Geometry on nodal curves

2002/10/15 by Ziv Ran, Ran, Ziv
Mathematics · #14D06 14H10 14N10 14N35 #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG #msc:14D06 #msc:14H10 #msc:14N10 #msc:14N35

paper · pdf · doi:10.48550/arxiv.math/0210209

arxiv created 2002/10/15 · arxiv updated 2009/11/30

Abstract

Given a family X/B of nodal curves, we construct canonically and compatibly with base-change, via an explicit blow-up of the Cartesian product X^r/B, a family W^r(X/B) parametrizing length-r subschemes of fibres of X/B (plus some additional data). Though W^r(X/B) is singular, the important sheaves on it are locally free, which allows us to study intersection theory on it and deduce enumerative applications, including some relative multiple point formulae, enumerating the length-r schemes contained simultaneously in some fibre of X/B and some fibre of a given map from X to a smooth variety.

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