Sheaves of low rank in three-dimensional projective space (Q6085499)

The classification of numerical invariants of Gieseker-semistable sheaves in the projective plane is a classical result, see [\textit{J. M. Drezet} and \textit{J. Le Potier}, Ann. Sci. Éc. Norm. Supér. (4) 18, 193--243 (1985; Zbl 0586.14007)]. However, we know significantly less for the case of threefolds. In this article, the author classifies the Chern characters of slope-semistable sheaves to rank four in the three-dimensional projective space. The results for rank less than or equal to three were known, and some were proved by classical methods. The author worked out the case of rank one and two in [\textit{E. Macrì} and \textit{B. Schmidt}, Algebr. Geom. 7, No. 2, 153--191 (2020; Zbl 1455.14039)], and [\textit{B. Schmidt}, ``Rank two sheaves with maximal third Chern character in three-dimensional projective space'', Preprint, \url{arXiv:1811.11951}]. The rank two and three cases were also solved via more classical methods in [\textit{C. Okonek} and \textit{H. Spindler}, Das Spektrum torsionsfreier Garben. II. Math. Gottingensis, Schriftenr. Sonderforschungsbereichs Geom. Anal. 17 (1985; Zbl 0581.14011)], and in [\textit{R. M. Miro-Roig}, Math. Ann. 276, 291--302 (1987; Zbl 0588.14011)]. The primary tool of the article is the notion of tilt stability in the derived category. As a highlight, the article's classification allows the author to give explicit descriptions of the moduli spaces either directly or with some standard uses of deformation theory. For instance, as a corollary of its results, the author shows that the moduli of spaces of semistable sheaves between rank zero and four with maximal third Chern character are smooth and irreducible.