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(P 2 ) to certain line arrangements. Our main result is Theorem 3.3 and the main open challenge is Problem 3.10. Problem 1.2 (Birational invariance of the BNC). Let X and Y be birationally equivalent projective surfaces. Does BNC hold for X if and only if it holds for Y ? In other words: is the bounded negativity property a birational invariant? Remark 1.3. Note, that a solution to the above problem is not known even if Y is the blow-up of X in a single point.Of course, if BNC is true in general, then the above problem has an affirmative solution. However, even in that situation, it is still of interest to know how the bounds b(X) and b(Y ) are related in terms of the complexity of a birational map between X and Y .

Abstract. The purpose of this note is to give defined over the real numbers counterexamples to a question relevant in the commutative algebra, concerning a containment relation between algebraic and symbolic powers of homogeneous ideal.

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In this note we find a bound for the so-called linear Harbourne constants for smooth hypersurfaces in P 3 C .