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Cuspidal curves, minimal models and Zaidenberg’s finiteness conjecture

  • Karol Palka EMAIL logo
Veröffentlicht/Copyright: 12. Juli 2016

Abstract

Let E2 be a complex rational cuspidal curve and let (X,D)(2,E) be the minimal log resolution of singularities. We prove that E has at most six cusps and we establish an effective version of the Zaidenberg finiteness conjecture (1994) concerning Eisenbud–Neumann diagrams of E. This is done by analyzing the Minimal Model Program run for the pair (X,12D). Namely, we show that 2E is **-fibred or for the log resolution of the minimal model the Picard rank, the number of boundary components and their self-intersections are bounded.

Funding source: Narodowe Centrum Nauki

Award Identifier / Grant number: 2012/05/D/ST1/03227

Funding statement: The author was supported by the National Science Centre Poland, Grant No. 2012/05/D/ST1/03227, and by the Foundation for Polish Science within the Homing Plus programme, cofinanced by the European Union, Regional Development Fund.

Acknowledgements

The author is grateful to Mariusz Koras for discussions concerning producing *’s contained in the affine part of the surface. His remark to reuse the BMY inequality at the final stage of the proof of Theorem 1.1 (4) (instead of using the bound obtained by Tono) led to improved numerical bounds in the inequalities. The author would also thank Mikhail Zaidenberg for discussing results in the literature.

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Received: 2015-02-10
Revised: 2015-11-23
Published Online: 2016-07-12
Published in Print: 2019-02-01

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