Mixed multiplicities, Segre numbers and Segre classes

Based on classical results of Rees and on multivariate Hilbert polynomials, we define new mixed multiplicities of two arbitrary ideals in a local ring (A, m) and use them to express the local degrees of all varieties appearing in the Gaffney–Gassler construction of Segre cycles. We prove that the classical mixed multiplicities of m and an arbitrary ideal I, which are a special case of the new ones, are equal to the generalized Samuel multiplicities of an ideal in the Rees algebra R_I(A). This equality is used to improve a result of Jeffries, Montaño and Varbaro on the degree of the fiber cone of an ideal. We conclude the paper with formulas (and their inverses) which express the degrees of Segre classes of subschemes of arbitrary projective varieties by generalized Samuel multiplicities or by classical mixed multiplicities. Using the mixed multiplicities of balanced rational normal scrolls, which have been computed by Hoang and Lam, we find the mixed multiplicities of all rational normal scrolls as well as their Segre classes and their generalized Samuel multiplicities.