The Chow ring of a sequence of point blow-ups

Abstract

Given a sequence of point blow-ups of smooth n-dimensional projective varieties $Z_{i}$ defined over an algebraically closed field k, $Z_{s}\rightarrow Z_{s-1}\rightarrow ...\rightarrow Z_{1}\rightarrow Z_{0}$, we give two presentations of the Chow ring of its sky $A^{\bullet}(Z_{s})$. The first one using the classes of the total transforms of the exceptional components as generators and the second one using the classes of the strict transforms ones. We prove that the skies of two sequences of point blow-ups of the same length have isomorphic Chow rings. Finally we give a characterization of final divisor of a sequence of point blow-ups in terms of some relations defined over the Chow group of zero-cycles of its sky $A_{0}(Z_{s})$.