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Universal polynomials for singular curves on surfaces

Part of: Projective and enumerative geometry Curves Cycles and subschemes

Published online by Cambridge University Press:  06 June 2014

Jun Li  and
Yu-jong Tzeng
Jun Li
Affiliation:
Department of Mathematics, Stanford University, California, CA 94305, USA email jli@math.stanford.edu
Yu-jong Tzeng
Affiliation:
Department of Mathematics, Harvard University, Cambridge, MA 02138, USA email ytzeng@math.harvard.edu
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Abstract

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}S$ 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.

Keywords

curve countinguniversal polynomialGöttsche’s conjecture

MSC classification

Secondary: 14N10: Enumerative problems (combinatorial problems) 14C20: Divisors, linear systems, invertible sheaves 14H20: Singularities, local rings

Information

Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 7 , July 2014 , pp. 1169 - 1182
Copyright
© The Author(s) 2014 

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