Enumeration of rational curves with cross-ratio constraints

Tyomkin, Ilya

doi:10.1016/j.aim.2016.10.010

Cited by 21 publications

(49 citation statements)

References 17 publications

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“…Moreover, they are independent of the exact nonzero lengths of the non-degenerated tropical cross-ratios µ ′ [l ′ ] . Theorem 1.34 (Correspondence Theorem 5.1 of [Tyo17]). Let ∆ m d (α, β) be a degree as in Notation 1.17.…”

Section: Correspondence Theorem and Previous Resultsmentioning

confidence: 99%

“…The first celebrated correspondence theorem was proved by Mikhalkin [Mik05]. Tyomkin [Tyo17] proved a correspondence theorem that involves cross-ratios. Using Tyomkin's correspondence theorem, question (1) can be rephrased as (2) Determine the weighted number of rational tropical curves in R m of a given degree that satisfy general positioned point conditions and tropical cross-ratio conditions.…”

Section: Introductionmentioning

confidence: 99%

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Counting tropical rational curves with cross-ratio constraints

Goldner

2020

Math. Z.

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This is a follow-up paper of [Gol18], where rational curves in surfaces that satisfy general positioned point and cross-ratio conditions were enumerated. A suitable correspondence theorem provided in [Tyo17] 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 R 3 . Additionally, socalled 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. Notation 1.1. We write [m] ∶= {1, . . . , m} if 0 ≠ m ∈ N, and if m = 0, then define [m] ∶= ∅.Underlined symbols indicate a set of symbols, e.g. n ⊂ [m] is a subset {1, . . . , m}. We may also use sets S of symbols as an index, e.g. p S , to refer to the set of all symbols p with indices taken from S, i.e. p S ∶= {p i i ∈ S}. The #-symbol is used to indicate the number of elements in a set, for example #[m] = m.

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Section: Correspondence Theorem and Previous Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Counting tropical rational curves with cross-ratio constraints

Goldner

2020

Math. Z.

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“…As a result, we obtain a general tropical Kontsevich's formula (Theorem 68) that recursively calculates the weighted number of rational plane tropical curves of degree d that satisfy point conditions, curve conditions and tropical cross-ratio conditions. In order to obtain a classical general Kontsevich's formula (Corollary 69), we apply Tyomkin's correspondence theorem [Tyo17]. Notice that Tyomkin's correspondence theorem only holds for point and cross-ratio conditions.…”

Section: Splitting Multiplicitiesmentioning

confidence: 99%

“…holds, where λ ′ j is the tropical cross-ratio associated to µ j for j ∈ [l] in the sense of [Tyo17].…”

Section: Corollary 27 ([Gol20]mentioning

confidence: 99%

“…In [Mik07] Mikhalkin introduced a tropical version of cross-ratios under the name "tropical double ratio" to embed the moduli space of n-marked abstract rational tropical curves M 0,n into R N in order to give it the structure of a balanced fan. Tyomkin proved a correspondence theorem [Tyo17] that involves cross-ratios, where the length of a tropical cross-ratio is related to a given classical cross-ratio via the valuation map. More precisely, Tyomkin's correspondence theorem states that the number of rational plane degree d curves satisfying point and cross-ratio conditions equals its tropical counterpart.…”

Section: Introductionmentioning

confidence: 99%

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Generalizing Tropical Kontsevich's Formula to Multiple Cross-Ratios

Goldner¹

2020

Electron. J. Combin.

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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.

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