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Geometry on nodal curves

Ziv Ran
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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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Ziv Ran. 2002-10-15. Geometry on nodal curves. https://arxiv.org/abs/math/0210209

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