The Chow ring of a sequence of point and rational curve blow-ups

Abstract

Given a sequence of point and rational curve blow-ups blow-ups of smooth $3-$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 $Z_{0}\cong\mathbb{P}^{3}$, we give an explicit presentations of the Chow ring $A^{\bullet}(Z_{s})$ of its sky. We prove that, in contrast to the case of sequences of point blow-ups, the skies of two sequences of point and rational curve blow-ups of the same length and even with the same proximity relations may have non-isomorphic Chow rings. Moreover, we explore some necessary conditions for the existence of such an isomorphism under some proximity configurations, and we apply the previous results in order to establish the allowed proximity type between two irreducible components of the exceptional divisor when both are regularly and projectively contractable.