Pencils on double coverings of curves

Let X be a smooth curve of genus g. When and d ≥ π−2g+1 we show the existence of a double covering where C a smooth curve of genus π with a base-point-free pencil of degree d which is not the pull-back of a pencil on X. Received: 7 February 2007; Revised: 1 July 2008     

Gonality for stable curves and their maps with a smooth curve as their target

Here we study the deformation theory of some maps f: X → ℙ ^r , r = 1, 2, where X is a nodal curve and f|T is not constant for every irreducible component T of X. For r = 1 we show that the “stratification by gonality” for any subset of with fixed topological type behaves like the stratification by gonality of M _g.      

Bounds for binary multiple covering codes

We study codes that are multiple coverings of the Hamming space and discuss lower and upper bounds onK(n, r, ), the minimum cardinality of a binary code of lengthn such that the Hamming spheres of radiusr centered at the codewords cover at least times. We also study the similar problem of multiple coverings containing repeated words. A table of bounds forn16,r4, 4 is given.     

The chromatic numbers of double coverings of a graph

If we fix a spanning subgraph H of a graph G, we can define a chromatic number of H with respect to G and we show that it coincides with the chromatic number of a double covering of G with co-support H. We also find a few estimations for the chromatic numbers of H with respect to G.     

Fold‐2‐covering triangular embeddings

For a graph G and a positive integer m, G_(m) is the graph obtained from G by replacing every vertex by an independent set of size m and every edge by m² edges joining all possible new pairs of ends. If G triangulates a surface, then it is easy to see from Euler's formula that G_(m) can, in principle, triangulate a surface. For m prime and at least 7, it has previously been shown that in fact G_(m) does triangulate a surface, and in fact does so as a “covering with folds” of the original triangulation. For m = 5, this would be a consequence of Tutte's 5‐Flow Conjecture. In this work, we investigate the case m = 2 and describe simple classes of triangulations G for which G₍₂₎ does have a triangulation that covers G “with folds,” as well as providing a simple infinite class of triangulations G of the sphere for which G₍₂₎ does not triangulate any surface. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 79–92, 2003     

具有双障碍的退缩抛物变分不等式解的存在性和唯一性(英)

本文研究具有双障碍的退缩抛物变分不等式．我们利用罚技巧，有限逼近和先验估计方法，得到一类退缩抛物变分不等式弱解的存在性，并在一定条件之下，建立了弱解的唯一性．本文结论对广泛的一类抛物型变分不等式成立．     

ω_（2，2g／（3＋2））~2的分类

本文给出在一般代数曲线Ｃ上由ｄｅｇ（Ｅ）＝ｄ，ｄｉｍＨ￣０（Ｃ，Ｅ）≥３的二维稳定向量丛Ｅ组成的解析子簇．ｄ在其维数为零时所含的元素个数．     

A generalization of the existence theorem of special line bundles

LetX be a generic smooth irreducible complex projective curve of genusg withg4. In this paper, we generalize the existence theorem of special divisors to high dimensional indecomposable vector bundles. We give a necessary and sufficient condition on the existence ofn-dimensional indecomposable vector bundlesE onX with det(E)=d, dimH ⁰(X,E)h. We also determine under what condition the set of all such vector bundles will be finite and how many elements it contains.Project partly supported by the National Natural Science Foundation of China.     

All double coverings of cyclotomic fields

A double covering of a Galois extension K/F in the sense of [3] is an extension /K of degree ≤2 such that /F is Galois. In this paper we determine explicitly all double coverings of any cyclotomic extension over the rational number field in the complex number field. We get the results mainly by Galois theory and by using and modifying the results and the methods in [2] and [3]. Project 10571097 supported by NSFC     

The moduli space of curves of genus two covering elliptic curves

We consider the moduli spaceS _n of curvesC of genus 2 with the property:C has a “maximal” mapf of degreen to an elliptic curveE. Here, the term “maximal” means that the mapf∶C→E doesn't factor over an unramified cover ofE. By Torelli mapS _n is viewed as a subset of the moduli spaceA ₂ of principally polarized abelian surfaces. On the other hand the Humbert surfaceH _Δ of invariant Δ is defined as a subvariety ofA ₂(C), the set of C-valued points ofA ₂. The purpose of this paper is to releaseS _n withH _Δ.     

Application of covering techniques to families of curves

Much success in finding rational points on curves has been obtained by using Chabauty's Theorem, which applies when the genus of a curve is greater than the rank of the Mordell-Weil group of the Jacobian. When Chabauty's Theorem does not directly apply to a curve , a recent modification has been to cover the rational points on by those on a covering collection of curves , obtained by pullbacks along an isogeny to the Jacobian; one then hopes that Chabauty's Theorem applies to each . So far, this latter technique has been applied to isolated examples. We apply, for the first time, certain covering techniques to infinite families of curves. We find an infinite family of curves to which Chabauty's Theorem is not applicable, but which can be solved using bielliptic covers, and other infinite families of curves which even resist solution by bielliptic covers. A fringe benefit is an infinite family of Abelian surfaces with non-trivial elements of the Tate-Shafarevich group killed by a bielliptic isogeny.     

On numerical semigroups related to covering of curves

We investigate arithmetical properties of a class of semigroups that includesthose appearing as Weierstrass semigroups at totally ramified points of coveringof curves.     

Divisibility of zeta functions of curves in a covering

We prove, as an analogy of a conjecture of Artin, that if is a finite flat morphism between two singular reduced absolutely irreducible projective algebraic curves defined over a finite field, then the numerator of the zeta function of X divides that of Y in . Then, we give some interpretations of this result in terms of semi-abelian varieties. Received: 23 July 2001     

Double perfect partitions

MacMahon [Combinatory Analysis, vols. I and II, Cambridge University Press, Cambridge, 1915, 1916 (reprinted, Chelsea, 1960)] introduced a perfect partition of positive integer n, which is a partition such that every positive integer less than or equal to n can be uniquely represented by the sum of its parts. We generalize perfect partition and find a relation with ordered factorizations.     

Hardy-Hilbert重级数定理的一个改进

利用Gram矩阵的正定性和Bernoulli不等式得到Holder不等式的一个加强的结果.由此建立了Hardy-Hilbert重级数定理的一个改进.特别,当p=2时,得到了经典的Hilbert重级数定理的一个很强的结果.