The Speciality Lemma, Rank 2 Bundles and Gherardelli-type Theorems for Surfaces in P 4

In the present paper we give a very short and easy proof of the speciality lemma for codimension 2 subvarieties, even those that are reducible or non-reduced, in P n . Furthermore we give cohomological conditions that force a subcanonical surface in P 4 to be a complete intersection and a rank 2 bundle to split, which generalize the classical First Theorem of Gherardelli.