On double coverings of hyperelliptic curves

We prove various properties of varieties of special linear systems on double coverings of hyperelliptic curves. We show and determine the irreducibility, generically reducedness and singular loci of the variety for bi-elliptic curves and double coverings of genus two curves. Similar results for double coverings of hyperelliptic curves of genus h≥3 are also presented.     

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     

On Weierstrass semigroups of double coverings of genus three curves

Let C be a complete non-singular curve of genus 3 over an algebraically closed field of characteristic 0. We determine all possible Wierstrass semigroups of ramification points on double coverings of C whose covering curves have genus greater than 8. Moreover, we construct double coverings with ramification points whose Weierstrass semigroups are the possible ones.     

Ramification points of double coverings of curves and Weierstrass points

Summary Let π: X→C be a double covering with X smooth curve and C elliptic curve. Let R(π)⊂X be the ramification locus of π. Every P∈R(π) is a Weierstrass point of X and we study the triples (C, π, X) for which the set of corresponding Weierstrass points have certain semigroups of non-gaps. We study the same problem also for triple cyclic coverings of C. Entrata in Redazione il 17 luglio 1998. The authors were partially supported by MURST and GNSAGA of CNR (Italy).     

The minimal degree of plane models of algebraic curves and double coverings

Let C be a smooth irreducible projective curve of genus g and s(C, 2) (or simply s(2)) the minimal degree of plane models of C. We show the non-existence of curves with s(2) = g for g ≥ 10, g ≠ 11. Another main result is determining the value of s(2) for double coverings of hyperelliptic curves. We also give a criterion for a curve with big s(2) to be a double covering.     

Numerical semigroups of genus eight and double coverings of curves of genus three

The authors determine all possible numerical semigroups at ramification points of double coverings of curves when the covered curve is of genus three and the covering curve is of genus eight. Moreover, it is shown that all of such numerical semigroups are actually of double covering type.     

On multiple coverings of irrational curves

The first named author was partially supported by MURST and GNSAGA of CNR (Italy).     

Projective-planar double coverings of graphs

We shall show that a connected graph G is projective-planar if and only if it has a projective-planar double covering and that any projective-planar double covering of a 2-connected nonplanar graph is planar.     

Double coverings of hyperelliptic real algebraic curves

We consider double and (possibly) branched coverings π:X→X^′ between real algebraic curves where X is hyperelliptic. We are interested in the topology of such coverings and also in describing them in terms of algebraic equations. In this article we completely solve these two problems. We first analyse the topological features and ramification data of such coverings. Second, for each isomorphism class of these coverings we then describe a representative, with defining polynomial equations for X and for X^′, a formula for the involution that generates the covering transformation group, and a rational formula for the covering projection π:X→X^′.     

On unramified normal coverings of hyperelliptic curves

It is well known that the number of unramified normal coverings of an irreducible complex algebraic curve C with a group of covering transformations isomorphic to Z₂⊕Z₂ is (2^4g−3⋅2^2g+2)/6. Assume that C is hyperelliptic, say . Horiouchi has given the explicit algebraic equations of the subset of those covers which turn out to be hyperelliptic themselves. There are of this particular type. In this article, we provide algebraic equations for the remaining ones.     

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     

A classification of unramified double coverings of hyperelliptic Riemann surfaces

Archiv der Mathematik -     

On the ordinariness of coverings of stable curves

In the present paper, we study the ordinariness of coverings of stable curves. Let $f:Y\to X$ be a morphism of stable curves over a discrete valuation ring R with algebraically closed residue field of characteristic $p>0$. Write S for Spec R and η (resp. s) for the generic point (resp. closed point) of S. Suppose that the generic fiber $X_{\eta }$ of X is smooth over η, that the morphism $f_{\eta }:Y_{\eta }\to X_{\eta }$ over η on the generic fiber induced by f is a Galois étale covering (hence $Y_{\eta }$ is smooth over η too) whose Galois group is a solvable group G, that the genus of the normalization of each irreducible component of the special fiber $X_s$ is ≥2, and that $Y_s$ is ordinary. Then we have that the morphism $f_s:Y_s\to X_s$ over s induced by f is an admissible covering. This result extends a result of M. Raynaud concerning the ordinariness of coverings to the case where $X_s$ is a stable curve. If, moreover, we suppose that G is a p-group, and that the p-rank of the normalization of each irreducible component of $X_s$ is ≥2, we can give a numerical criterion for the admissibility of $f_s$.     

Symplectic Lefschetz fibrations of curves as gonal coverings

We show that after a finite base change every symplectic Lefschetz fibration ${f \colon X \rightarrow B}$ of genus g >  3 curves over a closed oriented surface becomes a finite covering of degree ${\frac{g}{2} + 1}$ or ${\frac{g}{2} + \frac{3}{2}}$ of a family of spheres over a Riemann surface, with a branch locus admitting complex algebraic curves as local models. In the case of fibers of genus 4, it is shown that after a 2:1 base change the family admits a trigonal covering to a symplectic ruled surface, with symplectic branch locus.