On the cohomology of moduli spaces of (weighted) stable rational curves

We give a recursive algorithm for computing the character of the cohomology of the moduli space ${\overline{M}}_{0,n}$ of stable $n$ -pointed genus zero curves as a representation of the symmetric group $\mathbb{S }_n$ on $n$ letters. Using the algorithm we can show a formula for the maximum length of this character. Our main tool is connected to the moduli spaces of weighted stable curves introduced by Hassett.     

Equivariant cohomology of certain moduli of weighted pointed rational curves

We determine the action of the product of symmetric groups on the cohomology of certain moduli of weighted pointed rational curves. The moduli spaces that we study are of stable rational curves with m + n marked points where the first m marked points are distinct from all the others where as the last n may coincide among themselves. We give a recipe for calculating the equivariant Poincaré polynomials and list them for small m and n.     

Quantum cohomology of moduli spaces of genus zero stable curves

We investigate the (small) quantum cohomology ring of the moduli spaces of stable n-pointed curves of genus 0. In particular, we determine an explicit presentation in the case n = 5 and we outline a computational approach to the case n = 6. This research was partially supported by MIUR and GNSAGA of INdAM (Italy).     

On the rationality of moduli spaces of pointed curves

Motivated by several recent results on the geometry of the modulispaces of stable curves of genus g with n marked points, we determine the birational structureof these spaces for small values of g and n by exploiting suitableplane models of a general curve. More precisely, _g,n is shownto be rational for g = 2 and 1 n 12, g = 3 and 1 n 14, g= 4 and 1 n 15, and g = 5 and 1 n 12     

Permutohedral spaces and the Cox ring of the moduli space of stable pointed rational curves

We study the Cox ring of the moduli space of stable pointed rational curves, ${\overline{M}_{0,n}}$ , via the closely related permutohedral (or Losev-Manin) spaces ${\overline{L}_{n-2}}$ . Our main result establishes $\left(\begin{array}{ll} n \\ 2 \end{array}\right)$ polynomial subrings of ${{\rm Cox}(\overline{M}_{0,n})}$ , thus giving collections of boundary variables that intersect the ideal of relations of ${{\rm Cox}(\overline{M}_{0,n})}$ trivially. As applications, we give a combinatorial way to partially solve the Riemann-Roch problem for ${\overline{M}_{0,n}}$ , and we show that all relations in degrees of ${{\rm Cox}(\overline{M}_{0,6})}$ arising from certain pull-backs from projective spaces are generated by the Plücker relations.     

On the geometry of moduli spaces of symplectic structures

 An approach to a natural geometry of moduli spaces of symplectic structures is presented. Received: 21 March 2002 / Revised version: 28 August 2002 This work was partially supported by the Deutsche Forschungsgemeinschaft (DFG).     

The orbifold cohomology of moduli of genus 3 curves

In this work we study the additive orbifold cohomology of the moduli stack of smooth genus g curves. We show that this problem reduces to investigating the rational cohomology of moduli spaces of cyclic covers of curves where the genus of the covering curve is g. Then we work out the case of genus g =  3. Furthermore, we determine the part of the orbifold cohomology of the Deligne–Mumford compactification of the moduli space of genus 3 curves that comes from the Zariski closure of the inertia stack of ${\mathcal{M}3}$ .     

Towards the cohomology of moduli spaces of higher genus stable maps

We prove that the orbifold desingularization of the moduli space of stable maps of genus g = 1 recently constructed by Vakil and Zinger has vanishing rational cohomology groups in odd degree k < 11. Received: 29 January 2007     

Geometry of moduli spaces of rational curves in linear sections of Grassmannian Gr(2,5)

We prove that the moduli spaces of rational curves of degree at most 3 in linear sections of the Grassmannian $\mathrm{G}\mathrm{r}(2,5)$ are all rational varieties. We also study their compactifications and birational geometry.     

The second cohomology with symmplectic coefficients of the moduli space of smooth projcetive curves

Each finite dimensional irreducible rational representation V of the symplectic group Sp_2g(Q) determines a generically defined local system V over the moduli space M_g of genus g smooth projective curves. We study H² (M_g; V) and the mixed Hodge structure on it. Specifically, we prove that if g 6, then the natural map IH²(M~_g; V) H²(M_g; V) is an isomorphism where M~_{_g} is tfhe Satake compactification of M_g. Using the work of Saito we conclude that the mixed Hodge structure on H²(M_g; V) is pure of weight 2+r if V underlies a variation of Hodge structure of weight r. We also obtain estimates on the weight of the mixed Hodge structure on H²(M_g; V) for 3 g < 6. Results of this article can be applied in the study of relations in the Torelli group T_g.     

Chen-Ruan cohomology of some moduli spaces of parabolic vector bundles

Let (X,D) be an ?-pointed compact Riemann surface of genus at least two. For each point x∈D, fix parabolic weights such that . Fix a holomorphic line bundle ξ over X of degree one. Let PMξ denote the moduli space of stable parabolic vector bundles, of rank two and determinant ξ, with parabolic structure over D and parabolic weights . The group of order two line bundles over X acts on PMξ by the rule E_∗⊗L?E_∗⊗L. We compute the Chen-Ruan cohomology ring of the corresponding orbifold.