The Chow ring of a sequence of point blow-ups

TL;DR

This paper provides two explicit presentations of the Chow ring for a sequence of point blow-ups of smooth projective varieties and characterizes the final divisor using Chow group relations.

Contribution

It introduces two new presentations of the Chow ring for blown-up varieties and characterizes the final divisor via Chow group relations.

Findings

Chow rings of two sequences of point blow-ups of same length are isomorphic.

Provides explicit generators and relations for the Chow ring.

Characterizes the final divisor using zero-cycle Chow groups.

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 $A_{0}(Z_{s})$.