Even and odd instanton bundles on Fano threefolds

We define non-ordinary instanton bundles on Fano threefolds $X$ extending the notion of (ordinary) instanton bundles. We determine a lower bound for the quantum number of a non-ordinary instanton bundle, i.e. the degree of its second Chern class, showing the existence of such bundles for each admissible value of the quantum number when $i_X\ge 2$ or $i_X=1$, $\mathrm{Pic}(X)$ is cyclic and $X$ is ordinary. In these cases we deal with the component inside the moduli spaces of simple bundles containing the vector bundles we construct and we study their restriction to lines. Finally we give a monadic description of non-ordinary instanton bundles on $\mathbb{P}^3$ and the smooth quadric studying their loci of jumping lines, when of the expected codimension.