Let $Q$ be a quasi-projective $k$-scheme (not necessarily smooth), $X$ a smooth projective $k$-variety and $\mathcal E$ a family of (torsion free) sheaves on $X$ parametrized by $Q$. Suppose that $\mathcal E$ is flat over $Q$. Can we find, for any $i\geq 1$, a cycle $Z\in CH^i(Q\times X)$ of class $c_i(\mathcal E)$ which is flat over $Q$? Is it easier if we add that $\mathcal E_q$ is globally generated on $X$ for all $q\in Q$?
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$\begingroup$ There is a reduction to the case that $\mathcal{E}$ is locally free of finite rank using Hilbert's Syzygy Theorem. Also it suffices to prove the result for $\mathcal{E}(m)$ with $m\gg 0$ using the following identity in the K-group of the projective space $\mathbb{P}^n$ in which $X$ embeds: $\sum_r \binom{n+1}{r}(-1)^r[\mathcal{O}(m+r)] = 0$. Also, using Whitney sum, it suffices to prove the result for $\mathcal{F} = \mathcal{E}(m)\oplus \mathcal{O}(m)^{\oplus N}$ for $N\gg 0$. Now use Thom-Porteous. You need to prove, ala Bertini, that degeneracy loci of $\mathcal{F}$ are flat over $Q$. $\endgroup$– Jason StarrCommented Jan 27, 2016 at 11:28
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$\begingroup$ ... Finally, to prove the degeneracy loci of $\mathcal{F}$ are flat over $Q$ (for general choices of global sections of $\mathcal{F}$), you will probably use Eagon-Hochster. $\endgroup$– Jason StarrCommented Jan 27, 2016 at 13:02
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$\begingroup$ You might check if Gabber-Liu-Lorenzini have proved something about this; they proved results that sound similar to this. $\endgroup$– Jason StarrCommented Jan 27, 2016 at 13:04
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