S equivalence

Definition

Let X be a projective curve over an algebraically closed field k. A vector bundle on X can be considered as a locally free sheaf. Every semistable locally free E on X admits a Jordan-Hölder filtration with stable subquotients, i.e.

where E i are locally free sheaves on X and E i / E i − 1 are stable. Although the JH-Filtration is not unique, the subquotients are, which means that g r E = ⨁ i E i / E i − 1 is unique up to isomorphism.

Two semistable locally free sheaves E and F on X are S-equivalent if gr E ≅ gr F.