Moduli spaces of curves with effective 𝑟-spin structures

Abstract

Abstract. We introduce the moduli stack of pointed curves equipped with effective r-spin structures: these are effective divisors D such that rD is a canonical divisor modified at marked points. We prove that this moduli space is smooth and describe its connected components. We also prove that it always contains a component that projects birationally to the locus S 0 in the moduli space of r-spin curves consisting of r-spin structures L such that h 0 (L) = 0. Finally, we study the relation between the locus S 0 and Witten's virtual top Chern class.

Cohomology classes of strata of differentials

“…The divisors D 1 and D 2 can be of one of the four types described in Proposition 4. 32. We will prove this proposition by considering every possible cases.…”

“…The dimension of PA g,Z,P is equal to the dimension of its image in the moduli space of curves. Then the image of PA g,Z,P is of dimension 2g − 2 + n if P is empty (see [32]) and 2g − 3 + n + m otherwise (see [15]). By a simple count of dimension we can check that the proposition is valid in this specific case.…”

“…Proposition 4. 32. Let Z ′ be a completion of Z and let (Γ, I, l) be an admissible graph such that D = π(A Γ,I ) is a divisor of A R g,Z,P (where π is the forgetful map of the points), then D is necessarily of one of the four kinds:…”

The Moduli Space of Twisted Canonical Divisors

“…, p n ] ∈ M g,n for which the evaluation map H 0 (C, ω C ) → H 0 C, ω C|m 1 p 1 +···+mnpn is not injective, every component of H g (µ) has dimension at least 2g−2+n in M g,n by degeneracy loci considerations [13]. Polishchuk [25] has shown that H g (µ) is a nonsingular substack of M g,n of pure dimension 2g − 2 + n. In fact, the arguments of [25] can be extended to the case where the vector µ has negative parts. 1 In the strictly meromorphic case, H g (µ) is a nonsingular substack of dimension 2g−3+n in M g,n .…”

Moduli of roots of line bundles on curves

“…When we consider stable curves without markings (i.e., when n = 0), we denote by S L), so that the largest locus corresponds to "effective r-spin curves"; we refer to the recent paper of A. Polishchuk [21] for more details and open problems related to this interesting aspect.…”