Universal polynomials for singular curves on surfaces

Abstract

Let S be a complex smooth projective surface and L be a line bundle on S. For any given collection of isolated topological or analytic singularity types, we show the number of curves in the linear system |L| with prescribed singularities is a universal polynomial of Chern numbers of L and S, assuming L is sufficiently ample. Moreover, we define a generating series whose coefficients are these universal polynomials and discuss its properties. This work is a generalization of Gottsche's conjecture to curves with higher singularities.

12 pages

References in corpus (5)

• A Proof of the Göttsche-Yau-Zaslow Formula
• The Universal Severi Variety of Rational Curves on K3 Surfaces
• The Algebraic Proof of the Universality Theorem
• Good degeneration of Quot-schemes and coherent systems
• Algebraic cobordism of bundles on varieties

Cited by in corpus (11)

• Discriminants in the Grothendieck Ring
• Universal Polynomials for Tautological Integrals on Hilbert Schemes
• Enumeration of curves with one singular point
• Enumeration of rational curves in a moving family of ${\mathbb{P}}^2$
• Log BPS numbers of log Calabi-Yau surfaces
• Enumeration of singular varieties with tangency conditions
• Counting planar curves in ${\mathbb{P}}^3$ with degenerate singularities
• Higher rank flag sheaves on Surfaces and Vafa-Witten invariants
• The Kleiman-Piene Conjecture and node polynomials for plane curves in ${\mathbb{P}}^3$
• Hilbert Schemes, Donaldson-Thomas Theory, Vafa-Witten and Seiberg Witten theories
• Curve Counting à la Göttsche