Simple sheaves versus stable sheaves on algebraic surfaces (Q811490)

It is shown that if a torsion free rank-2 sheaf is simple but not stable with respect to any polarization on a minimal algebraic surface over \(\mathbb{C}\), then it sits in a canonical nontrivial extension. As applications, we prove that every simple locally free rank-2 sheaf on a rational (ruled) surface is stable and that simple locally free rank-2 sheaves on surfaces of Kodaira dimension zero satisfy an inequality of Bogomolov-type; moreover, we essentially confirm the following conjecture of \textit{A. N. Tyurin} [see line 5, page 23 in Duke Math. J. 54, 1-26 (1987; Zbl 0631.14009)]: The irreducible component in the moduli space of simple rank-2 sheaves which contains the tangent sheaf of a \(K3\)-surface \(X\) is isogenous to the Hilbert scheme \(\text{Hilb}^{45}(X)\).