2022/02/26 by Lisa Marquand, Marquand, Lisa · 2 citations
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2202.13213
openalex publication_date 2022/02/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
There are three types of involutions on a cubic fourfold; two of anti-symplectic type, and one symplectic. Here we show that cubics with involutions exhibit the full range of behaviour in relation to rationality conjectures. Namely, we show a general cubic fourfold with symplectic involution has no associated K3 surface and is conjecturely irrational. In contrast, we show a cubic fourfold with a particular anti-symplectic involution has an associated K3, and is in fact rational. We show such a cubic is contained in the intersection of all non-empty Hassett divisors; we call such a cubic Hassett maximal. We study the algebraic and transcendental lattices for cubics with an involution both lattice theoretically and geometrically.