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The universal Kummer threefold. (English) Zbl 1312.14103

A Kummer threefold is the quotient of a principally polarized abelian threefold by \(\pm 1_A\). It admits an natural embedding into \(\mathbb{P}^7\) by second order theta functions. Their moduli space lies naturally in a second \(\mathbb{P}^7\), taking the closure, this gives the universal Kummer threefold. It is a \(six\)-dimensional projective variety \({\mathcal K}_3\) naturally embedded in \(\mathbb{P}^7\times\mathbb{P}^7\) which lies over the Satake compactification of the moduli space \({\mathcal A}_3(2,4)\). The main object of this paper is the investigation of the ideal \(I_3\) of \({\mathcal K}_3\) in \(\mathbb{P}^7\times\mathbb{P}^7\).
The paper also studies variants of \({\mathcal K}_3\) over the Göpel variety as well as over the reflection representation of type \(E_7\). The paper contains very interesting relations between these classical objects as well as well as to a certain toric variety. Section 8 contains what is currently known about \(I_3\).

MSC:

14K10 Algebraic moduli of abelian varieties, classification
14K25 Theta functions and abelian varieties
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14T05 Tropical geometry (MSC2010)

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