On linear systems of $$\mathbb {P}^3$$ P 3 with nine base points

Abstract

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 the blown-up projective plane in points 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 $$2m+1$$ .