On the non-Transversality of the Hyperelliptic Locus and the Supersingular Locus for $g=3$

TL;DR

This paper establishes a criterion for identifying non-transversal intersections between the hyperelliptic and supersingular loci in genus 3 moduli space, demonstrating such points exist for infinitely many primes.

Contribution

It provides a new criterion for non-transversality in the intersection of these loci and proves the existence of such points for infinitely many primes.

Findings

Existence of non-transversal intersection points for infinitely many primes

A criterion to detect non-transversality in the moduli space

Implications for the geometry of hyperelliptic and supersingular loci

Abstract

This paper gives a criterion for a moduli point to be a point of non-transversal intersection of the hyperelliptic locus and the supersingular locus in the Siegel moduli stack $\mathfrak{A}_3 \times \mathbb{F}_p$. It is shown that for infinitely many primes $p$ there exists such a point.