## Duke Mathematical Journal

### 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
Revised: 5 February 2017
First available in Project Euclid: 13 July 2017

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

Digital Object Identifier
doi:10.1215/00127094-2017-0010

Mathematical Reviews number (MathSciNet)
MR3715805

Zentralblatt MATH identifier
06812215

#### 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

#### 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.