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

dmtcs:3093 - Discrete Mathematics & Theoretical Computer Science, January 1, 2012, DMTCS Proceedings vol. AR, 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012)
$q$-Floor Diagrams computing Refined Severi Degrees for Plane Curves

Authors: Block, Florian

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.


Volume: DMTCS Proceedings vol. AR, 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012)
Section: Proceedings
Published on: January 1, 2012
Submitted on: January 31, 2017
Keywords: $q$-analog,Severi degree, refined Severi degree, Gôttsche conjecture, floor diagram, node polynomial,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]


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