Masur-Veech volume of the gothic locus

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.