The Cycle Classes of Divisorial Maroni Loci

Abstract

Abstract. We determine the cycle classes of effective divisors in the compactified Hurwitz spaces H d,g of curves of genus g with a linear system of degree d, that extend the Maroni divisors on H d,g . Our approach uses Chern classes associated to a global-to-local evaluation map of a vector bundle over a generic P 1 -bundle over the Hurwitz space.

Divisors on Hurwitz Spaces: An Appendix to `The Cycle Classes of Divisorial Maroni Loci'

“…Therefore it follows that π * O(D) is a locally free sheaf on P. We denote byD the proper transform of D under the resolution map ν. Since ν * O(D) = ν * OỸ ⊗ O(D), we conclude by [4,Lemma 4.4] and the fact that R j ν * O Y = 0 for j ≥ 1, thatπ * O(D) = π * O(D). We can use the restriction of π * (O(D)) to the open part over H d,g to define a stratification by type of the bundle on P 1 just as for the Maroni stratification.…”

“…We recall the setting from [4]. We denote by H d,g the compactified Hurwitz space of admissible covers of degree d and genus g. We have…”

“…We resolve the singularities of P obtaining a space P and then let Y be the normalization of C × P P and let Y be the resolution of singularities of Y . We then find the basic diagram as in [4]:…”

“…where D is an appropriately chosen divisor of degree s with support in the ramification locus of γ. The cycle classes of an extension of these divisors to the compactified Hurwitz space H d,g can be calculated by using a global-to-local evaluation map p * p * V → V of a vector bundle V on an extension of the P 1 -fibration p : P → H d,g to the compactified Hurwitz space that is trivial on the generic fibre of p. The calculation and the answer are completely analogous to case of the cycle classes of the Maroni divisors calculated in [4]. The cycle classes are given in terms of an explicit sum of boundary divisors.…”

Syzygy divisors on Hurwitz spaces

“…It is an interesting (and challenging) problem to compute the class of the closure of µ i on a full compactification. This was carried out for the Maroni divisor for d = 3 in [6] and for higher d in [23]. It would also be interesting to find replacements for µ i when d does not divide g − 1.…”