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

Palka, Karol; Pełka, Tomasz (2020). Classification of planar rational cuspidal curves II. Log del Pezzo models. Proceedings of the London Mathematical Society, 120(5), pp. 642-703. Oxford University Press 10.1112/plms.12300

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Let E ⊆ P² be a complex curve homeomorphic to the projective line. The Negativity Conjecture asserts that the Kodaira–Iitaka dimension of K_X + 1/ 2 D , where (X, D) ⟶ (P², 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 complements admit no 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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