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

Block, Florian

Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AR, 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012), DMTCS Proceedings vol. AR, 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012) (2012).

Résumé

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.

Détail

DOI : 10.46298/dmtcs.3093