Instanton sheaves on Fano threefolds

Generalizing the definitions originally presented by Kuznetsov and Faenzi, we study (possibly non locally free) instanton sheaves of arbitrary rank on Fano threefolds. We classify rank 1 instanton sheaves and describe all curves whose structure sheaves are rank 0 instanton sheaves. In addition, we show that every rank 2 instanton sheaf is an elementary transformation of a locally free instanton sheaf along a rank 0 instanton sheaf. To complete the paper, we describe the moduli space of rank 2 instanton sheaves of charge 2 on a quadric threefold X and show that the full moduli space of rank 2 semistable sheaves on X with Chern classes \((c_1,c_2,c_3)=(-\,1,2,0)\) $(c_1,c_2,c_3)=(-1,2,0)$ is connected and contains, besides the instanton component, just one other irreducible component which is also fully described.