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Battistella, Luca; Carocci, Francesca

A smooth compactification of the space of genus two curves in projective space: via logarithmic geometry and Gorenstein curves. (English) Zbl 1521.14052

Geom. Topol. 27, No. 3, 1203-1272 (2023).
The boundary components of the space of stable maps to projective space may in higher genus have excess dimension. The goal of this paper is an explicit modular desingularisation of the main component of the moduli space of stable maps to projective space in genus two and degree \(d\geq 3\). More precisely, there exists a logarithmically smooth and proper DM stack \(\mathcal{VZ}_{2,n}(\mathbb P^r,d)\) mapping to the main component. Families of curves of genus two are described by double covers of families of rational curves. Besides the curves with isolated singularities studied earlier by the first author also non-reduced curves occur.
Reviewer: Jan Stevens (Göteborg)

Cited in 2 Documents

MSC:

14H10 Families, moduli of curves (algebraic)
14H45 Special algebraic curves and curves of low genus
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14T90 Applications of tropical geometry

Keywords:

curves of genus two; Gorenstein curves; desingularisation; moduli space; reduced Gromov-Witten invariants
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References:

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