Universal polynomials for singular curves on surfaces

Jun Li, Yu Jong Tzeng

Research output: Contribution to journalArticlepeer-review

11 Scopus citations

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. More generally, we show for vector bundles of any rank and smooth varieties of any dimension, similar universal polynomials also exist and equal the number of singular subvarieties cutting out by sections of the vector bundle. This work is a generalization of Göttsche's conjecture.

Original languageEnglish (US)
Pages (from-to)1169-1182
Number of pages14
JournalCompositio Mathematica
Volume150
Issue number7
DOIs
StatePublished - Jul 17 2014

Bibliographical note

Publisher Copyright:
© © The Author(s) 2014.

Keywords

  • Göttsche's conjecture
  • curve counting
  • universal polynomial

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