Some moduli spaces for rank $2$ stable reflexive sheaves on $\textbf {P}^ 3$

Abstract:

In [Ma], Maruyama proved that the set $M({c_1},{c_2},{c_3})$ of isomorphism classes of rank $2$ stable reflexive sheaves on ${{\mathbf {P}}^3}$ with Chern classes $({c_1},{c_2},{c_3})$ has a natural structure as an algebraic scheme. Until now, there are no general results about these schemes concerning dimension, irreducibility, rationality, etc. and only in a few cases a precise description of them is known. This paper is devoted to the following cases: (i) $M( - 1,{c_2},c_2^2 - 2r{c_2} + 2r(r + 1))$ with ${c_2} \geqslant 4$, $1 \leqslant r \leqslant ( - 1 + \sqrt {4{c_2} - 7} )/2$; and (ii) $M( - 1,{c_2},c_2^2 - 2(r - 1){c_2})$ with ${c_2} \geqslant 8$, $2 \leqslant r \leqslant ( - 1 + \sqrt {4{c_2} - 7} )/2$.