Asian Journal of Mathematics

Volume 17 (2013)

Number 2

Tautological module and intersection theory on Hilbert schemes of nodal curves

Pages: 193 – 264



Ziv Ran (Department of Mathematics, University of California at Riverside)


This paper presents the rudiments of Hilbert-Mumford Intersection (HMI) theory: intersection theory on the relative Hilbert scheme of a family of nodal (or smooth) curves, over a base of arbitrary dimension. We introduce an additive group of geometric cycles, called ‘tautological module’, generated by diagonal loci, node scrolls, and twists thereof. We determine recursively the intersection action on this group by the discriminant ( big diagonal) divisor and all its powers. We show that this suffices to determine arbitrary polynomials in Chern classes, in particular Chern numbers, for the tautological vector bundles on the Hilbert schemes, which are closely related to enumerative geometry of families of nodal curves.


Hilbert scheme, nodal curves, intersection theory, enumerative geometry

2010 Mathematics Subject Classification

14Hxx, 14Nxx

Published 5 July 2013