# Masur-Veech volume of the gothic locus

## TL;DR

This paper computes the Masur-Veech volume of the gothic locus in genus four and revisits volumes of related Prym loci using a novel approach based on Euler characteristics and lattice point counts.

## Contribution

It introduces a method leveraging Euler characteristic formulas to determine Masur-Veech volumes for specific loci in moduli spaces, including the gothic and Prym loci.

## Key findings

- Calculated the Masur-Veech volume of the gothic locus in genus four.
- Recalculated volumes of Prym loci in genus three and four.
- Established a new approach connecting Euler characteristics to volume computations.

## Abstract

We calculate the Masur-Veech volume of the gothic locus $\mathcal{G}$ in the stratum $\mathcal{H}(2^{3})$ of genus four. Our method is based on the use of the formulae for the Euler characteristics of gothic Teichm\"{u}ller curves to determine the number of lattice points of given area. We also use this method to recalculate the Masur-Veech volumes of the Prym loci $\mathcal{P}_{3}\subset\mathcal{H}(4)$ and $\mathcal{P}_{4}\subset\mathcal{H}(6)$ in genus three and four.