Duality and Serre's criterion Let $X$ be a projective scheme, $\mathcal{F}$ a coherent sheaf on $X$ which is $S_2$. Then under what additional conditions is its dual, $\mathcal{H}om_X(\mathcal{F},\mathcal{O}_X)$ also $S_2$?
 A: To me it seems that whether $F$ satisfies $S_2$ or not doesn't have much of an influence on the question if $\mathrm{\mathcal{H}\it{om}}_{\mathcal{O}_X}(F,\mathcal{O}_X)$ satisfies $S_2$. If $X$ satisfies $S_2$, then the dual of any coherent sheaf on $X$ also satisfies $S_2$. On the other hand, I have no idea how to check whether $\mathrm{\mathcal{H}\it{om}}_{\mathcal{O}_X}(F,\mathcal{O}_X)$ is $S_2$ unless I have information on the depth-behavoiur of $\mathcal{O}_X$. The most general thing I know about the depth of a $\mathrm{Hom}$-module is the following:
Let $A$ be a Noetherian local ring and $M,M'$ two finitely generated $A$-modules.
Then $$\mathrm{depth}(\mathrm{Hom}_A(M,M'))\geq \min(\mathrm{depth}(M'),2).\quad(\ast)$$
This easily gives that $\mathrm{Hom}_A(M,M')$ satisfies $S_2$ if so does $M'$.
Since you're asking for the dual, I'll simplify and take $M' = A$. The argument, however, is the same.
We take a finite presentation of $M$,
$$A^k\to A^m\to M\to 0,$$
and apply $\mathrm{Hom}_A(-,A)$ to get a short exact sequence
$$0\to \mathrm{Hom}_A(M,A)\to \mathrm{Hom}_A(A^m,A)\to N\to 0,$$
where $N\subset \mathrm{Hom}_A(A^k,A)$ is the image of $\mathrm{Hom}_A(M,A)\to \mathrm{Hom}_A(A^m,A)$.
From the long exact $\mathrm{Ext}$-sequence we see that $$\mathrm{depth}(\mathrm{Hom}_A(M,A))\geq \min(\mathrm{depth}(A),\mathrm{depth}(N)+1).$$
Suppose $\mathrm{depth}(N) = 0$. Then, since $N\subset \mathrm{Hom}_A(A^k,A)\cong A^k$, also $\mathrm{depth}(A) = 0$, hence $(\ast)$ is trivially true (where $M' = A$).
Consequently, in any case, $\mathrm{depth}(\mathrm{Hom}_A(M,A))\geq \min(\mathrm{depth}(A),2)$. Finally, if $A$ is $S_2$, then
$$\mathrm{depth}(\mathrm{Hom}_A(M,A))\geq \min(\dim(A),2)\geq \min(\dim(\mathrm{Hom}_A(M,A)),2),$$
so $\mathrm{Hom}_A(M,A)$ is $S_2$.
A: If $X$ is noetherian and integral, the the dual of a coherent sheaf is always reflexive. If $X$ is even normal then a coherent sheaf is reflexive if and only if it is $S_2$. 
You can read about that in Hartshornes paper about reflexive sheaves. Or there is one by Karl Schwede titled "Generalized divisors and reflexive sheaves" which can be found with Google.
