Chow Quotients and Projective Bundle Formulas for Euler-chow Series

Given a projective algebraic variety X, let Πp(X) denote the monoid of effective algebraic equivalence classes of effective algebraic cycles on X. The p-th Euler-Chow series of X is an element in the formal monoid-ring Z[[Πp(X)]] defined in terms of Euler characteristics of the Chow varieties Cp,α (X) of X, with α ∈ Πp(X). We provide a systematic treatment of such series, and give projective bundle formulas which generalize previous results by [LY87] and [Eli94]. The techniques used involve the Chow quotients introduced in [Kap93], and this allows the computation of various examples including some Grassmannians and flag varieties. There are relations between these examples and representation theory, and further results point to interesting connections between Euler-Chow series for certain varieties and the topology of the moduli spaces M0,n+1.