Cox rings of rational surfaces and applications of double point divisors

Abstract

This thesis consists of three parts; (1) Cox rings of rational surfaces, (2) Anticanonical models, and (3) Double point divisors. The first two parts are related by redundant blow-ups or redundant minimal model program, but the third part is independent of them.In the first part, we study redundant blow-ups based on Sakai`s work on the anticanonical models of rational surfaces. We show the existence of redundant blow-ups, and then, we classify them. It turns out that redundant blow-ups play a dominant role in studying morphisms between rational surfaces with finitely generated Cox rings. Furthermore, we prove that the finite generation of Cox rings are preserved under redundant blow-ups with some suitable assumptions. For this purpose, we directly control extremal rays of effective cones of rational surfaces. We also construct many new examples of rational surfaces with finitely generated Cox rings. In particular, we show that certain minimal resolutions of rational Q-homology projective planes and some general blow-ups of weighted projective planes have finitely generated Cox rings.To understand the structures of Fano type varieties, we study their anticanonical models and maps in the second part of this thesis. Our principal aim is to generalize Sakai`s work to higher dimensional varieties. Although canonical models have only canonical singularities, anticanonical models can have very bad singularities. Thus, we first characterize varieties whose anticanonical models have mild singularities. The canonical maps of varieties of general type are relatively well understood by recent developments of the minimal model program. Thus, we also study the structure of the anticanonical maps using so-called redundant minimal model program. We note that only the redundant blow-ups of terminal varieties with big anticanonical divisor preserve the anticanonical models, whereas every blow blow-ups of terminal varieties of general type preserves the canonical models. In surf...