppose that \(C\) is irreducible and has at least one triple point. Then, for \(p\geq 5\), there is no \({\mathcal D}_{2p}\) covering branched along \(C\). For \(p=3\), we give a sufficient condition for \(C\) to be the branch locus of a \({\mathcal D}_6\) covering. Several examples of such \(C\) are given. dihedral covering; elliptic surface; sextic Tokunaga, H, Dihedral coverings branched along maximizing sextics, Math. Ann., 308, 633-648, (1997) Coverings in algebraic geometry, Coverings of curves, fundamental group, Elliptic surfaces, elliptic or Calabi-Yau fibrations Dihedral coverings branched along maximizing sextics | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let Y be a projective manifold and D a smooth divisor on it. Assume that \(D\in | F^ k|\), where F is a line bundle on Y and \(k\geq 2\) is an integer. A k-cyclic covering X of Y branched along D is the hypersurface of F defined locally by \(w^ k_{\alpha}=\Phi_{\alpha},\) where \(w_{\alpha}\) is the fibre coordinate of F and \(\phi_{\alpha}=0\) is the local equation of D. In this paper, the author studies the local Torelli problem of such X. This is the question whether locally periods of holomorphic n-forms on X \((n=\dim X)\) determine X. Griffiths gave a cohomological criterion for solving this problem provided that the Kuranishi space of X is smooth. In the first part of the paper, the author gives a sufficient condition for the Kuranishi space of X to be smooth under some assumptions on Y and F, which one may regard as a generalization of a result of \textit{J. J. Wavrik} [Am. J. Math. 90, 926- 960 (1968; Zbl 0176.039)]. Next he shows that Griffiths' criterion is reduced to the verification of certain conditions on the canonical bundle of X and some cohomology groups on Y, using the decompositions of the cohomology groups appearing in Griffiths' criterion under the action of the Galois group Gal(X/Y). As an application, the local Torelli theorem for cyclic coverings of the Hirzebruch surfaces is proved. It should be remarked that there are serious mistakes in the proofs of this theorem given by \textit{C. A. M. Peters} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 3, 321-339 (1976; Zbl 0329.14011)] and by \textit{C. Peters, D. Lieberman} and \textit{R. Wilsker} [Math. Ann. 231, 39-45 (1977; Zbl 0367.14006)]. cyclic covering; local Torelli problem; periods of holomorphic n-forms; Hirzebruch surfaces Konno, K. : On Deformations and the local Torelli Problem of cyclic branched coverings . Math. Ann. 271 (1985) 601-617. Coverings in algebraic geometry, Ramification problems in algebraic geometry, Period matrices, variation of Hodge structure; degenerations, Jacobians, Prym varieties On deformations and the local Torelli problem of cyclic branched coverings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper studies symplectic automorphisms of Kummer surfaces. Recall that an automorphism \(\sigma\) of finite order of a \(K3\) surface \(X\) is called symplectic if the desingularization of the quotient \(X/\sigma\) is still \(K3\). Finite abelian groups of symplectic automorphisms of \(K3\) were classified by \textit{V.V. Nikulin} [Trans. Mosc. Math. Soc. 2, 71--135 (1980; Zbl 0454.14017)]. Unfortunately that paper has a small mistake, and this paper studies precisely how Nikulin's result needs to be adjusted, providing some examples not covered by Nikulin where the \(K3\) surface is Kummer and the group of symplectic automorphisms is not cyclic. \(K3\) surface; symplectic automorphism; Kummer surface Garbagnati A., Dedicata 145 pp 219-- (2010) \(K3\) surfaces and Enriques surfaces, Automorphisms of surfaces and higher-dimensional varieties, Coverings in algebraic geometry Symplectic automorphisms on Kummer 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 paper investigates the situation in which a normal (possibly singular) projective variety \(X\) has an endomorphism \(f\) with degree greater than one and a prime (Cartier) divisor \(V\) satisfying \(f^{-1}(V)=V\). The various hypotheses made for each of the main results in the paper all hold when the dimension of \(X\) is at least two, the Picard rank of \(X\) is one, and the singularities of \(X\) are restricted in a somewhat standard way; the author shows (Example 1.9) that there are many singular hypersurfaces in projective spaces of all large dimensions that satisfy these conditions. Some of the conclusions made in this case are: (1) there are at most \(\dim(X)+1\) prime divisors \(V\) satisfying \(f^{-1}(V)=V\); (2) \(X\) is rationally connected if there are at least \(\dim(X)\) such \(V\); (3) \(X\) is always rationally chain connected, as are each \(V\) and the normalization of each \(V\); and (4) \(X\) is simply connected, as is each \(V\).
The author takes great care to state the optimal generality under which each of the main conclusions holds. For example, if \(\dim(X)\) is replaced by \(\dim(X)+\rho(X)-1\), (1) and (2) remain true for \(X\) with Picard rank bigger than one so long as \(f\) is a polarized endomorphism and \(f^*\) is a multiple of the identity on \(\text{NS}(X)\). The author also gives examples to show that the bound in (1) is optimal.
The proofs of the theorems incorporate many existing results regarding singularities of normal projective varieties and \(\mathbb{Q}\)-Cartier divisors on normal (singular) projective varieties to achieve detailed descriptions of the pull-back actions of endomorphisms on such varieties. polarized endomorphisms; singular varieties; Cartier divisors; hypersurfaces; invariant sets Zhang, D-Q, Invariant hypersurfaces of endomorphisms of projective varieties, Adv. Math., 252, 185-203, (2014) Coverings in algebraic geometry, Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables, Hypersurfaces and algebraic geometry, Fano varieties, Singularities of surfaces or higher-dimensional varieties, Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets Invariant hypersurfaces of endomorphisms of projective varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper provides an overview of recent results on Galois extensions of function fields, which have been obtained using formal and rigid analytic methods. The starting point is a \(p\)-adic analogue to the inverse Galois problem in number theory. The basic definitions and facts relating to formal and rigid geometry which are needed to construct Galois coverings of algebraic curves are recalled, and a sketch is given of Harbater's proof that every finite group occurs as a Galois group over the \(p\)-adic field \(\mathbb{Q}_p\), using a \(p\)-adic analogue of the geometric approach to the inverse Galois problem. Additional results on Galois coverings obtained using rigid geometry are then presented, as well as descriptions of the basic principles on which their proofs rely.
This article provides a nice introduction to the ideas behind many of the important results in this area, as well as ample references to the literature for the reader who wants the full details. inverse Galois theory; Galois coverings; rigid analytic spaces; Galois extensions of function fields Inverse Galois theory, Coverings in algebraic geometry Rigid geometry and Galois extensions of function fields in one variable | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a compact Kähler manifold, and let \(f: X \rightarrow X\) be a surjective endomorphism. The \(i\)-th dynamical degree \(d_i(f)\) is defined as
\[
d_i(f) := \lim_{s \to \infty} \root{s}\of{\int_X (f^s)^* \omega^i \wedge \omega^{n - i}},
\]
where \(\omega\) is any Kähler form on \(X\). A similar definition works if \(f\) is only a meromorphic dominant endomorphism, [cf. \textit{T.-C. Dinh} and \textit{N. Sibony}, Ann. Sci. Éc. Norm. Supér. (4) 37, No. 6, 959--971 (2004; Zbl 1074.53058)]. The endomorphism \(f\) is said to be cohomologically hyperbolic in the sense of \textit{V. Guedj} [Propriétés ergodiques des applications rationnelles, \url{arXiv:math/0611302} (2006)] if there exists an \(l\) such that \(d_l(f)>d_i(f)\) for all \(i \neq l\). In his paper Guedj classified cohomologically hyperbolic meromorphic endomorphisms of surfaces and deduced from this classification that the Kodaira dimension of such a surface is not positive. He also conjectured that this should hold in arbitrary dimension. \newline In the paper under review the author proves Guedj's conjecture in arbitrary dimension for a holomorphic surjective endomorphism. The proof is not based on any classification and follows from results due to \textit{N. Nakayama} and the author [Building blocks of étale endomorphisms of complex projective manifolds, \url{arXiv:0903.3729} (2009)]. The main part of the paper is dedicated to the classification of smooth projective threefolds \(X\) admitting a surjective and cohomologically hyperbolic endomorphism that is \textit{étale}: up to birational equivalence and étale cover the variety \(X\) is a torus, weak Calabi-Yau, rationally connected, a product of an elliptic curve and a \(K3\) surface or a smooth fibration over an elliptic curve such that the general fibre is a rational surface with Picard number at least 11. As a corollary of this classification one obtains that the fundamental group of \(X\) is either finite or contains a finite-index subgroup isomorphic to \(\mathbb Z^{\oplus 2}\) or \(\mathbb Z^{\oplus 6}\). The proof of the classification uses earlier results due to Nakayama and the author as well as the minimal model program in dimension three. endomorphism; Calabi-Yau; rationally connected variety; dynamics DOI: 10.1142/S0129167X09005546 Coverings in algebraic geometry, Birational automorphisms, Cremona group and generalizations, \(3\)-folds, Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables Cohomologically hyperbolic endomorphisms of complex manifolds | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems \textit{W. D. Neumann} and \textit{J. Wahl} [Math. Ann. 326, No.1, 75--93 (2003; Zbl 1032.14010)] proved that the universal abelian cover of every quotient-cusp is a complete intersection and have conjectured that a similar results holds for any \(\mathbb Q\)-Gorenstein normal surface whose link is a rational homology sphere. Cusps are not included in the scope of this conjecture, nevertheless the author considers a similar question and exhibits a cusp with no Galois cover by a complete intersection. The main techniques are plumbing and study of monodromy for cusp singularities. link of a singularity Singularities in algebraic geometry, Coverings in algebraic geometry, Complex surface and hypersurface singularities, Milnor fibration; relations with knot theory A cusp singularity with no Galois cover by a complete intersection | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In the paper under review, the author studies the geometry of elliptic surfaces of rank one and its applications towards the existence of Zariski pairs.
Let \(\phi: S \rightarrow C\) be an elliptic surface over a smooth projective curve such that \(\phi\) is relatively minimal, there exists a section \(O: C \rightarrow S\), and there exists at least one degenerate fiber. We denote the set of sections of \(\phi: S \rightarrow C\) by \(\mathrm{MW}(S)\), the Mordell-Weil group, which is non-empty. Let \(E_{S}\) denote the generic fiber of \(\phi\) and we denote by \(E_{S}(\mathbb{C}(C))\) the set of \(\mathbb{C}(C)\) -rational points of \(E_{S}\) and thus \((E_{S},O)\) is an elliptic curve defined over \(\mathbb{C}(C)\) and \(E_{S}(\mathbb{C}(C))\) has the structure of a finitely generated abelian group. We say that an elliptic surface \(\phi:S \rightarrow C\) is an elliptic surface of rank one if \(\mathrm{rk} \,E_{S}(\mathbb{C}(C)) = 1\).
Let \(D\) be a divisor on \(S\), by restricting \(D\) to \(E_{S}\) we have a divisor \(\mathfrak{d}\) in \(E_{S}\) defined over \(\mathbb{C}(C)\). If we apply Abel's theorem to \(\mathfrak{d}\), we have \(P_{D} \in E_{S}(\mathbb{C}(C))\) and thus the corresponding section \(s(D) \in\mathrm{ MW}(S)\). In the paper under review the authors study \(n\)-divisiblity of \(P_{D}\) in the setting when \(E_{S}(\mathbb{C}(C)) = \mathbb{Z}P_{o} \oplus E_{S}(\mathbb{C}(C))_{\mathrm{tor}}\) for some \(P_{o} \in E_{S}(\mathbb{C}(C))\) and then they apply this result to consider the embedded topology of reducible curves. The first main result can be formulated as follows.
Theorem A. Suppose that \(\mathrm{rk} \, E_{S}(\mathbb{C}(C))=1\). Let \(n\) be an integer such that \(P_{D} = nP_{o} + P_{\tau}\), \(P_{\tau} \in E_{S}(\mathbb{C}(C))_{\mathrm{tor}}\). Then we have
\[n^{2} = - \frac{\phi_{o}(D) \cdot \phi_{o}(D)}{\langle P_{o},P_{o}\rangle}, \quad \quad n = - \frac{\phi_{o}(D) \cdot \phi(P_{o})}{\langle P_{o}, P_{o} \rangle},\]
where \(\cdot\) and \(\langle, \rangle\) mean the intersection and height pairing, respectively, and \(\phi, \phi_{o}\) are homomorphisms described in Section \(2\) therein.
Since properties of \(P_{D} \in E_{S}(\mathbb{C}(C))\) play important roles in order to study
the existence/non-existence of dihedral covers of \(\mathbb{P}^{2}\), one can use them in order to obtain an observation
for the embedded topology of plane curves which arise from \(D\).
Consider the following combinatorics. Let \(E\) be a nodal cubic and \(L_{i}\) with \(i \in \{0,1,2,3\}\) be four lines as below, and we put \(\mathcal{B} = E + \sum_{i=0}^{e}L_{i}\):
i) \(L_{0}\) is a transversal line to \(E\) and we put \(E\cap L_{0} = \{p_{1}, p_{2},p_{3}\}\),
ii) \(L_{i}\) is a line through \(p_{i}\) and tangent to \(E\) at a point \(q_{i}\) distinct from \(p_{i}\), for each \(i \in \{1,2,3\}\),
iii) \(L_{1},L_{2},L_{3}\) are not concurrent and we put \(L_{i}\cap L_{j} = \{r_{k}\}\) for \(\{i,j,k\} = \{1,2,3\}\).
For \(\mathcal{B}\) with the above combinatorics we call it Type I (Type II, respectively) if \(q_{1},q_{2},q_{3}\) are collinear (not collinear, respectively).
Theorem B. Let \((\mathcal{B}^{1}, \mathcal{B}^{2})\) be a pair of plane curve with the above combinatorics such that their Types are distinct. Then both of the fundamental groups \(\pi_{1}(\mathbb{P}^{2} \setminus \mathcal{B}^{j}, *)\) with \(j \in \{1,2\}\) are non-abelian and there exists no homomorphism between \((\mathbb{P}^{2}, \mathcal{B}^{1})\) and \((\mathbb{P}^{2},\mathcal{B}^{2})\). elliptic surfaces; Mordell-Weil rank one; multi-sections; topology of reducible plane curves Elliptic surfaces, elliptic or Calabi-Yau fibrations, Coverings in algebraic geometry, Low-dimensional topology of special (e.g., branched) coverings, Elliptic curves Elliptic surfaces of rank one and the topology of cubic-line arrangements | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Geometric considerations identify what properties we desire of the canonical sequence of finite groups that are used to define modular towers. For instance, we need the groups to have trivial center for the Hurwitz spaces in the modular tower to be fine moduli spaces. The Frattini series, constructed inductively, provides our sequence: each group is the domain of a canonical epimorphism, which has elementary abelian \(p\)-group kernel, having the previous group as its range. Besides satisfying the desired properties, this choice is readily analyzable with modular representation theory.
Each epimorphism between two groups induces (covariantly) a morphism between the corresponding Hurwitz spaces. Factoring the group epimorphism into intermediate irreducible epimorphisms simplifies determining how the Hurwitz-space map ramifies and when connected components have empty preimage. Only intermediate epimorphisms that have central kernel of order \(p\) matter for this. The most important such epimorphisms are those through which the universal central \(p\)-Frattini cover factors; the elementary abelian \(p\)-Schur multiplier classifies these.
This paper, the second of three in this volume on the topic of modular towers, reviews for arithmetic-geometers the relevant group theory, culminating with the current knowledge of the \(p\)-Schur multipliers of our sequence of groups. Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Modular representations and characters, Special subgroups (Frattini, Fitting, etc.), Limits, profinite groups The group theory behind modular towers | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The theory of congruence subgroups, more precisely the theory of the action of congruence subgroups of the modular group on the upper half plane, is an area of mathematics in which the mathematical structure is well understood and wonderfully intricate. In contrast, the arithmetic theory of subgroups which are not congruence subgroups is surprisingly little developed. In a neighbouring area, there is a beautiful corpus of recent work concerned with the arithmetic of Galois coverings of the projective line, ramified in a prescribed way above a finite set of places; a motivation for this has been its application to the inverse Galois problem. This essay is a summary of various talks given by the author on the subject during the past couple of years.
Theorem 1. For each positive integer \(n\), the following families of objects are in 1-1 correspondence:
(i) Triples \((\mathcal R,\varphi,O)\) where \(\mathcal R\) is an \(n\)-sheeted Riemann surface, \(\varphi:\mathcal R\to\overline\mathbb{C}\) is a covering map branched at most above \(\{\infty,0,k\}\), and \(O\) is a point of \(\mathcal R\) above 0;
(ii) Quadruples \((\sigma_\infty,\sigma_0,\sigma_1;\nu)\) where \(\sigma_\infty,\sigma_0,\sigma_1\) are permutations of \(S_n\) such that \(\sigma_\infty\sigma_0\sigma_1=\text{id}\) and such that the group generated by \(\sigma_0,\sigma_1\) is transitive on the symbols permuted by \(S_n\), and \(\nu\) is a marked cycle of \(\sigma_\infty\); all modulo equivalence corresponding to simultaneous conjugation by an element of \(S_n\);
(iii) Subgroups \(\Gamma\leq\Gamma(2)\) of index \(n\), modulo conjugacy by translation;
(iv) Drawings with \(n\) edges.
Here a drawing is a finite connected 1-complex which is the skeleton of an oriented 2-complex. It has two sets of vertices, coloured respectively black and white, and a set of edges. Each edge is incident to a single vertex of each colour, and for each vertex there is a cyclic ordering of the edges incident to it. These drawings are of course pretty much the same as what \textit{G. E. Shabat} and \textit{V. A. Voevodsky} [The Grothendieck Festschrift, Vol. III, Prog. Math. 88, 199--227 (1990; Zbl 0790.14026)] call a dessin: their dessins are a particular case of drawings. Conversely, every drawing may be made into a dessin by re-colouring all the vertices black, and putting a new white vertex in the middle of each edge.
Having set up his apparatus, the author lists three questions, essentially the ones asked by \textit{A.~O.~L. Atkin} and \textit{H. P. F. Swinnerton-Dyer} [in: Combinatorics, Proc. Symp. Pure Math. 19, 1--25 (1971; Zbl 0235.10015)].
Question 0. Describe how to list the subgroups of \(\Gamma(2)\), in order (say) of increasing index.
Question 1. The Riemann surfaces \({\mathcal R}\) are algebraic curves, and the projections \(\varphi\) are algebraic maps. What can we say about the equations of these curves and maps; in particular, what is their field of definition?
Question 2. For any subgroup \(\Gamma\) of finite index in \(\Gamma(1)\), and any integer \(k\geq 1\), we may define cusp forms of weight \(2k\) on \(\mathbb{H}/\Gamma\). Such a form \(f(z)\) has a Fourier expansion \(f(z)=\sum^\infty_1a_rq^r_\mu\), where \(q_\mu=\exp(2\pi iz/\mu)\) and \(z\to z+\mu\) is the least translation in \(\Gamma\). What can be said about the coefficients \(a_n\)? (In the congruence subgroup case, the Hecke algebra action implies that there is a basis of `eigenforms', whose coefficients are multiplicative.)
Question 0 has been answered in effect by (ii) of the theorem 1 since it is not too hard to train a computer to list pairs of permutations up to conjugacy. In section 2, it is recalled what is known in general about the nature of the curves \({\mathcal R}\) and the covering maps \(\varphi\). Almost the only answers to question 2 are due to \textit{A. J. Scholl} [cf. Invent. Math. 79, 49--77 (1985; Zbl 0553.10023) and J. Reine Angew. Math. 392, 1--15 (1988; Zbl 0647.10022)] recalling his results in section 3. Finally, in section 4, a little bit is said about calculation, and tables of examples are given. drawings; congruence subgroups; Galois coverings of the projective line; inverse Galois problem; Riemann surface; covering map B. Birch, ''Non-congruence subgroups, covers and drawings,'' in: \textit{The Grothendieck Theory of Dessins D'enfants}, Cambridge Univ. Press, Cambridge (1994), pp. 25-46. Structure of modular groups and generalizations; arithmetic groups, Arithmetic aspects of dessins d'enfants, Belyĭ theory, Fuchsian groups and their generalizations (group-theoretic aspects), Riemann surfaces; Weierstrass points; gap sequences, Coverings in algebraic geometry Noncongruence subgroups, covers and drawings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Author's abstract: During the last 10 years there have been several new results on the representation of real polynomials, positive on some semi-algebraic subset of \(\mathbb R^n\). These results started with a solution of the moment problem by Schmüdgen for compact semi-algebraic sets. Later, Wörmann realized that the same results could be obtained by the so-called ``Kadison-Dubois'' representation theorem.
The aim of our paper is to present this representation theorem together with its history, and to discuss its implication to the representation of positive polynomials. Also recent improvements of both topics by T. Jacobi and the author will be included. ring of continuous real-valued functions Prestel, A., Representation of real commutative rings, Expo. Math., 23, 89-98, (2005) Real algebra, Semialgebraic sets and related spaces, Functions of several variables, Rings and algebras of continuous, differentiable or analytic functions Representation of real commutative rings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a complex projective algebraic variety. An endomorphism \(f : X \rightarrow X\) is called polarized if \(f^{\ast}H\sim qH\), for some ample line bundle \(H\) and \(q>0\). In particular, any surjective endomorphism of a projective variety of Picard number one is polarized. In a previous paper, \textit{N. Nakayama} and the author [Math. Ann. 346, No. 4, 991--1018 (2010; Zbl 1189.14043)] have proved that any normal variety \(X\) with a polarized endomorphism which is not an isomorphism either has only canonical singularities and \(K_X\sim_{\mathbb{Q}} 0\), or it is uniruled. Moreover, there is a maximal rationally connected fibration \(f : X \dasharrow Y\) such that \(f\) descends to a polarized endomorphism \(f_Y\) of \(Y\) and therefore there are polarized endomorphisms of the rationally connected fibers \(\Gamma_i\) of the graph \(\Gamma(X/Y)\) over a dense set of \(f_Y\)-periodic points of \(Y\). Consequently, the study of polarized endomorphisms of uniruled varieties is reduced to the study of polarized endomorphisms of rationally connected varieties.
In the paper under review, the author studies polarized endomorphisms of a \(\mathbb{Q}\)-factorial rationally connected projective variety \(X\) of dimension 3 or 4 with log terminal singularities. Let \(f : X \rightarrow X\) be a polarized morphism of degree \(q^n >1\). According to the Minimal Model Program, which at the moment of this writing is known to work in dimension at most 4, there is a sequence of divisorial contractions and flips \(X=X_0 \dasharrow \cdots \dasharrow X_k\) such that \(X_k\) is a Mori fiber space, that is there is a morphism \(g_r : X_r \rightarrow Y\) such that \(-K_{X_r}\) is \(g_r\)-ample and \(\rho(X_r/Y)=1\). Since \(X\) is assumed rationally connected, so is \(Y\). It is shown that up to replacing \(f\) with a suitable power of it, the induced dominant rational maps \(f_i : X_i \dasharrow X_i\) are in fact morphisms preserving the exceptional set of \(X_i \rightarrow X_{i+1}\). Moreover, \(f_r\) is a polarized endomorphism of \(X_r\) and it factors through a polarized endomorphism of \(Y\). By running another Minimal Model Program on \(Y\) the author concludes that the study of polarized endomorphisms of rationally connected varieties is reduced to the study of polarized endomorphisms of \(\mathbb{Q}\)-Fano varieties of Picard number 1. He also shows that any smooth Fano 3-fold with a polarized endomorphism of degree greater than one is rational.
Finally the author produces some results about the structure of the cone of curves of a rationally connected \(\mathbb{Q}\)-factorial 3-fold \(X\) with log terminal singularities that has a polarized endomorphism \(f\) of degree greater than one. He shows that \(X\) has at most finitely many \(K_X\)-negative extremal rays and that if the degree of \(f\) is \(q^3>1\), then there is an \(s>0\) such that the induced map \((f^s)^{\ast}\) of \(f^s\) on \(N^1(X)\) is \(q^s 1_{N^1(X)}\), where \(1_{N^1(X)}\) is the identity map of \(N^1(X)\). Moreover, in the last case, \(X\) ir either rational or \(-K_X\) is big. polarized endomorphisms; uniruled variety; rational variety Zhang, D-Q, Polarized endomorphisms of uniruled varieties, Compos. Math., 146, 145-168, (2010) Coverings in algebraic geometry, Fano varieties, Rationality questions in algebraic geometry, Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables Polarized endomorphisms of uniruled varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This set of lectures given at the C.I.M.E. Summer School 2003 in Cetraro, Italy, provides an overview of S. Donaldson's approach to the study of symplectic 4-manifolds via the algebro-geometric concept of Lefschetz pencils for algebraic surfaces. After a brief introductory chapter on symplectic manifolds, almost-complex structures, pseudo-holomorphic curves, Gromov-Witten invariants, Lagrangian Floer homology, and the topology of symplectic 4-manifolds, the authors systematically survey and explain the following important developments of the past decade in the following four chapters:
(2) Symplectic-Lefschetz fibrations and the classification of symplectic 4-manifolds in algebraic terms (after S. Donaldson, D. Auroux, I. Smith, and others);
(3) Symplectic branched covers of \(\mathbb{CP}^2\) and their invariants (after D. Auroux, S. Donaldson, L. Katzarkov, M. Yotov, and others);
(4) Symplectic surfaces from symmetric products, with applications to C. Taubes's theorems (after S. Donaldson, I. Smith, and M. Usher);
(5) Fukaya categories and Lefschetz fibrations, with applications to vanishing cycles and mirror symmetry (after P. Seidel, D. Auroux, L. Katzarkov, and D. Orlov).
All together, the authors depict some of the most recent results on Lefschetz fibrations in symplectic geometry, an essential part of which is due to S. Donaldson and the authors themselves, in very vivid, lucid and enlightening a manner, thereby providing a highly valuable guide to the corresponding, just as recent research literature into the bargain. Lefschetz fibrations; Lefschetz pencils; coverings; symplectic manifolds; algebraic surfaces; Fukaya categories; mirror symmetry D. Auroux and I. Smith, ``Lefschetz pencils, branched covers and symplectic invariants'' in Symplectic 4-manifolds and Algebraic Surfaces (Cetraro, 2003), Lecture Notes in Math. 1938, Springer, Berlin, 2008, 1--53. Fibrations, degenerations in algebraic geometry, Coverings in algebraic geometry, Symplectic manifolds (general theory), Lagrangian submanifolds; Maslov index, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants), Pseudoholomorphic curves Lefschetz pencils, branched covers and symplectic 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 The author considers the Lamé operators (i.e. the differential operators of second-order with four singular points) having the form
\[
L_n= D^2+ {f'\over 2f} D- {n(n+ 1) x+ B\over f},\tag{\(*\)}
\]
and tries to give a method which allows to calculate the number of these operators having prescribed monodromy. In \((*)\) \(D\) stands for \(d/dx\) while \(n\in \mathbb{N}\) and \(B\in \mathbb{C}\) are constants. The function \(f\) is given by \(f= 4(x- e_1) (x- e_2) (x- e_3)\), where \(e_i\in \mathbb{C}\) are constant numbers which differ from each other. He shows that the number (up to homography) of equations whose monodromy group is a dihedral group of order \(2m\) \((m\in \mathbb{N})\) is finite for each \(n\in \mathbb{N}\) and \(m\in \mathbb{N}\). Notice that two Lamé operators are homographic if one of them can be transformed into the other by a homographic change of the independent variable. The number of the operators (up to homography) is calculated for the case where \(n= 1\). The case of operators with infinite monodromy is also considered. The results are given in one lemma, nine propositions, six theorems and one corollary. The paper includes also four examples. Lamé operators; prescribed monodromy; monodromy group Bruno Chiarellotto, On Lamé operators which are pull-backs of hypergeometric ones, Trans. Amer. Math. Soc. 347 (1995), no. 8, 2753 -- 2780. Special ordinary differential equations (Mathieu, Hill, Bessel, etc.), Coverings in algebraic geometry, Ordinary differential equations in the complex domain, Lamé, Mathieu, and spheroidal wave functions On Lamé operators which are pull-backs of hypergeometric ones | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(B\) be a plane curve in \(\mathbb{P}^2\) given by an equation \(F(X_0,X_1,X_2)=0\), \(X_0\), \(X_1\), \(X_2\) being homogeneous coordinates. Let \(B_a\) be the affine plane curve given by \(f(x,y)=F(1,x,y)=0\). Define an \(n\)-cyclic extension, \(K\), of the rational function field of \(\mathbb{P}^2\), \(\mathbb{C}(\mathbb{P}^2)= \mathbb{C}(x,y)\), by: \(K=\mathbb{C} (x,y)(\zeta)\), \(\zeta^n=f(x,y)\). Let \(S_n'\) be the \(K\)-normalization of \(\mathbb{P}^2\); and we denote its smooth model by \(S_n\). \(S_n\) is an \(n\)-fold cyclic covering of \(\mathbb{P}^2\) branched along \(B\) and possibly the line at infinity, \(X_0 =0\). It is called a cyclic multiple plane, of which investigation bas been done by many mathematicians. Their main interest has been to study the first Betti number, \(b_1(S_n)\), the Alexander polynomial of \(B\), and \(S_n\) itself. In this note we focus our attention on the image of the Albanese mapping \(\alpha_n: S_n \to\text{Alb}(S_n)\).
Definition. The number \(a(B_a): =\max_{n\in\mathbb{N}} \dim\alpha_n(S_n)\) is called the Albanese dimension of \(B_a\).
A priori, \(a(B_a)=0,1\), or 2. The purpose of this note is to give examples of irreducible affine plane curves, \(B_a\), with \(a(B_a)=2\). plane curve; cyclic covering; Albanese dimension Tokunaga, H., Irreducible plane curves with Albanese dimension \(2\), Proc. Amer. Math. Soc., 127, 1935-1940, (1999) Coverings of curves, fundamental group, Special algebraic curves and curves of low genus, Coverings in algebraic geometry Irreducible plane curves with the Albanese 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 As the author mentions, in [\textit{M. Namba}, J. Math. Soc. Japan 41, 391-403 (1989; Zbl 0701.14013)] a construction of a versal \(G\) - covering of dimension equal to the order of \(G\) was given, for any finite group \(G\). In this paper one generalizes partially a result of \textit{H. Tsuchihashi} [Kyushu J. Math. 57, 411--427 (2003; Zbl 1072.14018)], constructing a versal \(G\) - covering of dimension \(n\) for any subgroup \(G\) of GL\((n,{\mathbb Z})\).
The main result of the paper is the following: Let \(N\) be a free \(\mathbb Z\)-module, \(\Delta \) a fan in \(N_{\mathbb R}\) and \(X({\Delta})\) the toric variety associated to \(X({\Delta})\). If \(G\) is a subgroup of \(\Aut_{\mathbb Z}(N)\) for which \(\Delta \) is invariant , then \(X({\Delta})\to X({\Delta})/G\) is a versal G-covering. Bannai, S, Construction of versal Galois coverings using toric varieties, Osaka J. Math., 44, 139-146, (2007) Coverings in algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies Construction of versal Galois coverings using toric 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 \(f:{\mathcal C}\to \mathbb{P}^1\) be a ramified covering, where \({\mathcal C}\) is an algebraic curve, defined over an algebraic closure \(\overline{\mathbb{Q}}\) of \(\mathbb{Q}\). Let \(\sigma\) be an element of \(\text{Gal} (\overline{\mathbb{Q}}/\mathbb{Q})\), denote by \({}^\sigma f: {}^\sigma{\mathcal C}\to \mathbb{P}^1\) the covering obtained by base change. There are two ways of saying that the covering is ``stable'' under \(\sigma\): 1) there exists \(u_\sigma:{\mathcal C}\simeq {}^\sigma{\mathcal C}\), defined over \(\overline{\mathbb{Q}}\), such that \(f= {}^\sigma fu_\sigma\), 2) there exists \(u_\sigma\) as above and an automorphism \(v_\sigma\) of \(\mathbb{P}^1\), defined over \(\overline{\mathbb{Q}}\), such that \(v_\sigma f= {}^\sigma fu_\sigma\).
In both cases, the set of such \(\sigma\) is an open subgroup of \(\text{Gal} (\overline{\mathbb{Q}}/\mathbb{Q})\), then one can define ``fields of definition'' for the covering \(f\). The author compares these two natural notions of rationality, especially when they differ, when the two fields of definition are not the same. Explicit examples are given. Galois group; fields of definition; ramified covering; rationality D'costa, L. Pharamond Dit: Comparaison de deux notions de rationalité d'un dessin d'enfant. J. théor. Nombres Bordeaux 13, No. 2, 529-538 (2001) Arithmetic algebraic geometry (Diophantine geometry), Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Coverings of curves, fundamental group, Coverings in algebraic geometry Comparison of two notions of rationality for a dessin d'enfant | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(S\) (resp. \(\Sigma\)) be a normal (resp. smooth) projective surface defined over the complex number field. Then we consider a special type of finite morphism which is defined by the following;
(i) \(\pi : S\to \Sigma\) is a Galois covering;
(ii) the Galois group \(\text{Gal}(S/\Sigma)\) of \(\pi\) is isomorphic to the dihedral group \(\mathcal{D}_{2n}\). This covering is said to be a dihedral covering. In this paper the author consider the following problem: Give a sufficient condition of the existence of a dihedral covering \(\pi : S\to \Sigma\) such that the branch locus \(B\) of \(\pi\) is reduced and \(B\) has only at most simple singularities and the ramification index along the nonsingular locus of \(B\) is two.
By considering the above problem, the author gives an example of Zariski pair. (Let \(B_{1}\) and \(B_{2}\) be irreducible and reduced curves on \(\mathbb{P}^{2}\). Then the pair \((B_{1},B_{2})\) is said to be the Zariski pair if \(\pi_{1}(\mathbb{P}^{2}\backslash B_{1})\not\cong \pi_{1}(\mathbb{P}^{2}\backslash B_{2})\). [For more information about Zariski pairs, see \textit{O. Zariski}, Am. J. Math. 51, 305-328 (1928; JFM 55.0806.01) and \textit{H. Tokunaga}, Math. Z. 227, No.~3, 465-477 (1998; Zbl 0923.14012) or 230, No.~2, 389-400 (1999; Zbl 0930.14018).]
In this paper, the author gives a sufficient condition of the existence of a dihedral covering with the properties as in the problem above.
Let \(\Sigma\) be a smooth projective surface and let \(B\) be a reduced divisor on \(\Sigma\) such that \(B\) has at most simple singularities. Then by an earlier paper [\textit{H. Tokunaga}, Can. J. Math. 46, No. 6, 1299-1317 (1994; Zbl 0857.14009)] we may assume that there exists a double covering \(f'': \Sigma''\to \Sigma\) branched along \(B\). Let \(g: Z\to \Sigma''\) be a resolution of singularities of \(\Sigma''\) and let \(f=f''\circ g\). Assume that the Néron-Severi group \(\text{NS}(Z)\) is torsion free. Let \(T\) be a subgroup of \(\text{NS}(Z)\) such that \(T\) is generated by \(f^{*}\text{NS}(\Sigma)\) and the exceptional divisors of \(g\).
Main theorem: Assume that (1) \(\text{NS}(Z)/T\) has an \(n\)-torsion element,
(2) \(\text{gcd}(n,\text{disc}(f^{*}\text{NS}(\Sigma)))=1\).
Then there exists a dihedral covering branched along \(B\) such that the ramification index along \(B\) is two. (Here \(\text{disc}(f^{*}\text{NS}(\Sigma))\) denotes the determinant of intersection matrix of \(f^{*}\text{NS}(\Sigma)\)). dihedral covering; Zariski pair; resolution of singularities; JFM 55.0806.01; Néron-Severi group; ramification index Coverings in algebraic geometry, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Homotopy theory and fundamental groups in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties, Ramification problems in algebraic geometry Conditions for the existence of dihedral coverings of algebraic surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The idea of parity sheaves resembles the definition of perverse sheaves [\textit{A. A. Beilinson} et al., Astérisque 100, 172 p. (1982; Zbl 0536.14011)]. For a given stratification of a complex algebraic variety \(X=\bigsqcup_{\lambda\in\Lambda}X_\lambda\) one considers pariversity, i.e., a function \(\Lambda\to\mathbb Z/2\). Let \(k\) be a complete local principal ideal domain. For a complex of sheaves of \(k\)-modules \(\mathcal F\) , or rather an element of \(D(X)\), the bounded constructible derived category of \(k\)-sheaves and a fixed pariversity there is defined a condition via vanishing of stalk and costalk cohomology of \(\mathcal F\) along strata. The equivariant context when a reductive group \(G\) is acting on \(X\) is also treated in parallel. Here in contrast to perversity condition only parity of the degree matters. The sheaves satisfying suitable conditions are called parity sheaves. The effects related to the torsion cohomology in this context might be better traced. For example, the Decomposition Theorem for a proper push forward is proven for a preferred pariversity. On the other hand parity sheaves with a given support might not exist in general. They do exist in many cases of varieties coming from the representation theory (Schubert varieties, nilpotent cones), also for toric varieties. pairity sheaves; pervers sheaves; decomposition theorem; flag varieties D. Juteau, C. Mautner and G. Williamson, Parity sheaves. \textit{J. Amer. Math. Soc. }27 (2014), no. 4, 1169--1212.MR 3230821 Zbl 06591557 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Intersection homology and cohomology in algebraic topology, Modular representations and characters, Grassmannians, Schubert varieties, flag manifolds Parity sheaves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 nonsingular curve on a rational surface \(S\). In the case when the logarithmic 2 genus of \(C\) is equal to two, Iitaka proved that the geometric genus of \(C\) is either zero or one and classified such pairs \((S,C)\).
In this article, we prove the existence of these classes with geometric genus one in Iitaka's classification. The curve in the class is a singular curve on \(\mathbb{P}^2\) or the Hirzebruch surface \(\Sigma_d\) and its singularities are not in general position. For this purpose, we provide the arrangement of singular points by considering invariant curves under a certain automorphism of \(\Sigma_d\). plane curve; rational surface; double cover Rational and birational maps, Special algebraic curves and curves of low genus, Rational and ruled surfaces, Coverings in algebraic geometry On classes in the classification of curves on rational surfaces with respect to logarithmic plurigenera | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this article we will present some fundamental results concerning finite étale coverings of the unit disc \(\mathbb{D}\), respectively of the annulus \(A(r,R)\), over a \(p\)-adic field \(K\) which imply the analogon of Riemann's existence theorem in the \(p\)-adic case. \(p\)-adic field; étale coverings; Riemann's existence theorem Local ground fields in algebraic geometry, Coverings in algebraic geometry, Non-Archimedean function theory Étale coverings of \(p\)-adic discs | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We study the contribution of multiple covers of an irreducible rational curve \(C\) in a Calabi-Yau threefold \(Y\) to the genus 0 Gromov-Witten invariants in the following cases.
(1) If the curve \(C\) has one node and satisfies a certain genericity condition, we prove that the contribution of multiple covers of degree \(d\) is given by \(\sum_{n| d} \frac{1}{n^3}\).
(2) For a smoothly embedded contractable curve \(C\subset Y\) we define schemes \(C_i\) for \(1\leq i\leq l\) where \(C_i\) is supported on \(C\) and has multiplicity \(i\), the number \(l\in\{1,\dots,6\}\) being Kollar's invariant ``length''. We prove that the contribution of multiple covers of \(C\) of degree \(d\) is given by \(\sum_{n | d}\frac{k_{d/n}} {n^3}\), where \(k_i\) is the multiplicity of \(C_i\) in its Hilbert scheme (and \(k_i=0\) if \(i>l)\). In the latter case we also get a formula for arbitrary genus.
These results show that the curve \(C\) contributes an integer amount to the so-called instanton numbers, which are defined recursively in terms of the Gromov-Witten invariants and are conjectured to be integers. multiple covers of an irreducible rational curve; Calabi-Yan threefold; Gromov-Witten invariants; instanton numbers Bryan J., Katz S., Leung N.C.: Multiple covers and the integrality conjecture for rational curves in Calabi--Yau threefolds. J. Algebraic Geom. 10, 549 (2001) arXiv:math/9911056 Calabi-Yau manifolds (algebro-geometric aspects), Coverings in algebraic geometry, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Plane and space curves, \(3\)-folds, Coverings of curves, fundamental group Multiple covers and the integrality conjecture for rational curves in Calabi-Yau threefolds | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For the entire collection see Zbl 0626.00011.]
These lectures have as aim: to discuss some recent work related to the homotopy groups \(\pi_ i\) of algebraic varieties, with particular attention to \(\pi_ 0\) (connectivity) and \(\pi_ 1\) (fundamental groups):
I. Introduction. An attempt to unify classical theorems of Bézout, Bertini, Lefschetz, Zariski, and Barth leads to a general connectedness principle, which provides the theme of these lectures.
II. Morse theory on analytic spaces: A bit of the history, and an introduction to the work of \textit{M. Goresky} and \textit{R.MacPherson} [``Stratified Morse theory'' (1988; Zbl 0639.14012)].
III. Lefschetz and connectedness theorems: From this Morse theory one proves a version of Lefschetz' theorem, from which a general connectedness theorem follows.
IV. \(\pi_ 0\) and varieties of small codimension: Some remarkable applications by Zak and others to problems in classical projective geometry.
V. \(\pi_ 1\) and branched coverings: Applications to fundamental groups and branched coverings of projective space, with an introduction to the work of \textit{M. V. Nori} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 16, 305-344 (1983; Zbl 0527.14016)].
Some of this work has been discussed in other survey lectures. One purpose of these talks is to bring the survey by the author and \textit{R. Lazarsfeld} in Algebraic geometry, Proc. Conf., Chicago Circle 1980, Lect. Notes Math. 862, 26-91 (1981; Zbl 0484.14005) up to date. The bibliography includes many papers and recent preprints with related and overlapping results. Bibliography; homotopy groups; connectivity; fundamental groups; Morse theory; Lefschetz' theorem; varieties of small codimension; branched coverings [12] Fulton (W.).-- On the topology of algebraic varieties, Proc. Symp. Pure Math. 46, Amer. Math. Soc., Providence, RI, p. 15-46 (1987). &MR 9 | &Zbl 0703. Topological properties in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry, Coverings in algebraic geometry, Low codimension problems in algebraic geometry, Coverings of curves, fundamental group On the topology of algebraic varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The main result of this paper classifies all group actions on cycles covers which ramify over three distinct points of \({\mathbb P}^{1}.\) These covers are given by
\[
C : y^{n} = x^{a} (x - 1)^{b} (x + 1)^{c},
\]
where \(1 \leq a, b, c \leq n - 1, a + b + c \equiv 0 (mod n),\) as cyclic Belyi covers or generalized Lefschetz curves. \textbf{Theorem 1.} Let \(C : y^{n} = x^{a} (x - 1)^{b} (x + 1)^{c}\) be a cyclic \({\mathbb Z}_{n}\) Galois cover of the line with \(1 \leq a, b, c \leq n - 1, a + b + c \equiv 0 (mod n),\) and \([n, a, b, c] = 1.\) We fix \(n \geq 4\) and let \(G\) be the full automorphism group of \(C.\) Then \(G\) is completely determined by the values of \(n, a, b, c.\) \textbf{Theorem 2.} Let \(C : y^{p} = x^{a} (x + 1)\) be a Lefschetz curve and let \(G : = Aut (C).\) Then : (1) if \(a = 1, G\) is cyclic of order \(2p;\) (2) if \(p = 7\) and \(a = 2,\) then \(G = PSL(2, 7)\) is the simple group of order 168; (3) if \(p \equiv 1 (mod 3), p > 7\) and \(1 + a + a^{2} \equiv 0 (mod p),\) then \(G\) is the unique non-abelian group of order \(3p;\) (4) for all other cases, \(Aut (C) = {Z}_{p}.\) \textbf{Theorem 4.} Let \(C : y^{p} = f(x),\) where \(p\) is a prime and \(f(x)\) is a polynomial with \(r\) distinct roots, \(r > 2p.\) Then the automorphism group of \(C\) is an extension of \({\mathbb Z}_{p}\) by a polyhedral group. group actions on cycles covers; Belyi covers; generalized Lefschetz curves; full automorphism group; extension of \({\mathbb Z}_{p}\) S. Kallel and D. Sjerve, On the group of automorphisms of cyclic covers of the Riemann sphere, Mathematical Proceedings of the Cambridge Philosophical Society 138 (2005), 267--287. Compact Riemann surfaces and uniformization, Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization), Coverings in algebraic geometry, Algebraic functions and function fields in algebraic geometry, Automorphisms of curves On the group of automorphisms of cyclic covers of the Riemann sphere | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper, we approach the study of modules of constant Jordan type and equal images modules over elementary abelian \(p\)-groups \(E_r\) of rank \(r\geq 2\) by exploiting a functor from the module category of a generalized Beilinson algebra \(B(n,r)\), \(n\leq p\), to \(\text{mod\,}E_r\).
We define analogues of the above-mentioned properties in \(\text{mod\,}B(n,r)\) and give a homological characterization of the resulting subcategories via a \(\mathbb P^{r-1}\)-family of \(B(n,r)\)-modules of projective dimension 1. This enables us to apply general methods from Auslander-Reiten theory and thereby arrive at results that, in particular, contrast the findings for equal images modules of Loewy length 2 over \(E_2\) by \textit{J. F. Carlson, E. M. Friedlander} and \textit{A. Suslin} [Comment. Math. Helv. 86, No. 3, 609-657 (2011; Zbl 1229.20039)] with the case \(r>2\). Moreover, we give a generalization of the \(W\)-modules introduced by the aforementioned authors. categories of modules; generalized Beilinson algebras; group schemes; modules of constant Jordan type; \(W\)-modules; equal images property; modular representations Julia Worch, Categories of modules for elementary abelian \?-groups and generalized Beilinson algebras, J. Lond. Math. Soc. (2) 88 (2013), no. 3, 649 -- 668. Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers, Representations of quivers and partially ordered sets, Group rings of finite groups and their modules (group-theoretic aspects), Modular representations and characters, Group schemes Categories of modules for elementary abelian \(p\)-groups and generalized Beilinson algebras. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In the paper under review, the authors give a characterization of the degree \(d\) of the canonical \(\varphi_{K_S}\) of a regular surface of general type \(S\), in the case when \(\varphi_{K_S}\) is an abelian cover of \(\mathbb{P}^2\).
In particular, they are able to prove that \(d=2,3,4.6,8,9,16\). In addition, they give the equation of the found surfaces. Finally, the give the configuration of the branch divisor explicitly for the case \(d=16\). canonical map; abelian cover Du, R; Gao, Y, Canonical maps of surfaces defined on abelian covers, Asian J. Math., 18, 219-228, (2014) Surfaces of general type, Coverings in algebraic geometry Canonical maps of surfaces defined by 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 [For the entire collection see Zbl 0516.00012.]
The author studies a ''residue homomorphism'' res: \(\hat K_{q+1}(\hat M)\to \hat K_ q(k)\), where \(\hat K_*\) denotes a certain completion of Milnor's higher K-groups, k is a complete discrete valuation field, and \(\hat M\) is the field of series \(\sum_{n\in {\mathbb{Z}}}a_ n X^ n\) over k with \(ord_ k(a_ n)\) bounded below, and \(\lim_{n\to -\infty}a_ n=0.\)
The paper extends work of \textit{J.-L. Brylinski} [Ann. Inst. Fourier 33, No.3, 23-38 (1983; Zbl 0524.12008)], and provides a simple interpretation for the p-primary part of the reciprocity map \(K_ n(K)\to Gal(K^{ab}/K)\) used in the higher-dimensional class field theory of the author. Here K, a ''local field of dimension n'', is assumed to be of characteristic p. Some of the proofs make use of similar results for the Quillen K-groups; the residue map comes from a boundary map in the exact sequence for a localization. ''residue homomorphism''; completion of Milnor's higher K-groups; reciprocity map; higher-dimensional class field theory K. Kato : Residue homomorphisms in Milnor K-theory . Adv. St. Pure Math. 2 (1983) 153-172. \(K\)-theory of local fields, Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects), Class field theory; \(p\)-adic formal groups, Coverings in algebraic geometry Residue homomorphisms in Milnor K-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 A Weierstrass polynomial of degree \(n\geq 1\) over a topological space X is a polynomial function \(P: X\times {\mathbb{C}}\to {\mathbb{C}}\) of the form \(P(x,z)=z^ n+\sum^{n}_{i=1}a_ i(x)z^{n-i}\), where \(a_ 1,...,a_ n: X\to {\mathbb{C}}\) are continuous, complex valued functions. If P(x,z) has no multiple roots for any \(x\in X\), we call it a ``simple Weierstrass polynomial.'' We associate to a simple Weierstrass polynomial P(x,z) of degree n, an n-fold polynomial covering map \(\pi: E\to X,\) where \(E\subset X\times {\mathbb{C}}\) is the zero set for P(x,z) and \(\pi\) is the projection onto X. We say that two simple Weierstrass polynomials over X are ``topological equivalent'' if their associated polynomial covering maps are equivalent.
The first statement is that any simple Weierstrass polynomial over X is topologically equivalent to a simple Weierstrass polynomial of the form \(P(x,z)=z^ n+\sum^{n}_{i=1}\tilde a_ i(x)z^{n-i}\) with \(\tilde a_ 1(x)=0\) for all \(x\in X\). The most elementary type of Weierstrass polynomials over X are those of the form \(P(x,z)=z^ n-q(x)\) \(n\geq 1\), where \(q: X\to {\mathbb{C}}\) is a continuous function. By the above result, any simple Weierstrass polynomial of degree 2 over X is topologically equivalent to the above form.
In this paper the author tries to simplify a Weierstrass polynomial of the above form within its topological equivalence class. The statement is that a simple Weierstrass polynomial over X of the form \(P(x,z)=z^ n- q(x)\) is topologically equivalent to its discriminant radical. Here, the discriminant radical of P(x,z) is a Weierstrass polynomial defined by the discriminant function of P(x,z).
The author also studies topological structures on the associated polynomial covering maps. Weierstrass polynomial over a topological space; simple Weierstrass polynomial; polynomial covering map; discriminant radical V.L. Hansen, Algebra and topology of Weierstrass polynomials, Expositiones Mathematicae, to appear. Covering spaces and low-dimensional topology, Coverings in algebraic geometry, Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral) Algebra and topology of Weierstrass polynomials | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 addresses the unstable case in the GIT approach to the moduli of rank-2 tensors on a smooth projective variety \(X\), i.e.~ pairs \((E, \phi : E^{\otimes s} \to M)\) where \(E\) is a coherent torsion-free sheaf of rank \(2\) on \(X\) and \(M\) is a line bundle on \(X.\)
In this setting, a Harder-Narasimhan filtration of an unstable pair cannot be defined naïvely, as one cannot speak of quotient objects. \textit{T. Gómez} et al. proved in [Rev. Mat. Complut. 28, No. 1, 169--190 (2015; Zbl 1327.14059)] that the Harder-Narasimhan filtration of an unstable coherent sheaf \(E\) coincides with a filtration induced by a 1-parameter subgroup that maximally destabilizes \(E\); the latter arises from work of \textit{G. R. Kempf} [Ann. Math. (2) 108, 299--316 (1978; Zbl 0406.14031)] and is therefore called the \textit{Kempf filtration}. The main result of the paper under review (Theorem 4.2) ensures a well-defined Kempf filtration for rank-2 tensors, which in turn is used to define a reasonable Harder-Narasimhan filtration (Definition 7.3).
For sufficiently large \(m,\) there is a ``maximally unstable'' flag of subspaces of \(H^{0}(E(m))\) whose images under the evaluation map \(H^{0}(E(m)) \otimes \mathcal{O}(-m) \to E\) together with restrictions of \(\phi\) form the so-called \textit{\(m\)-Kempf filtration} of the pair \((E,\phi).\) Theorem 4.2 produces an \(m'\) such that the \(m\)- and \(m'\)-Kempf filtrations coincide for \(m \geq m'\); the Kempf filtration of \((E,\phi)\) can then be defined as its \(m\)-Kempf filtration for \(m \gg 0.\)
As an application of the previous results, the author gives a characterization (Theorem 8.6) of an unstable rank-2 tensor over a smooth projective curve in terms of the intersection theory of the associated ruled surface. Harder-Narasimhan filtration; geometric invariant theory; tensors; curve coverings; moduli space Zamora, A.: Harder-Narasimhan filtration for rank 2 tensors and stable coverings. arXiv:1306.5651, (2013) (submitted preprint) Algebraic moduli problems, moduli of vector bundles, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Geometric invariant theory, Coverings in algebraic geometry Harder-Narasimhan filtration for rank 2 tensors and stable coverings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author investigates the generalized Jacobian conjecture, i.e. whether an étale endomorphism \(\varphi\) of an algebraic variety \(X\) defined over an algebraically closed field of characteristic zero is an automorphism.
This generalized problem was first posted by \textit{M. Miyanishi} [Osaka J. Math. 22, No. 2, 345--364 (1985; Zbl 0614.14006)] who gave counterexamples and positive answers for certain varieties. Notably, these results seemed to underline the special role of \(\mathbb{C}^n\).
In the paper under review, the author treats the case of complements of affine plane curves
\(X={\mathbb{A}}^2\setminus C\) and shows that the generalized Jacobian conjecture is true except for
\(C=\{y^m-x^n =0\}\) with \(\gcd(m,n)=1\). In an appendix (written together with M. Miyanishi) it is proved that
for such curves there exist étale endomorphisms of arbirarily large degree. generalized Jacobian conjecture Aoki, Hisayo, Étale endomorphisms of smooth affine surfaces, J. Algebra, 226, 1, 15-52, (2000) Jacobian problem, Automorphisms of surfaces and higher-dimensional varieties, Coverings in algebraic geometry Étale endomorphisms of smooth affine surfaces. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper under review is a nice exposition on problems in affine algebraic geometry. The paper is based on lectures given in Bochum and Toulouse. Some problems in affine algebraic geometry are notoriously hard. The author gives an overview of the problems of affine space in the first section. Some famous open problems are the Zariski cancellation problem, the Jacobian conjecture, the linearization problem for the multiplicative group (Is every action of \(\mathbb{C}^\star\) on \(\mathbb{C}^n\) a linear one up to conjugation with an automorphism?), the embedding problem (Is every embedding of \(\mathbb{C}^k\) into \(\mathbb{C}^n\) equivalent to a linear embedding using the automorphism group of \(\mathbb{C}^n\)?). All these problems are related to the fact that the automorphism group of \(\mathbb{C}^n\) is complicated. For example, it is still unknown for \(n\geq 3\) whether every automorphism is a composition of certain elementary automorphisms. In the second section acyclic surfaces are studied. An important tool, the logarithmic Kodaira dimension, is discussed. One result is Ramanujam's theorem that a smooth acyclic surface that is simply connected at infinity must be isomorphic to \(\mathbb{C}^2\). Another result by Miyanishi, Sugie and Fujita states that an acyclic surface with logarithmic Kodaira dimension \(-\infty\) must be isomorphic to \(\mathbb{C}^2\).
In the third section exotic product structures are discussed. The Zariski cancellation problem asks whether \(X\times \mathbb{C}^k\cong \mathbb{C}^{n+k}\) implies \(X\cong \mathbb{C}^n\). Using results of the previous section one can show that the cancellation problem is true for \(n\leq 2\). The Zariski cancellation problem is open for \(n\geq 3\). A slightly more general problem is known to be false. Danielewski gave affine varieties \(X\) and \(Y\) such that \(X\times \mathbb{C}\cong Y\times \mathbb{C}\) but \(X\) and \(Y\) are not isomorphic. Also, a birational version of the cancellation problem is known to be false as well. If \(X\times \mathbb{C}^k\cong \mathbb{C}^{n+k}\) then \(X\) is contractible. It is therefore interesting to study contractible smooth affine varieties. Dimca and Ramanujam showed that a contractible smooth affine variety of (complex) dimension \(n\geq 3\) is always diffeomorphic to \(\mathbb{R}^{2n}\). Some constructions of nontrivial contractible affine varieties are discussed. There are now many examples of contractible affine varieties. It is not known for all examples whether they are isomorphic to \(\mathbb{C}^n\) or not. For some examples it can be shown that they are diffeomorphic to \(\mathbb{R}^{2n}\), but not isomorphic to \(\mathbb{C}^n\). Such an affine variety then gives an exotic complex algebraic structure on \(\mathbb{R}^{2n}\). Exotic algebraic structures also show up in the linearization problem of the multiplicative group. In order to prove that \(\mathbb{C}^\star\) actions on \(\mathbb{C}^3\) are linearizable, Koras and Russell constructed three dimensional contractible affine varieties with a \(\mathbb{C}^\star\) action. If those three dimensional varieties would be isomorphic to \(\mathbb{C}^3\) then they would provide counterexamples to the linearization problem. It was also shown that these varieties are the only possible counterexamples. Fortunately Makar-Limanov came up with a method of proving that the varieties in question are not isomorphic to \(\mathbb{C}^3\) using locally nilpotent derivations on the coordinate ring. These methods have been succesfully applied to all possible counterexamples, thus proving that \(\mathbb{C}^\star\) actions on \(\mathbb{C}^3\) are always linearizable. This is an excellent overview with a large bibliography. exotic structures; affine space; cancellation problem; embedding problem; jacobian conjecture; linearization problem; coverings; logarithmic Kodaira dimension; automorphism group Serre, J.-P.: Sur les modules projectifs. In: Séminaire Dubreil-Pisot (1960/61), vol. 14, Exposé 2. Sectrétariat Mathématique, Paris (1963) Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem), Group actions on affine varieties, Group actions on varieties or schemes (quotients), Coverings in algebraic geometry Exotic algebraic structures on affine 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 A Kummer cover is a map \(\pi_n: \mathbb P^2 \to \mathbb P^2\) given by \((x:y:z) \mapsto (x^n:y^n:z^n)\), for a choice of linear coordinates \((x,y,z)\) on the complex projective plane \(\mathbb P^2\). If \(D\) is a plane curve in \(\mathbb P^2\) and \(D_n=\pi_n^{-1}(D)\) is its Kummer cover, it is a natural question to relate the topological invariants of the complements of \(D\) and \(D_n\), e.g. fundamental groups, Alexander polynomials and so on. In this paper the authors concentrate on the very powerful invariant given by the generic braid monodromy. A number of interesting examples are treated with full details, including some classical line arrangements. Kummer covers; braid monodromy; plane curves Artal, E., Cogolludo-Agustín, J., Ortigas-Galindo, J.: Kummer covers and braid monodromy. J. Inst. Math. Jussieu 13 (3), 633--670 (2014). http://dx.doi.org/10.1017/S1474748013000297 Coverings of curves, fundamental group, Plane and space curves, Low-dimensional topology of special (e.g., branched) coverings, Coverings in algebraic geometry, Topological properties in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry Kummer covers and braid monodromy | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 reduced connected scheme and \(\mathcal{F}\) a vector bundle of rank \(m-1\) on \(X\). A multiple structure or more precisely a \(m\)-rope on \(X\) with conormal bundle \(\mathcal{F}\) is a scheme \(\overset{\sim}{X}\) with \(\overset{\sim}{X}_{red}=X\) such that \(\mathcal{I}^{2}_{X/\overset{\sim}{X}}=0\) and \(\mathcal{I}_{X/\overset{\sim}{X}}=\mathcal{F}\) as \(\mathcal{O}_{X}\)-modules, where \(\mathcal{I}_{X/\overset{\sim}{X}}\) is the ideal sheaf of the embedding of \(X\) in \( \overset{\sim}{X}\). A smoothing of multiple structures \(D\) is a flat family of schemes over a curve such that the central fibre is \(D\). It has been related to the deformations of morphisms in [\textit{F. J. Gallego} et al., Rev. Mat. Complut. 26, No. 1, 253--269 (2013; Zbl 1312.14009)] where Gallego et al. give a sufficient condition for a finite morphism to be deformed to an embedding. \newline In this paper the authors investigate the deformations of the composite of the universal covering map for an Enriques manifold \(Y\) of index \(d\) with an embedding of \(Y\) in a projective space \(\mathbb{P}^{N}\) using the infinitesimal condition described by Gallego et al.. The definition of an Enriques manifold used corresponds either to the one in the sense of [\textit{S. Boissière} et al., J. Math. Pures Appl. (9) 95, No. 5, 553--563 (2011; Zbl 1215.14046)] or [\textit{K. Oguiso} and \textit{S. Schröer}, J. Reine Angew. Math. 661, 215--235 (2011; Zbl 1272.14026)] depending on the index \(d\) and whether the universal cover of \(Y\) is a Calabi-Yau or a hyperkähler manifold. Given an embedding of an Enriques manifold \(Y\) of index \(d\) and dimension \(2n\) with canonical bundle \(K_{Y}\) in a large enough projective space, embedded \(d\)-ropes with conormal bundle \(\mathcal{E}=\displaystyle\bigoplus_{i=1}^{d-1}K_{Y}^{\otimes i}\) on \(Y\) are thoroughly studied and the dimension of the quasi-projective space parametrizing such \(d\)-ropes is computed. It turns out that \(d\)-ropes with conormal bundle \(\mathcal{E}\) appear naturally on Enriques manifolds of index \(d\) as flat limits of hyperkähler or Calabi-Yau manifolds. The main results on smoothing of such multiple structures are proven with the deformation results obtained in [\textit{F. J. Gallego} et al., Rev. Mat. Complut. 26, No. 1, 253--269 (2013; Zbl 1312.14009)]. These \(d\)-ropes on Enriques manifold are points in the Hilbert scheme of embedded Calabi-Yau or hyperkähler manifolds whose smoothness is proven for ropes with \(d=2\) as a consequence of smoothing. Moreover, in the Appendix the authors show that if an Enriques manifold \(Y\) of index \(d= 2\) whose universal cover is one of the known examples of hyperkähler six-folds is embedded inside \(\mathbb{P}^{N}\) then \(N\geq 13\).
Reviewer's remark: Small typos on the pages 1250 and 1253: ``will will'' should be ``we will'', and ``upto'' should be ``up to'' respectively. ropes; Enriques manifolds; hyper-Kähler manifolds; Calabi-Yau manifolds; smoothing of multiple structures; Hilbert scheme Holomorphic symplectic varieties, hyper-Kähler varieties, Calabi-Yau manifolds (algebro-geometric aspects), Parametrization (Chow and Hilbert schemes), Fibrations, degenerations in algebraic geometry, Formal methods and deformations in algebraic geometry, Coverings in algebraic geometry, Hyper-Kähler and quaternionic Kähler geometry, ``special'' geometry Smoothing of multiple structures on embedded Enriques manifolds | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The object of this paper is the classification of pairs \((X,L)\), where \(X\) is a smooth projective variety of dimension \(n\geq 4\) and \(L\) is a very ample (or ample) divisor, such that one of the smooth elements of the linear system \(|L|\) is a double cover of a smooth quadric. This problem may be regarded as a generalization of the classical question of which smooth projective surfaces admit a hyperelliptic curve as a hyperplane section, solved by \textit{A. J. Sommese} and \textit{A. Van de Ven} [Math. Ann. 278, 593-603 (1987; Zbl 0655.14001)] and by \textit{F. Serrano} [J. Reine Angew. Math. 381, 90-109 (1987; Zbl 0618.14001)].
The author solves the problem completely, and moreover he generalizes along the same lines an earlier result of his, namely the classification of pairs \((X,L)\), with \(L\) ample, such that one of the smooth elements of \(|L|\) is a double cover of \(\mathbb{P}^n\). The proof rests on Fujita's classification of varieties according to \(\Delta\)-genus. very ample divisor; \(\Delta\)-genus; double cover of a smooth quadric Lanteri, Double Covers of Smooth Hyperquadrics as Ample and Very Ample Divisors, Abh. Math. Sem. Univ. Hamburg 64 pp 97-- (1994) Divisors, linear systems, invertible sheaves, Coverings in algebraic geometry Double covers of smooth hyperquadrics as ample and very ample divisors | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The notion of support is a fundamental concept which provides a geometric approach for studying various algebraic structures. The prototype for this has been Quillen's description of the algebraic variety corresponding to the cohomology ring of a finite group, based on which Carlson introduced support varieties for modular representations. This has made it possible to apply methods of algebraic geometry to obtain representation theoretic information. Their work has inspired the development of analogous theories in various contexts, notably modules over commutative complete intersection rings, and over cocommutative Hopf algebras. The aim of this workshop has been to bring together experts from these fields and to stimulate interaction and exchange of ideas. Proceedings, conferences, collections, etc. pertaining to group theory, Proceedings, conferences, collections, etc. pertaining to associative rings and algebras, Collections of abstracts of lectures, Modular representations and characters, Homological methods in associative algebras, Homological methods in commutative ring theory, Homological algebra in category theory, derived categories and functors, Relevant commutative algebra, Varieties and morphisms, Representation theory of associative rings and algebras, Chain conditions, growth conditions, and other forms of finiteness for associative rings and algebras, Homological methods in Lie (super)algebras, Cohomology of groups Mini-workshop: Support varieties. Abstracts from the mini-workshop held February 15th -- February 21st, 2009. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general 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 \(G\) an infinitesimal group scheme over a field \(k\) of characteristic \(p>0\), the computation of its cohomological support variety is ``of considerable geometric complexity'', thereby making the study of its representations quite difficult. However, there is much value in being able to study these representations since, for example, the representations of the Frobenius kernels of a smooth connected algebraic group is equivalent to its rational representation theory. In the work under review, the authors consider \(1\)-parameter subgroups of \(G\), allowing for a new way to study representations that may be modified for non-infinitesimal groups.
Let \(k[G]\) denote the coordinate algebra of \(G\), and \(kG\) the linear dual to \(k[G]\). For each \(r>0\), the scheme of maps \(\mathbb G_{a(r)}\to G\), where \(\mathbb G_{a(r)}\) is the \(r\)-th Frobenius kernel of the additive group scheme, is denoted \(V_r(G)\) and is affine. The direct limit (under the closed immersion \(V_r(G)\hookrightarrow V_{r'}(G)\), \(r<r'\)), denoted \(V(G)\), is the scheme which parameterizes \(1\)-parameter subgroups of \(G\). If the height of \(G\) is bounded by \(r\), then \(V(G)\) is naturally identified with \(V_r(G)\). The primary construction in this paper is a global \(p\)-nilpotent \(k\)-linear map \(\Theta_G\colon k[G]\to k[V(G)]\). This operator, when viewed as an element of \(kG\otimes k[V(G)]\), describes all \(1\)-parameter subgroups of \(G\).
For any \(r\) we let \(\Theta_{G,r}\colon k[G]\to k[V_r(G)]\) be given as a natural composition \(k[G]\to k[V_r(G)]\otimes k[G]\to k[V_r(G)]\otimes k[\mathbb G_{a(r)}]\to k[V_r(G)]\). Then \(\Theta_G\) is the limit of the \(\Theta_{G,r}\), and is naturally identified with \(\Theta_{G,r}\) whenever \(r\) is at least the height of \(G\). One can then get an induced map \(\Theta_M\colon M\otimes k[G]\to M\otimes k[V(G)]\), where \(M\) is a \(k[G]\)-comodule. This operator allows for an effective way to investigate the local Jordan type of \(M\) when \(M\) is a finite-dimensional \(kG\)-module. Applications are given in which the rank and dimension of \(kG\) modules of constant Jordan type are bounded.
Readers of previous works of the authors, [e.g. Duke Math. J. 139, No. 2, 317-368 (2007; Zbl 1128.20031)], will be familiar with the concept of \(\pi\)-points for \(G\) a finite group scheme, useful in investigating the Jordan type of a finite \(kG\) module \(M\). Here, connections are made between \(1\)-parameter subgroups (when \(G\) is infinitesimal) and \(\pi\)-points. It is shown that a projectivization \(\widetilde\Theta_G\) of \(\Theta_G\) gives a condition for when \(M\) has constant \(j\)-rank: namely, if and only if the coherent sheaf \(\widetilde\Theta_G^j\) is locally free.
A useful feature of this paper is the large number of explicit examples, drawn from some very familiar infinitesimal groups. The groups under consideration are \(p\)-restricted Lie algebras, the \(r\)-th Frobenius kernel of \(\mathbb G_a\), the \(r\)-th Frobenius kernel of \(\text{GL}_n\), and the second Frobenius kernel of \(\text{SL}_2\). Rather than do all of the calculations at the end, the authors choose to develop these examples as each new concept is introduced. (Also submitted to MR.) infinitesimal group schemes; nilpotent operators; Jordan types; pi-points; support varieties; coordinate algebras; finite-dimensional modules; cohomology rings; Frobenius kernels Friedlander E. M. and Pevtsova J., Constructions for infinitesimal group schemes, Trans. Amer. Math. Soc. 363 (2011), no. 11, 6007-6061. Representation theory for linear algebraic groups, Group schemes, Cohomology theory for linear algebraic groups, Modular representations and characters, Representations of associative Artinian rings, Linear algebraic groups over finite fields Constructions for infinitesimal group schemes. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We mainly deal with the generic behaviour of Cartan invariants for finite groups of Lie type in this paper. Following \textit{L. Chastkofsky}'s paper [J. Algebra 103, 466-478 (1986; Zbl 0605.20046)], we describe the generic Cartan invariants for finite groups of Lie type by using the representation theory of Frobenius kernels, and show their symmetry properties. Moreover, we specialize to the case \(n=1\), and clarify the transition from \(n=1\) to general n. Finally, we explain the connection between Cartan invariants and generic Cartan invariants.
An abridged version of this paper was included in the proceedings of a conference in honor of L. K. Hua [Contemp. Math. 82, 235-241 (1989; Zbl 0673.20005)]. \textit{J. E. Humphrey}'s paper [J. Algebra 122, 345-352 (1989; Zbl 0674.20023)] is closer in spirit to the author's paper than to Chastkofsky's paper, but avoids the complicated calculations. He offers a more conceptual treatment, emphasizing the much simpler picture for the group schemes \(G_ nT\) associated with the Frobenius kernels \(G_ n\) in the ambient algebraic group. finite groups of Lie type; generic Cartan invariants; Frobenius kernels; group schemes Representation theory for linear algebraic groups, Modular representations and characters, Linear algebraic groups over finite fields, Representations of finite symmetric groups, Simple groups: alternating groups and groups of Lie type, Group schemes Cartan invariants of finite groups of Lie type. 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 Professor George Janelidze posed the problem whether the study of classical topological coverings via categorical Galois Theory could be extended to the setting of relational algebras (see, e.g., [\textit{G. Dzhanelidze}, J. Algebra 132, No. 2, 270--286 (1990; Zbl 0702.18006)]. It is the purpose of this paper to give a first contribution to the investigation of coverings in the realm of relational algebras. In particular the authors obtain new characterizations for effective descent maps in the categories of \(M\)-ordered sets, for a given monoid \(M,\) and of multi-ordered sets. relational algebra; effective descent morphism; Galois theory; covering; connected component Clementino, MM; Hofmann, D; Montoli, A, Covering morphisms in categories of relational algebras, appl, Categ. Structures, 22, 767-788, (2014) Categories of topological spaces and continuous mappings [See also 54-XX], Connected and locally connected spaces (general aspects), Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.), Coverings in algebraic geometry, Special maps on topological spaces (open, closed, perfect, etc.) Covering morphisms in categories of relational algebras | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(Y\to M\) be an infinite covering space of an \(n\)-dimensional projective manifold, \(n\geq 2.\) Suppose \(M\) is embedded into a projective space by sections of a very ample line bundle \(L.\)
The author shows that if \(L\) is sufficiently ample, then the restriction map \(\cdot|X\) is an isomorphism of the Hilbert spaces of holomorphic functions of slow growth. Using the isomorphism theorem, new examples of Riemann surfaces and domains of holomorphy in \(\mathbb C^n\) with corona are constructed. These easily defined surfaces have many symmetries, the characters in the corona have a simple description, and the corona is large in the sense that it contains a domain in euclidean space of arbitrarily high dimension. The results of Hörmander on generating algebras of holomorphic functions of exponential growth to the case of covering spaces are adapted. Consequently, the restriction map \(\cdot|X\) may fail to be an isomorphism if \(L\) is not sufficiently ample compared to the exhaustion.
A sampling and interpolation theorem for the weighted Bergman spaces on the unit ball \(\mathbb B\subset\mathbb C^n\) is obtained. For each weight, the restriction to \(X\) induces an isomorphism from the weighted Bergman space on \(\mathbb B\) to the one on \(X\) if \(L\) is sufficiently ample. Also, every bounded holomorphic function on \(X\) extends to a unique function of infraexponential growth on \(\mathbb B,\) i.e., a function in the intersection of all the nontrivial weighted Bergman spaces on \(\mathbb B,\) when \(L\) is sufficiently ample, for instance when \(L\) is the \(m\)-th tensor power of the canonical bundle \(K\) with \(m\geq 2.\) Assuming that the covering group is an arithmetic subgroup of the automorphism group \(PU(1,n)\) of \(\mathbb B,\) two dichotomies concerning the extension of bounded holomorphic functions from \(X\) to \(\mathbb B\) are established. One of them says that either every holomorphic function \(f\) continuous up to the boundary on the preimage of a \(K^{\otimes m}\)-curve in a finite covering of \(M\) extends to a continuous function on \(\overline{\mathbb B}\) which is holomorphic on \(\mathbb B,\) or the boundary functions of such functions \(f\) generate a dense subspace of the space of continuous functions \(\partial\mathbb B\to\mathbb C.\) An analogous dichotomy for harmonic functions is obtained. In contrast to the case of holomorphic functions, a harmonic function on \(X,\) continuous up to the boundary, generally does not extend to a plurisubharmonic function bounded above on \(\mathbb B.\) covering spaces of projective manifolds; ample line bundle; holomorphic functions of slow growth; Riemann surfaces with corona; generating Hörmander algebras on covering spaces; dichotomy; covering group; harmonic functions; weighted Bergman spaces Finnur Lárusson, Holomorphic functions of slow growth on nested covering spaces of compact manifolds, Canad. J. Math. 52 (2000), no. 5, 982 -- 998. Holomorphic functions of several complex variables, Coverings in algebraic geometry, Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects), Riemann surfaces, Continuation of analytic objects in several complex variables Holomorphic functions of slow growth on nested covering spaces of compact manifolds | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author proves here some statements concerning the structure of an abelian cover f: \(Y\to X\), with X smooth and Y normal. He proves that if the branch divisor B of f is irreducible and the cover is totally ramified, then f is a simple cyclic cover, namely it can be embedded in the total space of a line bundle L such that rL is linearly equivalent to B, r being the degree of the cover.
Specializing to the case in which X is an abelian variety and Y is of general type the author computes the cohomology of the structure sheaf on Y. - Finally he studies the singularities of a cyclic cover of \({\mathbb{P}}^ 2\) branched on the union of two smooth curves.
For more general results of the same type see also ``Abelian covers of algebraic varieties'' by the reviewer [J. Reine Angew. Math. 417, 191-213 (1991; Zbl 0721.14009)]. branch divisor; simple cyclic cover Tokunaga, H.: On a cyclic covering of a projective manifold. J. math. Kyoto univ. 30, 109-121 (1990) Coverings in algebraic geometry On a cyclic covering of a projective manifold | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems If \(Y\) is a normal variety and \(B\) is a finite union of irreducible hypersurfaces on \(Y\), then the canonical group homomorphism induced by the Hurewicz map gives a sequence of coverings branched along \(B\). The function which gives the first Betti numbers of these coverings (which is an invariant of the embedding \(B\subset Y\)) is known in several cases, and it exhibits both polynomial and periodic behaviour. This paper describes several techniques for studying this function, both in the general topological and algebraic geometry contexts. branched coverings; hypersurfaces; Hurewicz map; Betti numbers Coverings in algebraic geometry, Coverings of curves, fundamental group, Special algebraic curves and curves of low genus Multi-polynomial invariants for plane algebraic curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(L_k\) be the affine line over an algebraically closed field \(k\), with char \(k = p \neq 0\). In a previous paper it has been conjectured by the first named author [Am. J. Math. 79, 825-856 (1957; Zbl 0087.036)] that the algebraic fundamental group \(\pi_A (L_k)\) (i.e. the set of finite Galois groups of unramified coverings of \(L_k)\) is the set \(Q(p)\) of all quasi \(p\)-groups (i.e. finite groups which are generated by all their \(p\)-Sylow subgroups), and it was shown that \(\pi_A (L_k) \subseteq Q (p)\). -- In particular, the conjecture would imply that \(\pi_A (L_k) \supseteq A_n\), the alternating group, for all \(n \geq p\) when \(p > 2\), and for all \(n \neq 3,4\) when \(p = 2\).
Later this fact has been proved for \(p > 2\) and for \(p = 2\) and \(n \neq 3,4,6,7\) (see the preceding article by \textit{S. S. Abhyankar}, \textit{J. Ou} and \textit{A. Sathaye} in the same volume, reviewed above). This paper completes the study of the problem by giving explicit equations of unramified coverings of \(L_k\), with char \(k = 2\), which do have \(A_6\) and \(A_7\) as Galois groups.
As in the other article of the volume cited above, the equations in question are among the ones found in the paper by the first author quoted above, and the matter is to prove that they are what is needed; this is done by using a particular form of Jacobson's criterion which works in every characteristic. algebraic fundamental group; Galois groups of unramified coverings Abhyankar, S. S.; Yie, I.: Small degree coverings of the affine line in characteristic two. Discrete math. 133, 1-23 (1994) Coverings of curves, fundamental group, Coverings in algebraic geometry, Finite ground fields in algebraic geometry Small degree coverings of the affine line in characteristic two | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper we investigate whether the property of a variety having a generically large and large fundamental groups is stable under Kähler deformations. The main tools we used are collected in the three theorems the different nature of which give us extra flexibility to tackle these kind of problems. The statements of the theorems are also of independent interest and have applications outside this article. variety with a large fundamental group; Shafarevich uniformization conjecture; Kähler deformation; variety with Stein universal covering de Oliveira B., Katzarkov L., Ramachandran M.: Deformations of large fundamental groups. Geom. Funct. Anal. 12(4), 651--668 (2002) Homotopy theory and fundamental groups in algebraic geometry, Coverings in algebraic geometry, Surfaces and higher-dimensional varieties, Holomorphic convexity, Compact complex \(n\)-folds, Compact Kähler manifolds: generalizations, classification Deformations of large 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 The present work is inspired by that of \textit{D. J. Benson, J. F. Carlson}, and \textit{J. Rickard} [Math. Proc. Camb. Philos. Soc. 120, No. 4, 597-615 (1996; Zbl 0888.20003)] in the representation theory of finite groups. In [J. Pure Appl. Algebra 173, No. 1, 59-86 (2002; Zbl 1006.20035)] the author constructed the support cone of a possibly infinite dimensional representation of a Frobenius kernel. Now she introduces support cones for representations of arbitrary infinitesimal group schemes over an algebraically closed field of characteristic \(p>0\). This is not done in terms of cohomology, but in terms of the \(1\)-parameter subgroups of \textit{A. Suslin, E. M. Friedlander}, and \textit{C. P. Bendel} [J. Am. Math. Soc. 10, No. 3, 693-728 (1997; Zbl 0960.14023)]. Thus the support cone of a module \(M\) over an infinitesimal group \(G\) of height \(r\) is a subset of the affine scheme \(V_r(G)\) representing the functor of \(1\)-parameter subgroups of height \(r\) of \(G\). Eventually the author reproduces all the desirable features, like Rickard idempotents and detection of projectivity. But the arguments are necessarily different from those in the finite group setting. support cones; Rickard idempotents; Frobenius kernels; stable categories; infinitesimal group schemes; 1-parameter subgroups Pevtsova, Julia, Support cones for infinitesimal group schemes.Hopf algebras, Lecture Notes in Pure and Appl. Math. 237, 203\textendash213 pp., (2004), Dekker, New York Representation theory for linear algebraic groups, Group schemes, Cohomology theory for linear algebraic groups, Modular representations and characters Support cones for infinitesimal group schemes. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this note we introduce branched foldings between compact surfaces using a local description. We remark that if some branched folding has some particular local structure then it will be non-regular. branched coverings; orbifolds; Klein surfaces Compact Riemann surfaces and uniformization, Klein surfaces, Coverings in algebraic geometry, Low-dimensional topology of special (e.g., branched) coverings, Topology of Euclidean 2-space, 2-manifolds A note on regular branched foldings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Denote by \(G\) a reductive algebraic group over a field \(K\) of characteristic 0 and by \(T\) a maximal torus of \(G\). In this paper new functors
\[
H^i_{ab}:=\mathbb{H}^i(K,T^{(sc)}\to T)\quad(i\geq-1)
\]
are defined as the abelian Galois cohomology groups, where \(\mathbb{H}^i(K,T^{(sc)}\to T)\) is the Galois hypercohomology of the complex \(T^{sc}(\overline K)\to T(\overline K)\).
This definition allows the author to generalize the abelian cohomology to arbitrary reductive groups (and indeed to arbitrary algebraic groups, modulo the unipotent radical) over a field of characteristic zero; and to obtain even in this case some result proved by Sansuc for semisimple groups, by Kottwitz, Langlands for local fields and by Kneser, Harder and Chernousov for number fields. The definition of two abelianization maps
\[
ab^0\colon H^0(K,G)\to H^0_{ab}(K,G)\qquad ab^1\colon H^1(K,G)\to H^1_{ab}(K,G)
\]
relates the abelian cohomology to the usual one: for local non archimedean fields \(ab_0\) is a surjective homomorphism, and for number fields \(ab_1\) is surjective. Abelian Galois cohomology groups; algebraic fundamental groups; Galois hypercohomology; reductive groups M. Borovoi, \textit{Abelian Galois Cohomology of Reductive Groups} (Am. Math. Soc., Providence, RI, 1998), Mem. AMS 132 (626). Cohomology theory for linear algebraic groups, Galois cohomology of linear algebraic groups, Coverings in algebraic geometry, Nonabelian homological algebra (category-theoretic aspects) Abelian Galois cohomology of reductive groups | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The aim of the article under review is to establish a formula for the irregularity of abelian coverings of smooth projective surfaces by using mixed multiplier ideals.
Let \(X\) be a smooth projective surface and let \(G\) be a abelian group of the form \(\bigoplus_{j=1}^s \mathbb{Z} / n_j \mathbb{Z}\). Let \(\pi: S \rightarrow X\) be a standard \(G\)-abelian covering, and let \(\widetilde{S} \rightarrow S\) be a desingularisation of \(S\). The main theorem of the article is to give an explicit formula to calculate \(h^1 (\widetilde{S} , \mathcal{O}_{\widetilde{S}}) - h^1 (X, \mathcal{O}_X)\) by using the mixed multiplier ideals associated to the branch locus of \(\pi\). The formula can be seen as a full generalization of \textit{O. Zariski}'s paper [Ann. Math. (2) 32, 485--511 (1931; JFM 57.0440.03)] for cyclic coverings of the projective plane. In the last part of the article, the author gives several interesting applications of the formula. algebraic surface; abelian covering; mixed multiplier ideal D.Naie, Mixed multiplier ideals and the irregularity of abelian coverings of smooth projective surfaces. Expo. Math. 31 (2013), no. 1, 40-72. Divisors, linear systems, invertible sheaves, Coverings in algebraic geometry, Singularities of curves, local rings, Surfaces and higher-dimensional varieties Mixed multiplier ideals and the irregularity of abelian coverings of smooth projective surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper considers the phenomenon of inertia groups jumping at regular points of the branch locus of a normal cover of a regular scheme. It is shown that this cannot happen under certain circumstances, e.g. if the ramification is generically tame, or if no two ramification components intersect over the branch point in question.
An application is then given to the problem of finding the fundamental group of the affine line in finite characteristic. Namely, for a large class of finite groups \(G\), certain \(G\)-Galois branched covers of the projective line over \(p\)-adic rings are considered. It is shown that each of these covers gives rise either to a \(G\)-Galois unramified cover of the affine line modulo \(p\), or else to a covering of surfaces in which inertia groups must jump. Examples are given illustrating both possibilities. purity; inertia groups; branch locus of a normal cover of a regular scheme; fundamental group of the affine line in finite characteristic [Ha2] D. Harbater. On purity of inertia. Proc. Amer. Math. Soc.112, 311-319 (1991). Ramification problems in algebraic geometry, Coverings in algebraic geometry, Coverings of curves, fundamental group, Local ground fields in algebraic geometry, Arithmetic ground fields for curves On purity of inertia | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper proves results on Galois covers of affine varieties in characteristic \(p\), showing that they behave extremely well under embedding problems with \(p\)-group kernel. Namely, given such a connected Galois cover \(Y\rightarrow X\) with Galois group \(G= \Gamma/P\), where \(P\) is a \(p\)-group, is there a connected Galois cover \(Z\rightarrow X\) with group \(\Gamma\) that dominates \(Y\rightarrow X\)? Moreover, can \(Z\rightarrow X\) be chosen with presribed local behavior? For example, if \(X'\) is a closed subset of \(X\), and if the restriction \(Y'\rightarrow X'\) of \(Y\rightarrow X\) is dominated by a (possibly disconnected) \(\Gamma\)-Galois cover \(Z'\rightarrow X'\), then can \(Z\rightarrow X\) above be chosen so as to restrict to \(Z'\) over \(X'\)? This general type of problem is traditionally called an ``embedding problem'', since on the function field level a \(G\)-Galois extension is being embedded into a \(\Gamma\)-Galois extension. Over an arbitrary field of characteristic \(p>0\) (e.g. finite fields or Laurent series fields), this paper answers several questions of this type in the affirmative.
In particular, if \(Y\rightarrow X\) is étale, then in the situation above there is such a \(Z\) which is étale over \(Y\) and extends the given \(Z'\) (Theorem 3.11). Moreover this remains true even if \(Y\rightarrow X\) is ramified, provided that its degree is prime to \(p\) (Theorem 4.3). In the case of curves (where \(X'\) is a finite set of points), it is true even if \(Y\rightarrow X\) is merely assumed to be tamely ramified (Theorem 5.14). In this context the local condition corresponds to specifying the residue field extensions over a given finite set of points -- and this is a nontrivial condition if the base field \(k\) is not algebraically closed. An ``adelic'' version for curves, in which \(X'\) is taken to consist of spectra of local fields rather than points, is also shown (Theorem 5.6).
The structure of this paper is as follows: Section 2 is purely group theoretic and provides a cohomological criterion for solving (group-theoretic) embedding problems with \(p\)-group kernel and local conditions. The ideas for this section are related to ideas of Katz, Serre, and Raynaud. Sections 3 and 4 then apply this to embedding problems over affine varieties of arbitrary dimension in characteristic \(p\), by using that the appropriate fundamental groups have \(p\)-cohomological dimension 1 and infinite \(p\)-rank, and by showing an appropriate surjectivity on \(H^1\)'s. Section 5 turns to results for curves and shows the adelic result (Theorem 5.6) similarly. In order to show the result in the case that \(Y\rightarrow X\) is tamely ramified (Theorem 5.14), the strategy is reversed: Theorem 5.6 is combined with group-theoretic results to obtain Theorem 5.14, and from that it follows that the corresponding fundamental group has \(p\)-cohomological dimension~1. prime characteristic; Galois covers of affine varieties; fundamental groups; \(p\)-cohomological dimension D. Harbater, Embedding problems with local conditions, Israel Journal of Mathematics 118 (2000), 317--355. Inverse Galois theory, Coverings in algebraic geometry, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Fundamental groups and their automorphisms (group-theoretic aspects) Embedding problems with local conditions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For the entire collection see Zbl 0626.00011.]
In the appendix 2 to the 2nd edition of \textit{O. Zariski}'s book [``Algebraic surfaces'' (1st edition 1935; Zbl 0010.37103; 2nd edition 1971; Zbl 0219.14020), \textit{D. Mumford} remarked:
``The classification of plane curves C with d nodes and k cusps and the computation of \(\pi_ 1(P^ 2-C)\) has unfortunately not been pursued. Zariski's techniques are closely connected to certain techniques in knot theory. For instance the polynomial f(x), appearing in the proof of theorem 1, {\S} 3 (of chapter VIII) is analogous to the Alexander polynomial...''
Here we shall describe the progress made in the directions pointed out in this quotation. {\S} 2 presents the basics of the techniques of B. Moishezon of braid monodromies, which are closely related to the study of the fundamental groups of the complements of plane curves. {\S} 3 surveys the results on Alexander invariants of plane curves. In particular we give examples of the direct computation of Alexander modules for certain curves avoiding calculations of the fundamental groups. In {\S} 4 we describe \textit{M. V. Nori}'s theory [see Ann. Sci. Éc. Norm. Supér., IV. Sér. 16, 305-344 (1983; Zbl 0527.14016)] allowing establishment of the commutativity of \(\pi_ 1(P^ 2-C)\) in many cases. Zariski problem on abelianness of fundamental group; Yau-Miyaoka inequalities; braid monodromies; fundamental groups of the complements of plane curves; Alexander modules A. Libgober, ''Fundamental groups of the complements to plane singular curves,'' In:Proc. Symp. Pure. Math., Vol. 46 (1987). Coverings in algebraic geometry, Coverings of curves, fundamental group Fundamental groups of the complements to plane singular curves | 0 |
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