Families and moduli of covers with specified ramification

TL;DR

This paper investigates the existence and properties of branched covers of algebraic curves with fixed ramification, revealing characteristic-dependent behaviors and establishing conditions for the existence of families of such covers.

Contribution

It provides a necessary and sufficient condition for the existence of families of covers with fixed ramification in positive characteristic, and proves one direction unconditionally.

Findings

No non-constant families in characteristic 0

Condition for existence in positive characteristic

Partial proof in characteristic 2 and 3

Abstract

We study branched covers of curves with specified ramification points, under a notion of equivalence derived from linear series. In characteristic 0, no non-constant families of covers with fixed ramification points exist. In positive characteristic we formulate a necessary and sufficient condition for the existence of such a family. We unconditionally prove one direction of this conjecture, and by studying infinitesimal deformations show the other direction in characteristic 2 and 3.