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A very special EPW sextic and two IHS fourfolds

Maria Donten-Bury, Bert van Geemen, Grzegorz Kapustka, Michał Kapustka and Jarosław A Wiśniewski

Geometry & Topology 21 (2017) 1179–1230

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 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 X0 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 X0 is also related to the Debarre–Varley abelian fourfold.

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EPW sextics, IHS fourfolds, abelian varieties
Mathematical Subject Classification 2010
Primary: 14D06, 14J35, 14J70, 14K12, 14M07
Secondary: 14J50, 14J28
Received: 28 September 2015
Revised: 28 January 2016
Accepted: 3 March 2016
Published: 17 March 2017
Proposed: Richard Thomas
Seconded: Jim Bryan, Simon Donaldson
Maria Donten-Bury
Institute of Mathematics
Freie Universität Berlin
Arnimallee 3
D-14195 Berlin
Bert van Geemen
Department of Mathematics
University di Milano
I-20133 Milan
Grzegorz Kapustka
Institute of Mathematics of the Polish Academy of Sciences
ul. Sniadeckich 8
P.O. Box 21
00-656 Warszawa
Michał Kapustka
Department of Mathematics and Informatics
Jagiellonian University
Łojasiewicza 6
30-348 Kraków
Jarosław A Wiśniewski
Faculty of Mathematics
Informatics and Mechanics
University of Warsaw
Banacha 2
02-097 Warszawa