Gushel-Mukai varieties: Classification and birationalities

Abstract

We perform a systematic study of Gushel-Mukai varieties-quadratic sections of linear sections of cones over the Grassmannian Gr(2, 5). This class of varieties includes Clifford general curves of genus 6, Brill-Noether general polarized K3 surfaces of genus 6, prime Fano threefolds of genus 6, and their higher-dimensional analogues. We establish an intrinsic characterization of normal Gushel-Mukai varieties in terms of their excess conormal sheaves, which leads to a new proof of the classification theorem of Gushel and Mukai. We give a description of isomorphism classes of Gushel-Mukai varieties and their automorphism groups in terms of linear-algebraic data naturally associated with these varieties. We carefully develop the relation between Gushel-Mukai varieties and Eisenbud-Popescu-Walter (EPW) sextics introduced earlier by Iliev-Manivel and O'Grady. We describe explicitly all Gushel-Mukai varieties whose associated EPW sextics are isomorphic or dual (we call them period partners or dual varieties, respectively). Finally, we show that in dimension 3 and higher, period partners or dual varieties are always birationally isomorphic.

Zero-cycles on double EPW sextics

“…where the correspondences I, A and B are now generically defined with respect to M dEP W and thus satisfy all the requirements of (8). The constant α ∈ Q being non-zero (Theorem 2.24), this proves equality (8), and hence Proposition 3.2. This closes the proof of the theorem.…”

“…To prove equality (8), let us start with the relation given by Proposition 2.28 and reduce to algebraic equivalence:…”

“…The only thing left to worry about is the fact that A and B and I in (11) are generically defined with respect to M GM 4 , whereas in (8) we require them to be generically defined with respect to M dEP W . To remedy this, let us consider the base change…”

“…Gushel-Mukai varieties have been studied in depth by Debarre and Kuznetzov [7], [8], [9], [10], [11].…”

Stability conditions and moduli spaces for Kuznetsov components of Gushel–Mukai varieties

“…where CGr(2, 5) is the cone over the Grassmannian Gr(2, 5) embedded via the Plücker embedding in a 10-dimensional projective space P 10 , and Q is a quadric hypersurface in a linear subspace P n+4 ⊂ P 10 . If n = 2 then X is a Brill-Noether general degree 10 polarized K3 surface, while for n ≥ 3 GM varieties coincide with the class of smooth Fano varieties of Picard number 1, degree 10 and coindex 3 [Gus83,Muk89,DK18].…”