On multiplicities of graded sequences of ideals

We generalize a result from [L. Ein, et al., math. AG/0202303], proving that for an arbitrary graded sequence of zero-dimensional ideals, the multiplicity of the sequence is equal to its volume. This is done using a deformation to monomial ideals. As a consequence of our result, we obtain a formula which computes the multiplicity of an ideal I in terms of the multiplicities of the initial monomial ideals of the powers Im. We use this to give a new proof of the inequality between multiplicity and the log canonical threshold from [de Fernex, et al., J. Alg. Geom., to appear].