Communications of the
Korean Mathematical Society CKMS

Given a sequence of point blow-ups of smooth $n-$dimensional projective varieties $Z_{i}$ defined over an algebraically closed field $\mathit{k}$, $Z_{s}\xrightarrow{\pi_{s}} Z_{s-1}\xrightarrow{\pi_{s-1}}\cdot\cdot\cdot\xrightarrow{\pi_{2}} Z_{1}\xrightarrow{\pi_{1}} Z_{0}$, with $Z_{0}\cong\mathbb{P}^{n}$, we give two presentations of the Chow ring $A^{\bullet}(Z_{s})$ of its sky. The first one uses the classes of the total transforms of the exceptional components as generators and the second one uses 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 the final divisors of a sequence of point blow-ups in terms of some relations defined over the Chow group of zero-cycles $A_{0}(Z_{s})$ of its sky.

Keywords: Blow-ups, Chow ring, intersection theory

MSC numbers: Primary 14C15, 14C17, 14E05, 14N10

Supported by: This work was financially supported by PGC2018-096446-B-C21.