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Let $X$ be a smooth projective variety. If $E$ is a coherent sheaf on $X$, we write its Hilbert polynomial: $$P_E(m) = \alpha^E_dm^d + O(m^{d-1}).$$ We say $E$ is Gieseker stable if $E$ is pure and $P_F/\alpha^F_d < P_E/\alpha^E_d$ for all nontrivial $F \subset E$.

Suppose we have a family of coherent sheaves on $X$. If this family contains a Gieseker stable member, does it follow that the general member is also Gieseker stable?

I am also interested in the same question in the situation where all the sheaves are locally free and Gieseker stability is replaced with slope stability.

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    $\begingroup$ Yes: see The Geometry of Moduli Spaces of Sheaves by Huybrechts and Lehn, Proposition 2.3.1. $\endgroup$
    – abx
    Mar 12, 2017 at 18:00
  • $\begingroup$ Is this true for Bridgeland stability and complexes of sheaves as well? $\endgroup$
    – user100272
    Dec 14, 2017 at 2:06

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