  The Bounded Negativity Conjecture predicts that for any smooth complex
surface $X$ there exists a lower bound for the selfintersection of reduced
divisors on $X$. This conjecture is open. It is also not known if the existence
of such a lower bound is invariant in the birational equivalence class of $X$.
In the present note we introduce certain constants $H(X)$ which measure in
effect the variance of the lower bounds in the birational equivalence class of
$X$. We focus on rational surfaces and relate the value of $H({\mathbb P}^2)$
to certain line arrangements. Our main result is Theorem 3.3 and the main open
challenge is Problem 3.10.
