A gem of the modular universe

zbMATH Open1014.14022arXivalg-geom/9503018MaRDI QIDQ2736217

B. Hunt

Publication date: 29 August 2001

Abstract: We introduce one of the most beautiful algebraic varieties known, a quintic hypersurface in projective five-space, which is invariant under the action of the Weyl group of $E_6$. This variety is intricately related with many other moduli problems, some of which are: marked hyperelliptic curves of genus two, Picard curves of genus four with a $sqrt-3$-level structure, six points on the projective line, abelian surfaces with (1,3) polarisations, quartic surfaces invariant under the action of the Heisenberg group in projective three-space, K3-surfaces which are double covers of the projective plane branched along six lines, and last but not least, cubic surfaces in projective three-space. These relationships are developed in some detail, with particular care on the birational aspects.

Full work available at URL: https://arxiv.org/abs/alg-geom/9503018

zbMATH Keywords

threefoldfourfoldNieto quinticarithmetic quotients of bounded symmetric domainssubball quotients

Mathematics Subject Classification ID

Group actions on varieties or schemes (quotients) (14L30) (4)-folds (14J35) Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects) (32M15) (n)-folds ((n>4)) (14J40)

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