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Working paper

Gushel--Mukai varieties: classification and birationalities

Kuznetsov A. G., Debarre O.
This paper performs a systematic study of Gushel–Mukai varieties—Fano manifolds with Picard number 1, coindex 3, and degree 10 (higher-dimensional analogues of prime Fano threefolds of genus 6). We introduce a new approach to the classification of these varieties which includes mildly singular varieties, gives a criterion for an isomorphism of such varieties, and describes their automorphisms groups. We carefully develop the relation between Gushel–Mukai varieties and Eisenbud–Popescu– Walter 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/dual varieties are always birationally isomorphic.