A smooth compactification of the space of genus two curves in projective space via logarithmic geometry and Gorenstein curves

TL;DR

The paper develops a new compactification of the moduli space of genus two curves in projective space using logarithmic geometry, resolving singularities and enabling the definition of reduced Gromov-Witten invariants.

Contribution

It introduces a modular desingularisation of the main component of the space of genus two stable maps via logarithmic modifications, addressing Gorenstein singularities.

Findings

Successfully desingularised the main component of the moduli space.

Established a framework for reduced Gromov-Witten invariants in genus two.

Revealed natural occurrence of isolated and non-reduced singularities.

Abstract

We construct a modular desingularisation of $\overline{\mathcal{M}}_{2,n}(\mathbb{P}^r,d)^{\text{main}}$. The geometry of Gorenstein singularities of genus two leads us to consider maps from prestable admissible covers: with this enhanced logarithmic structure, it is possible to desingularise the main component by means of a logarithmic modification. Both isolated and non-reduced singularities appear naturally. Our construction gives rise to a notion of reduced Gromov-Witten invariants in genus two.