Acm Vector Bundles on Prime Fano Threefolds and Complete Intersection Calabi Yau Threefolds

In this paper we derive a list of all the possible indecomposable normalized rank–two vector bundles without intermediate cohomology on the prime Fano threefolds and on the complete intersection Calabi-Yau threefolds, say V , of Picard number ρ = 1. For any such bundle E, if it exists, we find the projective invariants of the curves C ⊂ V which are the zero–locus of general global sections of E. In turn, a curve C ⊂ V with such invariants is a section of a bundle E from our lists. This way we reduce the problem for existence of such bundles on V to the problem for existence of curves with prescribed properties contained in V. In part of the cases in our lists the existence of such curves on the general V is known, and we state the question about the existence on the general V of any type of curves from the lists. § 0. Introduction Let V be a smooth projective threefold, and let E be a vector bundle on V. We say that E has no intermediate cohomology if h i (V, E ⊗ L) = 0, i = 1, 2 for any line bundle L on V. We say that an indecomposable (i.e. not split as a direct sum of two line bundles) vector bundle on V with no intermediate cohomology is a Cohen Macaulay bundle (a CM bundle, for short). In the rank–two case we will call such a bundle arithmetically Cohen Macaulay (an ACM bundle, for short). In particular, let ρ(V) = 1 where ρ(V) is the the rank of the Picard group of V , and let D be the class of an ample generator of Pic(V) over Z, i.e. Pic(V) ∼ = Z[D]. For the line bundle L = nD on V denote by E(n) the twist of E by nD, i.e. E(n) = E ⊗ nD. Now, if the rank of E is two, the identity E ∨ = E(−c 1), where det E = c 1 D, together with the Serre duality, imply that E has no intermediate cohomology if and only if h 1 (V, E(n)) = 0 for any n ∈ Z. In this paper we will deal on ACM bundles on prime Fano threefolds and complete intersection Calabi Yau threefolds. To start with, by the criterion of Horrocks the rank–two vector bundle E on V = P …