Towards the weighted Bounded Negativity Conjecture for blow-ups of algebraic surfaces
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© The Author(s) 2019
Roberto Laface · Piotr Pokora
Towards the weighted Bounded Negativity Conjecture for blow-ups of algebraic surfaces Received: 13 December 2018 / Accepted: 10 October 2019 Abstract. In the present paper we focus on a weighted version of the Bounded Negativity Conjecture, which predicts that for every smooth projective surface in characteristic zero the self-intersection numbers of reduced and irreducible curves are bounded from below by a function depending on the intesection of curve with an arbitrary big and nef line bundle that is positive on the curve.We gather evidence for this conjecture by showing various bounds on the self-intersection number of curves in an algebraic surface. We focus our attention on blow-ups of algebraic surfaces, which have so far been neglected.
Introduction In the last years, negative curves on surfaces have been researched extensively because of their connection to many open problems. Among these, one cannot refrain from mentioning Nagata’s conjecture [11] or the SHGH conjecture [5]. The present paper is devoted to yet another open question in the geometry of complex surfaces: Conjecture 0.1. (Bounded Negativity Conjecture) For every smooth projective surface X over the complex numbers, there exists a nonnegative integer b(X ) ∈ Z such that C 2 ≥ −b(X ) for all integral curves C ⊂ X . The Bounded Negativity Conjecture (BNC in short) has a long oral tradition, and it seems to date back to F. Enriques. In some cases, the conjecture is known to hold true, for instance when the anti-canonical bundle is Q-effective or when the surface is equipped with a surjective endomorphism of degree d > 1. However, if one considers non-minimal surfaces, e.g. blow-ups of a surface for which BNC is R. Laface: Technische Universität München, Zentrum Mathematik - M11, Boltzmannstraße 3, 85478 Garching bei München, Germany. e-mail: [email protected] P. Pokora (B): Institut für Algebraische Geometrie, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany. Present Address P. Pokora: Department of Mathematics, Pedagogical University of Cracow, Podchora˛˙zych 2, 30-084 Kraków, Poland. e-mail: [email protected]; [email protected] Mathematics Subject Classification: Primary 14C20; Secondary 14J70 · 52C35 · 32S22
https://doi.org/10.1007/s00229-019-01157-2
R. Laface, P. Pokora
known to hold, then very little is known and the problem acquires a very different flavor. As it turns out, the BNC is equivalent to the statement in Conjecture 0.1 where one allows C to be any reduced curve in X [1, Proposition 3.8.2]. This has paved the way to the study of the BNC from the point of view of configurations of curves via the notion of H-constant [2]. The H-constant is an asymptotic invariant that has the potential of studying the BNC on all blow-ups of a given algebraic surface at all possible configurations of points on it simultaneously, see for instance [2,6,7,13–15]. In the present paper, we go back to focusing our attention on integral curves and bounding their negativit
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