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Cubic Fourfolds with an Involution
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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.
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Cited by 2 Pith papers
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Kuznetsov components ans transcendental motives of cubic fourfolds
For Fourier-Mukai partners X and Y among special cubic fourfolds, t(X) ≅ t(Y), with explicit descriptions in rational and conjecturally irrational cases plus an equivariant construction for order-3 automorphisms.
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Kuznetsov components and transcendental motives of cubic fourfolds
For special cubic fourfolds that are Fourier-Mukai partners, transcendental motives are isomorphic, with explicit descriptions in Hassett divisor families and for those with order-3 automorphisms.
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