On the Number of Irreducible Components of the Moduli Space of Semistable Reflexive Rank 2 Sheaves on the Projective Space

In 2017, Jardim, Markushevich, and Tikhomirov found a new infinite series of irreducible components of the moduli space of semistable nonlocally free reflexive rank 2 sheaves on the complex three-dimensional projective space with even first Chern class whose second and third Chern classes can be represented as polynomials of a special form in three integer variables. A similar series for reflexive sheaves with odd first Chern class was found in 2022 by Almeida, Jardim, and Tikhomirov. In this article, we prove the uniqueness of the components in these series for the Chern classes represented by the above-mentioned polynomials and propose some criteria for the existence of these components. We formulate a conjecture on the number of components of the moduli space of stable rank 2 sheaves on a three-dimensional projective space such that the generic points of these components correspond to isomorphism classes of reflexive sheaves with fixed Chern classes defined by the same polynomials.