Stability of Kernel Sheaves Associated to Rank One Torsion-Free Sheaves

We show the kernel sheaf associated to a sufficiently positive torsion-free sheaf of rank 1 is slope stable. Furthermore, we are able to give an explicit bound for “sufficiently positive.” This settles a conjecture of Ein-Lazarsfeld-Mustopa. The main technical lemma is a bound on the number of global sections of a torsion-free, globally generated sheaf in terms of its rank, degree, and invariants of the variety.