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Equational theories of unstable involution semigroups
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 |
Using Langer's variation on the Bogomolov-Miyaoka-Yau inequality, we provide some Hirzebruch-type inequalities for curve arrangements in the complex projective plane.
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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