## Duke Mathematical Journal

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

### The Coolidge–Nagata conjecture

#### Abstract

Let $E\subseteq {\mathbb{P}}^{2}$ 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 ${\mathbb{P}}^{2}$. The second author recently analyzed the log minimal model program run for the pair $(X,\frac{1}{2}D)$, where $(X,D)\to ({\mathbb{P}}^{2},E)$ is a minimal resolution of singularities, and as a corollary he proved the conjecture in the case when more than one irreducible curve in ${\mathbb{P}}^{2}\setminus E$ is contracted by the process of minimalization. We prove the conjecture in the remaining cases.

#### Article information

**Source**

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

**Dates**

Received: 3 November 2015

Revised: 5 February 2017

First available in Project Euclid: 13 July 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.dmj/1499911237

**Digital Object Identifier**

doi:10.1215/00127094-2017-0010

**Mathematical Reviews number (MathSciNet)**

MR3715805

**Zentralblatt MATH identifier**

06812215

**Subjects**

Primary: 14H50: Plane and space curves

Secondary: 14J17: Singularities [See also 14B05, 14E15] 14E07: Birational automorphisms, Cremona group and generalizations

**Keywords**

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

#### Citation

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