Finite Abelian Groups of K3 Surfaces with Smooth Quotient

The quotient space of a K3 surface by a finite group is an Enriques surface or a rational surface if it is smooth. Finite groups where the quotient space are Enriques surfaces are known. In this paper, by analyzing effective divisors on smooth rational surfaces, the author will study finite groups which act faithfully on K3 surfaces such that the quotient space are smooth. In particular, he will completely determine effective divisors on Hirzebruch surfaces such that there is a finite Abelian co...     

Moduli of Framed Parabolic Sheaves

This paper gives a construction of the moduli space of framed parabolic sheaves on a Riemann surface. This space serves as a universal, master, space for the well known moduli space of parabolic bundles, as well as moduli spaces of vector bundles, which can all be obtained from this space by torus quotients. The construction is given for the structure group SL(N, C), and indeed is adapted to this case. At the end of the paper, an approach is suggested for dealing with the case of arbitrary reductive groups, involving the associated loop group.     

Monodromy Groups of Irregular Elliptic Surfaces

Monodromy groups, i.e. the groups of isometries of the intersection lattice L _X H ₂/torsion generated by the monodromy action of all deformation families of a given surface, have been computed by the author for any minimal elliptic surface with p _g > q = 0. New and refined methods are now employed to address the cases of minimal elliptic surfaces with p _g q > 0. Thereby we get explicit families such that any isometry is in the group generated by their monodromies or does not respect the invariance of the canonical class or the spinor norm. The monodromy is also shown to act by the full symplectic group on the first homology modulo torsion.     

Marked Schottky Groups and Representations of Surface Groups

We give a complete condition for any n elements of PSL(2, R)to generate a Fuchsian group which is a Schottky group on thatset of generators, and apply it to a question of Bowditch onrepresentations of surface groups into PSL(2, R). 2000 MathematicalSubject Classification: 20H10, 32G15.     

Coxeter Groups, 2-Completion, Perimeter Reduction and Subgroup Separability

We show that all groups in a very large class of Coxeter groups are locally quasiconvex and have a uniform membership problem solvable in quadratic time. If a group in the class satisfies a further hypothesis it is subgroup separable and relevant homomorphisms are also calculable in quadratic time. The algorithm also decides if a finitely generated subgroup has finite index.     

Quasi-isometries of groups, graphs and surfaces

We give a characterization of virtual surface groups as groups quasi-isometric to complete simply-connected Riemannian surfaces. Results on the equivalence up to quasi-isometry of various bounded geometry conditions for Riemannian surfaces are also obtained. Received: January 18, 2000     

Topological triviality of smoothly knotted surfaces in -manifolds

Some generalizations of the Fintushel-Stern rim surgery are known to produce smoothly knotted surfaces. We show that if the fundamental groups of their complements are standard, then these surfaces are topologically unknotted.

     

The Subgroups of Direct Products of Surface Groups

A subgroup of a product of n surface groups is of type FP _n if and only if it contains a subgroup of finite index that is itself a product of (at most n) surface groups.     

Pinching Surface Groups in Complex Hyperbolic Plane

We construct first examples of discrete geometrically finite subgroups of PU(21) which contain parabolic elements, and are isomorphic to surface groups of genus 2.     

The Brauer group of an affine double plane associated to a hyperelliptic curve

We study the Brauer group of an affine double plane π:X→𝔸² defined by an equation of the form z² = f in two separate cases. In the first case, f is a product of n linear forms in k[x,y] and X is birational to a ruled surface ?¹×C, where C is rational if n is odd and hyperelliptic if n is even. In the second case, f = y²?p(x) is the equation of an affine hyperelliptic curve. For π as well as the unramified part of π, we compute the groups of divisor classes, the Brauer groups, the relative Brauer groups, and all of the terms in the sequences of Galois cohomology.     

有限群在曲面上的自由作用

本文借助于计算机编程给出了有限群在可定向闭曲面T~(nr 1)上反向自由作用个数的上界,同时决定了反向自由作用于小亏格闭曲面T~(nr 1)上的有限群以及p-1为素数时反向自由作用于闭曲面T_p上的有限群。     

The Groups of Symmetric Genus σ ≤ 8

Let G be a finite group. The symmetric genus σ (G) is the minimum genus of any compact Riemann surface on which G acts faithfully as a group of automorphisms. Here we classify the groups of symmetric genus σ, for all values of σ such that 4 ≤ σ ≤ 8. In addition, we obtain some general results about the partial presentations that groups acting on surfaces must have. We show that a group with even genus and no “large order” elements in its Sylow 2-subgroup has restrictions on its Sylow 2-subgroup. As a consequence, we show that if G is a 2-group with positive symmetric genus, then σ(G) is odd. The software package MAGMA was employed to help with the calculations, and the MAGMA library of small groups was essential to the classification.     

Surfaces in Three-Dimensional Lie Groups

We derive the Weierstrass (or spinor) representation for surfaces in the three-dimensional Lie groups Nil, \(\widetilde{SL}_2\), and Sol with Thurston's geometries and establish the generating equations for minimal surfaces in these groups. Using the spectral properties of the corresponding Dirac operators, we find analogs of the Willmore functional for surfaces in these geometries.     

On Lacunary Analogs of the Poincare Theta-Series and Their Applications

We consider Fuchsian groups of linear-fractional transformations such that each vertex of the fundamental polygon is common for an even or infinite number of fundamental congruent polygons meeting at this point. The whole collection of transformations splits into two disjoint sets. For these sets we introduce two lacunary kernels whose sum represents the well-known analog of Chibrikova and Sil'vestrov's kernel and study their properties. We introduce automorphic forms of dimension –4m that differ from the Poincare theta-series. We indicate an application of one of the constructed lacunary kernels which does not include the Cauchy kernel to solving some boundary value problem with a shift of the contour inside the domain.     

The Second Type Singularities of Symplectic and Lagrangian Mean Curvature Flows

This paper mainly deals with the type II singularities of the mean curvature flow from a symplectic surface or from an almost calibrated Lagrangian surface in a Kähler surface. The relation between the maximum of the Kähler angle and the maximum of |H|² on the limit flow is studied. The authors also show the nonexistence of type II blow-up flow of a symplectic mean curvature flow which is normal flat or of an almost calibrated Lagrangian mean curvature flow which is flat.     

Least-Squares Fitting of Algebraic Spline Surfaces

We present an algorithm for fitting implicitly defined algebraic spline surfaces to given scattered data. By simultaneously approximating points and associated normal vectors, we obtain a method which is computationally simple, as the result is obtained by solving a system of linear equations. In addition, the result is geometrically invariant, as no artificial normalization is introduced. The potential applications of the algorithm include the reconstruction of free-form surfaces in reverse engineering. The paper also addresses the generation of exact error bounds, directly from the coefficients of the implicit representation.     

Splittings of Surfaces

The main result characterizes small actions of surface groups on -trees.

     

On K1 and K2 of Algebraic Surfaces

In this paper we study higher Chow groups of smooth, projective surfaces over a field k of characteristic zero, using some new Hodge theoretic methods which we develop for this purpose. In particular we investigate the subgroup of CH ^r+1 (X,r) with r = 1,2 consisting of cycles that are supported over a normal crossing divisor Z on X. In this case, the Hodge theory of the complement forms an interesting variation of mixed Hodge structures in any geometric deformation of the situation. Our main result is a structure theorem in the case where X is a very general hypersurface of degree d in projective 3-space for d sufficiently large and Z is a union of very general hypersurface sections of X. In this case we show that the subgroup of CH ^r+1 (X,r) we consider is generated by obvious cycles only arising from rational functions on X with poles along Z. This can be seen as a generalization of the Noether–Lefschetz theorem for r = 0. In the case r = 1 there is a similar generalization by Müller-Stach, but our result is more precise than it, since it is geometric and not only cohomological. The case r = 2 is entirely new and original in this paper. For small d, we construct some explicit examples for r = 1 and 2 where the corresponding higher Chow groups are indecomposable, i.e. not the image of certain products of lower order groups. In an appendix Alberto Collino constructs even more indecomposable examples in CH ³ (X,2) which move in a one-dimensional family on the surface X.Contribution to appendix.     

Maximum orders of extendable actions on surfaces

We determine the maximum order E_g of finite groups G acting on the closed surface Σ_g of genus g which extends over(S~3, Σ_g) for all possible embeddings Σ_g → S~3, where g 1.     

Characterization of between-group inequality of longevity in European Union countries

Comparisons of differential survival by country are useful in many domains. In the area of public policy, they help policymakers and analysts assess how much various groups benefit from public programs, such as social security and health care. In financial markets and especially for actuaries, they are important for designing annuities and life insurance products. This paper presents a method for clustering information about differential mortality by country. The approach is then used to group mortality surfaces for European Union (EU) countries. The aim of this paper is to measure between-group inequality in mortality experience in EU countries through a range of mortality indicators. Additionally, the indicators permit the characterization of each group. It is important to take into account characteristics such as sex; therefore, this study differentiates between males and females in order to detect whether their patterns and characterizations are different. It is concluded that there are clear differences in mortality between the east and west of the EU that are more important than the traditional south–north division, with a significant disadvantage for Eastern Europe, and especially for males in Baltic countries. We find that the mortality indicators have evolved in all countries in such a way that the gap between groups has been maintained, both in terms of the differences in mortality levels and variability.