Vector bundles E on P^3 with homological dimension 2 and chi(End E)=1

Abstract

We find the complete integer solutions of the equation X2 + Y2 + Z2 − 4XY − 4YZ + 10XZ = 1. As an application, we prove that, for each solution (a, b, c) such that a > 0, b − 2a > 0 and (b − 2a)2 ≥ 4a, there is a vector bundle E on ℙ3 defined by a minimal linear resolution 0 → Oℙ3 (−2)a → Oℙ3 (−1)b → Oℙc3 → E → 0. In particular, E satisfies ?(End E) = 1.