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Let $L$ be a very ample line bundle on a variety $X$ of dimension $n$. The slope of a torsion free sheaf $E$ on $X$ is given by $$\mu_{L}(E) : = \dfrac{c_{1}(E).L^{n-1}}{rank(E)}$$
Let $\pi : \widetilde{\mathbb{P}^{3}} \rightarrow \mathbb{P}^{3}$ be the blowup of $\mathbb{P}^{3}$ along a regular curve $C$. What is an ample sheaf on $\widetilde{\mathbb{P}^{3}}$? Does it make sense to talk about the stability of tangent sheaf of $\widetilde{\mathbb{P}^{3}}$? I can't find any references in this matter.

Thanks in advance.

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    $\begingroup$ If $E$ is the exceptional divisor, the line bundle $\pi ^*\mathcal{O}_{\mathbb{P}}(k)(-E)$ is ample (and, indeed, very ample) for $k$ large enough, but for which $k$ depends very much on the curve. Nothing prevents you to look at the stability of the tangent bundle of $\widetilde{\mathbb{P}}^3 $, but of course you have to choose an ample line bundle and there is no canonical choice. $\endgroup$
    – abx
    Commented May 5, 2019 at 16:53
  • $\begingroup$ first: thanks for your answer. But, I did not understand why this line bundle is ample and also because k depends on the curve. And if the curve is $\mathbb{P}^{1}$ for example? Thank you very much. $\endgroup$
    – Allan Ramos
    Commented May 5, 2019 at 20:12
  • $\begingroup$ You can check the ampleness by the Kleiman criterion. $\endgroup$
    – Edward Teach
    Commented May 10, 2019 at 12:05
  • $\begingroup$ Ok. Thank you very much for your suggestion. I'm going to look for this result. thank you, Edward. $\endgroup$
    – Allan Ramos
    Commented May 10, 2019 at 12:16

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