Hyperelliptic curves among cyclic coverings of the projective line, I

In this note, we prove a necessary and sufficient condition for whether a d-cyclic covering of the complex projective line with 3 branch points has gonality 2 (i.e., is elliptic or hyperelliptic), where d is a positive integer.     

Monodromy of cyclic coverings of the projective line

We show that the image of the pure braid group under the monodromy action on the homology of a cyclic covering of degree d of the projective line is an arithmetic group provided the number of ramification points is sufficiently large compared to the degree d and the ramification degrees are co-prime to d.     

Mumford coverings of the projective line

A Mumford covering of the projective line over a complete non-archimedean valued field K is a Galois covering X? P¹_K X\rightarrow {\bf P}^1_K such that X is a Mumford curve over K. The question which finite groups do occur as Galois group is answered in this paper. This result is extended to the case where P¹_K {\bf P}^1_K is replaced by any Mumford curve over K.     

On cyclic covers of the projective line

We construct configuration spaces for cyclic covers of the projective line that admit extra automorphisms and we describe the locus of curves with given automorphism group. As an application we provide examples of arbitrary high genus that are defined over their field of moduli and are not hyperelliptic.     

Weierstrass points on cyclic covers of the projective line

We are interested in cyclic covers of the projective line which are totally ramified at all of their branch points. We begin with curves given by an equation of the form , where is a polynomial of degree . Under a mild hypothesis, it is easy to see that all of the branch points must be Weierstrass points. Our main problem is to find the total Weierstrass weight of these points, . We obtain a lower bound for , which we show is exact if and are relatively prime. As a fraction of the total Weierstrass weight of all points on the curve, we get the following particularly nice asymptotic formula (as well as an interesting exact formula):

where is the genus of the curve. In the case that (cyclic trigonal curves), we are able to show in most cases that for sufficiently large primes , the branch points and the non-branch Weierstrass points remain distinct modulo .

     

Cyclic coverings of the <Emphasis Type="Italic">p</Emphasis>-adic projective line by Mumford curves

Exact bounds for the positions of the branch points for cyclic coverings of the p-adic projective line by Mumford curves are calculated in two ways. Firstly, by using Fumiharu Kato’s *-trees, and secondly by giving explicit matrix representations of the Schottky groups corresponding to the Mumford curves above the projective line through combinatorial group theory.     

Secant planes of projective curves embedded by nonspecial line bundles

For a nonspecial line bundle ? on a smooth curve X we consider a presentation ??𝒦_X?D+E which is minimal with respect to deg E. If ? is very ample, then this minimality means that any n-points of φ_?(X) with n≤deg E?1 are in general position while φ_?(E) spans a (deg E?2)-plane. In this work, we investigate conditions on D and E for ??𝒦_X?D+E to be minimal. We also observe s-secant (s?k?1)-planes which are minimal with respect to the secant degree s for a given k. We apply minimal presentations to problems about the exactness of Green-Lazarsfeld’s conjecture on property (N_p).     

A Hyperelliptic Smoothness Test, II

This series of papers presents and rigorously analyzes a probabilisticalgorithm for finding small prime factors of an integer. Thealgorithm uses the Jacobian varieties of curves of genus 2 inthe same way that the elliptic curve method uses elliptic curves.This second paper in the series is concerned with the orderof the group of rational points on the Jacobian of a curve ofgenus 2 defined over a finite field. We prove a result on thedistribution of these orders. 2000 Mathematical Subject Classification:11Y05, 11G10, 11M20, 11N25.     

Galois coverings and endomorphisms of projective varieties

We prove that the vector bundle associated to a Galois covering of projective manifolds is ample (resp. nef) under very mild conditions. This results is applied to the study of ramified endomorphisms of Fano manifolds with b ₂ = 1. It is conjectured that is the only Fano manifold admitting an endomorphism of degree d ≥ 2, and we verify this conjecture in several cases. An important ingredient is a generalization of a theorem of Andreatta–Wisniewski, characterizing projective space via the existence of an ample subsheaf in the tangent bundle. Marian Aprodu was supported in part by a Humboldt Research Fellowship and a Humboldt Return Fellowship. He expresses his special thanks to the Mathematical Institute of Bayreuth University for hospitality during the first stage of this work. Stefan Kebekus and Thomas Peternell were supported by the DFG-Schwerpunkt “Globale Methoden in der komplexen Geometrie” and the DFG-Forschergruppe “Classification of Algebraic Surfaces and Compact Complex Manifolds”. A part of this paper was worked out while Stefan Kebekus visited the Korea Institute for Advanced Study. He would like to thank Jun-Muk Hwang for the invitation.     

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.     

On multiple coverings of irrational curves

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

Normally generated line bundles on general curves, II

Let C be a general curve of genus g≥3. Here we prove that there is a normally generated L∈Pic^d(C) such that h⁰(C,L)=r+1≥4 (i.e. a very ample line bundle which embeds C in P^r as a projectively normal curve) if and only if (r+1)h¹≤g≤r(r−1)/2+2h¹, where h¹?g+r−d=h¹(C,L).     

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     

Prym varieties of cyclic coverings

The Prym map of type (g, n, r) associates to every cyclic covering of degree n of a curve of genus g ramified at a reduced divisor of degree r the corresponding Prym variety. We show that the corresponding map of moduli spaces is generically finite in most cases. From this we deduce the dimension of the image of the Prym map.     

Hyperelliptic curves and values of Gaussian hypergeometric series

We express the number of ${\mathbb{F}_q}$ -points on the hyperelliptic curve ${\alpha{y}^2=\beta{x}^f + \gamma}$ in terms of Gaussian hypergeometric series. We also find some special values of ${{_{2}}F_1}$ -Gaussian hypergeometric series containing characters of order 3 and 4 as parameters.     

On the order of projective curves

Here we prove the existence of several componentsW of the Hilbert scheme of curves inP ⁿ such that the generalC W has Hartshorne-Rao module with order equal to its diameter.     

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 $f_s$.