  We study the curvature of the moduli space M_g of curves of genus g with the
Siegel metric induced by the period map. We give an explicit formula for the
holomorphic sectional curvature of M_g along a Schiffer variation at a point P
on the curve X, in terms of the holomorphic sectional curvature of A_g and the
second Gaussian map. Finally we extend the Kaehler form of the Siegel metric as
a closed current on the Deligne-Mumford compatification of M_g and we determine
its cohomology class as a multiple of the first Chern class of the Hodge
bundle.
