Gushel--Mukai varieties: classification and birationalities

TL;DR

This paper provides a comprehensive classification and analysis of Gushel-Mukai varieties, revealing their intrinsic properties, automorphisms, and birational relationships, and connecting them with Eisenbud-Popescu-Walter sextics.

Contribution

It introduces an intrinsic characterization of Gushel-Mukai varieties, offers a new proof of their classification, and explores their relations with EPW sextics and birational properties.

Findings

Intrinsic characterization of Gushel-Mukai varieties.

Classification theorem and automorphism group descriptions.

Birational equivalence of period partners and dual varieties in higher dimensions.

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 to these varieties. 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.