Moduli Space Of Spin Curves Research Articles

The moduli space of quasistable spin curves

We study a compactification of the moduli space of theta characteristics, giving a modular interpretation of the geometric points and describing the boundary stratification. This space is different from the moduli space of spin curves. The modular description and the boundary stratification of the new compactification are encoded by a tropical moduli space. We show that this tropical moduli space is a refinement of the moduli space of spin tropical curves. We describe explicitly the induced decomposition of its cones.

Tropicalizing the moduli space of spin curves

We study the tropicalization of the moduli space of algebraic spin curves, $$\overline{\mathcal {S}}_{g,n}$$. We exhibit its combinatorial stratification and prove that the strata are irreducible. We construct the moduli space of tropical spin curves $$\overline{S}_{g,n}^{{\text {trop}}}$$, prove that is naturally isomorphic to the skeleton of the analytification, $$\overline{S}_{g,n}^{{\text {an}}}$$, of $$\overline{\mathcal {S}}_{g,n}$$, and give a geometric interpretation of the retraction of $$\overline{S}_{g,n}^{{\text {an}}}$$ onto its skeleton in terms of a tropicalization map $$\overline{S}_{g,n}^{{\text {an}}}\rightarrow \overline{S}_{g,n}^{{\text {trop}}}$$.

Components of moduli spaces of spin curves with the expected codimension

We prove a conjecture of Gavril Farkas claiming that for all integers r \geq 2 and g \geq \binom{r+2}{2} there exists a component of the locus \mathcal{S}^r_g of spin curves with a theta characteristic L such that h^0(L) \geq r+1 and h^0(L)\equiv r+1 (mod 2) which has codimension \binom{r+1}{2} inside the moduli space \mathcal{S}_g of spin curves of genus g.

On the rational cohomology of moduli spaces of curves with level structures

We investigate low degree rational cohomology groups of smooth compactifications of moduli spaces of curves with level structures. In particular, we determine \({H^k\left({\bar S}_{g}, {\mathbb Q}\right)}\) for g ≥ 2 and k ≤ 3, where \({{\bar S}_{g}}\) denotes the moduli space of spin curves of genus g.

On the geometry of the moduli space of spin curves

We determine the smooth locus and the locus of canonical singularities in the Cornalba compactification S ¯ g \overline {S}_g of the moduli space S g S_g of spin curves, i.e., smooth curves of genus g g with a theta characteristic. Moreover, the following lifting result for pluricanonical forms is proved: Every pluricanonical form on the smooth locus of S ¯ g \overline {S}_g extends holomorphically to a desingularisation of S ¯ g \overline {S}_g .

Enriched spin curves on stable curves with two components

In [7], Mainò constructed a moduli space for enriched stable curves, by blowing-up the moduli space of Deligne–Mumford stable curves. We introduce enriched spin curves, showing that a parameter space for these objects is obtained by blowing-up the moduli space of spin curves.

A Genus-3 Topological Recursion Relation

In this paper, we give a new genus-3 topological recursion relation for Gromov-Witten invariants of compact symplectic manifolds. This formula also applies to intersection numbers on moduli spaces of spin curves. A by-product of the proof of this formula is a new relation in the tautological ring of the moduli space of 1-pointed genus-3 stable curves.

Moduli of Prym curves

Here we focus on the compactification of the moduli space of curves of genus g together with an unramified double cover, constructed by Arnaud Beauville in order to compactify the Prym mapping. We present an alternative description of it, inspired by the moduli space of spin curves of Maurizio Cornalba, and we discuss in detail its main features, both from a geometrical and a combinatorial point of view.

Super Atiyah Classes and Obstructions to Splitting of Supermoduli Space

The first obstruction to splitting a supermanifold S is one of the three components of its super Atiyah class, the two other components being the ordinary Atiyah classes on the reduced space M of the even and odd tangent bundles of S. We evaluate these classes explicitly for the moduli space of super Riemann surfaces (super moduli space) and its reduced space, the moduli space of spin curves. These classes are interpreted in terms of certain extensions arising from line bundles on the square of the varying (super) Riemann surface. These results are used to give a new proof of the non- projectedness of Mg,1, the moduli space of super Riemann surfaces with one puncture.