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Ìn the book "Vector bundles on complex projective spaces" the authors prove in Chapter 2 Lemma 1.2.5 that, if $E$ is a rank $2$ reflexive sheaf on $\mathbb{P}^n$, then $E$ is stable if, and only if, $H^0(\mathbb{P}^n, E_{norm}) = 0$, where $E_{norm}$ is the sheaf $E$ normalized.

The proof of this Lemma relies on the fact that any rank 1 subsheaf of a reflexive sheaf whose quotient is a torsion free sheaf is also reflexive.

My question is that if it is possible to obtain examples of normalized rank 2 torsion free sheaves $E$ such that $H^0(\mathbb{P}^n, E) = 0 =H^0(\mathbb{P}^n, E^{\vee}) = 0$ and $E$ is not stable.

Additionally, in Remark 1.2.6 the authors says that if a normalized rank $r$ sheaf on $\mathbb{P}^n$ is stable, then $H^0(\mathbb{P}^n, E) = 0 =H^0(\mathbb{P}^n, E^{\vee}) = 0$, where $E^{\vee}$ is the dual of the sheaf $E$, and since this is not an ''if and only if'' kind of result, this gives the idea that such example that I am asking for indeed exists.

EDIT: Some details on the question were fixed, after the abx's comments.

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    $\begingroup$ Is your looked for torsion free sheaf of rank 2, as the title suggests? And normalized? $\endgroup$
    – abx
    Commented Oct 24, 2018 at 19:43
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    $\begingroup$ If your question is about non-refexive sheaves, just take $\mathcal{O}_{\mathbb{P}}(-1)\oplus \mathfrak{m}_p$, where $\mathfrak{m}_p$ is the ideal sheaf of a point. $\endgroup$
    – abx
    Commented Oct 24, 2018 at 20:00
  • $\begingroup$ Dear @abx, yes I was looking for rank 2 torsion free sheaves. I also forgot to mention that I wanted the sheaf and the dual of the sheaf to not have global sections. I am not sure whether the example that you gave me above works in this case. Thank you for pointing that out. $\endgroup$
    – User43029
    Commented Oct 24, 2018 at 20:06

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