For simple normal crossings divisors, logarithmic and local/naive Gromov-Witten invariants with maximal contacts differ, and this paper gives the first counterexamples plus a blowup formula measuring the difference.
Curve counting in genus one: elliptic singularities & relative geometry
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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.
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Gromov-Witten theory with maximal contacts
For simple normal crossings divisors, logarithmic and local/naive Gromov-Witten invariants with maximal contacts differ, and this paper gives the first counterexamples plus a blowup formula measuring the difference.