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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 \( 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.

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