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 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 Wisniewski [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.

A very special EPW sextic and two IHS fourfolds / M. Donten Bury, B. van Geemen, G. Kapustka, M. Kapustka, J. Wiśniewski. - In: GEOMETRY & TOPOLOGY. - ISSN 1364-0380. - 21:2(2017), pp. 1179-1230. [10.2140/gt.2017.21.1179]

A very special EPW sextic and two IHS fourfolds

B. van Geemen;
2017

Abstract

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 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 Wisniewski [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.
surfaces; varieties; manifolds
Settore MAT/03 - Geometria
GEOMETRY & TOPOLOGY
Article (author)
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2434/489079
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