The universal Kummer threefold

Qingchun Ren; Steven V. Sam; Gus Schrader; Bernd Sturmfels

doi:10.1080/10586458.2013.816206

The universal Kummer threefold

Qingchun Ren, Steven V. Sam, Gus Schrader, Bernd Sturmfels

Mathematics

Research output: Contribution to journal › Article › peer-review

16 Scopus citations

Abstract

The universal Kummer threefold is a 9-dimensional variety that represents the total space of the 6-dimensional family of Kummer threefolds in P7. We compute defining polynomials for three versions of this family, over the Satake hypersurface, over the Göpel variety, and over the reflection representation of type E7. We develop classical themes such as theta functions and Coble's quartic hypersurfaces using current tools from combinatorics, geometry, and commutative algebra. Symbolic and numerical computations for genus-3 moduli spaces appear alongside toric and tropical methods.

Original language	English (US)
Pages (from-to)	327-362
Number of pages	36
Journal	Experimental Mathematics
Volume	22
Issue number	3
DOIs	https://doi.org/10.1080/10586458.2013.816206
State	Published - 2013

Funding

We thank Melody Chan, Igor Dolgachev, Jan Draisma, Bert van Geemen, Sam Grushevsky, Thomas Kahle, Daniel Plau-mann, and Riccardo Salvati Manni for helpful communications. We made extensive use of the software packages Sage, Macaulay2, and GAP. Qingchun Ren was supported by a Berkeley fellowship. Gus Schrader was supported by a Fulbright fellowship. Steven Sam was supported by an NDSEG fellowship and a Miller research fellowship. Bernd Sturmfels was partially supported by NSF grant DMS-0968882.

Keywords

Abelian varieties
Moduli spaces
Theta functions
Toric varieties

ASJC Scopus subject areas

General Mathematics

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10.1080/10586458.2013.816206

Cite this

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title = "The universal Kummer threefold",

abstract = "The universal Kummer threefold is a 9-dimensional variety that represents the total space of the 6-dimensional family of Kummer threefolds in P7. We compute defining polynomials for three versions of this family, over the Satake hypersurface, over the G{\"o}pel variety, and over the reflection representation of type E7. We develop classical themes such as theta functions and Coble's quartic hypersurfaces using current tools from combinatorics, geometry, and commutative algebra. Symbolic and numerical computations for genus-3 moduli spaces appear alongside toric and tropical methods.",

keywords = "Abelian varieties, Moduli spaces, Theta functions, Toric varieties",

author = "Qingchun Ren and Sam, {Steven V.} and Gus Schrader and Bernd Sturmfels",

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doi = "10.1080/10586458.2013.816206",

language = "English (US)",

volume = "22",

pages = "327--362",

journal = "Experimental Mathematics",

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AU - Ren, Qingchun

AU - Sam, Steven V.

AU - Schrader, Gus

AU - Sturmfels, Bernd

N1 - Funding Information: We thank Melody Chan, Igor Dolgachev, Jan Draisma, Bert van Geemen, Sam Grushevsky, Thomas Kahle, Daniel Plau-mann, and Riccardo Salvati Manni for helpful communications. We made extensive use of the software packages Sage, Macaulay2, and GAP. Qingchun Ren was supported by a Berkeley fellowship. Gus Schrader was supported by a Fulbright fellowship. Steven Sam was supported by an NDSEG fellowship and a Miller research fellowship. Bernd Sturmfels was partially supported by NSF grant DMS-0968882.

PY - 2013

Y1 - 2013

N2 - The universal Kummer threefold is a 9-dimensional variety that represents the total space of the 6-dimensional family of Kummer threefolds in P7. We compute defining polynomials for three versions of this family, over the Satake hypersurface, over the Göpel variety, and over the reflection representation of type E7. We develop classical themes such as theta functions and Coble's quartic hypersurfaces using current tools from combinatorics, geometry, and commutative algebra. Symbolic and numerical computations for genus-3 moduli spaces appear alongside toric and tropical methods.

AB - The universal Kummer threefold is a 9-dimensional variety that represents the total space of the 6-dimensional family of Kummer threefolds in P7. We compute defining polynomials for three versions of this family, over the Satake hypersurface, over the Göpel variety, and over the reflection representation of type E7. We develop classical themes such as theta functions and Coble's quartic hypersurfaces using current tools from combinatorics, geometry, and commutative algebra. Symbolic and numerical computations for genus-3 moduli spaces appear alongside toric and tropical methods.

KW - Abelian varieties

KW - Moduli spaces

KW - Theta functions

KW - Toric varieties

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The universal Kummer threefold

Abstract

Funding

Keywords

ASJC Scopus subject areas

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