# Prelog Chow groups of self-products of degenerations of cubic threefolds.

@article{Bohning2019PrelogCG, title={Prelog Chow groups of self-products of degenerations of cubic threefolds.}, author={Christian Bohning and Hans-Christian Graf von Bothmer and Michel van Garrel}, journal={arXiv: Algebraic Geometry}, year={2019} }

It is unknown whether very general cubic threefolds have an (integral Chow-theoretic) decomposition of the diagonal, or whether they are stably rational or not. As a first step towards making progress on these questions, we compute the (saturated numerical) prelog Chow group of the self-product of a certain degeneration of cubic threefolds.

## Figures from this paper

## 3 Citations

Matrix factorizations and intermediate Jacobians of cubic threefolds

- Mathematics
- 2021

Results due to Druel and Beauville show that the blowup of the intermediate Jacobian of a smooth cubic threefold X in the Fano surface of lines can be identified with a moduli space of semistable…

The diagonal of quartic fivefolds

- Mathematics
- 2021

We show that a very general quartic hypersurface in P over a field of characteristic different from 2 does not admit a decomposition of the diagonal, hence is not retract rational. This generalizes a…

Prelog Chow rings and degenerations

- Mathematics
- 2019

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…

## References

SHOWING 1-10 OF 22 REFERENCES

On the rationality problem for quadric bundles

- MathematicsDuke Mathematical Journal
- 2019

We classify all positive integers n and r such that (stably) non-rational complex r-fold quadric bundles over rational n-folds exist. We show in particular that for any n and r, a wide class of…

Stably irrational hypersurfaces of small slopes

- Computer ScienceJournal of the American Mathematical Society
- 2019

It is shown that a very general hypersurface hypersurfaces can be an uncountable field of characteristic different from two and degree at least <inline-formula content-type="math/mathml" xmlns:mml="http://www.w3.org/1998/Math/MathML".

The motivic nearby fiber and degeneration of stable rationality

- MathematicsInventiones mathematicae
- 2019

We prove that stable rationality specializes in regular families whose fibers are integral and have at most ordinary double points as singularities. Our proof is based on motivic specialization…

Abel-Jacobi map , integral Hodge classes and decomposition of the diagonal

- Mathematics
- 2018

Given a smooth projective n-fold Y , with H(Y ) = 0, the Abel-Jacobi map induces a morphism from each smooth variety parameterizing codimension 2-cycles in Y to the intermediate Jacobian J(Y ), which…

3264 and All That: A Second Course in Algebraic Geometry

- Mathematics
- 2016

Introduction 1. Introducing the Chow ring 2. First examples 3. Introduction to Grassmannians and lines in P3 4. Grassmannians in general 5. Chern classes 6. Lines on hypersurfaces 7. Singular…

Hypersurfaces quartiques de dimension 3 : non rationalité

- stable, Ann. Sci. École Norm. Sup. (2)
- 2016

Hypersurfaces that are not stably rational

- Mathematics
- 2015

A fundamental problem of algebraic geometry is to determine which varieties are rational, that is, isomorphic to projective space after removing lower-dimensional subvarieties from both sides. In…

CHOW GROUPS, CHOW COHOMOLOGY, AND LINEAR VARIETIES

- MathematicsForum of Mathematics, Sigma
- 2014

Abstract We compute the Chow groups and the Fulton–MacPherson operational Chow cohomology ring for a class of singular rational varieties including toric varieties. The computation is closely related…

Hypersurfaces quartiques de dimension 3 : non rationalit\'e stable

- Mathematics
- 2014

Inspir\'es par un argument de C. Voisin, nous montrons l'existence d'hypersurfaces quartiques lisses dans ${\bf P}^4_{\mathbb C}$ qui ne sont pas stablement rationnelles, plus pr\'ecis\'ement dont le…

On the universal $CH_0$ group of cubic hypersurfaces

- Mathematics
- 2014

We study the existence of a Chow-theoretic decomposition of the diagonal of a smooth cubic hypersurface, or equivalently, the universal triviality of its ${\rm CH}_0$-group. We prove that for odd…