Let $\mathcal{F}$ be a coherent sheaf on a variety $X$, and assume $\mathcal{F}$ has generic rank $n$. I expect (see e.g. here) that this actually puts no conditions on its Chern classes $c_1(\mathcal{F}),c_2(\mathcal{F}),...$ . Constrast this with the vector bundle case, where all Chern classes above the $n$th will vanish. However, I have no idea how to construct examples with this non-vanishing property.

**Question:** Fix $n\ge 0$. For each positive integer $k$, is there a nice example of a variety $X_k$ with a coherent sheaf $\mathcal{F}_k$ of generic rank $n$ and $c_k(\mathcal{F}_k)\ne 0$?

Even the rank $n=1$ case seems unclear to me. I imagine it might be possible to answer the question using for $X_k$ projective spaces or partial flag varieties, but I'm not sure.