The article addresses a difficult conjecture in algebraic geometry, namely the Bounded Negativity Conjecture, which states that for any arbitrary smooth complex projective surface \(X\), it exists a constant \(b(X)\in \mathbb{N} \) such that for every reduced curve \(C\subset X\) we have: \(C^2 \geq -b(X)\).
For many surfaces the conjecture is known to hold, but we don’t know if the property remains after blowing up a few points; it is not even known if the conjecture holds for \(\mathbb{P}^2\) blown up at 10 generic points.
In view of proving the conjecture, it is useful to know the behavior of particular configurations of curves on surfaces and also what happens after blowing up \(X\) at points. In this paper the case of configurations of conics in the plane is studied, and it is proved that if \(C\) is a curve of degree \(2k\) composed by \(k\) smooth conic such that all of them do intersect transversally and they have no point common to all of them; then on the blow up of the plane at Sing(\(C\)), the strict transform \(\tilde C\) is such that \(\tilde C^2 \geq -4.5 s\), where \(s\) is the number of the points blown up. (a bound is given also for configurations of conics having 2,3 or 4 points in common).
In order to prove this result Harbourne constants are used, which are related to the negativity of self-intersection of strict transform of curves on a surface.