Moduli Stack Of Stable Curves Research Articles

The (almost) integral Chow ring of ℳ̅3

Abstract This paper is the fourth in a series of four papers aiming to describe the (almost integral) Chow ring of M ̄ 3 \overline{\mathcal{M}}_{3} , the moduli stack of stable curves of genus 3. In this paper, we finally compute the Chow ring of M ̄ 3 \overline{\mathcal{M}}_{3} with Z ⁢ [ 1 / 6 ] \mathbb{Z}[1/6] -coefficients.

Equivariant Gromov-Witten Invariants of Algebraic GKM Manifolds

An algebraic GKM manifold is a non-singular algebraic variety equipped with an algebraic action of an algebraic torus, with only finitely many torus fixed points and finitely many 1-dimensional orbits. In this expository article, we use virtual localization to express equivariant Gromov-Witten invariants of any algebraic GKM manifold (which is not necessarily compact) in terms of Hodge integrals over moduli stacks of stable curves and the GKM graph of the GKM manifold.

The tropicalization of the moduli space of curves

We show that the skeleton of the Deligne-MumfordKnudsen moduli stack of stable curves is naturally identified with the moduli space of extended tropical curves, and that this is compatible with the “naive” set-theoretic tropicalization map. The proof passes through general structure results on the skeleton of a toroidal Deligne-Mumford stack. Furthermore, we construct tautological forgetful, clutching, and gluing maps between moduli spaces of extended tropical curves and show that they are compatible with the analogous tautological maps in the algebraic setting.

Matrix factorizations and cohomological field theories

Abstract We give a purely algebraic construction of a cohomological field theory associated with a quasihomogeneous isolated hypersurface singularity W and a subgroup G of the diagonal group of symmetries of W. This theory can be viewed as an analogue of the Gromov–Witten theory for an orbifoldized Landau–Ginzburg model for W/G. The main geometric ingredient for our construction is provided by the moduli of curves with W-structures introduced by Fan, Jarvis and Ruan. We construct certain matrix factorizations on the products of these moduli stacks with affine spaces which play a role similar to that of the virtual fundamental classes in the Gromov–Witten theory. These matrix factorizations are used to produce functors from the categories of equivariant matrix factorizations to the derived categories of coherent sheaves on the Deligne–Mumford moduli stacks of stable curves. The structure maps of our cohomological field theory are then obtained by passing to the induced maps on Hochschild homology. We prove that for simple singularities a specialization of our theory gives the cohomological field theory constructed by Fan, Jarvis and Ruan using analytic tools.

Compactified Picard stacks over the moduli stack of stable curves with marked points

In this paper we give a construction of algebraic (Artin) stacks endowed with a modular map onto the moduli stack of stable curves of genus g with n marked points. The stacks we construct are smooth, irreducible and have dimension 4g−3+n, yielding a geometrically meaningful compactification of the universal Picard stack parametrizing n-pointed smooth curves together with a line bundle.

Pontrjagin–Thom maps and the homology of the moduli stack of stable curves

We study the singular homology (with field coefficients) of the moduli stack \({\overline{\mathfrak{M}}_{g, n}}\) of stable n-pointed complex curves of genus g. Each irreducible boundary component of \({\overline{\mathfrak{M}}_{g, n}}\) determines via the Pontrjagin–Thom construction a map from \({\overline{\mathfrak{M}}_{g, n}}\) to a certain infinite loop space whose homology is well understood. We show that these maps are surjective on homology in a range of degrees proportional to the genus. This detects many new torsion classes in the homology of \({\overline{\mathfrak{M}}_{g, n}}\).

Fundamental groups of moduli stacks of stable curves of compact type

Let $\widetilde{\cal M}_{g,n}$, for $2g-2+n>0$, be the moduli stack of $n$-pointed, genus $g$, stable complex curves of compact type. Various characterizations and properties are obtained of both the algebraic and topological fundamental groups of the stack $\widetilde{\cal M}_{g,n}$. Let $\Gamma_{g,n}$, for $2g-2+n>0$, be the Teichm\"uller group associated with a compact Riemann surface of genus $g$ with $n$ points removed $S_{g,n}$, i.e. the group of homotopy classes of diffeomorphisms of $S_{g,n}$ which preserve the orientation of $S_{g,n}$ and a given order of its punctures. Let $K_{g,n}$ be the normal subgroup of $\Gamma_{g,n}$ generated by Dehn twists along separating circles on $S_{g,n}$. As a first application of the above theory, a characterization of $K_{g,n}$ is given for all $n\geq 0$ (for $n=0,1$, this was done by Johnson). Let then ${\cal T}_{g,n}$ be the Torelli group, i.e. the kernel of the natural representation $\Gamma_{g,n}\ra Sp_{2g}(Z)$. The abelianization of ${\cal T}_{g,n}$ is determined for all $g\geq 1$ and $n\geq 1$, thus completing classical results by Johnson and Mess.

The One-Dimensional Stratum in the Boundary of the Moduli Stack of Stable Curves

It is well-known that the moduli space of Deligne-Mumford stable curves of genus g admits a stratification by the loci of stable curves with a fixed number i of nodes, where 0 ≤ i ≤ 3g - 3. There is an analogous stratification of the associated moduli stack .

On the homotopy type of the Deligne–Mumford compactification

An old theorem of Charney and Lee says that the classifying space of the category of stable nodal topological surfaces and isotopy classes of degenerations has the same rational homology as the Deligne-Mumford compactification. We give an integral refinement: the classifying space of the Charney-Lee category actually has the same homotopy type as the moduli stack of stable curves, and the etale homotopy type of the moduli stack is equivalent to the profinite completion of the classifying space of the Charney-Lee category.

Néron models and compactified Picard schemes over the moduli stack of stable curves

We construct modular Deligne-Mumford stacks representable over parametrizing Néron models of Jacobians as follows. Let B be a smooth curve and K its function field, let be a smooth genus-g curve over K admitting stable minimal model over B. The Néron model → B is then the base change of via the moduli map B → of f , i.e.: B. Moreover is compactified by a Deligne-Mumford stack over , giving a completion of Néron models naturally stratified in terms of Néron models.

Bredon-style homology, cohomology and Riemann–Roch for algebraic stacks

One of the main obstacles for proving Riemann–Roch for algebraic stacks is the lack of cohomology and homology theories that are closer to the K-theory and G-theory of algebraic stacks than the traditional cohomology and homology theories for algebraic stacks. In this paper we study in detail a family of cohomology and homology theories which we call Bredon-style theories that are of this type and in the spirit of the classical Bredon cohomology and homology theories defined for the actions of compact topological groups on topological spaces. We establish Riemann–Roch theorems in this setting: it is shown elsewhere that such Riemann–Roch theorems provide a powerful tool for deriving formulae involving virtual fundamental classes associated to dg-stacks, for example, moduli stacks of stable curves provided with a virtual structure sheaf associated to a perfect obstruction theory. We conclude the present paper with a brief application of this nature.

Extending families of curves over log regular schemes

Abstract In this paper, we generalize to the “log regular case” a result of de Jong and Oort which states that any morphism satisfying certain conditions from the complement of a divisor with normal crossings in a regular scheme to a moduli stack of stable curves extends over the entire regular scheme. The proof uses the theory of “regular log schemes” — i.e., schemes with singularities like those of toric varieties – due to K. Kato ([12]). We then use this extension theorem to prove that under certain natural conditions any scheme which is a successive fibration of smooth hyperbolic curves may be compactified to a successive fibration of stable curves.