Intersections of Q-Divisors on Kontsevich's Moduli Space $\bar{M}_{0,n}(P^r,d)$ and Enumerative Geometry

Let $\bar{M}_{0,n}(G(r,V), d)$ be the coarse moduli space of stable degree $d$ maps from $n$-pointed genus $0$ curves to a Grassmann variety $G(r,V)$. We provide a recursive method for the computation of the Hodge numbers and the Betti numbers of $\bar{M}_{0,n}(G(r,V), d)$ for all $n$ and $d$. Our method is a generalization of Getzler and Pandharipande's work for maps to projective spaces.

The odd dimensional projective space $\mathbb{P}^{2n-1}$ admits a contact structure arising from a non integrable distribution of hyperplanes determined by a symplectic form in $\mathbb{C}^{2n}$. Our object of interest is the set of rational curves of degree d which are tangent to that contact distribution in $\mathbb{P}^3$. Such curves are called contact curves or legendrian curves. To explore the geometry of contact curves, we construct the parameter space $\mathcal{L}_d$ u...

The rational Chow ring A?(S[n],Q) of the Hilbert scheme S[n] parametrising the length n zero-dimensional subschemes of a toric surface S can be described with the help of equivariant techniques. In this paper, we explain the general method and we illustrate it through many examples. In the last section, we present results on the intersection theory of graded Hilbert schemes.

We consider the locus of irreducible nonsingular rational curves of degree d Pn, n>2, meeting a generic collection of linear subspaces. When this locus is 0 (resp 1)- dimensional, we compute (recursively) its degree (resp. geometric genus). The method is completely elementary and similar to that of (alg-geom/9704004, alg-geom/9708013), where the case n=2 was considered.

We consider the interplay of point counts, singular cohomology, \'etale cohomology, eigenvalues of the Frobenius and the Grothendieck ring of varieties for two families of varieties: spaces of rational maps and moduli spaces of marked, degree $d$ rational curves in $\mathbb{P}^n$. We deduce as special cases algebro-geometric and arithmetic refinements of topological computations of Segal, Cohen--Cohen--Mann--Milgram, Vassiliev and others.

This note is about invariants of moduli spaces of curves. It includes their intersection theory and cohomology. Our main focus in on the distinguished piece containing the so called tautological classes. These are the most natural classes on the moduli space. We give a review of known results and discuss their conjectural descriptions.

In this paper, we ask: for which $(g, n)$ is the rational Chow or cohomology ring of $\overline{\mathcal{M}}_{g,n}$ generated by tautological classes? This question has been fully answered in genus $0$ by Keel (the Chow and cohomology rings are tautological for all $n$) and genus $1$ by Belorousski (the rings are tautological if and only if $n \leq 10$). For $g \geq 2$, work of van Zelm shows the Chow and cohomology rings are not tautological once $2g + n \geq 24$, leaving fi...

In this paper, we prove formulas that represent two-pointed Gromov-Witten invariant <O_{h^a}O_{h^b}>_{0,d} of projective hypersurfaces with d=1,2 in terms of Chow ring of Mbar_{0,2}(P^{N-1},d), the moduli spaces of stable maps from genus 0 stable curves to projective space P^{N-1}. Our formulas are based on representation of the intersection number w(O_{h^a}O_{h^b})_{0,d}, which was introduced by Jinzenji, in terms of Chow ring of Mp_{0,2}(N,d), the moduli space of quasi maps...

In this work, we introduce the moduli stack $\widetilde{\mathcal{M}}_{g,n}^r$ of $n$-pointed, $A_r$-stable curves of genus $g$ and use it to compute the Chow ring of $\overline{\mathcal{M}}_3$. As a byproduct, we also compute the Chow ring of $\widetilde{\mathcal{M}}_3^7$. All the Chow rings are assumed to be with coefficients in $\mathbb{Z}[1/6]$.

The moduli space of stable curves of genus $g$ with $n$ marked points, $\overline{\rm{M}}_{g,n}$, is a central object in algebraic geometry, and plays a crucial role in $2$-dimensional conformal field theory. In this paper, we apply the sheaf of coinvariants and conformal block divisors to study the geometry of $\overline{\rm{M}}_{0,n}$. The main theorem characterizes the line bundles on certain contractions of $\overline{\rm{M}}_{0,n}$ via F-curves, using Fakhruddin's basis ...