A geographical study of $\overline{\mathcal M}_{2}(\mathbb{P}^2,4)^{\text{main}}$

TL;DR

This paper investigates the conditions under which certain genus two, degree four stable maps to the projective plane can be smoothed, utilizing advanced geometric techniques to understand their moduli space.

Contribution

It introduces criteria for smoothability of genus two, degree four stable maps to the projective plane, applying modular desingularization via logarithmic geometry and Gorenstein singularities.

Findings

Criteria for smoothability of stable maps established

Application of modular desingularization techniques

Enhanced understanding of the moduli space structure

Abstract

We discuss criteria for a stable map of genus two and degree $4$ to the projective plane to be smoothable, as an application of our modular desingularisation of $\overline{\mathcal M}_{2,n}(\mathbb{P}^r,d)^{\text{main}}$ via logarithmic geometry and Gorenstein singularities.