Node polynomials for families: results and examples

2001/11/29 by Steven L. Kleiman, S. Kleiman, Kleiman, S. +3 · 2 citations
Mathematics · #14C20 #14K05 #14N10 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #math.AG #msc:14C20 #msc:14K05 #msc:14N10

paper · pdf · doi:10.48550/arxiv.math/0111299

23 pages, plain TeX

arxiv created 2001/11/29 · openalex publication_date 2001/11/29 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We continue the development of methods for enumerating nodal curves on smooth complex surfaces, stressing the range of validity. We illustrate the new methods in three important examples. First, for up to eight nodes, we confirm Göttsche's conjecture about plane curves of low degree. Second, we justify Vainsencher's enumeration of irreducible six-nodal plane curves on a general quintic threefold in four-space. Third, we supplement Bryan and Leung's enumeration of nodal curves in a given homology class on an Abelian surface of Picard number one.

Citations

• Equisingular Deformations of Plane Curve and of Sandwiched Singularities
• Enumeration of n-fold tangent hyperplanes to a surface

Cited by

• On the Göttsche Threshold
• Curve Counting à la Göttsche

Related

• Node polynomials for curves on surfaces
• The Kleiman-Piene Conjecture and node polynomials for plane curves in ℙ³
• Some Geometry of Nodal Curves
• Computing Node Polynomials for Plane Curves
• Enumeration of n-fold tangent hyperplanes to a surface