The generalized Franchetta conjecture for some hyper-Kähler varieties, II

Abstract

We prove the generalized Franchetta conjecture for the locally complete family of hyper-Kähler eightfolds constructed by Lehn-Lehn-Sorger-van Straten (LLSS). As a corollary, we establish the Beauville-Voisin conjecture for very general LLSS eightfolds. The strategy consists in reducing to the Franchetta property for relative fourth powers of cubic fourfolds, by using the recent description of LLSS eightfolds as moduli spaces of semistable objects in the Kuznetsov component of the derived category of cubic fourfolds, together with its generalization to the relative setting due to Bayer-Lahoz-Macrì-Nuer-Perry-Stellari. As a by-product, we compute the Chow motive of the Fano variety of lines on a smooth cubic hypersurface in terms of the Chow motive of the cubic hypersurface.

27 pages; improved version thanks to referees' comments

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Cited by in corpus (9)

• Multiplicative Chow-Künneth decompositions and varieties of cohomological K3 type
• Algebraic cycles and Gushel-Mukai fivefolds
• Motives and the Pfaffian-Grassmannian equivalence
• Algebraic cycles and intersections of three quadrics
• Algebraic cycles and intersections of 2 quadrics
• On the Chow ring of Fano varieties on the Fatighenti-Mongardi list
• On the tautological ring of Humbert curves
• Zero-cycles and the Cayley-Oguiso automorphism
• On the motive of codimension 2 linear sections of $Gr(3,6)$