Tautological Module and Intersection Theory on Hilbert Schemes of Nodal Curves

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. CONTENTS 0. Overview 2 0.1. Setting 2 0.2. Tautological module: motivation 3 0.3. Tautological module: elements 3 0.4. Tautological module: Discriminant action 5 0.5. Tautological module: transfer 5 0.6. Computation 5 0.7. Punctual transfer 5 0.8. Applications 5 1. Preliminaries 6 1.1. Graph enumeration, generating functions 6 1.2. Products, diagonals, partitions 8 1.3. Diagonal operators on tensors 10 1.4. Discriminant operator 13 1.5. (Half-) discriminant 15 1.6. Norm 16 1.7. Boundary data 17 2. The tautological module 18 2.1. The small diagonal 19 2.2. Monoblock and polyblock digaonals: ordered case 24 2.3. Monoblock and polyblock diagonals: unordered case 30 Date: March 15, 2012. 1991 Mathematics Subject Classification. 14N99, 14H99.