scholarly journals Refined curve counting with tropical geometry

2015 ◽  
Vol 152 (1) ◽  
pp. 115-151 ◽  
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.

scholarly journals Universal Polynomials for Severi Degrees of Toric Surfaces

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Federico Ardila ◽  
Florian Block
Keyword(s):  
Tropical Geometry ◽  
Large Family ◽  
Plane Curves ◽  
Toric Surfaces ◽  
Combinatorial Objects ◽  
Severi Varieties ◽  
Severi Variety ◽  
Nous Montrons ◽  
International Audience ◽  
Floor Diagrams

International audience The Severi variety parametrizes plane curves of degree $d$ with $\delta$ nodes. Its degree is called the Severi degree. For large enough $d$, the Severi degrees coincide with the Gromov-Witten invariants of $\mathbb{CP}^2$. Fomin and Mikhalkin (2009) proved the 1995 conjecture that for fixed $\delta$, Severi degrees are eventually polynomial in $d$. In this paper, we study the Severi varieties corresponding to a large family of toric surfaces. We prove the analogous result that the Severi degrees are eventually polynomial as a function of the multidegree. More surprisingly, we show that the Severi degrees are also eventually polynomial "as a function of the surface". Our strategy is to use tropical geometry to express Severi degrees in terms of Brugallé and Mikhalkin's floor diagrams, and study those combinatorial objects in detail. An important ingredient in the proof is the polynomiality of the discrete volume of a variable facet-unimodular polytope. La variété de Severi paramétrise les courbes planes de degré $d$ avec $\delta$ nœuds. Son degré s'appelle le degré de Severi. Pour $d$ assez grand, les degrés de Severi coïncident avec les invariants de Gromov-Witten de $\mathbb{CP}^2$. Fomin et Mikhalkin (2009) ont prouvé une conjecture de 1995 que pour $\delta$ fixé, les degrés de Severi sont à terme des polynômes en $d$. Nous étudions les variétés de Severi correspondant à une large famille de surfaces toriques. Nous prouvons le résultat analogue que les degrés de Severi sont à terme des fonctions polynomiales du multidegré. De manière plus surprenante, nous montrons que les degrés de Severi sont à terme des polynômes en tant que "fonction de la surface''. Notre stratégie est d'utiliser la géométrie tropicale pour exprimer les degrés de Severi en fonction des "floor diagrams" de Brugallé et Mikhalkin, et d'utiliser ces objets combinatoires en détail. Un autre ingrédient important de la preuve est la polynomialité du volume discret d'un polytope face-unimodulaire variable.

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

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.

scholarly journals $q$-Floor Diagrams computing Refined Severi Degrees for Plane Curves

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Florian Block
Keyword(s):  
Plane Curves ◽  
Combinatorial Description ◽  
Severi Variety ◽  
International Audience ◽  
Floor Diagrams

International audience The Severi degree is the degree of the Severi variety parametrizing plane curves of degree $d$ with $\delta$ nodes. Recently, Göttsche and Shende gave two refinements of Severi degrees, polynomials in a variable $q$, which are conjecturally equal, for large $d$. At $q=1$, one of the refinements, the relative Severi degree, specializes to the (non-relative) Severi degree. We give a combinatorial description of the refined Severi degrees, in terms of a $q$-analog count of Brugallé and Mikhalkin's floor diagrams. Our description implies that, for fixed $\delta$, the refined Severi degrees are polynomials in $d$ and $q$, for large $d$. As a consequence, we show that, for $\delta \leq 4$ and all $d$, both refinements of Göttsche and Shende agree and equal our $q$-count of floor diagrams. Le degré de Severi est le degré de la variété de Severi paramétrisant les courbes planes de degré $d$ à $\delta$ nœuds. Récemment, Göttsche et Shende ont donné deux raffinements des degrés de Severi, polynomiaux en la variable $q$, qui sont conjecturalement égaux pour $d$ assez grand. Pour $q=1$, un des ces raffinements, le degré de Severi relatif, se spécialise en le degré de Severi (non relatif). Nous donnons une description combinatoire des degrés de Severi raffinés, en fonction d'un comptage $q$-analogue des "floor diagrams'' de Brugallé et Mikhalkin. Notre description implique que, pour $\delta$ fixé, les degrés de Severi raffinés sont polynomiaux en $d$ et $q$, pour $d$ grand. On montre que, par conséquent, pour $\delta \leq 4$ et pour tout $d$, les deux raffinements de Göttsche et Shende coïncident et sont égaux à notre $q$-analogue de "floor diagrams''.

scholarly journals MODULI SPACES OF RATIONAL WEIGHTED STABLE CURVES AND TROPICAL GEOMETRY

2016 ◽  
Vol 4 ◽  
Author(s):  
RENZO CAVALIERI ◽  
SIMON HAMPE ◽  
HANNAH MARKWIG ◽  
DHRUV RANGANATHAN
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Tropical Geometry ◽  
Stable Curves ◽  
Tropical Curves ◽  
Graphic Matroid ◽  
Fibre Products ◽  
Algebraic Counterpart

We study moduli spaces of rational weighted stable tropical curves, and their connections with Hassett spaces. Given a vector $w$ of weights, the moduli space of tropical $w$-stable curves can be given the structure of a balanced fan if and only if $w$ has only heavy and light entries. In this case, the tropical moduli space can be expressed as the Bergman fan of an explicit graphic matroid. The tropical moduli space can be realized as a geometric tropicalization, and as a Berkovich skeleton, its algebraic counterpart. This builds on previous work of Tevelev, Gibney and Maclagan, and Abramovich, Caporaso and Payne. Finally, we construct the moduli spaces of heavy/light weighted tropical curves as fibre products of unweighted spaces, and explore parallels with the algebraic world.

Bend, break and count II: elliptics, cuspidals, linear genera

1999 ◽  
Vol 127 (1) ◽  
pp. 7-12
Author(s):  
Z. RAN
Keyword(s):  
Ruled Surfaces ◽  
Plane Curves ◽  
Recursive Procedure ◽  
Rational Plane ◽  
Tangent Direction

In [R2] we showed how elementary considerations involving geometry on ruled surfaces may be used to obtain recursive enumerative formulae for rational plane curves. Here we show how similar considerations may be used to obtain further enumerative formulae, as follow. First some notation. As usual we denote by Ngd the number of irreducible plane curves of degree d and genus g through 3d+g−1 general points. Also, we denote by Ngd→ (resp. Ngd×) the number of such curves passing through general points A1, …, A3d+g−2 and having a given tangent direction (resp. a node) at A1. As is well known and easy to see, we haveformula hereFor any d, g, these numbers are computed in [R1] as part of a more general recursive procedure. For N0d, N1d, relatively simple recursions have been given by Kontsevich–Manin (see [FP]) and Eguchi–Hori–Xiong–Getzler (see [P1]), respectively.

scholarly journals The moduli space of Harnack curves in toric surfaces

2021 ◽  
Vol 9 ◽  
Author(s):  
Jorge Alberto Olarte
Keyword(s):  
Moduli Space ◽  
Newton Polygon ◽  
Lattice Points ◽  
Toric Surface ◽  
Toric Surfaces ◽  
Tropical Curves ◽  
Secondary Polytope

Abstract In 2006, Kenyon and Okounkov Kenyon and Okounkov [12] computed the moduli space of Harnack curves of degree d in ${\mathbb {C}\mathbb {P}}^2$ . We generalise their construction to any projective toric surface and show that the moduli space ${\mathcal {H}_\Delta }$ of Harnack curves with Newton polygon $\Delta $ is diffeomorphic to ${\mathbb {R}}^{m-3}\times {\mathbb {R}}_{\geq 0}^{n+g-m}$ , where $\Delta $ has m edges, g interior lattice points and n boundary lattice points. This solves a conjecture of Crétois and Lang. The main result uses abstract tropical curves to construct a compactification of this moduli space where additional points correspond to collections of curves that can be patchworked together to produce a curve in ${\mathcal {H}_\Delta }$ . This compactification has a natural stratification with the same poset as the secondary polytope of $\Delta $ .

scholarly journals Labeled floor diagrams for plane curves

2010 ◽  
pp. 1453-1496 ◽  
Author(s):  
Sergey Fomin ◽  
Grigory Mikhalkin
Keyword(s):  
Plane Curves ◽  
Floor Diagrams

scholarly journals TBA type equations and tropical curves

2016 ◽  
Vol 27 (07) ◽  
pp. 1640005 ◽  
Author(s):  
Sara Angela Filippini ◽  
Jacopo Stoppa
Keyword(s):  
Integrable System ◽  
Tropical Geometry ◽  
Tropical Curves ◽  
Abelian Categories ◽  
Stability Data

The Joyce integrable system and the corresponding Bridgeland–Toledano-Laredo connections are fundamental objects associated with suitable abelian categories or, more generally, with a class of continuous families of stability data. We offer an overview of some of our work, mostly joint with M. Garcia Fernandez, focusing on equations of TBA type as a useful tool in the analysis of these objects and their deformations, and as a means to establish a connection with tropical geometry.

scholarly journals Irreducible Cycles and Points in Special Position in Moduli Spaces for Tropical Curves

10.37236/2862 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Andreas Gathmann ◽  
Franziska Schroeter
Keyword(s):  
Moduli Spaces ◽  
Plane Curves ◽  
Special Position ◽  
Tropical Curves ◽  
Point Configurations ◽  
Rational Plane

In the first part of this paper, we discuss the notion of irreducibility of cycles in the moduli spaces of $n$-marked rational tropical curves. We prove that Psi-classes and vital divisors are irreducible, and that locally irreducible divisors are also globally irreducible for $ n \le 6 $. In the second part of the paper, we show that the locus of point configurations in $({\mathbb R}^2)^n $ in special position for counting rational plane curves (defined in two different ways) can be given the structure a tropical cycle of codimension $1$. In addition, we compute explicitly the weights of this cycle.

scholarly journals Genus 0 characteristic numbers of the tropical projective plane

2013 ◽  
Vol 150 (1) ◽  
pp. 46-104 ◽  
Author(s):  
Benoît Bertrand ◽  
Erwan Brugallé ◽  
Grigory Mikhalkin
Keyword(s):  
Projective Plane ◽  
Tropical Geometry ◽  
Classical Problem ◽  
Integer Number ◽  
Hurwitz Numbers ◽  
Formula One ◽  
Characteristic Numbers ◽  
Pass Through ◽  
Finite Integer ◽  
Correspondence Theorem

AbstractFinding the so-called characteristic numbers of the complex projective plane$ \mathbb{C} {P}^{2} $is a classical problem of enumerative geometry posed by Zeuthen more than a century ago. For a given$d$and$g$one has to find the number of degree$d$genus$g$curves that pass through a certain generic configuration of points and at the same time are tangent to a certain generic configuration of lines. The total number of points and lines in these two configurations is$3d- 1+ g$so that the answer is a finite integer number. In this paper we translate this classical problem to the corresponding enumerative problem of tropical geometry in the case when$g= 0$. Namely, we show that the tropical problem is well posed and establish a special case of the correspondence theorem that ensures that the corresponding tropical and classical numbers coincide. Then we use the floor diagram calculus to reduce the problem to pure combinatorics. As a consequence, we express genus 0 characteristic numbers of$ \mathbb{C} {P}^{2} $in terms of open Hurwitz numbers.

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