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Register a new userCounting tropical rational curves with cross-ratio constraints
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Christoph Goldner
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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.
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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.
In a previous paper, we announced a formula to compute Gromov-Witten and Welschinger invariants of some toric varieties, in terms of combinatorial objects called floor diagrams. We give here detailed proofs in the tropical geometry framework, in the case when the ambient variety is a complex surface, and give some examples of computations using floor diagrams. The focusing on dimension 2 is motivated by the special combinatoric of floor diagrams compared to arbitrary dimension. We treat a general toric surface case in this dimension: the curve is given by an arbitrary lattice polygon and include computation of Welschinger invariants with pairs of conjugate points. See also cite{FM} for combinatorial treatment of floor diagrams in the projective case.
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.
We study generating series of Gromov-Witten invariants of $Etimesmathbb{P}^1$ and their tropical counterparts. Using tropical degeneration and floor diagram techniques, we can express the generating series as sums of Feynman integrals, where each summand corresponds to a certain type of graph which we call a pearl chain. The individual summands are --- just as in the case of mirror symmetry of elliptic curves, where the generating series of Hurwitz numbers equals a sum of Feynman integrals --- complex analytic path integrals involving a product of propagators (equal to the Weierstrass-$wp$-function plus an Eisenstein series). We also use pearl chains to study generating functions of counts of tropical curves in $E_{mathbb{T}}timesmathbb{P}^1_mathbb{T}$ of so-called leaky degree.
This is a sequel to our work in tropical Hodge theory. Our aim here is to prove a tropical analogue of the Clemens-Schmid exact sequence in asymptotic Hodge theory. As an application of this result, we prove the tropical Hodge conjecture for smooth projective tropical varieties which are rationally triangulable. This provides a partial answer to a question of Kontsevich who suggested the validity of the tropical Hodge conjecture could be used as a test for the validity of the Hodge conjecture.
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