Related papers: Correspondences acting on constant cycle curves on…

In the context of arithmetic surfaces, Bost defined a generalized Arithmetic Chow Group (ACG) using the Sobolev space L^2_1. We study the behavior of these groups under pull-back and push-forward and we prove a projection formula. We use…

We study a generalization of a conjecture made by Beauville on the Chow ring of hyper-K\"ahler algebraic varieties. Namely we prove in a number of cases that polynomial cohomological relations involving only CH^1(X) and the Chern classes of…

Let $C$ be an irreducible algebraic curve defined over a number field and inside an algebraic torus of dimension at least 3. We partially answer a question posed by Levin on points on $C$ for which a non-trivial power lies again on $C$. Our…

We construct motivic cohomology cycles in the group $H^3_{\mathcal M}(Z,{\mathbb Q}(2))$ where $Z$ is a K3 surface obtained as a double cover of a del Pezzo surface $X$ branched at a curve in $|-2K_X|$. The construction uses (-1) curves on…

For a curve C, viewed as a cycle in its Jacobian, we study its image n_*C under multiplication by n on JC. We prove that the subgroup generated by these cycles, in the Chow group modulo algebraic equivalence, has rank at most d-1, where d…

We study tautological cycle classes on the Jacobian of a curve. We prove a new result about the ring of tautological classes on a general curve that allows, among other things, easy dimension calculations and leads to some general results…

Given a smooth del Pezzo surface $X_d \subseteq \mathbb{P}^{d}$ of degree $d,$ we show that a smooth irreducible curve $C$ on $X_d$ represents the first Chern class of an Ulrich bundle on $X_d$ if and only if its kernel bundle $M_C$ admits…

This paper presents two approaches to reducing problems on $2$-cycles on a smooth cubic hypersurface $X$ over an algebraically closed field of characteristic $\neq 2$, to problems on $1$-cycles on its variety of lines $F(X)$. The first one…

We prove that for a system of indeterminates (X_a) indiced by the P^2(2), the projective plane over F_2, there exists a 3-3 correspondance compatible with the incidence structures of P^2(2), such that (X_a) is one of the orbits of it. We…

We construct higher Chow cycles of type (2,1) on some families of K3 surfaces with non-symplectic automorphisms of order 3 and prove that our cycles are indecomposable for very general members. The proof is a combination of some…

Let \(X\subset \mathbb{P}^{n+1}\) be a smooth cubic hypersurface, and let \(F(X)\) be the variety of lines on \(X\). We prove the surjectivity of the cylinder maps on the Chow groups of \(F(X)\) and \(X\) if \(X\) contains a one-cycle of…

We extend the Horrocks correspondence between vector bundles and cohomology modules on the projective plane to the product of two projective lines. We introduce a set of invariants for a vector bundle on the product of two projective lines,…

We construct a twin correspondence between graphs with prescribed mean curvature in three-dimensional Riemannian Killing submersions and spacelike graphs with prescribed mean curvature in three-dimensional Lorentzian Killing submersions.…

According to the Bloch-Beilinson conjectures, an automorphism of a K3 surface X that acts as the identity on the transcendental lattice should act trivially on CH^2(X). We discuss this conjecture for symplectic involutions and prove it in…

This survey is based on my talk at the conference `Classical algebraic geometry today' at the MSRI. Some new results on the action of symplectomorphisms on the Chow group are added.

In this note, we consider K3 surfaces X with an action by the alternating group A_5. We show that if a cyclic extension A_5 . C_n acts on X then n = 1, 2, or 4. We also determine the A_5-invariant sublattice of the K3 lattice and its…

Over any complex cubic hypersurface of dimension at least 2, the Chow group of 1-dimensional cycles is spanned by the lines lying on the hypersurface. The smooth case has already been given several other proofs. -- On montre que sur toute…

We propose a conjectural correspondence between the set of rigid indecomposable modules over the path algebras of acyclic quivers and the set of certain non-self-intersecting curves on Riemann surfaces, and prove the correspondence for the…

We show that, for each $n>0$, there is a family of elliptic surfaces which are covered by the square of a curve of genus $2n+1$, and whose Hodge structures have an action by ${\mathbb Q}(\sqrt{-n})$. By considering the case $n=3$, we show…

We study the interplay of the moduli of curves and the moduli of K3 surfaces via the virtual class of the moduli spaces of stable maps. Using Getzler's relation in genus 1, we construct a universal decomposition of the diagonal in Chow in…