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

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.

arXiv comment47 pages, 13 figures, comments are welcome! v2: minor expository improvements throughout the paper. v3: the main construction (Section 3) has been split into two steps, the second of which addresses the hyperelliptic components, with the advantage of simplifying the singularities that we have to introduce. Revised version to appear in Geometry & Topology

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Cited by in corpus (5)

• Curve counting in genus one: elliptic singularities & relative geometry
• Hyperelliptic Gorenstein curves and logarithmic differentials
• Gorenstein curve singularities of genus three
• A geographical study of ${\overline{\mathcal{M}}}_2({\mathbb{P}}^2,4{)}^{\text{main}}$
• The dual complex of ${\mathcal{M}}_{1,n}({\mathbb{P}}^r,d)$ via the geometry of the Vakil--Zinger moduli space