The Universal Kummer Threefold

The universal Kummer threefold is a 9-dimensional variety that represents the total space of the 6-dimensional family of Kummer threefolds in . 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.

[1]  A. Khaled.  Projective normality and equations of Kummer varieties , 1995 .

[2]  F. Piazza,et al.  Siegel modular forms and finite symplectic groups , 2008, 0804.3769.

[3]  Z. Teitler,et al.  TORIC VARIETIES , 2010 .

[4]  Nathan Pflueger,et al.  Tropical Curves , 2015 .

[5]  A. Cayley,et al.  Algorithm for the characteristics of the triple -functions. , 1879 .

[6]  È. Vinberg,et al.  THE WEYL GROUP OF A GRADED LIE ALGEBRA , 1976 .

[7]  G. Ottaviani,et al.  On the hypersurface of Luroth quartics , 2009, 0903.5149.

[8]  Bernd Sturmfels,et al.  How to Draw Tropical Planes , 2008, Electron. J. Comb..

[9]  J. P. Glass.  Theta constants of genus three , 1980 .

[10]  Bernd Sturmfels,et al.  Tropicalization of Classical Moduli Spaces , 2013, Math. Comput. Sci..

[11]  Tsukuba J. Math.  REMARK ON SOME COMBINATORIAL CONSTRUCTION OF RELATIVE INVARIANTS , 1981 .

[12]  Caroline J. Klivans,et al.  The Bergman complex of a matroid and phylogenetic trees , 2006, J. Comb. Theory, Ser. B.

[13]  D. Eisenbud.  Commutative Algebra: with a View Toward Algebraic Geometry , 1995 .

[14]  S. Grushevsky,et al.  On the Coble quartic , 2012, 1212.1895.

[15]  Herbert Lange,et al.  Complex Abelian Varieties , 1992 .

[16]  A. Beauville.  The Coble hypersurfaces , 2003, math/0306097.

[17]  B. Geemen,et al.  Siegel modular forms vanishing on the moduli space of curves , 1984 .

[18]  Aaas News,et al.  Book Reviews , 1893, Buffalo Medical and Surgical Journal.

[19]  D. Mumford.  On the equations defining abelian varieties. I , 1966 .

[20]  R. Stanley.  Hilbert functions of graded algebras , 1978 .

[21]  Bernard Deconinck,et al.  Computing Riemann theta functions , 2002, Math. Comput..

[22]  I. G. MacDonald.  Some Irreducible Representations of Weyl Groups , 1972 .

[23]  Tatsuo Kimura.  Remark on some combinatorial construction of relative inveriants , 1981 .

[24]  Alicia Dickenstein,et al.  Tropical Discriminants , 2005, math/0510126.

[25]  R. Green.  CHARACTERS OF FINITE COXETER GROUPS AND IWAHORI–HECKE ALGEBRAS (London Mathematical Society Monographs: New Series 21) By MEINOLF GECK and GÖTZ PFEIFFER: 446 pp., £65.00 (LMS members' price £45.50), ISBN 0-19-850250-8 (Clarendon Press, Oxford, 2000). , 2001 .

[26]  Andrew Snowden,et al.  The ideal of relations for the ring of invariants of "n" points on the line , 2009 .

[27]  Jun-ichi Igusa,et al.  Equations Defining Abelian Varieties , 1972 .

[28]  Meinolf Geck,et al.  Characters of Finite Coxeter Groups and Iwahori-Hecke Algebras , 2000 .

[29]  E. Freitag,et al.  The modular variety of hyperelliptic curves of genus three , 2007, 0710.5920.

[30]  C. Bramble.  A Collineation Group Isomorphic with the Group of the Double Tangents of the Plane Quartic , 1918 .

[31]  Vector bundles on curves and theta functions , 2005, math/0502179.

[32]  D. Ortland,et al.  Point sets in projective spaces and theta functions , 1988 .

[33]  D. Speyer,et al.  The Tropical Totally Positive Grassmannian , 2003, math/0312297.

[34]  C. Pauly.  Self-duality of Coble's quartic hypersurface and applications , 2001, math/0109218.

[35]  R. Manni.  MODULAR VARIETIES WITH LEVEL 2 THETA STRUCTURE , 1994 .

[36]  J. Müller.  Computing canonical heights on Jacobians , 2010 .

[37]  Melody Chan,et al.  Combinatorics of the tropical Torelli map , 2010, 1012.4539.

[38]  G. B. Mathews,et al.  Kummer's Quartic Surface , 1990 .

[39]  S. Kondō.  Moduli of Plane Quartics, Göpel Invariants and Borcherds Products , 2009, 0906.2598.

[40]  H. Barcelo,et al.  Lattices of Parabolic Subgroups in Connection with Hyperplane Arrangements , 1999 .

[41]  S. Ramanan,et al.  Moduli of Abelian varieties , 1996 .

[42]  I. Nakamura,et al.  Moduli Spaces and Arithmetic Geometry , 2007 .

[43]  R. Manni,et al.  On the Coble quartic and Fourier-Jacobi expansion of theta relations , 2013, 1304.7659.

[44]  Steven V. Sam,et al.  Moduli of Abelian Varieties, Vinberg θ-Groups, and Free Resolutions , 2012, 1203.2575.

[45]  G. Geer,et al.  Kummer Varieties and the Moduli Spaces of Abelian Varieties , 1986 .

[46]  B. V. Geemen.  Some equations for the universal Kummer variety , 2013, 1307.2463.

[47]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[48]  H. P..  Algebraic Geometry and Theta Functions , 1930, Nature.

[49]  S. Tsuyumine.  Factorial property of a ring of automorphic forms , 1986 .

[50]  Victor Reiner,et al.  Bergman complexes, Coxeter arrangements, and graph associahedra. , 2005 .

[51]  Paul Hacking,et al.  Stable pair, tropical, and log canonical compactifications of moduli spaces of del Pezzo surfaces , 2007 .

[52]  Grigory Mikhalkin,et al.  Tropical curves, their Jacobians and Theta functions , 2006 .

[53]  Siegel Modular Forms , 2006, math/0605346.

[54]  M. Chan,et al.  Tropical Teichmuller and Siegel spaces , 2012, 1207.2443.

[55]  P. Orlik,et al.  Unitary reflection groups and cohomology , 1980 .

[56]  Configurations of lines and models of Lie algebras , 2005, math/0507118.

[57]  I. Dolgachev,et al.  Lectures on Invariant Theory , 2003 .

[58]  Winfried Kohnen,et al.  On Siegel modular forms , 1996 .

[59]  Arthur Cayley,et al.  The Collected Mathematical Papers: Algorithm for the characteristics of the triple ϑ-functions , 2009 .

[60]  S. Grushevsky,et al.  The Scorza correspondence in genus 3 , 2010, 1009.0375.

[61]  Bert van Geemen,et al.  Del Pezzo Moduli via Root Systems , 2007 .