Unexpected surfaces singular on lines in P3

Abstract

We study linear systems of surfaces in P3 singular along general lines. Our purpose is to identify and classify special systems of such surfaces, i.e., those non-empty systems where the conditions imposed by the multiple lines are not independent. We prove the existence of four surfaces arising as (projective) linear systems with a single reduced member. Till now no such examples have been known. These are unexpected surfaces in the sense of recent work of Cook II, Harbourne, Migliore, and Nagel. It is an open problem if our list is complete, i.e., if it contains all reduced and irreducible unexpected surfaces based on lines in P3. As an application we find Waldschmidt constants of six general lines in P3 and an upper bound for this invariant for seven general lines.