The Ring of Projective Invariants of Eight Points on the Line via Representation Theory

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 S8, by an ideal I8, so the modular fivefold (P1)8// GL(2) is Proj(Sym•(V )/I8). 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 I8. 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 [Koi], 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 P1, with arbitrary weighting, in [HMSV3], completing the program of [HMSV1]. 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.