S-equivalence explained

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=E₀\subseteqE₁\subseteq\ldots\subseteqE_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=oplus_iE_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.