Chow quotients and projective bundle formulas for Euler-Chow series

Details Chow quotients and projective bundle formulas for Euler-Chow series Chow quotients and projective bundle formulas for Euler-Chow series

1998-04-02

arxiv.org/abs/math/9804012v2

Mathematics

Algebraic Geometry

Replaced by new version, which corrects erroneous references. (The "Chow quotient" was first introduced by Kapranov, Sturmfels

Scientific paper

Given a projective algebraic variety $X$, let $\Pi_p(X)$ 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 $Z[[\Pi_p(X)]]$ defined in terms of Euler characteristics of the Chow varieties $\cvpd{p}{\alpha}{X}$ of $X$, with $\alpha \in\Pi_p(X)$. We provide a systematic treatment of such series, and give projective bundle formulas which generalize previous results by B. Lawson and S.S.Yau and Elizondo. The techniques used involve the Chow quotients introduced by Kapranov, 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 closure of moduli spaces $M_{0,n+1}$.

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