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

TL;DR

This paper advances the classification of complex rational cuspidal curves in the projective plane by proving structure theorems related to the Negativity Conjecture, using the minimal model program, and identifying new bicuspidal curve series.

Contribution

It provides the first classification of such curves satisfying the Negativity Conjecture and introduces a new series of bicuspidal curves, confirming the Strong Rigidity Conjecture.

Findings

Proved structure theorems for curves satisfying the Negativity Conjecture.

Classified these curves up to projective equivalence.

Discovered a new series of bicuspidal curves.

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 $\mathbb{C}^{**}$-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.