Cubic curves and totally geodesic subvarieties of moduli space

In this paper we present the first example of a primitive, totally geodesic subvariety F subset of M-g,M-n with dim(F) > 1. The variety we consider is a surface F subset of M-1,M-3 defined using the projective geometry of plane cubic curves. We also obtain a new series of Teichmuller curves in M-4, and new SL2(R)-invariant varieties in the moduli spaces of quadratic differentials and holomorphic 1-forms.