Virasoro constraints and the Chern classes of the Hodge bundle

We analyse the consequences of the Virasoro conjecture of Eguchi, Hori and Xiong for Gromov-Witten invariants, in the case of zero degree maps to the manifolds P-1 and P-2 (or more generally, smooth projective curves and smooth simply connected projective surfaces). We obtain predictions involving intersections of psi and lambda classes on (M) over bar(g,n). In particular, we show that the Virasoro conjecture for P-2 implies the numerical part of Faber's conjecture on the tautological Chow ring of M-g. (C) 1998 Published by Elsevier Science B.V.