Moduli spaces of rank-2 ACM bundles on prime Fano threefolds
A prime Fano threeefold \(X\) is a threefold whose Picard group is generated by an ample line bundle \(H\), and the canonical bundle \(K\) is equal to \(-H\). The authors study the existence of rank \(2\) vector bundles \(E\), over prime Fano threefolds, which are Arithmetically Cohen-Macaulay, i.e. their intermediate cohomology groups \(H^1E(nH)\) vanish, for all \(n\). After twisting by a multiple of \(H\), one can always assume that the first Chern class of \(E\) satisfies \(c_1(E)=0,H\). In this situation, the authors classify all possible second Chern classes that may occur. When \(c_1(E)=0\), the second Chern class is necessarily \(2\) or \(4\). When \(c_1(E)=H\), then \(c_2(E)\) can be either \(1\) or any number between \(1+g/2\) and \(g+3\) (\(g\) being the genus of \(X\)). The existence part, which is the most difficult, is achieved by starting with non locally free sheaves and using deformations. When \(c_1(E)=H\) and \(c_2(E)\neq 1+g/2\), the existence is only proved on threefolds \(X\) containing a line \(L\) with normal bundle equal to \(\mathcal O_L\oplus\mathcal O_L(-H)\).
scientific article