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# Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie

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1435-5345
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Volume 2019, Issue 747

# Cuspidal curves, minimal models and Zaidenberg’s finiteness conjecture

Karol Palka
Published Online: 2016-07-12 | DOI: https://doi.org/10.1515/crelle-2016-0021

## Abstract

Let $E\subseteq {ℙ}^{2}$ be a complex rational cuspidal curve and let $\left(X,D\right)\to \left({ℙ}^{2},E\right)$ 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 $\left(X,\frac{1}{2}D\right)$. Namely, we show that ${ℙ}^{2}\setminus E$ is ${ℂ}^{**}$-fibred or for the log resolution of the minimal model the Picard rank, the number of boundary components and their self-intersections are bounded.

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Revised: 2015-11-23

Published Online: 2016-07-12

Published in Print: 2019-02-01

Funding Source: Narodowe Centrum Nauki

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

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.

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2019, Issue 747, Pages 147–174, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102,

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