Relative cone of curves and extremal contractions of a successive blowup

RESUMO

Let $X$ be a normal variety, and let $\pi\colon\tilde X\to X$ be the successive blowup along subvarieties $Z_1,\dotsc,Z_n\subseteq X$ of codimension at least two that have simple normal crossings and satisfy $Z_h\not\supseteq Z_i$ whenever $h<I$. We prove that the relative cone of curves $\overline{\operatorname{NE}}(\tilde X/X)$ is generated by the classes of finitely many elementary curves, and that every face admits a contraction over $X$. We describe the exceptional loci of extremal ray contractions, and prove that every small extremal ray contraction admits a $D$-flip for every $\mathbb R$-Cartier divisor $D$ negative on the corresponding ray.