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 one 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 globally generated, torsion-free sheaf in terms of its rank, degree, and invariants of the variety.