Let $X$ be a smooth projective 4-fold (over $\mathbb{C}$). Let $Z \subset X$ be a codimension two subscheme. Let $I_{Z}$ denote the ideal sheaf of $Z$. How does one compute the Chern classes of $I_Z^{\vee}$ from those of $I_{Z}$, where $I_Z^{\vee}$ denotes the dual of $I_{Z}$.
Note that (unlike vector bundles), it is not necessarily true that if $F$ is a torsion free sheaf, $$ c_{k}(F^{\vee}) = (-1)^k c_k(F). $$ As an example, let $I_{p}$ denote the ideal sheaf of a point on a surface $S$. The double dual of $I_p$ is the structure sheaf $\mathcal{O}_{S}$, which has zero Chern classes. But $c_2(I_p)$ is non zero. Hence, I was wondering, what is the general procedure to compute the Chern class of the dual sheaf, if we know the Chern classes of the original Sheaf.
The specific example that I have in mind is to apply it to the case where $I$ is the ideal sheaf of the diagonal inside $\mathbb{CP}^2 \times \mathbb{CP}^2$.
