S-equivalence

Template:More citations needed S-equivalence is an equivalence relation on the families of semistable vector bundles on an algebraic curve.

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.

$0=E_0\subseteq E_1\subseteq \dots \subseteq E_n=E$

where $E_i$ are locally free sheaves on X and $E_i/E_{i-1}$ are stable. Although the Jordan-Hölder filtration is not unique, the subquotients are, which means that $grE=\underset{i}{⨁}{E}_i/E_{i-1}$ is unique up to isomorphism.

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