Moduli of rank two semistable sheaves on rational Fano threefolds of the main series

In this paper we investigate the moduli spaces of semistable coherent sheaves of rank two on the projective space P³ and the following rational Fano manifolds of the main series - the three-dimensional quadric X₂, the intersection of two 4-dimensional quadrics X₄ and the Fano manifold X₅ of degree 5. For the quadric X₂, the boundedness of the third Chern class c₃ of rank two semistable objects in Dᵇ(X₂), including sheaves, is proved. An explicit description is given of all the moduli spaces of semistable sheaves of rank two on X₂, including reflexive ones, with a maximal third class c₃≥0. These spaces turn out to be irreducible smooth rational manifolds in all cases, except for the following two: (c₁,c₂,c₃)=(0,2,2) or (0,4,8). The first example of a disconnected module space of semistable rank two sheaves with fixed Chern classes on a smooth projective variety is found -- this is the second of these exceptional cases (c₁,c₂,c₃)= (0,4,8) on the quadric X₂. Several new infinite series of rational components of the moduli spaces of semistable sheaves of rank two on P³, X₂, X₄ and X₅ are constructed, as well as a new infinite series of irrational components on X₄. The boundedness of the class c₃ is proved for c₁=0 and any c₂>0 for stable reflexive sheaves of general type on manifolds X₄ and X₅.