Some unlikely intersections between the Torelli locus and Newton strata in $\mathcal{A}_g$

TL;DR

This paper investigates the Newton polygons of algebraic curves with specific symmetries in characteristic p, revealing unlikely intersections with the Torelli locus and providing evidence for conjectures about Newton polygon amalgamation.

Contribution

It constructs explicit examples of curves with prescribed Newton polygons and demonstrates asymptotic bounds, advancing understanding of Newton stratification in moduli spaces.

Findings

Existence of curves with specific Newton polygons for given genus and parameters

Construction of families of curves with Newton polygons bounded below by quadratic functions

Evidence supporting Oort's conjecture on Newton polygon amalgamation

Abstract

Let $p$ be an odd prime. What are the possible Newton polygons for a curve in characteristic $p$? Equivalently, which Newton strata intersect the Torelli locus in $\mathcal{A}_g$? In this note, we study the Newton polygons of certain curves with $\mathbb{Z}/p\mathbb{Z}$-actions. Many of these curves exhibit unlikely intersections between the Torelli locus and the Newton stratification in $\mathcal{A}_g$. Here is one example of particular interest: fix a genus $g$. We show that for any $k$ with $\frac{2g}{3}-\frac{2p(p-1)}{3}\geq 2k(p-1)$, there exists a curve of genus $g$ whose Newton polygon has slopes $\{0,1\}^{g-k(p-1)} \sqcup \{\frac{1}{2}\}^{2k(p-1)}$. This provides evidence for Oort's conjecture that the amalgamation of the Newton polygons of two curves is again the Newton polygon of a curve. We also construct families of curves $\{C_g\}_{g \geq 1}$, where $C_g$ is a curve of genus $g$, whose Newton polygons have interesting asymptotic properties. For example, we construct a family of curves whose Newton polygons are asymptotically bounded below by the graph $y=\frac{x^2}{4g}$. The proof uses a Newton-over-Hodge result for $\mathbb{Z}/p\mathbb{Z}$-covers of curves due to the author, in addition to recent work of Booher-Pries on the realization of this Hodge bound.