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Register a new userGeneralizing tropical Kontsevichs formula to multiple cross-ratios
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Christoph Goldner
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Kontsevichs formula is a recursion that calculates the number of rational degree $d$ curves in $mathbb{P}_{mathbb{C}}^2$ passing through $3d-1$ general positioned points. Kontsevich proved it by considering curves that satisfy extra conditions besides the given point conditions. These crucial extra conditions are two line conditions and a condition called cross-ratio. This paper addresses the question whether there is a general Kontsevichs formula which holds for more than one cross-ratio. Using tropical geometry, we obtain such a recursive formula. For that we use a correspondence theorem arXiv:1509.07453 that relates the algebro-geometric numbers in question to tropical ones. It turns out that the general tropical Kontsevichs formula we obtain is capable of not only computing the algebro-geometric numbers we are looking for, but also of computing further tropical numbers for which there is no correspondence theorem yet. We show that our recursive general Kontsevichs formula implies the original Kontsevichs formula and that the initial values are the numbers Kontsevichs fomula provides and purely combinatorial numbers, so-called cross-ratio multiplicities.
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We enumerate rational curves in toric surfaces passing through points and satisfying cross-ratio constraints using tropical and combinatorial methods. Our starting point is arXiv:1509.07453, where a tropical-algebraic correspondence theorem was proved that relates counts of rational curves in toric varieties that satisfy point conditions and cross-ratio constraints to the analogous tropical counts. We proceed in two steps: based on tropical intersection theory we first study tropical cross-ratios and introduce degenerated cross-ratios. Second we provide a lattice path algorithm that produces all tropical curves satisfying such degenerated conditions explicitly. In a special case simpler combinatorial objects, so-called cross-ratio floor diagrams, are introduced which can be used to determine these enumerative numbers as well.
This is a follow-up paper of arXiv:1805.00115, where rational curves in surfaces that satisfy general positioned point and cross-ratio conditions were enumerated. A suitable correspondence theorem provided in arXiv:1509.07453 allowed us to use tropical geometry, and, in particular, a degeneration technique called floor diagrams. This correspondence theorem also holds in higher dimension. In the current paper, we introduce so-called cross-ratio floor diagrams and show that they allow us to determine the number of rational space curves that satisfy general positioned point and cross-ratio conditions. Moreover, graphical contributions are introduced which provide a novel and structured way of understanding multiplicities of floor decomposed curves in $mathbb{R}^3$. Additionally, so-called condition flows on a tropical curve are used to reflect how conditions imposed on a tropical curve yield different types of edges. This concept is applicable in arbitrary dimension.
The goal of this article is to classify unramified covers of a fixed tropical base curve $Gamma$ with an action of a finite abelian group G that preserves and acts transitively on the fibers of the cover. We introduce the notion of dilated cohomology groups for a tropical curve $Gamma$, which generalize simplicial cohomology groups of $Gamma$ with coefficients in G by allowing nontrivial stabilizers at vertices and edges. We show that G-covers of $Gamma$ with a given collection of stabilizers are in natural bijection with the elements of the corresponding first dilated cohomology group of $Gamma$.
A biconvex polytope is a convex polytope that is also tropically convex. It is well known that every bounded cell of a tropical linear space is a biconvex polytope, but its converse has been a conjecture. We classify biconvex polytopes, and prove the conjecture by constructing a matroid subdivision dual to a biconvex polytope. In particular, we construct matroids from bipartite graphs, and establish the relationship between bipartite graphs and faces of a biconvex polytope. We also show that there is a bijection between monomials and a maximal set of vertices of a biconvex polytope.
In this paper we prove that the cohomology of smooth projective tropical varieties verify the tropical analogs of three fundamental theorems which govern the cohomology of complex projective varieties: Hard Lefschetz theorem, Hodge-Riemann relations and monodromy-weight conjecture. On the way to establish these results, we introduce and prove other results of independent interest. This includes a generalization of the results of Adiprasito-Huh-Katz, Hodge theory for combinatorial geometries, to any unimodular quasi-projective fan having the same support as the Bergman fan of a matroid, a tropical analog for Bergman fans of the pioneering work of Feichtner-Yuzvinsky on cohomology of wonderful compactifications (treated in a separate paper, recalled and used here), a combinatorial study of the tropical version of the Steenbrink spectral sequence, a treatment of Kahler forms in tropical geometry and their associated Hodge-Lefschetz structures, a tropical version of the projective bundle formula, and a result in polyhedral geometry on the existence of quasi-projective unimodular triangulations of polyhedral spaces.
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