On multiple coverings of irrational curves

Abstract

) 0. Introduction. In Section 1, we investigate the problem of the existence of base point free pencils of relatively low degree on a double covering X of genus 9 of a general curve C of genus q > 0. Such problem is classical and the picture is rather well known for the degree range close to the genus 9. On the other hand, by a simple application of the Castelnuovo-Severi inequality one can easily see that there does not exist a base point free pencil of degree less than or equal to g -2 q other than pull-backs from the base curve C; while for the degree beyond this range not many things have been known about the existence of such a pencil which is not composed with the given involution. Here we prove the following result. In the subsequent sections of this paper we consider the Brill-Noether theory for sufficiently general coverings of curves of genus q > 0. In Section 2 we use [12] to check the full Brill-Noether-Gieseker-Petri package for a general genus 9 covering of degree k of a general curve of genus q under very strong restrictions on 9, namely if kq<=g<=kq + (k-1).In Section 3 we consider the Brill-Noether theory for g~'s. Among the results we stress Theorem 3.1, Corollary 3.3, Theorem 3.4, and Corollary 3.5.In Section 4 we consider the Brill-Noether-Gieseker-Petri problem for 92 for a general curve Y in a "large" (i.e. of dimension at least g + 1) subvariety of ~0, when the g~ is base point free and induces a birational morphism from Y to IP 2. We prove the following result.

On double coverings of hyperelliptic curves

“…(i) For d ≤ g − 2, it is clear by Remark 2.8 (2). By Theorem 2.10 and the fact that W 2 g−2 (X ) = π * W 2 4 (C) + W g−10 (X ), we have…”

“…In the articles [2] and [3], the variety of pencils on a double covering X of a general curve C (in the sense of Brill-Noether) has been studied extensively. In this paper, we treat double covering of a very special curve, i.e.…”

Double coverings of hyperelliptic real algebraic curves

“…They are also permuted by u since the rotation z → ze iθ commutes with u. Therefore, we may assume from the beginning that S can be represented as in (3).…”

Pencils on double coverings of curves

“…2. If X is a generic curve and π ≥ 5g + 1 then there is a base-point-free pencil of any degree d ≥ π − g which is not composed with the given double covering (see [1]). 3.…”