Curve counting in genus one: elliptic singularities & relative geometry

Abstract

We construct and study the reduced, relative, genus one Gromov--Witten theory of very ample pairs. These invariants form the principal component contribution to relative Gromov--Witten theory in genus one and are relative versions of Zinger's reduced Gromov--Witten invariants. We relate the relative and absolute theories by degeneration of the tangency conditions, and the resulting formulas generalise a well-known recursive calculation scheme put forward by Gathmann in genus zero. The geometric input is a desingularisation of the principal component of the moduli space of genus one logarithmic stable maps to a very ample pair, using the geometry of elliptic singularities. Our study passes through general techniques for calculating integrals on logarithmic blowups of moduli spaces of stable maps, which may be of independent interest.

arXiv comment38 pages, 11 figures, comments welcome! v2: exposition improved thanks to the referee's comments. Final version to appear in Algebraic Geometry

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Cited by in corpus (6)

• Gromov-Witten theory with maximal contacts
• A smooth compactification of the space of genus two curves in projective space via logarithmic geometry and Gorenstein curves
• Tangent curves to degenerating hypersurfaces
• Hyperelliptic Gorenstein curves and logarithmic differentials
• Gorenstein curve singularities of genus three
• The dual complex of ${\mathcal{M}}_{1,n}({\mathbb{P}}^r,d)$ via the geometry of the Vakil--Zinger moduli space