Duke Mathematical Journal

Exceptional isomorphisms between complements of affine plane curves

Jérémy Blanc, Jean-Philippe Furter, and Mattias Hemmig

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

Received: 19 October 2017
Revised: 21 December 2018
First available in Project Euclid: 24 August 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

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

birational geometry surfaces complements of curves affine algebraic geometry


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

Export citation


  • [1] S. S. Abhyankar, W. Heinzer, and P. Eakin, On the uniqueness of the coefficient ring in a polynomial ring, J. Algebra 23 (1972), no. 2, 310–342.
  • [2] E. Artal-Bartolo and P. Cassou-Noguès, One remark on polynomials in two variables, Pacific J. Math. 176 (1996), no. 2, 297–309.
  • [3] E. Artal-Bartolo, P. Cassou-Noguès, and A. Dimca, “Sur la topologie des polynômes complexes” in Singularities (Oberwolfach, 1996), Progr. Math. 162, Birkhäuser, Basel, 1998, 317–343.
  • [4] J. Blanc, The correspondence between a plane curve and its complement, J. Reine Angew. Math. 633 (2009), 1–10.
  • [5] J. Blanc and F. Mangolte, “Cremona groups of real surfaces” in Automorphisms in Birational and Affine Geometry, Springer Proc. Math. Stat. 79, Springer, Cham, 2014, 35–58.
  • [6] J. Blanc and I. Stampfli, Automorphisms of the plane preserving a curve, Algebr. Geom. 2 (2015), no. 2, 193–213.
  • [7] M. Borodzik and H. Żoładek, Complex algebraic plane curves via Poincaré-Hopf formula, II: Annuli, Israel J. Math. 175 (2010), no. 1, 301–347.
  • [8] P. Cassou-Noguès and D. Daigle, “Rational polynomials of simple type: a combinatorial proof” in Algebraic Varieties and Automorphism Groups, Adv. Stud. Pure Math. 75, Math. Soc. Japan, Tokyo, 2017, 7–28.
  • [9] P. Cassou-Noguès, M. Koras, and P. Russell, Closed embeddings of $\mathbb{C}^{*}$ in $\mathbb{C}^{2}$, I, J. Algebra 322 (2009), no. 9, 2950–3002.
  • [10] P. Costa, New distinct curves having the same complement in the projective plane, Math. Z. 271 (2012), no. 3–4, 1185–1191.
  • [11] D. Daigle, Birational endomorphisms of the affine plane, J. Math. Kyoto Univ. 31 (1991), no. 2, 329–358.
  • [12] R. Ganong, Kodaira dimension of embeddings of the line in the plane, J. Math. Kyoto Univ. 25 (1985), no. 4, 649–657.
  • [13] M. H. Gizatullin and V. I. Danilov, Automorphisms of affine surfaces, I, Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), no. 3, 523–565, 703.
  • [14] S. Kaliman, Rational polynomials with a $\mathbb{C}^{*}$-fiber, Pacific J. Math. 174 (1996), no. 1, 141–194.
  • [15] T. Kambayashi, Automorphism group of a polynomial ring and algebraic group action on an affine space, J. Algebra 60 (1979), no. 2, 439–451.
  • [16] M. Koras, K. Palka, and P. Russell, The geometry of sporadic $\mathbb{C}^{*}$-embeddings into $\mathbb{C}^{2}$, J. Algebra 456 (2016), 207–249.
  • [17] M. Koras and P. Russell, “Some properties of $\mathbb{C}^{*}$ in $\mathbb{C}^{2}$” in Affine Algebraic Geometry, World Sci., Hackensack, NJ, 2013, 160–197.
  • [18] H. Kraft, Challenging problems on affine n-space, Astérisque 237 (1996), 29–317, Séminaire Bourbaki 1994/1995, no. 802.
  • [19] N. Mohan Kumar and M. P. Murthy, Curves with negative self-intersection on rational surfaces, J. Math. Kyoto Univ. 22 (1983), no. 4, 767–777.
  • [20] W. D. Neumann and P. Norbury, Rational polynomials of simple type, Pacific J. Math. 204 (2002), no. 1, 177–207.
  • [21] P.-M. Poloni, Counterexamples to the complement problem, to appear in Comment. Math. Helv., preprint, arXiv:1605.05169 [math.AG].
  • [22] P. Russell, Forms of the affine line and its additive group, Pacific J. Math. 32 (1970), no. 2, 527–539.
  • [23] J.-P. Serre, Arbres, amalgames, $\mathrm{SL}_{2}$, Astérisque 46, Soc. Math. France, Paris, 1977.