×

zbMATH — the first resource for mathematics

Bounded negativity, Harbourne constants and transversal arrangements of curves. (Négativité bornée, constantes de Harbourne et arrangements transverses de courbes.) (English. French summary) Zbl 1404.14014
The paper under review is devoted to birational geometry of complex algebraic surfaces. It is motivated by the Bounded Negativity Conjecture, which stiputales that for every smooth complex projective surface \(X\), there exists a number \(b(X)\), which bounds the self-intersection of an arbitrary reduced divisor on \(X\) from below (it is clear that no such upper bound can exist). The authors provide lower bound on the self-intersection of certain classes of divisors on blow-ups of surfaces with non-negative Kodaira dimension (Theorem A) and on blow-ups of \(\mathbb{P}^2\) (Theorem B).
The paper is to a far extent self-contained, the discussion is streamlined and the arguments are transparent, even if ocassionally technical. The main tools are variants of the Miyaoka-Yau inequality and the analysis of some numerical invariants, here Harbourne constants.

MSC:
14C20 Divisors, linear systems, invertible sheaves
14J70 Hypersurfaces and algebraic geometry
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Thomas Bauer, Sandra Di Rocco, Brian Harbourne, Jack Huizenga, Anders Lundman, Piotr Pokora & Tomasz Szemberg, “Bounded negativity and arrangements of lines”, Int. Math. Res. Not. (2015) no. 19, p. 9456-9471 · Zbl 1330.14007
[2] Thomas Bauer, Brian Harbourne, Andreas Leopold Knutsen, Alex Küronya, Stefan Müller-Stach, Xavier Roulleau & Tomasz Szemberg, “Negative curves on algebraic surfaces”, Duke Math. J.162 (2013) no. 10, p. 1877-1894 · Zbl 1272.14009
[3] Josef G. Dorfmeister, “Bounded Negativity and Symplectic 4-Manifolds”, , 2016
[4] Gert-Martin Greuel, Christoph Lossen & Eugenii Shustin, “Castelnuovo function, zero-dimensional schemes and singular plane curves”, J. Algebr. Geom.9 (2000) no. 4, p. 663-710 · Zbl 1037.14009
[5] Brian Harbourne, “Global aspects of the geometry of surfaces”, Ann. Univ. Paedagog. Crac. Stud. Math.9 (2010), p. 5-41 · Zbl 1247.14006
[6] John C. Hemperly, “The parabolic contribution to the number of linearly independent automorphic forms on a certain bounded domain”, Am. J. Math.94 (1972), p. 1078-1100 · Zbl 0259.32010
[7] Friedrich Hirzebruch, Arrangements of lines and algebraic surfaces, Arithmetic and geometry, Vol. II: Geometry, Progress in Mathematics 36, Birkhäuser, 1983, p. 113-140 · Zbl 0527.14033
[8] Friedrich Hirzebruch, Singularities of algebraic surfaces and characteristic numbers, The Lefschetz centennial conference, Part I (Mexico City, 1984), Contemporary Mathematics 58, American Mathematical Society, 1986, p. 141-155
[9] Yoichi Miyaoka, “The maximal number of quotient singularities on surfaces with given numerical invariants”, Math. Ann.268 (1984) no. 2, p. 159-171 · Zbl 0521.14013
[10] Yoichi Miyaoka, “The orbibundle Miyaoka-Yau-Sakai inequality and an effective Bogomolov-McQuillan theorem”, Publ. Res. Inst. Math. Sci.44 (2008) no. 2, p. 403-417 · Zbl 1162.14026
[11] Makoto Namba, Branched coverings and algebraic functions, Pitman Research Notes in Mathematics Series 161, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1987 · Zbl 0706.14017
[12] Piotr Pokora & Halszka Tutaj-Gasińska, “Harbourne constants and conic configurations on the projective plane”, Math. Nachr.289 (2016) no. 7, p. 888-894 · Zbl 1343.14008
[13] Xavier Roulleau, “Bounded negativity, Miyaoka-Sakai inequality and elliptic curve configurations”, Int. Math. Res. Not.2017 (2017) no. 8, p. 2480-2496
[14] Fumio Sakai, “Semi-stable curves on algebraic surfaces and logarithmic pluricanonical maps”, Math. Ann.254 (1980) no. 2, p. 89-120 · Zbl 0431.14011
[15] Li Zhong Tang, “Algebraic surfaces associated to arrangements of conics”, Soochow J. Math.21 (1995) no. 4, p. 427-440 © Annales de L’Institut Fourier - ISSN (électronique) : 1777-5310 · Zbl 0862.14024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.