Mistretta–Stoppino equivalence for incomplete generated linear series

Establish whether, for every smooth curve C of genus g and every generated linear series (L,V) of type (d,r+1) with V properly contained in H^0(L), the condition d≤kr, where k is the gonality of C, guarantees that linear (semi)stability of (L,V) is equivalent to slope-(semi)stability of the associated syzygy bundle V,L.

Background

The paper studies generated linear series (L,V) on smooth projective curves and the associated syzygy bundle V,L, defined as the kernel of the evaluation map V⊗O_C→L. The first Mistretta–Stoppino conjecture concerns incomplete linear series, meaning that V is a proper subspace of H^0(L). It predicts an equivalence between the geometric notion of linear (semi)stability for (L,V) and slope-(semi)stability for V,L under the numerical degree bound d≤kr, with k equal to the gonality of C.

The paper proves this equivalence for several families, including general curves under additional codimension hypotheses and curves on principally polarized K3 surfaces under specified degree bounds, but the conjecture is not established in full generality.