Resumen de Modifications of Hodge bundles and enumerative geometry: the stable hyperelliptic locus

Ziv Ran

• We study the stable hyperelliptic locus, i.e. the closure, in the Deligne–Mumford moduli of stable curves, of the locus of smooth hyperelliptic curves. Working on a suitable blowup of the relative Hilbert scheme (of degree 2) associated with a family of stable curves, we construct a bundle map (‘degree-2 Brill–Noether’) from a modification of the Hodge bundle to a tautological bundle, whose degeneracy locus is the natural lift of the stable hyperelliptic locus plus a simple residual scheme. Using intersection theory on Hilbert schemes and Fulton–MacPherson residual intersection theory, the class of the structure sheaf and various other sheaves supported on the stable hyperelliptic locus can be computed by the Porteous formula and similar tools.