Towards the weighted Bounded Negativity Conjecture for blow-ups of algebraic surfaces

Roberto Laface,Piotr Pokora

doi:10.1007/s00229-019-01157-2

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.

Highlights

In the last years, negative curves on surfaces have been researched extensively because of their connection to many open problems
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 C2 ≥ −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

Summary

Introduction

Negative curves on surfaces have been researched extensively because of their connection to many open problems. (Bounded Negativity Conjecture) For every smooth projective surface X over the complex numbers, there exists a nonnegative integer b(X ) ∈ Z such that C2 ≥ −b(X ) for all integral curves C ⊂ X. (Weighted BNC) For every smooth projective surface X over the complex numbers, there exists a nonnegative integer bw ∈ Z such that C2 ≥ −bw(X ) · (C.H ) for all integral curves C ⊂ X and all big and nef line bundles H for which C.H > 0. Conjecture 0.2 is asking for a bound on the self-intersection of all integral curves on X that depends on both X and the degree of the curve C with respect of every big and nef line bundle over which the curve is positive. We provide bounds for the self-intersection numbers of irreducible and reduced curves on blow-ups of algebraic surfaces at mutually distinct points.

Generalization of a result of Sakai and Orevkov–Zaidenberg

Bounding negativity on blow-ups of P2

Bounding negativity on blow-ups of Hirzebruch surfaces