Classification of planar rational cuspidal curves. II. Log del Pezzo models

Abstract

Let $E\subseteq {\mathbb{P}}^2$ be a complex curve homeomorphic to the projective line. The Negativity Conjecture asserts that the Kodaira-Iitaka dimension of $K_X+\frac{1}{2}D$, where $(X,D)\to ({\mathbb{P}}^2,E)$ is a minimal log resolution, is negative. We prove structure theorems for curves satisfying this conjecture and we finish their classification up to a projective equivalence by describing the ones whose complement admits no ${\mathbb{C}}^{\ast \ast }$-fibration. As a consequence, we show that they satisfy the Strong Rigidity Conjecture of Flenner-Zaidenberg. The proofs are based on the almost minimal model program. The obtained list contains one new series of bicuspidal curves.

50 pages