The Locus of Plane Quartics with A Hyperflex

TL;DR

This paper explicitly characterizes the locus of plane quartics with hyperflexes using modular forms, enabling precise divisor class computations and boundary descriptions within the moduli space of genus 3 curves.

Contribution

It provides explicit modular forms for hyperflex and Clebsch quartic loci, advancing the understanding of their geometric and divisor properties in moduli space.

Findings

Explicit modular form for hyperflex locus in ar{\u2113}_3

Computed divisor class of hyperflex locus in ar{\u2113}_3

Described boundary of hyperflex locus and included banana curves

Abstract

Using the results of Dalla Piazza, Fiorentino and Salvati Manni, we determine an explicit modular form defining the locus of plane quartics with a hyperflex among all plane quartics. As a result, we provide a direct way to compute the divisor class of the locus of plane quartics with a hyperflex within $\overline{\mathcal{M}_3}$, first obtained by Cukierman in 1989. Moreover, the knowledge of such an explicit modular form also allows us to describe explicitly the boundary of the hyperflex locus in $\overline{\mathcal{M}_3}$. As an example, we show that the locus of banana curves (two irreducible components intersecting at two nodes) is contained in the closure of the hyperflex locus. We also identify an explicit modular form defining the locus of Clebsch quartics and use it to recompute the class of this divisor, first obtained by Ottaviani and Sernesi.