Duke Mathematical Journal

The Coolidge–Nagata conjecture

Mariusz Koras and Karol Palka

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

Abstract

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

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


Export citation

References

  • [1] S. S. Abhyankar and T. T. Moh, Embeddings of the line in the plane, J. Reine Angew. Math. 276 (1975), 148–166.
  • [2] W. P. Barth, K. Hulek, C. A. M. Peters, and A. Van de Ven, Compact Complex Surfaces, 2nd ed., Ergeb. Math. Grenzgeb. (3) 4, Springer, Berlin, 2004.
  • [3] M. Borodzik and C. Livingstone, Heegaard Floer homology and rational cuspidal curves, Forum Math. Sigma 2 (2014), no. e28.
  • [4] M. Borodzik and H. Zoła̧dek, Complex algebraic plane curves via Poincaré-Hopf formula, II: Annuli, Israel J. Math. 175 (2010), 301–347.
  • [5] P. Cassou-Nogues, M. Koras, and P. Russell, Closed embeddings of $\mathbb{C}^{*}$ in $\mathbb{C}^{2}$, I, J. Algebra 322 (2009), 2950–3002.
  • [6] J. L. Coolidge, A Treatise on Algebraic Plane Curves, Dover, New York, 1959.
  • [7] J. F. de Bobadilla, I. Luengo-Velasco, A. Melle-Hernández, and A. Némethi, On rational cuspidal projective plane curves, Proc. Lond. Math. Soc. (3) 92 (2006), 99–138.
  • [8] J. F. de Bobadilla, I. Luengo-Velasco, A. Melle-Hernández, and A. Némethi, “Classification of rational unicuspidal projective curves whose singularities have one Puiseux pair” in Real and Complex Singularities, Trends Math., Birkhäuser, Basel, 2007, 31–45.
  • [9] T. Fenske, Rational 1- and 2-cuspidal plane curves, Beiträge Algebra Geom. 40 (1999), 309–329.
  • [10] T. Fenske, Rational cuspidal plane curves of type $(d,d-4)$ with $\chi(\Theta_{V}\langle D\rangle)\leq0$, Manuscripta Math. 98 (1999), 511–527.
  • [11] H. Flenner and M. Zaidenberg, “$\mathbf{Q}$-acyclic surfaces and their deformations” in Classification of Algebraic Varieties (L’Aquila, 1992), Contemp. Math. 162, Amer. Math. Soc., Providence, 1994, 143–208.
  • [12] H. Flenner and M. Zaidenberg, On a class of rational cuspidal plane curves, Manuscripta Math. 89 (1996), 439–459.
  • [13] H. Flenner and M. Zaidenberg, Rational cuspidal plane curves of type $(d,d-3)$, Math. Nachr. 210 (2000), 93–110.
  • [14] T. Fujita, On the topology of noncomplete algebraic surfaces, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982), 503–566.
  • [15] R. V. Gurjar and M. Miyanishi, On contractible curves in the complex affine plane, Tohoku Math. J. (2) 48 (1996), 459–469.
  • [16] H. Kashiwara, Fonctions rationnelles de type $(0,1)$ sur le plan projectif complexe, Osaka J. Math. 24 (1987), 521–577.
  • [17] R. Kobayashi, S. Nakamura, and F. Sakai, A numerical characterization of ball quotients for normal surfaces with branch loci, Proc. Japan Acad. Ser. A Math. Sci. 65 (1989), 238–241.
  • [18] M. Koras, “On contractible plane curves” in Affine Algebraic Geometry, Osaka Univ. Press, Osaka, 2007, 275–288.
  • [19] M. Koras, “$\mathbb{C}^{*}$ in $\mathbb{C}^{2}$ is birationally equivalent to a line” in Affine Algebraic Geometry, CRM Proc. Lecture Notes 54, Amer. Math. Soc., Providence, 2011, 165–191.
  • [20] A. Langer, Logarithmic orbifold Euler numbers of surfaces with applications, Proc. Lond. Math. Soc. (3) 86 (2003), 358–396.
  • [21] T. Liu, On planar rational cuspidal curves, Ph.D. dissertation, Massachusetts Institute of Technology, Cambridge, Mass., 2014.
  • [22] T. Matsuoka and F. Sakai, The degree of rational cuspidal curves, Math. Ann. 285 (1989), 233–247.
  • [23] M. Mella and E. Polastri, Equivalent birational embeddings, II: Divisors, Math. Z. 270 (2012), 1141–1161.
  • [24] M. Miyanishi, Open Algebraic Surfaces, CRM Monogr. Ser. 12, Amer. Math. Soc., Providence, 2001.
  • [25] M. Miyanishi and S. Tsunoda, Absence of the affine lines on the homology planes of general type, J. Math. Kyoto Univ. 32 (1992), 443–450.
  • [26] T. K. Moe, Rational cuspidal curves, preprint, http://folk.uio.no/torgunnk.
  • [27] N. Mohan Kumar and M. P. Murthy, Curves with negative self-intersection on rational surfaces, J. Math. Kyoto Univ. 22 (1982/83), 767–777.
  • [28] M. Nagata, On rational surfaces, I: Irreducible curves of arithmetic genus $0$ or $1$, Mem. Coll. Sci. Univ. Kyoto Ser. A Math. 32 (1960), 351–370.
  • [29] S. Y. Orevkov, On rational cuspidal curves, I: Sharp estimate for degree via multiplicities, Math. Ann. 324 (2002), 657–673.
  • [30] K. Palka, Exceptional singular $\mathbb{Q}$-homology planes, Ann. Inst. Fourier (Grenoble) 61 (2011), 745–774.
  • [31] K. Palka, The Coolidge-Nagata conjecture, part I, Adv. Math. 267 (2014), 1–43.
  • [32] K. Palka, A new proof of the theorems of Lin-Zaidenberg and Abhyankar-Moh-Suzuki, J. Algebra Appl. 14 (2015), no. 1540012.
  • [33] K. Palka, The Coolidge-Nagata conjecture holds for curves with more than four cusps, preprint, arXiv:1202.3491v1 [math.AG].
  • [34] K. Palka, Cuspidal curves, minimal models and Zaidenberg’s finiteness conjecture, J. Reine Angew. Math., published online 12 July 2016. DOI 10.1515/crelle-2016-0021.
  • [35] J. Piontkowski, On the number of cusps of rational cuspidal plane curves, Experiment. Math. 16 (2007), 251–256.
  • [36] P. Russell, Hamburger-Noether expansions and approximate roots of polynomials, Manuscripta Math. 31 (1980), 25–95.
  • [37] F. Sakai and K. Tono, Rational cuspidal curves of type $(d,d-2)$ with one or two cusps, Osaka J. Math. 37 (2000), 405–415.
  • [38] M. Suzuki, Propriétés topologiques des polynômes de deux variables complexes, et automorphismes algébriques de l’espace $\textbf{C}^{2}$, J. Math. Soc. Japan 26 (1974), 241–257.
  • [39] K. Tono, Defining equations of certain rational cuspidal curves, I, Manuscripta Math. 103 (2000), 47–62.
  • [40] K. Tono, “Rational unicuspidal plane curves with $\bar{\kappa}=1$” in Newton Polyhedra and Singularities (Kyoto, 2001), RIMS Kôkyûroku Bessatsu 1233, 2001, 82–89.
  • [41] K. Tono, On a new class of rational cuspidal plane curves with two cusps, preprint, arXiv:1205.1248v1 [math.AG].
  • [42] S. Tsunoda, The complements of projective plane curves, RIMS Kokyuroku 446 (1981), 48–56.
  • [43] I. Wakabayashi, On the logarithmic Kodaira dimension of the complement of a curve in $P^{2}$, Proc. Japan Acad. Ser. A Math. Sci. 54 (1978), 157–162.
  • [44] M. Zaidenberg and V. Y. Lin, An irreducible, simply connected algebraic curve in $\bf{C}^{2}$ is equivalent to a quasihomogeneous curve (in Russian), Dokl. Akad. Nauk SSSR 271 (1983), no. 5, 1048–1052.