On T-divisors and intersections in $\overline{M}_{1,3}$

TL;DR

This paper investigates the structure of the moduli space of certain stable surfaces, revealing intersections of components, new boundary divisors, and their intersection properties, advancing understanding of the space's geometry.

Contribution

It demonstrates the transversal intersection of known components and introduces new boundary divisors in the moduli space of stable surfaces with specific invariants.

Findings

Two known components intersect transversally in a divisor

Identification of new boundary divisors

Analysis of intersection patterns among divisors

Abstract

The moduli space of stable surfaces with $K_X^2 = 1$ and $\chi(X) = 3$ has at least two irreducible components that contain surfaces with T-singularities. We show that the two known components intersect transversally in a divisor. Moreover, we exhibit other new boundary divisors and study how they intersect one another.