PUB - Publikationen an der Universität Bielefeld

The generalized Franchetta conjecture for hyper-Kahler varieties predicts that an algebraic cycle on the universal family of certain polarized hyper-Keller varieties is fiberwise rationally equivalent to zero if and only if it vanishes in cohomology fiberwise. We establish Franchetta-type results for certain low (Hilbert) powers of low degree K3 surfaces, for the Beauville-Donagi family of Fano varieties of lines on cubic fourfolds and its relative square, and for 0-cycles and codimension-2 cycles for the Lehn-Lehn-Sorger-van Straten family of hyper-Kahler eightfolds. We also draw many consequences in the direction of the Beauville-Voisin conjecture as well as Voisin's refinement involving coisotropic subvarieties. In the appendix, we establish a new relation among tautological cycles on the square of the Fano variety of lines of a smooth cubic fourfold and provide some applications. (C) 2019 Elsevier Masson SAS. All rights reserved.