A very special EPW sextic and two IHS fourfolds

Donten-Bury, Maria¹ ; van Geemen, Bert² ; Kapustka, Grzegorz³ ; Kapustka, Michał⁴ ; Wiśniewski, Jarosław⁵

¹ Institute of Mathematics, Freie Universität Berlin, Arnimallee 3, D-14195 Berlin, Germany
² Department of Mathematics, University di Milano, I-20133 Milan, Italy
³ Institute of Mathematics of the Polish Academy of Sciences, ul. Sniadeckich 8, P.O. Box 21, 00-656 Warszawa, Poland
⁴ Department of Mathematics and Informatics, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland
⁵ Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland

Geometry & topology, Tome 21 (2017) no. 2, pp. 1179-1230.

Résumé

We show that the Hilbert scheme of two points on the Vinberg $K3$ surface has a two-to-one map onto a very symmetric EPW sextic $Y$ in ${\mathbb{P}}^5$. The fourfold $Y$ is singular along $60$ planes, $20$ of which form a complete family of incident planes. This solves a problem of Morin and O’Grady and establishes that $20$ is the maximal cardinality of such a family of planes. Next, we show that this Hilbert scheme is birationally isomorphic to the Kummer-type IHS fourfold $X_0$ constructed by Donten-Bury and Wiśniewski [On 81 symplectic resolutions of a 4–dimensional quotient by a group of order $32$, preprint (2014)]. We find that $X_0$ is also related to the Debarre–Varley abelian fourfold.

Détail

DOI : 10.2140/gt.2017.21.1179

Classification : 14D06, 14J35, 14J70, 14K12, 14M07, 14J50, 14J28
Keywords: EPW sextics, IHS fourfolds, abelian varieties

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