Duke Mathematical Journal

Exceptional isomorphisms between complements of affine plane curves

Jérémy Blanc, Jean-Philippe Furter, and Mattias Hemmig

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Abstract

This article describes the geometry of isomorphisms between complements of geometrically irreducible closed curves in the affine plane A2, over an arbitrary field, which do not extend to an automorphism of A2. 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 A1, 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 A2, except in the case where the curve is isomorphic to the affine line A1 or to the punctured line A1{0}. If one curve is isomorphic to A1, 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 A1{0} 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 A2 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 n2. 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


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