Huy Dang

## Institution:

University of Virginia

## Time:

Thursday, January 9, 2020 - 3:00pm to 4:00pm

## Location:

RH 306
An Artin-Schreier curve is a $\mathbb{Z}/p$-branched cover of the projective line over a field of characteristic $p>0$. A unique aspect of positive characteristic is that there exist flat deformations of a wildly ramified cover that change the number of branch points but fix the genus. In this talk, we introduce the notion of Hurwitz tree. It is a combinatorial-differential object that is endowed with essential degeneration data of a deformation. We then show how the existence of a deformation between two covers with different branching data equates to the presence of a Hurwitz tree with behaviors determined by the branching data. One application of this result is to prove that the moduli space of Artin-Schreier covers of fixed genus g is connected when g is sufficiently large. If time permits, we will describe a generalization of the technique for all cyclic covers and the lifting problem.