  We study special linear systems of surfaces of $\mathbb{P}^3$ interpolating
nine points in general position having a quadric as fixed component. By
performing degenerations in the blown-up space, we interpret the quadric
obstruction in terms of linear obstructions for a quasi-homogeneous class. By
degeneration we also prove a Nagata type result for $\mathbb{P}^2$ that implies
a base locus lemma for the quadric. As an application we establish
Laface-Ugaglia Conjecture for linear systems with multiplicities bounded by 8
and for homogeneous linear systems with multiplicity m and degree up to 2m+1.
