Rank Two Sheaves With Low Discriminant on the Fano Threefold of Index 2 and Degree 5

We describe rank 2 Gieseker semistable sheaves $E$ on the Fano threefold $X_5$ of index 2 and degree 5 with maximal third Chern class $c_3(E)$ for all possible low values of discriminant $\overlineΔ_H(E)\le 40$. The work uses the theory of tilt-stability and Bridgeland stability conditions on smooth projective threefolds. We also make a conjecture about the rank 2 sheaves on $X_5$ with maximal $c_3$ and discriminant big enough.