The moduli space $\bar{M}_A$ of weighted pointed stable curves of genus zero is stratified according to the degeneration types of such curves. We show that the homology groups of the moduli space $\bar{M}_A$ are generated by the strata of $\bar{M}_A$ and give all additive relations between them. We also observe that the Chow groups $A_i \bar{M}_A$ and the homology groups $H_{2i} \bar{M}_A$ are isomorphic. This generalizes Kontsevich-Manin's and Losev-Manin's theorems to arbitrary weight data $A$.
In this paper we give a complete description of the integral Chow ring of the stack $\mathscr{H}_{g,1}$ of 1-pointed hyperelliptic curves, lifting relations and generators from the Chow ring of $\mathscr{H}_g$. We also give a geometric interpretation for the generators.
We find a new presentation of the stack of hyperelliptic curves of odd genus as a quotient stack and we use it to compute its integral Chow ring by means of equivariant intersection theory.
We compute the integral Chow ring of the stack of smooth, non-hyperelliptic curves of genus three. We obtain this result by computing the integral Chow ring of the stack of smooth plane quartics, by means of equivariant intersection theory.
In the paper, we study the GIT construction of the moduli space of Chow semistable curves of genus 4 in P^3. By using the GIT method developed by Mumford and a deformation theoretic argument, we give a modular description of this moduli space. We classify Chow stable or Chow semistable curves when they are irreducible or nonreduced. Then we work out the case when a curve has two components. Our classification provides some clues to understand the birational map from the moduli space of stable curves of genus 4 to the moduli space of Chow semistable curves of genus 4 in P^3.
We determine the Chow ring (with Q-coefficients) of M_6 by showing that all Chow classes are tautological. In particular, all algebraic cohomology is tautological, and the natural map from Chow to cohomology is injective. To demonstrate the utility of these methods, we also give quick derivations of the Chow groups of moduli spaces of curves of lower genus. The genus 6 case relies on the particularly beautiful Brill-Noether theory in this case, and in particular on a rank 5 vector bundle "relativizing" a baby case of a celebrated construction of Mukai, which we interpret as a subbundle of the …
After recalling several constructions of the moduli space of curves of genus zero by different people we give our alternative construction of the moduli space. This gives a simple description of the intersection ring of this space. We give a basis for the Chow groups and an explicit duality between the Chow groups in complementary degrees. There is a recursive description of this algebra by S. Keel. Our presentation is simpler in the sense that there are fewer generators and fewer relations and our description is explicit.
The Chow group of zero cycles in the moduli space of stable pointed curves of genus zero is isomorphic to the integer additive group. Let $M$ be monomial in this Chow group. If no two factors of $M$ fulfill a particular quadratic relation, then the monomial can be represented equivalently by a specific tree; otherwise, $M$ is mapped to zero under the stated isomorphism. Starting from this tree representation, we introduce a graphical algorithm for computing the corresponding integer for $M$ under the aforementioned isomorphism. The algorithm is linear with respect to the size of the tree.
We determine the rational Chow ring of the moduli space $\mathcal{H}_{g,n}$ of $n$-pointed smooth hyperelliptic curves of genus $g$ when $n \leq 2g+6$. We also show that the Chow ring of the partial compactification $\mathcal{I}_{g,n}$, parametrizing $n$-pointed irreducible nodal hyperelliptic curves, is generated by tautological divisors. Along the way, we improve Casnati's result that $\mathcal{H}_{g,n}$ is rational for $n \leq 2g+8$ to show $\mathcal{H}_{g,n}$ is rational for $n \leq 3g+5$.
The Chow ring of the moduli space of marked rational curves is generated by Keel's divisor classes. The top graded part of this Chow ring is isomorphic to the integers, generated by the class of a single point. In this paper, we give an algorithm for computing the intersection degree of tuples of Keel's divisor classes. This computation is a concrete but complicated algorithmic question in the field. Also, we give a simple complexity argument for the algorithm. Additionally, we introduce three identities on multinomial coefficients, as well as proofs for them.
The moduli space $\bar{M}_S^σ(R)$ parameterizes the isomorphism classes of $S$-pointed stable real curves of genus zero which are invariant under relabeling by the involution $σ$. This moduli space is stratified according to the degeneration types of $σ$-invariant curves. The degeneration types of $σ$-invariant curves are encoded by their dual trees with additional decorations. We construct a combinatorial graph complex generated by the fundamental classes of strata of $\bar{M}_S^σ(R)$. We show that the homology of $\bar{M}_S^σ(R)$ is isomorphic to the homology of our graph complex. We also gi…
We show that prime Fano threefolds $Y$ of genus 8 have a multiplicative Chow-Künneth decomposition, in the sense of Shen-Vial. As a consequence, a certain tautological subring of the Chow ring of powers of $Y$ injects into cohomology.
We introduce a smooth projective variety $T_{d,n}$ which compactifies the space of configurations of $n$ distinct points on affine $d$-space modulo translation and homothety. The points in the boundary correspond to $n$-pointed stable rooted trees of $d$-dimensional projective spaces, which for $d = 1$, are $(n+1)$-pointed stable rational curves. In particular, $T_{1,n}$ is isomorphic to $\bar{M}_{0,n+1}$, the moduli space of such curves. The variety $T_{d,n}$ shares many properties with $\bar{M}_{0,n}$. For example, as we prove, the boundary is a smooth normal crossings divisor whose componen…
In this work we compute the Chen--Ruan cohomology and the stringy Chow ring of the moduli spaces of smooth and stable $n$-pointed curves of genus 1. We suggest a definition for an Orbifold Tautological Ring in genus 1, which is both a subring of the Chen--Ruan cohomology and of the stringy Chow ring.
We give a presentation for the Chow ring of the moduli space of degree two stable maps from two-pointed rational curves to P^1. Also, integrals of of all degree four monomials in the hyperplane pullbacks and boundary divisors of this ring are computed using equivariant intersection theory. Finally, the presentation is used to give a new computation of the (previously known) values of the genus zero, degree two, two-pointed gravitational correlators of P^1. This article is a sequel to math.AG/0501322, although the only information truly needed from that article is the Poincare polynomial of the…
In this paper we explicit the rational Chow ring of the stack consisting of nodal curves of genus 0 with at most 3 nodes: it is a Q-algebra with 10 generators and 11 relations.
The goal of this paper is to compute the rational Chow ring of the stack consisting of nodal curves of genus 0 with at most 3 nodes: it is a Q-algebra with 10 generators and 11 relations.
Title: Indecomposable Higher Chow Cycles on Low Dimensional Jacobians Authors: Alberto Collino Comments: AMS-TeX, 10 pages Subj-class: Algebraic Geometry MSC-class: 14C30 ;19E15 There is a basic indecomposable higher cycle K in Bloch's higher Chow group CH^{g}(J(C),1) on the Jacobian J(C) of a general hyperelliptic curve C of genus g. Consider K(t) the translation of K associated with a point t in C, we prove that in general K - K(t) is indecomposable if the genus is at least 3. Our tool is Lewis' condition for indecomposability. We show next that on the jacobian J(C) of a general curve C of g…
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 $\mathbb{Z}[1/6]$.
In this paper, we show the existence of a Chow--Kuenneth decomposition for the moduli stack of stable curves of genus g with r marked points, for low values of g,r. We also look at the moduli space R of double covers of genus 3 curves, branched along 4 distinct points. We obtain a birational model of the moduli space R as a group quotient of a product of two Grassmanian varieties. This provides a Chow-Kuenneth decomposition over an open subset of R. The question of rationality of R is also discussed.
We introduce the moduli space $R \bar{M}_{2k,l}$ of pointed real curves of genus zero and give its natural stratification. The strata of $R \bar{M}_{2k,l}$ correspond to real curves of genus zero with different degeneration types and are encoded by their dual trees with certain decorations. By using this stratification, we calculate the first Stiefel-Whitney class of $R \bar{M}_{2k,l}$ and construct the orientation double cover $R \widetilde{M}_{2k,l}$$ of $R \bar{M}_{2k,l}$.
We construct moduli spaces of representations of quivers over arbitrary schemes and show how moduli spaces of pointed curves of genus zero like the Grothendieck-Knudsen moduli spaces $\overline{M}_{0,n}$ and the Losev-Manin moduli spaces $\overline{L}_n$ can be interpreted as inverse limits of moduli spaces of representations of certain bipartite quivers. We also investigate the case of more general Hassett moduli spaces $\overline{M}_{0,a}$ of weighted pointed stable curves of genus zero.
We prove that the moduli space of polarized $K3$ surfaces of genus eleven with $n$ marked points is unirational when $n\leq 6$ and uniruled when $n\leq7$. As a consequence, we settle a long standing but not proved assertion about the unirationality of $\cal{M}_{11,n}$ for $n\leq6$. We also prove that the moduli space of polarized $K3$ surfaces of genus eleven with $9$ marked points has non-negative Kodaira dimension.
In this work we describe the Chen-Ruan cohomology of the moduli stacks of smooth and stable genus 2 pointed curves, and its algebraic counterpart: the stringy Chow ring. In the first half of the paper we compute the additive structure of the Chen-Ruan cohomology ring for the moduli stack of stable pointed genus 2 curves, describing it as a rationally graded vector space. In the second part we give generators for the even Chen--Ruan cohomology ring as an algebra on the ordinary cohomology.
The Chow ring of $\mathcal{M}_g$ is known to be generated by tautological classes for $g \leq 9$. Meanwhile, the first example of a non-tautological class on $\mathcal{M}_{g}$ is the fundamental class of the bielliptic locus in $\mathcal{M}_{12}$, due to van Zelm. It remains open if the Chow rings of $\mathcal{M}_{10}$ and $\mathcal{M}_{11}$ are generated by tautological classes. In these cases, a natural first place to look is at the bielliptic locus. In genus $10$, it is already known that classes supported on the bielliptic locus are tautological. Here, we prove that all classes supported o…
We show that the moduli space of stable n-pointed rational curves can be flatly degenerated to a projective toric variety. We arrive at this by showing that the Chow quotients of the Grassmannians admit toric degenerations, which in turn, follows from a theorem that we prove for toric degenerations of more general Chow quotients. Along the way, we also argued that GIT quotient of a flat family is again flat. In particular, all GIT quotients of flag varieties by maximal tori can be flatly degenerated to projective toric varieties ([9] and [12]).
We give a combinatorial description of the rational cohomology of the moduli spaces of pointed genus 1 curves with $n$ marked points and level $N$ structures. More precisely, we explicitly describe the $E_2$ term of the Leray spectral sequence of the forgetful mapping $\EuScript{M}_{1,n}(N)\to\EuScript{M}_{1,1}(N)$ and show that the result is isomorphic to the rational cohomology of $\EuScript{M}_{1,n}(N)$ as a rational mixed Hodge structure equipped with an action of the symmetric group $\mathfrak{S}_n$. The classical moduli space $\EuScript{M}_{1,n}$ is the particular case N=1.
In this note we prove that the kernel of the push-forward homomorphism on $d$-cycles modulo rational equivalence, induced by the closed embedding of an ample divisor linearly equivalent to some multiple of the theta divisor inside the Jacobian variety $J(C)$ is trivial. Here $C$ is a smooth projective curve of genus $g$.
We compute the Mori cone of curves of the moduli space \M_{g,n} of stable n-pointed curves of genus g in the case when g and n are relatively small. For instance, we show that for g<14 every curve in \M_g is numerically equivalent to an effective sum of 1-strata (loci of curves with 3g-4 nodes). We also prove that the nef cone of \M_{0,6} is composed of 11 natural subcones all contained in the convex hull of boundary classes. We apply this result to classify the fibrations of the moduli space of rational curves with n<7 marked points.
We study the Chow group of 1-cycles of the moduli space of semistable parabolic vector bundles of fixed rank, determinant and a generic weight over a nonsingular projective curve over $\mathbb{C}$ of genus at least 3. We show that, the Chow group of 1-cycles remains isomorphic as we vary the generic weight. As a consequence, we can give an explicit description of the Chow group in the case of rank 2 and determinant $\mathcal{O}(x)$, where $x\in X$ is a fixed point.
The purpose of this dissertation is to study the intersection theory of the moduli spaces of stable maps of degree two from two-pointed, genus zero nodal curves to arbitrary-dimensional projective space. Toward this end, first the Betti numbers of \bar{M}_{0,2}(P^r,2) are computed using Serre polynomials and equivariant Serre polynomials. Then, specializing to the space \bar{M}_{0,2}(P^1,2), generators and relations for the Chow ring are given. Chow rings of simpler spaces are also described, and the method of localization and linear algebra is developed. Both tools are used in finding the rel…
In this paper we investigate Murre's conjecture on the Chow--Künneth decomposition for two classes of examples. We look at the universal families of smooth curves over spaces which dominate the moduli space $\cM_g$, in genus at most 8 and show existence of a Chow--Künneth decomposition. The second class of examples include the representation varieties of a finitely generated group with one relation. This is done in the setting of equivariant cohomology and equivariant Chow groups to get equivariant Chow--Künneth decompositions.
We prove by Hilbert-Mumford criterion that a slope stable polarized weighted pointed nodal curve is Chow asymptotic stable. This generalizes the result of Caporaso on stability of polarized nodal curves, and of Hasset on weighted pointed stable curves polarized by the weighted dualizing sheaves. It also solved a question raised by Mumford and Gieseker to prove the Chow asymptotic stability of stable nodal curves by Hilbert-Mumford criterion.
We begin a study of the intersection theory of the moduli spaces of degree two stable maps from two-pointed rational curves to arbitrary-dimensional projective space. First we compute the Betti numbers of these spaces using Serre polynomial and equivariant Serre polynomial methods developed by E. Getzler and R. Pandharipande. Then, via the excision sequence, we compute an additive basis for their Chow rings in terms of Chow rings of nonlinear Grassmannians, which have been described by Pandharipande. The ring structure of one of these Chow rings is addressed in a sequel to this paper.
Let X_n := \bar M_{0,n}, the moduli space of n-pointed stable genus zero curves, and let X_{n,m} be the quotient of X_n by the action of the symmetric group S_{n-m} on the last n-m marked points. The cones of effective divisors of X_{n,m}, m = 0,1,2, are calculated. Using this, upper bounds for the cones Mov(X_{n,m}) generated by divisors with moving linear systems are calculated, m = 0,1, along with the induced bounds on the cones of ample divisors of \bar M_g and \bar M_{g,1}. As an application, the cone of effective divisors of \bar M_{2,1} is analyzed in detail.
We consider the moduli space $\Hh_{g,n}$ of $n$-pointed smooth hyperelliptic curves of genus $g$. In order to get cohomological information we wish to make $\s_n$-equivariant counts of the numbers of points defined over finite fields of this moduli space. We find recurrence relations in the genus that these numbers fulfill. Thus, if we can make $\s_n$-equivariant counts of $\Hh_{g,n}$ for low genus, then we can do this for every genus. Information about curves of genus 0 and 1 is then found to be sufficient to compute the answers for $\Hh_{g,n}$ for all $g$ and for $n \leq 7$. These results ar…
In this paper we study the moduli stack ${\mathcal U}_{1,n}^{ns}$ of curves of arithmetic genus 1 with n marked points, forming a nonspecial divisor. In arXiv:1511.03797 this stack was realized as the quotient of an explicit scheme $\widetilde{\mathcal U}_{1,n}^{ns}$, affine of finite type over ${\Bbb P}^{n-1}$, by the action of ${\Bbb G}_m^n$ . Our main result is an explicit description of the corresponding GIT semistable loci in $\widetilde{\mathcal U}_{1,n}^{ns}$. This allows us to identify some of the GIT quotients with some of the modular compactifications of ${\mathcal M}_{1,n}$ defined …
In this article we consider the moduli space of smooth $n$-pointed non-hyperelliptic curves of genus 3. In the pursuit of cohomological information about this space, we make $\mathbb{S}_n$-equivariant counts of its numbers of points defined over finite fields for $n \leq 7$. Combining this with results on the moduli spaces of smooth pointed curves of genus 0, 1 and 2, and the moduli space of smooth hyperelliptic curves of genus 3, we can determine the $\mathbb{S}_n$-equivariant Galois and Hodge structure of the ($\ell$-adic respectively Betti) cohomology of the moduli space of stable curves of…
Given a collection of boundary divisors in the moduli space of stable genus-zero n-pointed curves, Giansiracusa proved that their intersection is nonempty if and only if all pairwise intersections are nonempty. We give a complete classification of the pairs (g,n) for which the analogous statement holds in the moduli space of n-pointed curves of genus g.
In this note, we revisit the modified diagonal cycle of Gross and Schoen. We look at degenerations of this cycle, induced by a degeneration of the curve C, and explain how the specialization map with respect to the central fiber produces a higher Chow cycle. When C is non-hyperelliptic of genus three, and degenerates to a nodal curve, the degeneration of the cycle corresponds to an indecomposable higher Chow cycle on a curve of genus two.
In this paper we determine the structure of the Chow ring of the Delaunay-Voronoi compactification $\tilde{\cal A}_3$ of the moduli space of principally polarized abelian threefolds. This compactification was introduced by Namikawa and studied by Tsushima. We use equivariant classes on level coverings of $\tilde{\cal A}_3$. We also compare this ring with the Chow ring of the moduli space of stable genus 3 curves as determined by Faber.
