Construction of rank 2 semistable torsion free sheaves which are not limit of vector bundles
Let \(M(2;c_ 1,c_ 2,0)\) be a coarse moduli scheme of semistable coherent rank 2 sheaves \(E\) on \(\mathbb{P}_ 3\) with given Chern classes \(c_ 1(E) = c_ 1\), \(c_ 2(E) = c_ 2\) and \(c_ 3 (E) \neq 0\). It has an open subscheme \(M^ 0(2;c_ 1,c_ 2,0) \hookrightarrow M(2;c_ 1,c_ 2,0)\) which is a coarse moduli scheme of stable rank 2 vector bundles on \(\mathbb{P}_ 3\) with given Chern classes. The main result of the paper gives a construction for all \(c_ 2>1\), \(c_ 2 \neq 5,6,7, 8,9, 10,12,13,16\), of a family of rank 2 semistable torsion free sheaves on \(\mathbb{P}_ 3\) with Chern classes \((0,c_ 2,0)\) which are not limits of rank 2 stable vector bundles on \(\mathbb{P}_ 3\) with Chern classes \((0,c_ 2,0)\). Hence, \(M(2;0,c_ 2,0)\) is not irreducible and the moduli scheme \(M^ 0(2;0,c_ 2,0)\) is a no-dense open subset of \(M(2;0,c_ 2,0)\). The same holds for the case of \(M(2;-1,c_ 2,0)\) if \(c_ 2 \geq 2\) and \(c_ 2 \neq 11,14\). The proof uses simple dimensional counting based on mapping the well known extremal reflexive sheaves onto suitable sheaves supported along a zero dimensional subscheme of \(\mathbb{P}_ 3\) thus killing the third Chern class.
scientific article; zbMATH DE number 492809