Curve counting in genus one: Elliptic singularities and 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.

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

“…Naive computations of what we may now expect to coincide with our reduced invariants have made seldom appearances in the literature, for example, with Zinger's enumeration of genus two curves with a fixed complex structure in P 2 and P 3 [81], and the computation of characteristic numbers of plane curves due to T Graber, J Kock and R Pandharipande [33]. To make the relation with the latter work precise, we should first extend our methods to the analysis of relative and logarithmic stable maps (compare with Battistella, Nabijou and Ranganathan [13] and Ranganathan, Santos-Parker and Wise [64] in genus one). VZ 2;n .X; ˇ/ is only the main component of a moduli space of aligned admissible maps A 2;n .X; ˇ/, which dominates M 2;n .X; ˇ/ and is virtually birational to it.…”