Spin curves and Scorza quartics

TL;DR

This paper demonstrates that certain quartic hypersurfaces are equivalent to Scorza quartics associated with trigonal curves and theta characteristics, confirming a conjecture and exploring related moduli spaces.

Contribution

It proves the equivalence of specific quartics with Scorza quartics for general pairs of trigonal curves and theta characteristics, confirming a conjecture and describing related moduli spaces.

Findings

Quartics coincide with Scorza quartics for general pairs of trigonal curves.

Confirmed the conjecture of Dolgachev and Kanev on existence of Scorza quartics.

Described moduli spaces of trigonal even spin curves.

Abstract

In the previous paper, we construct new subvarieties in the varieties of power sums for certain quartic hypersurfaces. In this paper, we show that these quartics coincide with the Scorza quartics of general pairs of trigonal curves and ineffective theta characteristics. Among other applications, we give an affirmative answer to the conjecture of Dolgachev and Kanev on the existence of the Scorza quartics for any general pairs of curves and ineffective theta characteristics. We also give descriptions of the moduli spaces of trigonal even spin curves. For curves of genus 4, we deepen this description in the next paper.