  The ring of projective invariants of eight ordered points on the line is a
quotient of the polynomial ring on V, where V is a fourteen-dimensional
representation of S_8, by an ideal I_8, so the modular fivefold (P^1)^8 //
GL(2) is Proj(Sym* (V)/I_8). We show that there is a unique cubic hypersurface
S in PV whose equation s is skew-invariant, and that the singular locus of S is
the modular fivefold. In particular, over Z[1/3], the modular fivefold is cut
out by the 14 partial derivatives of s. Better: these equations generate I_8.
In characteristic 3, the cubic s is needed to generate the ideal. The existence
of such a cubic was predicted by Dolgachev. Over Q, we recover the 14 quadrics
found by computer calculation by Koike, and our approach yields a conceptual
representation-theoretic description of the presentation. Additionally we find
the graded Betti numbers of a minimal free resolution in any characteristic.
  The proof over Q is by pure thought, using Lie theory and commutative
algebra. Over Z, the assistance of a computer was necessary. This result will
be used as the base case describing the equations of the moduli space of an
arbitrary number of points on P^1, with arbitrary weighting, in a later paper,
completing the program of our previous paper (Duke Math. J.). The modular
fivefold, and corresponding ring, are known to have a number of special
incarnations, due to Deligne-Mostow, Kondo, and Freitag-Salvati Manni, for
example as ball quotients or ring of modular forms respectively.
