## Duke Mathematical Journal

- Duke Math. J.
- Volume 168, Number 12 (2019), 2235-2297.

### Exceptional isomorphisms between complements of affine plane curves

Jérémy Blanc, Jean-Philippe Furter, and Mattias Hemmig

#### Abstract

This article describes the geometry of isomorphisms between complements of geometrically irreducible closed curves in the affine plane ${\mathbb{A}}^{2}$, over an arbitrary field, which do not extend to an automorphism of ${\mathbb{A}}^{2}$. We show that such isomorphisms are quite exceptional. In particular, they occur only when both curves are isomorphic to open subsets of the affine line ${\mathbb{A}}^{1}$, with the same number of complement points, over any field extension of the ground field. Moreover, the isomorphism is uniquely determined by one of the curves, up to left composition with an automorphism of ${\mathbb{A}}^{2}$, except in the case where the curve is isomorphic to the affine line ${\mathbb{A}}^{1}$ or to the punctured line ${\mathbb{A}}^{1}\setminus \left\{0\right\}$. If one curve is isomorphic to ${\mathbb{A}}^{1}$, then both curves are equivalent to lines. In addition, for any positive integer $n$, we construct a sequence of $n$ pairwise nonequivalent closed embeddings of ${\mathbb{A}}^{1}\setminus \left\{0\right\}$ with isomorphic complements. In characteristic $0$ we even construct infinite sequences with this property.

Finally, we give a geometric construction that produces a large family of examples of nonisomorphic geometrically irreducible closed curves in ${\mathbb{A}}^{2}$ that have isomorphic complements, answering negatively the complement problem posed by Hanspeter Kraft. This also gives a negative answer to the holomorphic version of this problem in any dimension $n\ge 2$. The question had been raised by Pierre-Marie Poloni.

#### Article information

**Source**

Duke Math. J., Volume 168, Number 12 (2019), 2235-2297.

**Dates**

Received: 19 October 2017

Revised: 21 December 2018

First available in Project Euclid: 24 August 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.dmj/1566612024

**Digital Object Identifier**

doi:10.1215/00127094-2019-0012

**Mathematical Reviews number (MathSciNet)**

MR3999446

**Subjects**

Primary: 14E07: Birational automorphisms, Cremona group and generalizations

Secondary: 14J26: Rational and ruled surfaces 14R10: Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) 32M17: Automorphism groups of Cn and affine manifolds

**Keywords**

birational geometry surfaces complements of curves affine algebraic geometry

#### Citation

Blanc, Jérémy; Furter, Jean-Philippe; Hemmig, Mattias. Exceptional isomorphisms between complements of affine plane curves. Duke Math. J. 168 (2019), no. 12, 2235--2297. doi:10.1215/00127094-2019-0012. https://projecteuclid.org/euclid.dmj/1566612024