S-equivalence

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.

${\displaystyle 0=E_{0}\subseteq E_{1}\subseteq \ldots \subseteq E_{n}=E}$

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

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