Generalizing Tropical Kontsevich's Formula to Multiple Cross-Ratios

Abstract

Kontsevich's formula is a recursion that calculates the number of rational degree $d$ curves in $\mathbb{P}_{\mathbb{C}}^2$ passing through $3d-1$ points in general position. 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 Kontsevich's 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 of Tyomkin that relates the algebro-geometric numbers in question to tropical ones. It turns out that the general tropical Kontsevich's 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 Kontsevich's formula implies the original Kontsevich's formula and that the initial values are the numbers Kontsevich's fomula provides and purely combinatorial numbers, so-called cross-ratio multiplicities.

Counting tropical rational space curves with cross-ratio constraints

“…Remark 2.16. Notice that Proposition 2.14 can also be shown the way Corollary 2.31 in [15] was shown, where Corollary 2.31 follows from Proposition 2.1 of [15]. Since Proposition 2.1 of [15] is actually a stronger statement than Proposition 2.14 there is is no need to evoke the machinery developed in [15].…”

“…suitable cross-ratio floor diagrams are introduced that yield a combinatorial solution of question (2), see Theorem 4.1. It turns out that the multiplicities needed for such cross-ratio floor diagrams are exactly the numbers general Kontsevich's formula [15] provides (Corollary 3.7), i.e. in order to answer question (2) for m = 3, it is necessary to know its answer in case of m = 2.…”

“…the submatrices B, M marked above are equal. Therefore There is a general Kontsevich's formula [15] which recursively calculates the weighted number of rational tropical curves in R 2 that satisfy point, multi-line and cross-ratio conditions. As a consequence, the multiplicity of a cross-ratio floor diagram can be determined recursively.…”

“…In case of plane curves (i.e. m = 2) question (2) (and therefore question (1)) was answered exhaustively by giving a recursive formula [15] and by explicitly constructing plane rational tropical curves that satisfy the given conditions [16]. The present paper determines the numbers of questions (1), (2) in case of space curves, i.e.…”

“…Defining the multiplicity of such a cross-ratio floor diagram is done via a similar approach as in [15]. On the other hand, so-called condition flows are used to reflect how conditions imposed on a tropical curve yield different types of edges.…”