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Cuspidal curves, minimal models and Zaidenberg’s finiteness conjecture

  • Karol Palka EMAIL logo

Abstract

Let E2 be a complex rational cuspidal curve and let (X,D)(2,E) be the minimal log resolution of singularities. We prove that E has at most six cusps and we establish an effective version of the Zaidenberg finiteness conjecture (1994) concerning Eisenbud–Neumann diagrams of E. This is done by analyzing the Minimal Model Program run for the pair (X,12D). Namely, we show that 2E is **-fibred or for the log resolution of the minimal model the Picard rank, the number of boundary components and their self-intersections are bounded.

Funding source: Narodowe Centrum Nauki

Award Identifier / Grant number: 2012/05/D/ST1/03227

Funding statement: The author was supported by the National Science Centre Poland, Grant No. 2012/05/D/ST1/03227, and by the Foundation for Polish Science within the Homing Plus programme, cofinanced by the European Union, Regional Development Fund.

Acknowledgements

The author is grateful to Mariusz Koras for discussions concerning producing *’s contained in the affine part of the surface. His remark to reuse the BMY inequality at the final stage of the proof of Theorem 1.1 (4) (instead of using the bound obtained by Tono) led to improved numerical bounds in the inequalities. The author would also thank Mikhail Zaidenberg for discussing results in the literature.

References

[1] M. Borodzik and C. Livingstone, Heegaard Floer homology and rational cuspidal curves, Forum Math. Sigma 2 (2014), Article ID 28. 10.1017/fms.2014.28Search in Google Scholar

[2] J. L. Coolidge, A treatise on algebraic plane curves, Dover Publications, New York 1959. Search in Google Scholar

[3] J. Fernández de Bobadilla, I. Luengo, A. Melle-Hernández and A. Némethi, On rational cuspidal plane curves, open surfaces and local singularities, Singularity theory, World Scientific, Hackensack (2007), 411–442. 10.1142/9789812707499_0015Search in Google Scholar

[4] T. Fenske, Rational cuspidal plane curves of type (d,d-4) with χ(ΘVD)0, Manuscripta Math. 98 (1999), no. 4, 511–527. 10.1007/s002290050158Search in Google Scholar

[5] H. Flenner and M. Zaidenberg, -acyclic surfaces and their deformations, Classification of algebraic varieties (L’Aquila 1992), Contemp. Math. 162, American Mathematical Society, Providence (1994), 143–208. 10.1090/conm/162/01532Search in Google Scholar

[6] H. Flenner and M. Zaidenberg, On a class of rational cuspidal plane curves, Manuscripta Math. 89 (1996), no. 4, 439–459. 10.1007/BF02567528Search in Google Scholar

[7] G. Freudenburg and P. Russell, Open problems in affine algebraic geometry, Affine algebraic geometry, Contemp. Math. 369, American Mathematical Society, Providence (2005), 1–30. 10.1090/conm/369/06801Search in Google Scholar

[8] T. Fujita, On the topology of noncomplete algebraic surfaces, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982), no. 3, 503–566. Search in Google Scholar

[9] J.  Kollár and S.  Kovács, Birational geometry of log surfaces, preprint (1994), https://web.math.princeton.edu/~kollar/FromMyHomePage/BiratLogSurf.ps. Search in Google Scholar

[10] J. Kollár and S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Math. 134, Cambridge University Press, Cambridge 1998. 10.1017/CBO9780511662560Search in Google Scholar

[11] M. Koras and K. Palka, The Coolidge–Nagata conjecture, preprint (2015), http://arxiv.org/abs/1502.07149. 10.1215/00127094-2017-0010Search in Google Scholar

[12] A. Langer, Logarithmic orbifold Euler numbers of surfaces with applications, Proc. Lond. Math. Soc. (3) 86 (2003), no. 2, 358–396. 10.1112/S0024611502013874Search in Google Scholar

[13] R. Lazarsfeld, Positivity in algebraic geometry. II, Ergeb. Math. Grenzgeb. (3) 49, Springer, Berlin 2004. 10.1007/978-3-642-18810-7Search in Google Scholar

[14] T. Liu, On planar rational cuspidal curves, ProQuest LLC, Ann Arbor 2014; Ph.D. thesis, Massachusetts Institute of Technology, 2014. Search in Google Scholar

[15] K. Matsuki, Introduction to the Mori program, Universitext, Springer, New York 2002. 10.1007/978-1-4757-5602-9Search in Google Scholar

[16] T. Matsuoka and F. Sakai, The degree of rational cuspidal curves, Math. Ann. 285 (1989), no. 2, 233–247. 10.1007/BF01443516Search in Google Scholar

[17] M. Miyanishi, Open algebraic surfaces, CRM Monogr. Ser. 12, American Mathematical Society, Providence 2001. 10.1090/crmm/012Search in Google Scholar

[18] M. Miyanishi and T. Sugie, Homology planes with quotient singularities, J. Math. Kyoto Univ. 31 (1991), no. 3, 755–788. 10.1215/kjm/1250519728Search in Google Scholar

[19] M. Miyanishi and S. Tsunoda, Absence of the affine lines on the homology planes of general type, J. Math. Kyoto Univ. 32 (1992), no. 3, 443–450. 10.1215/kjm/1250519486Search in Google Scholar

[20] N. Mohan Kumar and M. Pavaman Murthy, Curves with negative self-intersection on rational surfaces, J. Math. Kyoto Univ. 22 (1982/83), no. 4, 767–777. 10.1215/kjm/1250521679Search in Google Scholar

[21] 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. 10.1215/kjm/1250776405Search in Google Scholar

[22] S. Y. Orevkov, On rational cuspidal curves, Math. Ann. 324 (2002), no. 4, 657–673. 10.1007/s002080000191Search in Google Scholar

[23] K. Palka, Exceptional singular -homology planes, Ann. Inst. Fourier (Grenoble) 61 (2011), no. 2, 745–774. 10.5802/aif.2628Search in Google Scholar

[24] K. Palka, The Coolidge–Nagata conjecture, part I, Adv. Math. 267 (2014), 1–43. 10.1016/j.aim.2014.07.038Search in Google Scholar

[25] K. Tono, Defining equations of certain rational cuspidal curves. I, Manuscripta Math. 103 (2000), no. 1, 47–62. 10.1007/s002290070028Search in Google Scholar

[26] K. Tono, On the number of the cusps of cuspidal plane curves, Math. Nachr. 278 (2005), no. 1–2, 216–221. 10.1002/mana.200310236Search in Google Scholar

[27] M. G. Zaidenberg and S. Y. Orevkov, On rigid rational cuspidal plane curves, Uspekhi Mat. Nauk 51 (1996), no. 1(307), 149–150. 10.1070/RM1996v051n01ABEH002770Search in Google Scholar

Received: 2015-02-10
Revised: 2015-11-23
Published Online: 2016-07-12
Published in Print: 2019-02-01

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