Galois branched covers with fixed ramification locus

Abstract

We examine conditions under which there exists a non-constant family of Galois branched covers of curves over an algebraically closed field $k$ of fixed degree and fixed ramification locus, under a notion of equivalence derived from considering linear series on a fixed smooth proper source curve $X$. We show such a family exists precisely when the following conditions are satisfied: $\mathrm{char}(k)=p>0$, $X$ is isomorphic to ${\mathbb{P}}_k^1$, there is a unique ramification point, and the Galois group is $(\mathbb{Z}/p\mathbb{Z}{)}^m$ for some integer $m>0$.

arXiv comment14 pages