Duke Mathematical Journal

The Coolidge–Nagata conjecture

Mariusz Koras and Karol Palka

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Let EP2 be a complex rational cuspidal curve contained in the projective plane. The Coolidge–Nagata conjecture asserts that E is Cremona-equivalent to a line, that is, it is mapped onto a line by some birational transformation of P2. The second author recently analyzed the log minimal model program run for the pair (X,12D), where (X,D)(P2,E) is a minimal resolution of singularities, and as a corollary he proved the conjecture in the case when more than one irreducible curve in P2E is contracted by the process of minimalization. We prove the conjecture in the remaining cases.

Article information

Duke Math. J., Volume 166, Number 16 (2017), 3085-3145.

Received: 3 November 2015
Revised: 5 February 2017
First available in Project Euclid: 13 July 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14H50: Plane and space curves
Secondary: 14J17: Singularities [See also 14B05, 14E15] 14E07: Birational automorphisms, Cremona group and generalizations

cuspidal curve rational curve Cremona transformation Coolidge–Nagata conjecture log minimal model program almost minimal model


Koras, Mariusz; Palka, Karol. The Coolidge–Nagata conjecture. Duke Math. J. 166 (2017), no. 16, 3085--3145. doi:10.1215/00127094-2017-0010. https://projecteuclid.org/euclid.dmj/1499911237

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