Tautological module and intersection theory on Hilbert schemes of nodal curves.

*(English)*Zbl 1282.14097This paper continues a long line of research by the author in which he investigates enumerative geometry of nodal curves via Hilbert schemes rather than the (now) more ubiquitous approach of stable maps pioneered by Kontsevich. This paper is mostly foundational, developing machinery to perform various intersection-theoretic computations in this setting. Along the way there are various nice concrete results, and the author concludes by re-deriving Harris-Mumford’s formula for the fundamental class of the hyperelliptic locus in the moduli space of curves (at least, he presents this computation in the case of genus \(g=3\)) and he claims that this is a step toward a program to the compute the fundamental class of all gonal loci in the moduli space of curves, an ambitious project. The reader is suggested to read the introduction of the paper for more details on the results contained therein, though the reader is also warned that this paper builds heavily on the author’s earlier work (a sequence of solo-authored papers) so it may be difficult to jump in and understand what’s going on without first putting time into these earlier papers.

Reviewer: Noah Giansiracusa (Zürich)