A gem of the modular universe

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.