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I have a very specific question (quite elementary, sorry!)

Let $G$ be a rank $2$ torsion free sheaf on an algebraic surface $X$ (normal maybe?)

Let $L\otimes \mathcal{I}_Z$ be a Gieseker destabilizing subsheaf of $G$ where $Z$ is a $0$ dimensional subscheme.

Is it true that $l(Z)<\frac{1}{2}c_2(G)$ where $l(Z)$ is the length of $Z$?

If not, can we at least say $\leq$ or something? Thank you.

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  • $\begingroup$ Did you try comparing Hilbert polynomials? $\endgroup$
    – Sasha
    Commented Sep 14, 2016 at 21:15
  • $\begingroup$ @Sasha Yeah, I made a mistake in my calculation of the Hilbert polynomials earlier today. I think for the case I'm thinking about this is quite easy. Maybe I will delete this post. $\endgroup$
    – HLC
    Commented Sep 15, 2016 at 3:27

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