Bounded Negativity Conjecture

Prove that for every smooth projective surface X, there exists an integer b(X) greater than or equal to zero such that every curve C contained in X satisfies C^2 greater than or equal to -b(X).

Background

The paper identifies the Bounded Negativity Conjecture as a major unresolved problem in the theory of algebraic surfaces. It asks whether the self-intersections of curves on any fixed smooth projective surface are bounded below by a surface-dependent constant.

The conjecture is presented as motivation for the paper’s discussion of the bounded cohomology property, since the bounded cohomology property is known to imply bounded negativity.