Brill-Noether theory for cyclic covers
Algebraic Geometry
2018-11-16 v2
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.
Cite
@article{arxiv.1603.05084,
title = {Brill-Noether theory for cyclic covers},
author = {Irene Schwarz},
journal= {arXiv preprint arXiv:1603.05084},
year = {2018}
}