Abstract:
Through this talk I will review some of the principal results of my thesis. The main object of our research is the study of the exceptional divisor $E$ living in the sky $Z_{s}$ of a sequence of blow-ups at smooth centers: $Z_{s}\xrightarrow{\pi_{s}} Z_{s-1}\xrightarrow{\pi_{s-1}}\cdot\cdot\cdot\xrightarrow{\pi_{2}} Z_{1}\xrightarrow{\pi_{1}} Z_{0}$. More precisely, we are interested in giving if possible an explicit presentation of the Chow ring $A^{\bullet}(Z_{s})$ as well as a numerical characterization of final divisors, that is irreducible components of $E$ that can be regularly contracted, by using the numerical data encoded by the intersection form on divisors with exceptional support $\mathcal{I}_{E,Z_{s}}$. We will present results for sequences of point blow-ups in arbitrary dimension and for sequences of point and rational curve blow-ups in dimension $3$.