  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 global constant. 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.
