Florian Block - 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) - https://doi.org/10.46298/dmtcs.3093
q-Floor Diagrams computing Refined Severi Degrees for Plane CurvesConference paper

Authors: Florian Block 1

  • 1 Warwick Mathematics Institute

The Severi degree is the degree of the Severi variety parametrizing plane curves of degree d with δ 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 δ, the refined Severi degrees are polynomials in d and q, for large d. As a consequence, we show that, for δ4 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
Imported 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]
Funding:
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