Unexpected surfaces singular on lines in P³

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.