The weighted Bounded Negativity Conjecture

Let $X$ be a smooth projective surface over the complex numbers, let $C\subset X$ be an integral curve, and let $H$ be a big and nef line bundle on $X$ such that $C.H>0$. Write $C^2$ for the self-intersection of $C$.

Weighted Bounded Negativity Conjecture. There exists a nonnegative integer $b_w\in\mathbb Z$ such that

$$C^2\geq-b_w(X)\cdot(C.H)^2$$

for all integral curves $C\subset X$ and all big and nef line bundles $H$ for which $C.H>0$.

This weighted variant refines bounded negativity by measuring the lower bound using the intersection with a big and nef line bundle. The paper presents it as an open question and develops evidence through bounds for curves, especially on blow-ups of algebraic surfaces.