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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.

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  • $\begingroup$ Why does the title of this question differ from the title on the list of new questions? $\endgroup$
    – Jason Starr
    Apr 1, 2021 at 20:56
  • $\begingroup$ @JasonStarr I edited it to improve the title and I think it takes a minute to update. $\endgroup$
    – Pulcinella
    Apr 1, 2021 at 21:05
  • $\begingroup$ Surely there's a typo in the current title? Should be "making"? $\endgroup$
    – paul garrett
    Apr 1, 2021 at 23:05
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    $\begingroup$ For $n=1$ and any $k$ you can consider the ideal sheaf of a subvariety of codimension $k$. $\endgroup$
    – naf
    Apr 2, 2021 at 2:50

1 Answer 1

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Take any affine variety $X$ of dimension $k$ which has a vector bundle $E$ of rank $k$ with $c_k(E)\neq 0$. If $n\geq k$ take $F=E\oplus O_X^{n-k}$. If $n<k$ take $F$ to be the quotient of $E$ by $k-n$ general sections of $E$.

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