Current results on Newton polygons of curves

TL;DR

This paper surveys the current understanding of Newton polygons and Ekedahl-Oort types for Jacobians of smooth curves in positive characteristic, including new results on supersingular curves and stratification geometry.

Contribution

It introduces new results on the existence of supersingular curves for infinitely many genera and reduces questions about stratification geometry to the case of p-rank zero.

Findings

Existence of supersingular curves for infinitely many genera in each odd prime characteristic.

Sketch of proof for the geometry of the p-rank stratification.

Reduction of stratification questions to p-rank zero cases.

Abstract

There are open questions about which Newton polygons and Ekedahl-Oort types occur for Jacobians of smooth curves of genus $g$ in positive characteristic $p$. In this chapter, I survey the current state of knowledge about these questions. I include a new result, joint with Karemaker, which verifies, for each odd prime $p$, that there exist supersingular curves of genus $g$ defined over $\overline{{\mathbb F}}_p$ for infinitely many new values of $g$. I sketch a proof of Faber and Van der Geer's theorem about the geometry of the $p$-rank stratification of the moduli space of curves. The chapter ends with a new theorem, in which I prove that questions about the geometry of the Newton polygon and Ekedahl-Oort strata can be reduced to the case of $p$-rank $0$.