Intersections of the automorphism and the Ekedahl-Oort strata in $M_2$

TL;DR

This paper analyzes the intersection of automorphism and Ekedahl-Oort strata in the moduli space of genus two curves, providing explicit descriptions, an algorithm, and dimension computations.

Contribution

It introduces an explicit description of automorphism groups, an algorithm for Ekedahl-Oort strata, and computes intersection dimensions and components.

Findings

Explicit automorphism group descriptions for genus two curves

Algorithm for Ekedahl-Oort stratum computation

Dimension and irreducible component counts of intersections

Abstract

We compute the intersections between the automorphism strata and the pullback by the Torelli map of the Ekedahl-Oort strata inside the moduli space of genus two curves. We first describe explicitly which possible automorphism groups a genus two curve can have over a field of positive characteristic, and parametrise the families of curves with a prescribed automorphism group. Then, we describe an algorithm to compute the strata of genus two curves whose Jacobian variety has a fixed Ekedahl-Oort type. Finally, we compute the dimension and number of irreducible components of the intersections between the strata.