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

Karol Palka; Tomasz Pełka

doi:10.1112/plms.12300

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arXiv · 1810.08180

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

Karol Palka·Tomasz Pełka

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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.

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BibTeX RIS

Karol Palka, Tomasz Pełka. 2018-10-18. Classification of planar rational cuspidal curves. II. Log del Pezzo models. https://doi.org/10.1112/plms.12300

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