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Suppose $C$ is a complete algebraic curve.

Define a coherent locally free sheaf $\mathcal{F}$ over $C$ to be stable if $\mu(\mathcal{E})<\mu(\mathcal{F})$ for any subsheaf $\mathcal{E}$, where $\mu(\mathcal{E})=\text{deg}(\mathcal{E})/\text{rank}(\mathcal{E})$.

Suppose $\tilde{C}\to C$is a degree $d$ covering, I read about that the pull back of a semi-stable sheaf is semi-stable, but the case is not true for stable, is there any examples?

Thanks for comments and reference books!

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Let $f \colon \tilde{E} \to E$ be an isogeny of degree $r$ between two elliptic curves, and let $\mathscr{L}$ be a degree $d$ line bundle on $\tilde{E}$, where $(r, \, d)=1$. Then $\mathscr{E} = f_* \mathscr L$ is stable vector bundle on $E$ of rank $r$ and degree $d$. Moreover $$f^* \mathscr{E} = \bigoplus_{\sigma \in G} t_\sigma^* \mathscr{L},$$
where $G \subset \tilde{E}$ is the kernel of $f$ and $t_\sigma \colon \tilde E \to \tilde E$ denotes the translation by $\sigma \in G$.

In particular, $f^* \mathscr{E}$ is semistable but not stable.

A reference is Ishida's paper Bounds for the relative Euler-Poincaré characteristic of certain hyperelliptic fibrations, Manuscripta Mathematica 118 (2005), look in particular at Theorem 2.2.

See also the classical paper by M. F. Atiyah Vector bundles on an elliptic curve, Proc. London Math. Soc. (3) 7 (1957), 414–452.

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