arXiv subjects
Publications and source records attributed to Andreas Gathmann.
We construct moduli spaces of rational covers of an arbitrary smooth tropical curve in R^r as tropical varieties. They are contained in the balanced fan parametrizing tropical stable maps of the appropriate degree to R^r. The weights of the top-dimensional polyhedra are given in terms of certain lattice indices and local Hurwitz numbers.
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.
Let X be a plane in a torus over an algebraically closed field K, with tropicalization the matroidal fan Sigma. In this paper we present an algorithm which completely solves the question whether a given one-dimensional balanced polyhedral complex in Sigma is relatively realizable, i.e. whether it is the tropicalization of an algebraic curve, over the field of Puiseux series over K, in X. The algorithm implies that the space of all such relatively realizable curves of fixed degree is an abstract polyhedral set. In the case when X is a general plane in 3-space, we use the idea of this algorithm to prove some necessary and some sufficient conditions for relative realizability. For 1-dimensional polyhedral complexes in Sigma that have exactly one bounded edge, passing through the origin, these necessary and sufficient conditions coincide, so that they give a complete non-algorithmic solution of the relative realizability problem.
In this paper we introduce broccoli curves, certain plane tropical curves of genus zero related to real algebraic curves. The numbers of these broccoli curves through given points are independent of the chosen points - for arbitrary choices of the directions of the ends of the curves, possibly with higher weights, and also if some of the ends are fixed. In the toric Del Pezzo case we show that these broccoli invariants are equal to the Welschinger invariants (with real and complex conjugate point conditions), thus providing a proof of the independence of Welschinger invariants of the point conditions within tropical geometry. The general case gives rise to a tropical Caporaso-Harris formula for broccoli curves which suffices to compute all Welschinger invariants of the plane.
In the first part of this paper, we discuss the notion of irreducibility of cycles in the moduli spaces of n-marked rational tropical curves. We prove that Psi-classes and vital divisors are irreducible, and that locally irreducible divisors are also globally irreducible for n \leq 6. In the second part of the paper, we show that the locus of point configurations in (\R^2)^n in special position for counting rational plane curves (defined in two different ways) can be given the structure a tropical cycle of codimension 1. In addition, we compute explicitly the weights of this cycle.
Let E be a plane in an algebraic torus over an algebraically closed field. Given a balanced 1-dimensional fan C in the tropicalization of E, i.e. in the Bergman fan of the corresponding matroid, we give a complete algorithmic answer to the question whether or not C can be realized as the tropicalization of an algebraic curve contained in E. Moreover, in the case of realizability the algorithm also determines the dimension of the moduli space of all algebraic curves in E tropicalizing to C, a concrete simple example of such a curve, and whether C can also be realized by an irreducible algebraic curve in E. In the first important case when E is a general plane in a 3-dimensional torus we also use our algorithm to prove some general criteria for C that imply its realizability resp. non-realizability. They include and generalize the main known obstructions by Brugalle-Shaw and Bogart-Katz coming from tropical intersection theory.
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.
We give a rigorous definition of tropical fans (the "local building blocks for tropical varieties") and their morphisms. For such a morphism of tropical fans of the same dimension we show that the number of inverse images (counted with suitable tropical multiplicities) of a point in the target does not depend on the chosen point - a statement that can be viewed as the beginning of a tropical intersection theory. As an application we consider the moduli spaces of rational tropical curves (both abstract and in some R^r) together with the evaluation and forgetful morphisms. Using our results this gives new, easy, and unified proofs of various tropical independence statements, e.g. of the fact that the numbers of rational tropical curves (in any R^r) through given points are independent of the points.
Using Gromov-Witten theory the numbers of complex plane rational curves of degree d through 3d-1 general given points can be computed recursively with Kontsevich's formula that follows from the so-called WDVV equations. In this paper we establish the same results entirely in the language of tropical geometry. In particular this shows how the concepts of moduli spaces of stable curves and maps, (evaluation and forgetful) morphisms, intersection multiplicities and their invariance under deformations can be carried over to the tropical world.
Some years ago Caporaso and Harris have found a nice way to compute the numbers N(d,g) of complex plane curves of degree d and genus g through 3d+g-1 general points with the help of relative Gromov-Witten invariants. Recently, Mikhalkin has found a way to reinterpret the numbers N(d,g) in terms of tropical geometry and to compute them by counting certain lattice paths in integral polytopes. We relate these two results by defining an analogue of the relative Gromov-Witten invariants and rederiving the Caporaso-Harris formula in terms of both tropical geometry and lattice paths.
Recently, Baker and Norine have proven a Riemann-Roch theorem for finite graphs. We extend their results to metric graphs and thus establish a Riemann-Roch theorem for divisors on (abstract) tropical curves.
Tropical algebraic geometry is an active new field of mathematics that establishes and studies some very general principles to translate algebro-geometric problems into purely combinatorial ones. This expository paper gives an introduction to these new techniques with a special emphasis on the recent applications to problems in enumerative geometry.
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 Mikhalkin's 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.
We state and prove a topological recursion relation that expresses any genus-g Gromov-Witten invariant of a projective manifold with at least a (3g-1)-st power of a cotangent line class in terms of invariants with fewer cotangent line classes. For projective spaces, we prove that these relations together with the Virasoro conditions are sufficient to calculate the full Gromov-Witten potential. This gives the first computationally feasible way to determine the higher genus Gromov-Witten invariants of projective spaces.
Given a general plane curve Y of degree d, we compute the number n_d of irreducible plane conics that are 5-fold tangent to Y. This problem has been studied before by Vainsencher using classical methods, but it could not be solved there because the calculations received too many non-enumerative correction terms that could not be analyzed. In our current approach, we express the number n_d 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 P^2 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 P^2, and compare this number to the corresponding Yau-Zaslow K3 invariant. This gives an example of such a K3 invariant for a non-primitive homology class.
Let X be a smooth complex projective variety, and let Y in X be a smooth very ample hypersurface such that -K_Y is nef. Using the technique of relative Gromov-Witten invariants, we give a new short and geometric proof of (a version of) the "mirror formula", i.e. we show that the generating function of the genus zero 1-point Gromov-Witten invariants of Y can be obtained from that of X by a certain change of variables (the so-called "mirror transformation"). Moreover, we use the same techniques to give a similar expression for the (virtual) numbers of degree-d plane rational curves meeting a smooth cubic at one point with multiplicity 3d, which play a role in local mirror symmetry.
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.
In the first part of the paper, we give an explicit algorithm to compute the (genus zero) Gromov-Witten invariants of blow-ups of an arbitrary convex projective variety in some points if one knows the Gromov-Witten invariants of the original variety. In the second part, we specialize to blow-ups of P^r and show that many invariants of these blow-ups can be interpreted as numbers of rational curves on P^r having specified global multiplicities or tangent directions in the blown-up points. We give various numerical examples, including a new easy way to determine the famous multiplicity d^{-3} for d-fold coverings of rational curves on the quintic threefold, and, as an outlook, two examples of blow-ups along subvarieties, whose Gromov-Witten invariants lead to classical multisecant formulas.