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arXiv · 2008.13506

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

Luca Battistella·Francesca Carocci
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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.

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Luca Battistella, Francesca Carocci. 2020-08-31. A smooth compactification of the space of genus two curves in projective space via logarithmic geometry and Gorenstein curves. https://doi.org/10.2140/gt.2023.27.1203

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