THE MODULI SPACE OF TWISTED CANONICAL DIVISORS

By G. Farkas, R. Pandharipande. Journal of the Institute of Mathematics of Jussieu, 2015

92 citations recorded by the source corpus.

The moduli space of canonical divisors (with prescribed zeros and poles) on nonsingular curves is not compact since the curve may degenerate. We define a proper moduli space of twisted canonical divisors in $\overline{{\mathcal{M}}}_{g,n}$ which includes the space of canonical divisors as an open subset. The theory leads to geometric/combinatorial constraints on the closures of the moduli spaces of canonical divisors. In case the differentials have at least one pole (the strictly meromorphic case), the moduli spaces of twisted canonical divisors on genus $g$ curves are of pure codimension $g$ in $\overline{{\mathcal{M}}}_{g,n}$ . In addition to the closure of the canonical divisors on nonsingular curves, the moduli spaces have virtual components. In the Appendix A, a complete proposal relating the sum of the fundamental classes of all components (with intrinsic multiplicities) to a formula of Pixton is proposed. The result is a precise and explicit conjecture in the tautological ring for the weighted fundamental class of the moduli spaces of twisted canonical divisors. As a consequence of the conjecture, the classes of the closures of the moduli spaces of canonical divisors on nonsingular curves are determined (both in the holomorphic and meromorphic cases).

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/77c69342fc5c6cfc29f102523c092e6fb0c6b393

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Citations in the corpus

The moduli space of multi-scale differentials. (2019, unknown): “ree of each vertex is even and at least −2, and (ii) (Stability) the valence of each vertex of degree −2 is at least three. Our notion of enhancement is equivalent to the notion of twist used e.g. in [FP18] or [CMSZ19]. The main example is the enhanced level graph Γ+ X of a twisted diﬀerential (X,z,”
Masur–Veech volumes and intersection theory on moduli spaces of Abelian differentials (2019, background): “aph. For the reader familiar with related results of compactiﬁcations of strata of Abelian diﬀerentials, the above deﬁnition characterizes twisted diﬀerentials (or canonical divisors) in [BCG+18] and [FP18] (regarding the mi as the zero orders and pj + 1 as the pole orders of twisted diﬀerentials on”
Compactification of strata of Abelian differentials (2016, methodology): “ zero of ηto the pole, and leaving it unoriented if the diﬀerential has a simple pole at both branches. In this way, we obtain a partial order on the vertices vof Γ, with equality permitted (see also [FP16]). It turns out that a partial order is insuﬃcient to characterize actual limits of abelian di”
The Chern classes and Euler characteristic of the moduli spaces of Abelian differentials (2020, methodology): “cles for computation in the moduli space of curves ([DSZ20]). Specically, the evaluation of tautological classes below is performed using the formula for fundamental classes of strata conjectured in [FP18] and [Sch18] and proven recently in [BHPSS20] based on results from [HS19]. The algorithms in t”