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

TL;DR

This work addresses the problem of when the Newton polygon locus defined by supersingularity intersects the Torelli/Hyperelliptic locus non-transversally in genus $3$; it develops a criterion linking such non-transversal intersections to a geometric condition on a conic–line configuration arising from Li–Oort’s polarized flag-type quotients and the canonical embedding of hyperelliptic curves. The main method combines Dieudonné theory, the Kodaira–Spencer map, and explicit analysis of Li–Oort families to translate deformation-theoretic data into a tangible geometric test: a line $l$ touching a conic $Q$ at a point $P$ corresponds to non-transversality. In the special case $a(\mathrm{Jac}(C))=1$, the criterion simplifies to a Cartier–Manin matrix condition, enabling an explicit, algorithmic construction of examples, including infinite CM-reduction families. The results illuminate how non-transversal intersections arise already in the simplest nontrivial case $g=3$, with potential implications for understanding the interaction between Newton polygon strata and curve loci in higher genus and providing practical tools for constructing and verifying such points via $p$-adic and CM techniques.

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.

Figures (2)

• Figure 1: Generic picture
• Figure 2: CM touchpoint

Theorems & Definitions (46)

• Theorem A
• Theorem B
• Lemma 2.1
• proof
• Definition 2.2
• Theorem 2.3
• proof
• Definition 2.4
• Theorem 2.5
• proof