Gushel-Mukai varieties: classification and birationalities.

*(English)*Zbl 1408.14053By definition, a Gushel-Mukai variety (abbreviated GM variety) is a dimensionally transverse intersection of a cone over the Grassmannian \(\mathrm{Gr}(2,5)\) with a linear space and a quadric hypersurface.

Interest in these varieties comes from results of N. P. Gushel’ [Math. USSR, Izv. 21, 445–459 (1983; Zbl 0554.14014); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 46, No. 6, 1159–1174 (1982)], who showed that any smooth prime Fano threefold with index \((-K)^3=10\) is of this kind, and of S. Mukai [Proc. Natl. Acad. Sci. USA 86, No. 9, 3000–3002 (1989; Zbl 0679.14020)], who extended these results to various other contexts.

In the present paper, the authors give a detailed study of Gushel-Mukai varieties in general, including singular cases. Among many results they prove the following:

1. (Theorem 2.3) A characterisation of normal polarized GM varieties among all normal polarized varieties in terms of data including the polarization degree and simplicity of the so-called twisted excess conormal sheaf; in Theorem 2.16 and Lemma 2.29 this characterisation is simplified in case the singularities of the variety are sufficiently mild.

2. (Theorem 2.9) An equivalence of groupoids between the groupoid of normal polarized GM varieties of fixed dimension \(n\) and the groupoid of so-called GM data sets, which are defined as certain sets of linear-algebraic data of a particular type.

3. An extension of the result of A. Iliev and L. Manivel [Ann. Sci. Éc. Norm. Supér. (4) 44, No. 3, 393–426 (2011; Zbl 1258.14050)] (relating smooth GM varieites to so-called EPW sextics) to the locally complete intersection case.

4. (Sections 3.6 and 3.7; Theorem 3.25, Theorem 3.27) Notions of period partners and duality for GM varieties, together with explicit descriptions in terms of linear-algebraic data of the sets of period partners and dual varieties of a given GM variety.

5. (Section 4) Rationality results for high-dimensional smooth GM varieties, and birationality of period partners and dual varieties in dimensions 3 and 4.

Interest in these varieties comes from results of N. P. Gushel’ [Math. USSR, Izv. 21, 445–459 (1983; Zbl 0554.14014); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 46, No. 6, 1159–1174 (1982)], who showed that any smooth prime Fano threefold with index \((-K)^3=10\) is of this kind, and of S. Mukai [Proc. Natl. Acad. Sci. USA 86, No. 9, 3000–3002 (1989; Zbl 0679.14020)], who extended these results to various other contexts.

In the present paper, the authors give a detailed study of Gushel-Mukai varieties in general, including singular cases. Among many results they prove the following:

1. (Theorem 2.3) A characterisation of normal polarized GM varieties among all normal polarized varieties in terms of data including the polarization degree and simplicity of the so-called twisted excess conormal sheaf; in Theorem 2.16 and Lemma 2.29 this characterisation is simplified in case the singularities of the variety are sufficiently mild.

2. (Theorem 2.9) An equivalence of groupoids between the groupoid of normal polarized GM varieties of fixed dimension \(n\) and the groupoid of so-called GM data sets, which are defined as certain sets of linear-algebraic data of a particular type.

3. An extension of the result of A. Iliev and L. Manivel [Ann. Sci. Éc. Norm. Supér. (4) 44, No. 3, 393–426 (2011; Zbl 1258.14050)] (relating smooth GM varieites to so-called EPW sextics) to the locally complete intersection case.

4. (Sections 3.6 and 3.7; Theorem 3.25, Theorem 3.27) Notions of period partners and duality for GM varieties, together with explicit descriptions in terms of linear-algebraic data of the sets of period partners and dual varieties of a given GM variety.

5. (Section 4) Rationality results for high-dimensional smooth GM varieties, and birationality of period partners and dual varieties in dimensions 3 and 4.

Reviewer: Artie Prendergast-Smith (Loughborough)

##### MSC:

14E07 | Birational automorphisms, Cremona group and generalizations |

14J45 | Fano varieties |

14J60 | Vector bundles on surfaces and higher-dimensional varieties, and their moduli |