Is the condition that a module is reflexive an open condition? 


That is, if $X$ is a smooth projective complex variety, $T$ a quasi-projective variety, and $F$ a finitely presented module on $X \times T$ that is $T$-flat, then we can form the locus $T' \subset T$ of points $t$ such that the restriction of $F$ to $X \times t$ is reflexive.


Is $T' \subset T$ open?


If not, is it locally closed?


Recall that a coherent sheaf $F$ is said to be reflexive if the natural map $F \to (F^{\vee})^{\vee}$ to the double dual is an isomorphism.



Is reflexivity an open condition?
This locus is indeed open. I will explain why using Kollar's "Hulls and Husks" (arXiv:0805.0576). More generally, this article studies in great detail when taking the double dual commutes with base change.


First, we may restrict to the open locus of $T$ where $F\_t$ is torsion-free (because reflexive sheaves are torsion-free).
 Then, we choose an ample line bundle $H$ on $X$, and we compute the Hilbert polynomials relatively to $H$. The Hilbert polynomial $P(F\_t)$ of $F\_t$ is constant by flatness. From the exact sequence $0\to F\_t\to F^{\vee\vee}\_t\to F^{\vee\vee}\_t/F\_t\to 0$, we see that $F\_t$ is reflexive exactly when $P(F^{\vee\vee}\_t)=P(F\_t)$, i.e. exactly when $P(F^{\vee\vee}\_t)$ takes its minimal value. But the polynomial $P(F^{\vee\vee}\_t)$ is constructible and upper semicontinuous by Proposition 28 (3) of Hulls ans Husks. This proves that this locus is open.


