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 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 $q$-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 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 Conjectures on Stably Newton Degenerate Singularities

2021 ◽  
Author(s):  
Jan Stevens
Keyword(s):  
Characteristic Zero ◽  
Arbitrary Number ◽  
Milnor Number ◽  
Plane Curves ◽  
Newton Diagram ◽  
Finite Characteristic ◽  
Vanishing Cycles

AbstractWe discuss a problem of Arnold, whether every function is stably equivalent to one which is non-degenerate for its Newton diagram. We argue that the answer is negative. We describe a method to make functions non-degenerate after stabilisation and give examples of singularities where this method does not work. We conjecture that they are in fact stably degenerate, that is not stably equivalent to non-degenerate functions.We review the various non-degeneracy concepts in the literature. For finite characteristic, we conjecture that there are no wild vanishing cycles for non-degenerate singularities. This implies that the simplest example of singularities with finite Milnor number, $$x^p+x^q$$ x p + x q in characteristic p, is not stably equivalent to a non-degenerate function. We argue that irreducible plane curves with an arbitrary number of Puiseux pairs (in characteristic zero) are stably non-degenerate. As the stabilisation involves many variables, it becomes very difficult to determine the Newton diagram in general, but the form of the equations indicates that the defining functions are non-degenerate.

On an area-preserving inverse curvature flow of convex closed plane curves

2021 ◽  
Vol 280 (8) ◽  
pp. 108931
Author(s):  
Laiyuan Gao ◽  
Shengliang Pan ◽  
Dong-Ho Tsai
Keyword(s):  
Curvature Flow ◽  
Plane Curves ◽  
Area Preserving ◽  
Closed Plane

scholarly journals On 3-syzygy and unexpected plane curves

2021 ◽  
Author(s):  
Grzegorz Malara ◽  
Piotr Pokora ◽  
Halszka Tutaj-Gasińska
Keyword(s):  
Projective Plane ◽  
Plane Curves ◽  
Complex Projective Plane ◽  
Geometric Properties ◽  
Complex Projective

AbstractIn this note we study curves (arrangements) in the complex projective plane which can be considered as generalizations of free curves. We construct families of arrangements which are nearly free and possess interesting geometric properties. More generally, we study 3-syzygy curve arrangements and we present examples that admit unexpected curves.

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