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

Luca Battistella and Francesca Carocci

Geometry & Topology 27 (2023) 1203–1272
Abstract

We construct a modular desingularisation of ¯2,n(r,d)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.

Keywords
curves of genus two, Gorenstein curves, Vakil–Zinger, desingularisation, moduli space, reduced Gromov–Witten invariants
Mathematical Subject Classification
Primary: 14H10
Secondary: 14H45, 14N35, 14T90
References
Publication
Received: 10 January 2021
Revised: 12 August 2021
Accepted: 10 September 2021
Published: 8 June 2023
Proposed: Dan Abramovich
Seconded: Mark Gross, Richard P Thomas
Authors
Luca Battistella
Humboldt-Universität zu Berlin
Institut für Mathematik
Berlin
Germany
Francesca Carocci
Institute of Mathematics
École Polytechnique Fédérale de Lausanne
Lausanne
Switzerland

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