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

TL;DR

This paper investigates special linear systems of surfaces in projective three-space with nine base points, focusing on fixed quadrics, and proves results related to obstructions and conjectures in algebraic geometry.

Contribution

It introduces a degeneration approach to interpret quadrics as obstructions and proves a Nagata type result, advancing understanding of base loci in these systems.

Findings

Interpretation of quadric obstructions via degenerations

Proof of a Nagata type result for 

Verification of Laface-Ugaglia Conjecture for specific multiplicities

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 $\mathbb{P}^2$ 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.