# American Institute of Mathematical Sciences

February 2017, 24: 21-27. doi: 10.3934/era.2017.24.003

## The orbifold Langer-Miyaoka-Yau Inequality and Hirzebruch-type inequalities

 1 Instytut Matematyki, Pedagogical University of Cracow, Podchorążych 2, PL-30-084 Kraków, Poland 2 Institut für Algebraische Geometrie, Leibniz Universität Hannover, Welfengarten 1, D-30167 Hannover, Germany

To Professor Kamil Rusek, on the occasion of his 70th birthday.

Received  December 26, 2016 Published  April 2017

Fund Project: I am very grateful to Adrian Langer for stimulating conversations, useful comments about the content of the note, and for pointing out [2]. Finally, I would like to thank the anonymous referees for valuable comments that allowed to improve the note. The author is partially supported by National Science Centre Poland Grant 2014/15/N/ST1/02102.

Using Langer's variation on the Bogomolov-Miyaoka-Yau inequality, we provide some Hirzebruch-type inequalities for curve arrangements in the complex projective plane.

Citation: Piotr Pokora. The orbifold Langer-Miyaoka-Yau Inequality and Hirzebruch-type inequalities. Electronic Research Announcements, 2017, 24: 21-27. doi: 10.3934/era.2017.24.003
##### References:
 [1] G. Barthel, F. Hirzebruch and Th. Höfer, Geradenkonfigurationen und Algebraische Flächen, Aspects of Mathematics, D4, Vieweg, Braunschweig, 1987. doi: 10.1007/978-3-322-92886-3. [2] R. Bojanowski, Zastosowania Uogólnionej Nierówno÷ci Bogomolova-Miyaoka-Yau, Master Thesis (in Polish), 2003. Available from: http://www.mimuw.edu.pl/%7Ealan/postscript/bojanowski.ps. [3] E. Brieskorn and H. Knörrer, Ebene Algebraische Kurven, Birkhäuser Verlag, Basel u. a., 1981. [4] P. Cassou-Noguè and A. Płoski, Invariants of plane curve singularities and Newton diagrams, Univ. Iagel. Acta Math., 49 (2011), 9-34. [5] F. Hirzebruch, Arrangements of lines and algebraic surfaces, in Arithmetic and Geometry, Vol. II, Progr. Math., 36, Birkhäuser, Boston, Mass., 1983,113-140. [6] F. Hirzebruch, Singularities of algebraic surfaces and characteristic numbers, in The Lefschetz Centennial Conference, Part I (Mexico City, 1984), Contemp. Math., 58, Amer. Math. Soc., Providence, RI, 1986,141-155. doi: 10.1090/conm/058.1/860410. [7] V. Klee and S. Wagon, Old and New Unsolved Problems in Plane Geometry and Number Theory, The Mathematical Association of America, 1991. [8] A. Langer, Logarithmic orbifold Euler numbers with applications, Proc. London Math. Soc., 86 (2003), 358-396. doi: 10.1112/S0024611502013874. [9] Y. Miyaoka, On the Chern numbers of surfaces of general type, Invent. Math., 42 (1977), 225-237. doi: 10.1007/BF01389789. [10] P. Pokora, X. Roulleau and T. Szemberg, Bounded negativity, Harbourne constants and transversal arrangements of curves, to appear in Ann. Inst. Fourier Grenoble, arXiv: 1602.02379. [11] H. Schenck and S. Tohaneanu, Freeness of Conic-Line arrangements in ℙ2, Commentarii Mathematici Helvetici, 84 (2009), 235-258. doi: 10.4171/CMH/161. [12] L. Tang, Algebraic surfaces associated to arrangements of conics, Soochow Journal of Mathematics, 21 (1995), 427-440. [13] Z. Han, A note on the weak Dirac conjecture, The Electronic Journal of Combinatorics, 24 (2017). Available from: http://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i1p63.

show all references

##### References:
 [1] G. Barthel, F. Hirzebruch and Th. Höfer, Geradenkonfigurationen und Algebraische Flächen, Aspects of Mathematics, D4, Vieweg, Braunschweig, 1987. doi: 10.1007/978-3-322-92886-3. [2] R. Bojanowski, Zastosowania Uogólnionej Nierówno÷ci Bogomolova-Miyaoka-Yau, Master Thesis (in Polish), 2003. Available from: http://www.mimuw.edu.pl/%7Ealan/postscript/bojanowski.ps. [3] E. Brieskorn and H. Knörrer, Ebene Algebraische Kurven, Birkhäuser Verlag, Basel u. a., 1981. [4] P. Cassou-Noguè and A. Płoski, Invariants of plane curve singularities and Newton diagrams, Univ. Iagel. Acta Math., 49 (2011), 9-34. [5] F. Hirzebruch, Arrangements of lines and algebraic surfaces, in Arithmetic and Geometry, Vol. II, Progr. Math., 36, Birkhäuser, Boston, Mass., 1983,113-140. [6] F. Hirzebruch, Singularities of algebraic surfaces and characteristic numbers, in The Lefschetz Centennial Conference, Part I (Mexico City, 1984), Contemp. Math., 58, Amer. Math. Soc., Providence, RI, 1986,141-155. doi: 10.1090/conm/058.1/860410. [7] V. Klee and S. Wagon, Old and New Unsolved Problems in Plane Geometry and Number Theory, The Mathematical Association of America, 1991. [8] A. Langer, Logarithmic orbifold Euler numbers with applications, Proc. London Math. Soc., 86 (2003), 358-396. doi: 10.1112/S0024611502013874. [9] Y. Miyaoka, On the Chern numbers of surfaces of general type, Invent. Math., 42 (1977), 225-237. doi: 10.1007/BF01389789. [10] P. Pokora, X. Roulleau and T. Szemberg, Bounded negativity, Harbourne constants and transversal arrangements of curves, to appear in Ann. Inst. Fourier Grenoble, arXiv: 1602.02379. [11] H. Schenck and S. Tohaneanu, Freeness of Conic-Line arrangements in ℙ2, Commentarii Mathematici Helvetici, 84 (2009), 235-258. doi: 10.4171/CMH/161. [12] L. Tang, Algebraic surfaces associated to arrangements of conics, Soochow Journal of Mathematics, 21 (1995), 427-440. [13] Z. Han, A note on the weak Dirac conjecture, The Electronic Journal of Combinatorics, 24 (2017). Available from: http://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i1p63.
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