Andreas Gathmann

Absolute and relative Gromov-Witten invariants of very ample hypersurfaces

For any smooth complex projective variety X and smooth very ample hypersurface Y in X, we develop the technique of genus zero relative Gromov-Witten invariants of Y in X in algebro-geometric terms. We prove an equality of cycles in the Chow groups of the moduli spaces of relative stable maps that relates these relative invariants to the Gromov-Witten invariants of X and Y. Given the Gromov-Witten invariants of X, we show that these relations are sufficient to compute all relative invariants, as well as all genus zero Gromov-Witten invariants of Y whose homology and cohomology classes are induced by X.

The numbers of tropical plane curves through points in general position

Abstract We show that the number of tropical curves of given genus and degree through some given general points in the plane does not depend on the position of the points. In the case when the degree of the curves contains only primitive integral vectors this statement has been known for a while now, but the only known proof was indirect with the help of Mikhalkins Correspondence Theorem that translates this question into the well-known fact that the numbers of complex curves in a toric surface through some given points do not depend on the position of the points. This paper presents a direct proof entirely within tropical geometry that is in addition applicable to arbitrary degree of the curves.

Psi-floor diagrams and a Caporaso-Harris type recursion

Floor diagrams are combinatorial objects which organize the count of tropical plane curves satisfying point conditions. In this paper we introduce Psi-floor diagrams which count tropical curves satisfying not only point conditions but also conditions given by Psi-classes (together with points). We then generalize our definition to relative Psi-floor diagrams and prove a Caporaso-Harris type formula for the corresponding numbers. This formula is shown to coincide with the classical Caporaso-Harris formula for relative plane descendant Gromov-Witten invariants. As a consequence, we can conclude that in our case relative descendant Gromov-Witten invariants equal their tropical counterparts.

The number of plane conics that are five-fold tangent to a given curve

Given a general plane curve Y of degree d, we compute the number nd of irreducible plane conics that are five-fold tangent to Y . This problem has been studied before by Vainsencher using classical methods, but it could not be solved because the calculations produced too many non-enumerative correction terms that could not be analyzed. In our current approach, we express the number nd in terms of relative Gromov–Witten invariants that can then be directly computed. As an application, we consider the K3 surface given as the double cover of P2 branched along a sextic curve. We compute the number of rational curves in this K3 surface in the homology class that is the pull-back of conics in P2, and compare this number with the corresponding Yau–Zaslow K3 invariant. This gives an example of such a K3 invariant for a non-primitive homology class. Let Y ⊂ P2 be a generic plane curve of degree d 5. We want to consider smooth plane conics that are five-fold tangent to Y . As the space of all plane conics is five-dimensional and each tangency imposes one condition on the curves, we expect a finite number of such five-fold tangent conics. It will be easy to see that this number is indeed finite; let us call it nd. The goal of this paper is to compute it. Of course this is a classical problem, and attempts have been made to solve it using classical methods of enumerative geometry. Vainsencher [Vai98] tried to use various blow-ups of the ordinary P5 of conics as moduli spaces, but the intersection of the five tangency conditions in this moduli space always resulted in a scheme with many non-enumerative and non-reduced components whose geometry was so complicated that the problem could not be solved that way. In this paper we use different moduli spaces, namely moduli spaces of relative stable maps, to solve the problem. There is a well-defined compact moduli space M̄Y (2,2,2,2,2)(P 2, 2) ⊂ M̄0,5(P, 2) that parametrizes rational stable maps to P2 of degree 2 (i.e. conics) with five marked points such that the stable map is tangent to Y at all of these points. It comes equipped with a zero-dimensional virtual fundamental class, whose degree Nd can be computed explicitly using the methods of [Gat02]. We can interpret the number Nd as the ‘virtual number’ of conics that are five-fold tangent to Y . It is only virtual because it contains, just as in Vainsencher’s classical computations, nonenumerative contributions from the ‘boundary’ of the moduli space. These contributions are quite simple however. It is not hard to see that the only degree-2 rational stable maps f : C → P2 that satisfy the tangency conditions at the five marked points are all double covers of a bitangent of Y , and have the marked points distributed in one of the following two ways: Received 5 August 2003, accepted in final form 22 March 2004, published online 10 February 2005. 2000 Mathematics Subject Classification 14N35.

Tropical Moduli Spaces of Stable Maps to a Curve

We construct moduli spaces of rational covers of an arbitrary smooth tropical curve in \({\mathbb R}^r\) as tropical varieties. They are contained in the balanced fan parametrizing tropical stable maps of the appropriate degree to \({\mathbb R}^r\). The weights of the top-dimensional polyhedra are given in terms of certain lattice indices and local Hurwitz numbers.

Moduli Spaces of Curves in Tropical Varieties

We describe a framework to construct tropical moduli spaces of rational stable maps to a smooth tropical hypersurface or curve. These moduli spaces will be tropical cycles of the expected dimension, corresponding to virtual fundamental classes in algebraic geometry. As we focus on the combinatorial aspect, we take the weights on certain basic 0-dimensional local combinatorial curve types as input data, and give a compatibility condition in dimension 1 to ensure that this input data glues to a global well-defined tropical cycle. As an application, we construct such moduli spaces for the case of lines in surfaces, and in a subsequent paper for stable maps to a curve [Gathmann et al., Tropical moduli spaces of stable maps to a curve, in Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory, ed. by G. Bockle, W. Decker, G. Malle (Springer, Heidelberg, 2018). https://doi.org/10.1007/978-3-319-70566-8_12].