Chow Ring Research Articles

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.

Chow rings of matroids are Koszul

Chow rings of matroids were instrumental in the resolution of the Heron-Rota-Welsh Conjecture by Adiprasito, Huh, and Katz and in the resolution of the Top-Heavy Conjecture by Braden, Huh, Matherne, Proudfoot, and Wang. The Chow ring of a matroid is a commutative, graded, Artinian, Gorenstein algebra with linear and quadratic relations defined by the matroid. Dotsenko conjectured that the Chow ring of any matroid is Koszul. The purpose of this paper is to prove Dotsenko's conjecture. We also show that the augmented Chow ring of a matroid is Koszul. As a corollary, we show that the Chow rings and augmented Chow rings of matroids have rational Poincar\'{e} series.

Varieties of tropical ideals are balanced

Tropical ideals, introduced in [12], define subschemes of tropical toric varieties. We prove that the top-dimensional parts of their varieties are balanced polyhedral complexes of the same dimension as the ideal. This means that every subscheme of a tropical toric variety defined by a tropical ideal has an associated class in the Chow ring of the toric variety. A key tool in the proof is that specialization of variables in a tropical ideal yields another tropical ideal; this plays the role of hyperplane sections in the theory. We also show that elimination theory (projection of varieties) works for tropical ideals as in the classical case. The matroid condition that defines tropical ideals is crucial for these results.

Moduli of genus one curves with two marked points as a weighted blow-up

We give an explicit description of \(\overline{{\mathcal {M}}}_{1,2}\) as a weighted blow-up of a weighted projective stack. We use this description to compute the Brauer group of \(\overline{{\mathcal {M}}}_{1,2;S}\) over any base scheme S where 6 is invertible, and the integral Chow rings of \(\overline{{\mathcal {M}}}_{1,2}\) and \({\mathcal {M}}_{1,2}\).

A semi-small decomposition of the Chow ring of a matroid

We give a semi-small orthogonal decomposition of the Chow ring of a matroid M. The decomposition is used to give simple proofs of Poincaré duality, the hard Lefschetz theorem, and the Hodge–Riemann relations for the Chow ring, recovering the main result of [1]. We also introduce the augmented Chow ring of M and show that a similar semi-small orthogonal decomposition holds for the augmented Chow ring.

Logarithmic intersections of double ramification cycles

We describe a theory of logarithmic Chow rings and tautological subrings for logarithmically smooth algebraic stacks, via a generalisation of the notion of piecewise-polynomial functions. Using this machinery we prove that the double-double ramification cycle lies in the tautological subring of the (classical) Chow ring of the moduli space of curves, and that the logarithmic double ramification cycle is divisorial (as conjectured by Molcho, Pandharipande, and Schmitt).

An algebraic construction of sum-integral interpolators

This paper presents an algebraic construction of Euler-Maclaurin formulas for polytopes. The formulas obtained generalize and unite the previous lattice point formulas of Morelli and Pommersheim-Thomas, and the Euler-Maclaurin formulas of Berline-Vergne While the approach of this paper originates in the theory of toric varieties, and recovers previous results about characteristic classes of toric varieties, the present paper is self-contained and does not rely on results from toric geometry. We aim in particular to exhibit in a combinatorial way ingredients such as such Todd classes and cycle-level intersections in Chow rings, that first entered the theory of polytopes from algebraic geometry.

The intersection theory of the moduli stack of vector bundles on

AbstractWe determine the integral Chow and cohomology rings of the moduli stack $\mathcal {B}_{r,d}$ of rank r, degree d vector bundles on $\mathbb {P}^1$ -bundles. We work over a field k of arbitrary characteristic. We first show that the rational Chow ring $A_{\mathbb {Q}}^*(\mathcal {B}_{r,d})$ is a free $\mathbb {Q}$ -algebra on $2r+1$ generators. The isomorphism class of this ring happens to be independent of d. Then, we prove that the integral Chow ring $A^*(\mathcal {B}_{r,d})$ is torsion-free and provide multiplicative generators for $A^*(\mathcal {B}_{r,d})$ as a subring of $A_{\mathbb {Q}}^*(\mathcal {B}_{r,d})$ . From this description, we see that $A^*(\mathcal {B}_{r,d})$ is not finitely generated as a $\mathbb {Z}$ -algebra. Finally, when $k = \mathbb {C}$ , the cohomology ring of $\mathcal {B}_{r,d}$ is isomorphic to its Chow ring.

Chow rings of low-degree Hurwitz spaces

Abstract While there is much work and many conjectures surrounding the intersection theory of the moduli space of curves, relatively little is known about the intersection theory of the Hurwitz space ℋ k , g {\mathcal{H}_{k,g}} parametrizing smooth degree k, genus g covers of ℙ 1 {\mathbb{P}^{1}} . Let k = 3 , 4 , 5 {k=3,4,5} . We prove that the rational Chow rings of ℋ k , g {\mathcal{H}_{k,g}} stabilize in a suitable sense as g tends to infinity. In the case k = 3 {k=3} , we completely determine the Chow rings for all g. We also prove that the rational Chow groups of the simply branched Hurwitz space ℋ k , g s ⊂ ℋ k , g {\mathcal{H}^{s}_{k,g}\subset\mathcal{H}_{k,g}} are zero in codimension up to roughly g k {\frac{g}{k}} . In [S. Canning and H. Larson, The Chow rings of the moduli spaces of curves of genus 7 , 8 {7,8} and 9, preprint 2021, arXiv:2104.05820], results developed in this paper are used to prove that the Chow rings of ℳ 7 , ℳ 8 , {\mathcal{M}_{7},\mathcal{M}_{8},} and ℳ 9 {\mathcal{M}_{9}} are tautological.

Poincaré duality for tautological Chern subrings of orthogonal grassmannians

Let $X$ be an orthogonal grassmannian of a nondegenerate quadratic form $q$ over a field. Let $C$ be the subring in the Chow ring $\text {CH}(X)$ generated by the Chern classes of the tautological vector bundle on $X$. We prove Poincaré duality for $C$. For $q$ of odd dimension, the result was already known due to an identification between $C$ and the Chow ring of certain symplectic grassmannian. For $q$ of even dimension, such an identification is not available.

Prelog Chow rings and degenerations

For a simple normal crossing variety X, we introduce the concepts of prelog Chow ring, saturated prelog Chow group, as well as their counterparts for numerical equivalence. Thinking of X as the central fibre in a (strictly) semistable degeneration, these objects can intuitively be thought of as consisting of cycle classes on X for which some initial obstruction to arise as specializations of cycle classes on the generic fibre is absent. Cycle classes in the generic fibre specialize to their prelog counterparts in the central fibre, thus extending to Chow rings the method of studying smooth varieties via strictly semistable degenerations. After proving basic properties for prelog Chow rings and groups, we explain how they can be used in an envisaged further development of the degeneration method by Voisin et al. to prove stable irrationality of very general fibres of certain families of varieties; this extension would allow for much more singular degenerations, such as toric degenerations as occur in the Gross–Siebert programme, to be usable. We illustrate that by looking at the example of degenerations of elliptic curves, which, although simple, shows that our notion of prelog decomposition of the diagonal can also be used as an obstruction in cases where all components in a degeneration and their mutual intersections are rational. We also compute the saturated prelog Chow group of degenerations of cubic surfaces.

Matroid psi classes.

Motivated by the intersection theory of moduli spaces of curves, we introduce psi classes in matroid Chow rings and prove a number of properties that naturally generalize properties of psi classes in Chow rings of Losev-Manin spaces. We use these properties of matroid psi classes to give new proofs of (1) a Chow-theoretic interpretation for the coefficients of the reduced characteristic polynomials of matroids, (2) explicit formulas for the volume polynomials of matroids, and (3) Poincaré duality for matroid Chow rings.

On the Chow ring of some Lagrangian fibrations

On the Chow ring of some Lagrangian fibrations

The Chow ring of the classifying space of the unitary group

The Chow ring of the classifying space of the unitary group

On the structure of the algebra generated by the rational equivalence classes of Brill–Noether loci in the Chow ring of the moduli space of semistable bundles on elliptic curve

On the structure of the algebra generated by the rational equivalence classes of Brill–Noether loci in the Chow ring of the moduli space of semistable bundles on elliptic curve

On product identities and the Chow rings of holomorphic symplectic varieties

For a moduli space \({\mathsf M}\) of stable sheaves over a K3 surface X, we propose a series of conjectural identities in the Chow rings \(CH_\star ({\mathsf M}\times X^\ell ),\, \ell \ge 1,\) generalizing the classic Beauville–Voisin identity for a K3 surface. We emphasize consequences of the conjecture for the structure of the tautological subring \(R_\star ({\mathsf M}) \subset CH_\star ({\mathsf M}).\) The conjecture places all tautological classes in the lowest piece of a natural filtration emerging on \(CH_\star ({\mathsf M})\), which we also discuss. We prove the proposed identities when \({\mathsf M}\) is the Hilbert scheme of points on a K3 surface.

Valuations and the Hopf Monoid of Generalized Permutahedra

AbstractThe goal of this paper is to show that valuation theory and Hopf theory are compatible on the class of generalized permutahedra. We prove that the Hopf structure $\textbf {GP}^+$ on these polyhedra descends, modulo the inclusion-exclusion relations, to an indicator Hopf monoid $\mathbb {I}(\textbf {GP}^+)$ of generalized permutahedra that is isomorphic to the Hopf monoid of weighted ordered set partitions. This quotient Hopf monoid $\mathbb {I}(\textbf {GP}^+)$ is cofree. It is the terminal object in the category of Hopf monoids with polynomial characters; this partially explains the ubiquity of generalized permutahedra in the theory of Hopf monoids. This Hopf theoretic framework offers a simple, unified explanation for many new and old valuations on generalized permutahedra and their subfamilies. Examples include, for matroids: the Chern–Schwartz–MacPherson cycles, Eur’s volume polynomial, the Kazhdan–Lusztig polynomial, the motivic zeta function, and the Derksen–Fink invariant; for posets: the order polynomial, Poincaré polynomial, and poset Tutte polynomial; for generalized permutahedra: the universal Tutte character and the corresponding class in the Chow ring of the permutahedral variety. We obtain several algebraic and combinatorial corollaries; for example, the existence of the valuative character group of $\textbf {GP}^+$ and the indecomposability of a nestohedron into smaller nestohedra.

Algebraic cycles and Fano threefolds of genus 8

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.

The Chern classes and Euler characteristic of the moduli spaces of Abelian differentials

AbstractFor the moduli spaces of Abelian differentials, the Euler characteristic is one of the most intrinsic topological invariants. We give a formula for the Euler characteristic that relies on intersection theory on the smooth compactification by multi-scale differentials. It is a consequence of a formula for the full Chern polynomial of the cotangent bundle of the compactification.The main new technical tools are an Euler sequence for the cotangent bundle of the moduli space of multi-scale differentials and computational tools in the Chow ring, such as a description of normal bundles to boundary divisors.

On the Chow ring of the classifying stack of algebraic tori

We investigate the structure of the Chow ring of the classifying stacks $BT$ of algebraic tori, as it has been defined by B. Totaro. Some previous work of N. Karpenko, A. Merkurjev, S. Blinstein and F. Scavia has shed some light on the structure of such rings. In particular Karpenko showed the absence of torsion classes in the case of permutation tori, while Merkurjev and Blinstein described in a very effective way the second Chow group $A^2(BT)$ in the general case. Building on this work, Scavia exhibited an example where $A^2(BT)_\text{tors}\neq 0$. Here, by making use of a very elementary approach, we extend the result of Karpenko to special tori and we completely determine the Chow ring $A^*(BT)$ when $T$ is an algebraic torus admitting a resolution with special tori $0\rightarrow T\rightarrow Q\rightarrow P$. In particular we show that there can be torsion in the Chow ring of such tori.