The geometry and topology of rational surfaces with an anticanonical cycle

Let \(Y\) be a smooth rational surface and let \(D\) be an effective divisor linearly equivalent to \(-K_Y\), such that \(D\) is a cycle of smooth rational curves. Such pairs \((Y,D)\) arise in many contexts, for example in the study of degenerations of \(K3\) surfaces or in the theory of deformations of minimally elliptic singularities. Deformation types of such pairs come with two extra pieces of structure: the “generic” ample cone, i.e. the ample cone of a generic small deformation, and a distinguished set of divisor classes of square \(-2\) orthogonal to the components of the cycle. The goal of this talk is to describe these structures as well as their applications to various questions involving geometric, topological and lattice-theoretic properties of such pairs.