Chern classes of rank 3 stable reflexive sheaves
The paper contains the characterization of the set of integers \(c_ 1, c_ 2, c_ 3\) such that there exists a rank 3 stable reflexive sheaf on \({\mathbb{P}}^ 3\) with Chern classes \(c_ 1, c_ 2\) and \(c_ 3\). The proof consists of two parts. First of all, we construct some good families of rank 3 stable reflexive sheaves on \({\mathbb{P}}^ 3\) and we analyze which values of \(c_ 1, c_ 2\) and \(c_ 3\) appear. Then we prove that the undecided cases are indeed non-existent. The characterization of the triples \((c_ 1,c_ 2,c_ 3)\in {\mathbb{Z}}^ 3\) such that there exists a rank 3 stable reflexive sheaf E on \({\mathbb{P}}^ 3\) with \(c_ iE=c_ i\) is a step toward the problem of characterizing the 4-uples \((c_ 1,c_ 2,c_ 3,s)\in {\mathbb{Z}}^ 4\) such that there exists a rank 3 stable reflexive sheaf E on \({\mathbb{P}}^ 3\) with \(c_ iE=c_ i\) and having s singular points. For \(s=0\), this problem amounts to characterizing the Chern classes of rank 3 stable vector bundles on \({\mathbb{P}}^ 3\), settled recently by \textit{I. Coandă} [Math. Ann. 273, 65-79 (1985; see the preceding review)].
rank 3 stable reflexive sheaf on projective 3-space
scientific article; zbMATH DE number 3944012