On the geometry of anticanonical pairs

TL;DR

This paper surveys the geometry and deformation theory of rational surfaces with anticanonical cycles, highlighting recent advances and proving new results about their diffeomorphism and deformation types, as well as root characterizations.

Contribution

It introduces new proofs linking diffeomorphism and deformation types and offers a novel characterization of roots in the context of anticanonical pairs.

Findings

Diffeomorphism type equals deformation type for pairs (Y,D).

New characterization of roots as classes of square -2 becoming rational curves.

Survey of birational geometry and deformation theory of anticanonical pairs.

Abstract

The systematic study of rational surfaces $Y$ with an anticanonical cycle $D$ dates back to a fundamental paper of Looijenga in 1981. Recently, Gross, Hacking and Keel have introduced new ideas into the subject. The goal of this mainly expository paper is to survey some results about such surfaces, old and new. We discuss the birational geometry and deformation theory of such pairs as well as the behavior of nef and big linear systems. We prove a theorem of Torelli type due to Gross-Hacking-Keel and describe some consequences. Among the new results in this paper are (1) a proof that the diffeomorphism type of a pair $(Y,D)$ is the same as its deformation type, and (2) a new characterization of the roots of the pair, i.e. the integral classes of square $-2$ in $H^2(Y)$ orthogonal to the components of $D$ which become the class of a smooth rational curve in some deformation.