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

Battistella, Luca¹ ; Carocci, Francesca²

¹ Humboldt-Universität zu Berlin, Institut für Mathematik, Berlin, Germany
² Institute of Mathematics, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

Geometry & topology, Tome 27 (2023) no. 3, pp. 1203-1272.

Résumé

We construct a modular desingularisation of ${\overline{\mathcal{ℳ}}}_{2,n}{({\mathbb{P}}^r,d)}^{\mathrm{main}}$. 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 nonreduced singularities appear naturally. Our construction gives rise to a notion of reduced Gromov–Witten invariants in genus two.

Détail

DOI : 10.2140/gt.2023.27.1203

Keywords: curves of genus two, Gorenstein curves, Vakil–Zinger, desingularisation, moduli space, reduced Gromov–Witten invariants

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