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!



Example for pullback of stable sheaf not stable
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*](http://link.springer.com/article/10.1007/s00229-005-0599-5), 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. 


