Some moduli spaces for rank 2 stable reflexive sheaves on 𝑃³

Abstract

In [Ma], Maruyama proved that the set M ( c 1 , c 2 , c 3 ) M({c_1},{c_2},{c_3}) of isomorphism classes of rank 2 2 stable reflexive sheaves on P 3 {{\mathbf {P}}^3} with Chern classes ( c 1 , c 2 , c 3 ) ({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 − 2 r c 2 + 2 r ( r + 1 ) ) M( - 1,{c_2},c_2^2 - 2r{c_2} + 2r(r + 1)) with c 2 ⩾ 4 {c_2} \geqslant 4 , 1 ⩽ r ⩽ ( − 1 + 4 c 2 − 7 ) / 2 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 ) M( - 1,{c_2},c_2^2 - 2(r - 1){c_2}) with c 2 ⩾ 8 {c_2} \geqslant 8 , 2 ⩽ r ⩽ ( − 1 + 4 c 2 − 7 ) / 2 2 \leqslant r \leqslant ( - 1 + \sqrt {4{c_2} - 7} )/2 .