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Article

Gushel-Mukai varieties: Classification and birationalities

Algebraic Geometry. 2018. Vol. 5. No. 1. P. 15-76.
Kuznetsov A. G., Debarre O.

This paper performs a systematic study of Gushel–Mukai varieties —Fano mani- folds with Picard number 1, coindex 3, and degree 10 (higher-dimens ional analogues of prime Fano threefolds of genus 6). We introduce a new approach to the c lassification of these va- rieties which includes mildly singular varieties, gives a criterion for an iso morphism of such varieties, and describes their automorphisms groups. We carefully develop the relation between Gushel–Mukai varieties an d Eisenbud–Popescu– Walter sextics introduced earlier by Iliev–Manivel and O’Grady. We de scribe explicitly all Gushel–Mukai varieties whose associated EPW sextics are isomorph ic or dual (we call them period partners or dual varieties respectively). Finally, we show th at in dimension 3 and higher, period partners/dual varieties are always birationally isomo rphic.