Related papers: The diagonal of (3,3) fivefolds

We show that a very general quartic hypersurface in $\mathbb P^6 $ over a field of characteristic different from 2 does not admit a decomposition of the diagonal, hence is not retract rational. This generalizes a result of Nicaise--Ottem,…

We study the question of the existence of a decomposition of the diagonal for very general quartic and $(2,3)$-complete intersection $n$-folds. Using cycle-theoretic techniques of Lange, Pavic and Schreieder we reduce the question via a…

We apply the specialization technique based on the decomposition of the diagonal to find an explicit example over $\mathbb{Q}$ of a quadric and cubic hypersurface in $\mathbb{P}^6$ such that their intersection is a smooth stably irrational…

We study the existence of a decomposition of the diagonal for bidegree hypersurfaces in a product of projective spaces. Using a cycle theoretic degeneration technique due to Lange, Pavic and Schreieder, we develop an inductive procedure…

We consider the connections among algebraic cycles, abelian varieties, and stable rationality of smooth projective varieties in positive characteristic. Recently Voisin constructed two new obstructions to stable rationality for rationally…

We prove that the general quartic double solid with $k\leq 7$ nodes does not admit a Chow theoretic decomposition of the diagonal, or equivalently has a nontrivial universal ${\rm CH}_0$ group. The same holds if we replace in this statement…

We generalize a recent result of Pavic--Schreieder regarding the surjectivity of the obstruction morphism defined in [PS23]. As a consequence of this result, we show that geometrically (retract) rational varieties over a Laurent field of…

We use the motivic obstruction to stable rationality introduced by Shinder and the first-named author to establish several new classes of stably irrational hypersurfaces and complete intersections. In particular, we show that very general…

We show that a very general hypersurface of degree d at least 4 and dimension at most $(d+1)2^{d-4}$ over a field of characteristic different from 2 does not admit a decomposition of the diagonal; hence, it is neither stably nor retract…

We adapt for algebraically closed fields $k$ of characteristic greater than $2$ two results of Voisin, on the decomposition of the diagonal of a smooth cubic hypersurface $X$ of dimension $3$ over $\mathbb C$, namely: the equivalence…

We describe explicit birational maps from some rational complete intersections of three quadrics in $\mathbb{P}^7$ to some prime Fano manifolds together with their Sarkisov decomposition via a single Secant Flop, allowing us to recover the…

Let $Y$ be a smooth complete intersection of a quadric and a cubic in $\mathbb{P}^n$, with $n$ even. We show that $Y$ has a multiplicative Chow-K\"unneth decomposition, in the sense of Shen-Vial. As a consequence, the Chow ring of (powers…

This work begins the process of using the decomposition of the diagonal as a tool for studying the rationality of invariant fields of finite groups $G$. Our ground field must be characteristic 0 because of the use we make of Bertini…

The main aim of this paper is to show that a cyclic cover of $\mathbb{P}^n$ branched along a very general divisor of degree $d$ is not stably rational provided that $n \ge 3$ and $d \ge n+1$. This generalizes the result of…

We prove that a complete intersection of $c$ very general hypersurfaces of degree at least two in $N$-dimensional complex projective space is not ruled (and therefore not rational) provided that the sum of the degrees of the hypersurfaces…

We show that the natural nc-space attached to an intersection of three quadrics in P^7 is truly non-commutative. In particular, its associated numerical K-lattice is not isomorphic to the K-lattice of any smooth projective surface, so the…

On one hand, for a general Calabi-Yau complete intersection X, we establish a decomposition, up to rational equivalence, of the small diagonal in X^3, from which we deduce that any decomposable 0-cycle of degree 0 is in fact rationally…

Let $Y$ be a smooth complete intersection of three quadrics, and assume the dimension of $Y$ is even. We show that $Y$ has a multiplicative Chow-K\"unneth decomposition, in the sense of Shen-Vial. As a consequence, the Chow ring of (powers…

We study rationality properties of smooth complete intersections of three quadrics in $\mathbb{P}^7$. We exhibit a smooth family of such intersections with both rational and non-rational fibers.

We study twists of the Burkhardt quartic threefold over non-algebraically closed base fields of characteristic different from 2,3,5. We show they all admit quartic models in projective four-space. We identify a Galois-cohomological…