2020/02/29 by Lie Fu, Robert Laterveer, Charles Vial · 24 citations
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Combinatorics #Conjecture #Geometry and complex manifolds #Humanities #Hypersurface #Mathematics #Philosophy #Pure mathematics #math.AG #msc:14C15 #msc:14C25 #msc:14D20 #msc:14F08 #msc:14H10 #msc:14J28 #msc:14J42 #msc:14J70
paper · pdf · doi:10.5802/jep.166
published in Journal de l’École polytechnique — Mathématiques 8, 1065-1097 · 27 pages; improved version thanks to referees' comments
openalex publication_date 2021/05/25 · arxiv created 2021/06/01 · arxiv updated 2021/06/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 Bridgeland 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.