Related papers: A threefold violating a local-to-global principle …
We use the Clemens-Griffiths method to construct smooth projective threefolds, over any field $k$ admitting a separable quadratic extension, that are $k$-unirational and $\bar{k}$-rational but not $k$-rational. When $k=\mathbb{R}$, we can…
We implement two-cover descent for plane quartics over Q with all 28 bitangents rational and show that on a significant collection of test cases, it resolves the existence of rational points. We also review a classical description of the…
In this paper the notion of rational simple connectedness for the quintic Fano threefold $V_5\subset \mathbb{P}^6$ is studied and unirationality of the moduli spaces $\overline{M}_{0,0}^{\text{bir}}(V_5,d)$, with $d \ge 1$, is proved. Many…
Given a projective intersection of two quadrics X in at least 9 variables, the quantitative behaviour of the rational points on X is investigated under the assumption that X contains a pair of conjugate singular points defined over the…
We determine the algebraic and transcendental lattices of a general cubic fourfold with a symplectic automorphism of prime order. We prove that cubic fourfolds admitting a symplectic automorphism of order at least three are rational, and we…
A well known conjecture asserts that a cubic fourfold X is rational if it has a cohomologically associated K3 surface. G.Ouchi proved that if X admits a finite group G of symplectic automorphisms, whose order is different from 2, then X has…
We classify rational, irreducible quartic symmetroids in projective 3-space. They are either singular along a line or a smooth conic section, or they have a triple point or a tacnode.
For any field k of characteristic at most 5 we exhibit an explicit smooth quartic surface in projective threespace over k with trivial automorphism group over the algebraic closure of k. We also show how this can be extended to higher…
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…
It is known that the smooth rational threefolds of P^5 having a rational non-special surface of P^4 as general hyperplane section have degree d=3,... ,7. We study such threefolds X from the point of view of linear systems of surfaces in…
Let $R$ be the field of real Puiseux series. It is a real closed field. We construct the first examples of smooth intersections of two quadrics in $\mathbb{P}_R^5$ and smooth cubic hypersurfaces in $\mathbb{P}_R^4$ which are not stably…
We examine the existence of rational divisors on modular curves of $\mathcal{D}$-elliptic sheaves and on Atkin-Lehner quotients of these curves over local fields. Using a criterion of Poonen and Stoll, we show that in infinitely many cases…
The paper contains a general construction which produces new examples of non simply-connected smooth projective surfaces. We analyze the resulting surfaces and their fundamental groups. Many of these fundamental groups are expected to be…
We show that a very general (3,3) complete intersection in $\mathbb{P}^7$ over an algebraically closed uncountable field of characteristic different from 2 admits no decomposition of the diagonal, in particular it is not retract rational.…
We pose some questions about spaces parametrizing rational curves on rationally connected varieties. We give a partial answer for cubic threefolds. Many of our results were previously proved by Iliev, Markushevich and Tikhimirov by…
The Beilinson-Bloch type conjectures predict that the low degree rational Chow groups of intersections of quadrics are one dimensional. This conjecture was proved by Otwinowska. Making use of homological projective duality and the recent…
Let X -> Y be a fibration whose fibers are complete intersections of two quadrics. We develop new categorical and algebraic tools---a theory of relative homological projective duality and the Morita invariance of the even Clifford algebra…
In this research announcement we associate to each convex polytope, possibly nonrational and nonsimple, a family of compact spaces that are stratified by quasifolds, i.e. the strata are locally modelled by $\R^k$ modulo the action of a…
In this work, we show that for a certain class of threefolds in positive characteristics, rational-chain-connectivity is equivalent to supersingularity. The same result is known for K3 surfaces with elliptic fibrations. And there are…
In this paper we define the notion of a hyperk\"ahler manifold (potentially) of Jacobian type. If we view hyperk\"ahler manifolds as "abelian varieties", then those of Jacobian type should be viewed as "Jacobian varieties". Under a minor…