Nonexistence of Negative Curves

Abstract

Abstract Let $X$ be the blowup of a weighted projective plane ${\mathbb {P}}(a,b,c)$ at a general point. The Kleiman–Mori cone of $X$ is two-dimensional with one ray generated by the class of the exceptional curve $E$. It is not known if the second extremal ray is always generated by the class of a curve. We construct an infinite family of projective toric surfaces of Picard number one such that their blowups $X$ at a general point have half-open Kleiman–Mori cones: there is no negative curve generating the other boundary ray of the cone.