Second Gaussian map and curvature of the moduli space of curves

We study the second Gaussian map for a curve X of genus g, in relation with the second fundamental form of the period map. We show that the holomorphic sectional curvature of M_g along a Schiffer variation at a point P on the curve X can be expressed in terms of the holomorphic sectional curvature of A_g and the second Gaussian map.. We exhibit a class of infinitely many curves with surjective second Gaussian map. We compute its rank on the hyperelliptic and trigonal locus and we study rank properties of the second Gaussian map for curves on Abelian surfaces and for curves on K3 surfaces. These results are obtained in collaboration with Elisabetta Colombo.