Dynamical invariants and intersection theory on the flex and gothic loci

TL;DR

This paper studies special loci in moduli space related to plane cubics and quadratic differentials, determining divisor classes and intersection numbers, and confirming a conjecture on Lyapunov exponents in Teichmüller dynamics.

Contribution

It computes divisor classes and intersection numbers for the flex and gothic loci, providing new insights into their geometric and dynamical properties.

Findings

Divisor class of the flex locus determined

Intersection numbers on the gothic locus computed

Numerical confirmation of a conjecture on Lyapunov exponents

Abstract

The flex locus parameterizes plane cubics with three collinear cocritical points under a projection, and the gothic locus arises from quadratic differentials with zeros at a fiber of the projection and with poles at the cocritical points. The flex and gothic loci provide the first example of a primitive, totally geodesic subvariety of moduli space and new ${\rm SL}_2(\mathbb{R})$-invariant varieties in Teichm\"uller dynamics, as discovered by McMullen-Mukamel-Wright. In this paper we determine the divisor class of the flex locus as well as various tautological intersection numbers on the gothic locus. For the case of the gothic locus our result confirms numerically a conjecture of Chen-M\"oller-Sauvaget about computing sums of Lyapunov exponents for ${\rm SL}_2(\mathbb{R})$-invariant varieties via intersection theory.