Stable rank three vector bundles on image with Chern classes (c1,c2,c3)=(−2,4,0)

The stable rank three vector bundles on projective fourspace with Chern classes (−2,4,0) whose first twists have no non-zero global sections form an open subset of the moduli space of all stable rank three vector bundles in this class. Furthermore, these bundles are precisely those associated with elliptic conic bundles in projective fourspace. We prove that the family of these bundles is irreducible, non-singular of dimension 28. As a corollary, we show that the family of elliptic conic bundles is irreducible of dimension 36.