Chow Ring Research Articles

A tree-based algorithm for the integration of monomials in the Chow ring of the moduli space of stable marked curves of genus zero

The Chow ring of the moduli space of stable marked 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 equivalent graphical characterization on the monomials in this Chow ring, as well as the characterization on the algebraic reduction on such monomials. Moreover, we provide an algorithm for computing the intersection degree of tuples of Keel's divisor classes — we call it the forest algorithm; the complexity of which is O(n3) in the worst case, where n refers to the number of marks on the stable curves in the ambient moduli space. Last but not least, we introduce three identities on multinomial coefficients which naturally came into play, showing that they are all equivalent to the correctness of the base case of the forest algorithm.

Some motivic properties of Gushel–Mukai sixfolds

AbstractGushel–Mukai (GM) sixfolds are an important class of so‐called Fano‐K3 varieties. In this paper, we show that they admit a multiplicative Chow–Künneth decomposition modulo algebraic equivalence and that they have the Franchetta property. As side results, we show that double Eisenbud‐Popescu‐Walter (EPW) sextics and cubes have the Franchetta property, modulo algebraic equivalence, and some vanishing results for the Chow ring of GM sixfolds.

Envelopes and classifying spaces

AbstractFor a split semisimple algebraic group H with its split maximal torus S, let be the restriction homomorphism of the Chow rings of the classifying spaces of H and S, where W is the Weyl group. A constraint on the image of f, given by the Steenrod operations, has been applied to the spin groups in a previous paper. Here, we describe and apply to the spin groups another constraint, which is given by the reductive envelopes of H. We also recover in this way some older results on orthogonal groups.

The Chow rings of moduli spaces of elliptic surfaces over ${\mathbb P}^1$

The Chow rings of moduli spaces of elliptic surfaces over ${\mathbb P}^1$

Maximal disjoint Schubert cycles in rational homogeneous varieties

AbstractIn this paper, we study properties of the Chow ring of rational homogeneous varieties of classical type, more concretely, effective zero divisors of low codimension, and a related invariant called effective good divisibility. This information is then used to study the question of (non)existence of nonconstant maps among these varieties, generalizing previous results for projective spaces and Grassmannians.

Intersection theory on moduli of smooth complete intersections

We provide a general method for computing rational Chow rings of moduli of smooth complete intersections. We specialize this result in different ways: to compute the integral Picard group of the associated stack; to obtain an explicit presentation of rational Chow rings of moduli of smooth complete intersections of codimension two; to prove old and new results on moduli of smooth curves of genus le 5 and polarized K3 surfaces of degree le 8.

Simplicial generation of Chow rings of matroids

We introduce a presentation of the Chow ring of a matroid by a new set of generators, called “simplicial generators.” These generators are analogous to nef divisors on projective toric varieties, and admit a combinatorial interpretation via the theory of matroid quotients. Using this combinatorial interpretation, we (i) produce a bijection between a monomial basis of the Chow ring and a relative generalization of Schubert matroids, (ii) recover the Poincaré duality property, (iii) give a formula for the volume polynomial, which we show is log-concave in the positive orthant, and (iv) recover the validity of Hodge–Riemann relations in degree 1, which is the part of the Hodge theory of matroids that currently accounts for all combinatorial applications of the work of Adiprasito et al. (2018). Our work avoids the use of “flips,” the key technical tool employed in that work.

The Chow rings of the moduli spaces of curves of genus 7, 8, and 9

The rational Chow ring of the moduli space M g \mathcal {M}_g of curves of genus g g is known for g ≤ 6 g \leq 6 . Here, we determine the rational Chow rings of M 7 , M 8 \mathcal {M}_7, \mathcal {M}_8 , and M 9 \mathcal {M}_9 by showing they are tautological. The key ingredient is intersection theory on Hurwitz spaces of degree 4 4 and 5 5 covers of P 1 \mathbb {P}^1 via their associated vector bundles. The main focus of this paper is a detailed geometric analysis of special tetragonal and pentagonal covers whose associated vector bundles on P 1 \mathbb {P}^1 are highly unbalanced, expanding upon previous work of the authors in the more balanced case. In genus 9 9 , we use work of Mukai to present the locus of hexagonal curves as a global quotient stack, and, using equivariant intersection theory, we show its Chow ring is generated by restrictions of tautological classes.

Chow rings of stacks of prestable curves II

Abstract We continue the study of the Chow ring of the moduli stack 𝔐 g , n {\mathfrak{M}_{g,n}} of prestable curves begun in [Y. Bae and J. Schmitt, Chow rings of stacks of prestable curves I, Forum Math. Sigma 10 2022, Paper No. e28]. In genus 0, we show that the Chow ring of 𝔐 0 , n {\mathfrak{M}_{0,n}} coincides with the tautological ring and give a complete description in terms of (additive) generators and relations. This generalizes earlier results by Keel and by Kontsevich and Manin for the spaces of stable curves. Our argument uses the boundary stratification of the moduli stack together with the study of the first higher Chow groups of the strata, in particular providing a new proof of the results of Kontsevich and Manin.

The rational Chow rings of moduli spaces of hyperelliptic curves with marked points

The rational Chow rings of moduli spaces of hyperelliptic curves with marked points

Tropical fans and normal complexes: Putting the “volume” back in “volume polynomials”

Associated to any divisor in the Chow ring of a simplicial tropical fan, we construct a family of polytopal complexes, called normal complexes, which we propose as an analogue of the well-studied notion of normal polytopes from the setting of complete fans. We describe certain closed convex polyhedral cones of divisors for which the “volume” of each divisor in the cone—that is, the degree of its top power—is equal to the volume of the associated normal complexes. For the Bergman fan of any matroid with building set, we prove that there exists an open family of such cones of divisors with nonempty interiors. We view the theory of normal complexes developed in this paper as a polytopal model underlying the combinatorial Hodge theory pioneered by Adiprasito, Huh, and Katz.

Hodge Theory for Polymatroids

Abstract We construct a Leray model for a discrete polymatroid with arbitrary building set and we prove a generalized Goresky–MacPherson formula. The first row of the model is the Chow ring of the polymatroid; we prove Poincaré duality, Hard Lefschetz, and Hodge–Riemann theorems for the Chow ring. Furthermore, we provide a relative Lefschetz decomposition with respect to the deletion of an element.

Flat Families of Point Schemes for Connected Graded Algebras

We study truncated point schemes of connected graded algebras as families over the parameter space of varying relations for the algebras, proving that the families are flat over the open dense locus where the point schemes achieve the expected (i.e., minimal) dimension. When the truncated point scheme is zero-dimensional, we obtain its number of points counted with multiplicity via a Chow ring computation. This latter application in particular confirms a conjecture of Brazfield that a generic two-generator two-relation algebra has seventeen truncated point modules of length six.

The Hodge bundle, the universal 0-section, and the log Chow ring of the moduli space of curves

We bound from below the complexity of the top Chern class $\lambda _g$ of the Hodge bundle in the Chow ring of the moduli space of curves: no formulas for $\lambda _g$ in terms of classes of degrees 1 and 2 can exist. As a consequence of the Torelli map, the 0-section over the second Voronoi compactification of the moduli of principally polarized abelian varieties also cannot be expressed in terms of classes of degree 1 and 2. Along the way, we establish new cases of Pixton's conjecture for tautological relations. In the log Chow ring of the moduli space of curves, however, we prove $\lambda _g$ lies in the subalgebra generated by logarithmic boundary divisors. The proof is effective and uses Pixton's double ramification cycle formula together with a foundational study of the tautological ring defined by a normal crossings divisor. The results open the door to the search for simpler formulas for $\lambda _g$ on the moduli of curves after log blow-ups.

Tautological classes on low-degree Hurwitz spaces

Abstract Let ${\mathcal {H}}_{k,g}$ be the Hurwitz stack parametrizing degree $k$, genus $g$ covers of ${\mathbb {P}}^{1}$. We define the tautological ring of ${\mathcal {H}}_{k,g}$ and we show that all Chow classes, except possibly those supported on the locus of “factoring covers,” are tautological up to codimension roughly $g/k$ when $k \leq 5$. The set-up developed here is also used in our subsequent work [6], wherein we prove new results about the structure of the Chow ring for $k \leq 5$.

Noninjectivity of the cycle class map in continuous -adic cohomology

AbstractJannsen asked whether the rational cycle class map in continuous$\ell $-adic cohomology is injective, in every codimension for all smooth projective varieties over a field of finite type over the prime field. As recently pointed out by Schreieder, the integral version of Jannsen’s question is also of interest. We exhibit several examples showing that the answer to the integral version is negative in general. Our examples also have consequences for the coniveau filtration on Chow groups and the transcendental Abel-Jacobi map constructed by Schreieder.

On Bloch’s map for torsion cycles over non-closed fields

Abstract We generalize Bloch’s map on torsion cycles from algebraically closed fields to arbitrary fields. While Bloch’s map over algebraically closed fields is injective for zero-cycles and for cycles of codimension at most two, we show that the generalization to arbitrary fields is only injective for cycles of codimension at most two but, in general, not for zero-cycles. Our result implies that Jannsen’s cycle class map in integral $\ell $ -adic continuous étale cohomology is, in general, not injective on torsion zero-cycles over finitely generated fields. This answers a question of Scavia and Suzuki.

Wonderful compactifications and rational curves with cyclic action

Abstract We prove that the moduli space of rational curves with cyclic action, constructed in our previous work, is realizable as a wonderful compactification of the complement of a hyperplane arrangement in a product of projective spaces. By proving a general result on such wonderful compactifications, we conclude that this moduli space is Chow-equivalent to an explicit toric variety (whose fan can be understood as a tropical version of the moduli space), from which a computation of its Chow ring follows.

The integral Chow ring of , for odd

For any odd integer $d$, we give a presentation for the integral Chow ring of the stack $\mathcal {M}_{0}(\mathbb {P}^r, d)$, as a quotient of the polynomial ring $\mathbb {Z}[c_1,c_2]$. We describe an efficient set of generators for the ideal of relations, and compute them in generating series form. The paper concludes with explicit computations of some examples for low values of $d$ and $r$, and a conjecture for a minimal set of generators.

Motivic cohomology and infinitesimal group schemes

For $k$ a perfect field of characteristic $p>0$ and $G/k$ a split reductive group with $p$ a non-torsion prime for $G,$ we compute the mod $p$ motivic cohomology of the geometric classifying space $BG_{(r)}$, where $G_{(r)}$ is the $r$th Frobenius kernel of $G.$ Our main tool is a motivic version of the Eilenberg-Moore spectral sequence, due to Krishna. For a flat affine group scheme $G/k$ of finite type, we define a cycle class map from the mod $p$ motivic cohomology of the classifying space $BG$ to the mod $p$ \'etale motivic cohomology of the classifying stack $\mathcal{B}G.$ This also gives a cycle class map into the Hodge cohomology of $\mathcal{B}G.$ We study the cycle class map for some examples, including Frobenius kernels.