Counting tropical rational space curves with cross-ratio constraints

manuscripta mathematica ◽

10.1007/s00229-021-01317-3 ◽

2021 ◽

Author(s):

Christoph Goldner

Keyword(s):

Tropical Geometry ◽

Rational Curves ◽

Cross Ratio ◽

Current Paper ◽

Higher Dimension ◽

Rational Space ◽

Space Curves ◽

Correspondence Theorem ◽

Floor Diagrams

AbstractThis is a follow-up paper of Goldner (Math Z 297(1–2):133–174, 2021), where rational curves in surfaces that satisfy general positioned point and cross-ratio conditions were enumerated. A suitable correspondence theorem provided in Tyomkin (Adv Math 305:1356–1383, 2017) 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. The multiplicities of such cross-ratio floor diagrams can be calculated by enumerating certain rational tropical curves in the plane.

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Counting curves on Hirzebruch surfaces: tropical geometry and the Fock space

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004120000171 ◽

2021 ◽

pp. 1-41

Author(s):

RENZO CAVALIERI ◽

PAUL JOHNSON ◽

HANNAH MARKWIG ◽

DHRUV RANGANATHAN

Keyword(s):

Fock Space ◽

Tropical Geometry ◽

Geometric Interpretation ◽

Toric Surfaces ◽

Tropical Curves ◽

Hirzebruch Surfaces ◽

Witten Theory ◽

Toric Degenerations ◽

Correspondence Theorem ◽

Floor Diagrams

We study the stationary descendant Gromov–Witten theory of toric surfaces by combining and extending a range of techniques – tropical curves, floor diagrams and Fock spaces. A correspondence theorem is established between tropical curves and descendant invariants on toric surfaces using maximal toric degenerations. An intermediate degeneration is then shown to give rise to floor diagrams, giving a geometric interpretation of this well-known bookkeeping tool in tropical geometry. In the process, we extend floor diagram techniques to include descendants in arbitrary genus. These floor diagrams are then used to connect tropical curve counting to the algebra of operators on the bosonic Fock space, and are showno coincide with the Feynman diagrams of appropriate operators. This extends work of a number of researchers, including Block–Göttsche, Cooper–Pandharipande and Block–Gathmann–Markwig.

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Generalizing Tropical Kontsevich's Formula to Multiple Cross-Ratios

The Electronic Journal of Combinatorics ◽

10.37236/9422 ◽

2020 ◽

Vol 27 (4) ◽

Author(s):

Christoph Goldner

Keyword(s):

General Position ◽

Tropical Geometry ◽

Recursive Formula ◽

Cross Ratio ◽

Initial Values ◽

Multiple Cross ◽

Correspondence Theorem ◽

The Given ◽

Rational Degree

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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Refined curve counting with tropical geometry

Compositio Mathematica ◽

10.1112/s0010437x1500754x ◽

2015 ◽

Vol 152 (1) ◽

pp. 115-151 ◽

Cited By ~ 10

Author(s):

Florian Block ◽

Lothar Göttsche

Keyword(s):

Tropical Geometry ◽

Ruled Surfaces ◽

Plane Curves ◽

Toric Surfaces ◽

Formula One ◽

Tropical Curves ◽

Tropical Curve ◽

Severi Variety ◽

Floor Diagrams

The Severi degree is the degree of the Severi variety parametrizing plane curves of degree $d$ with ${\it\delta}$ nodes. Recently, Göttsche and Shende gave two refinements of Severi degrees, polynomials in a variable $y$, which are conjecturally equal, for large $d$. At $y=1$, one of the refinements, the relative Severi degree, specializes to the (non-relative) Severi degree. We give a tropical description of the refined Severi degrees, in terms of a refined tropical curve count for all toric surfaces. We also refine the equivalent count of floor diagrams for Hirzebruch and rational ruled surfaces. Our description implies that, for fixed ${\it\delta}$, the refined Severi degrees are polynomials in $d$ and $y$, for large $d$. As a consequence, we show that, for ${\it\delta}\leqslant 10$ and all $d\geqslant {\it\delta}/2+1$, both refinements of Göttsche and Shende agree and equal our refined counts of tropical curves and floor diagrams.

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