Brill-Noether theory for cyclic covers

TL;DR

This paper extends Brill-Noether theory to cyclic covers of algebraic curves, providing new necessary conditions for the existence of linear series on these covers using degeneration techniques.

Contribution

It introduces a novel application of degeneration and limit linear series to cyclic covers, refining existence criteria for linear series on these curves.

Findings

Plücker formula gives necessary conditions for linear series existence

Conditions are nearly sharp, closely matching known sufficient conditions

Results apply to all degrees, ranks, and cyclic cover degrees

Abstract

The Brill-Noether Theorem gives necessary and sufficient conditions for the existence of a linear series. Here we consider a general n-fold, etale cyclic cover p of a curve C of genus g and investigate for which numbers r,d a linear series of dimension r and degree d exists on the covering curve. For r=1 this gives gonality. Using degeneration to a special singular example (containing a Castelnuovo canonical curve) and the theory of of limit linear series for tree-like curves we show that the Pl\"ucker formula yields a necessary condition for the existence of a linear series (of dimension r, degree d) which is only slightly weaker than the sufficient condition given by the result of Kleimann and Laksov, for all n,r,d.