For my research, I would like to know whether a certain Quot scheme is reduced. Reading the thread How To Tell Whether A Scheme Is Reduced From Its Functor, I was disappointed to find that there's no easy way to read this information off the functor in general.

Instead, how about this: are there standard situations in which it is known and proven that a Quot scheme is reduced? If $E$ is a vector bundle on a $k$-scheme $X$, and $Quot_{E/X}^{P,L}$ is the Quot scheme of quotients $Q$ of $E$ such that $Q$ has Hilbert polynomial $P$ with the respect to the line bundle $L$, then what are some restrictions on the data {$E,X,P,L$} that would allow one to conclude reducedness? For instance, reducedness holds for the Grassmannian: $X=Spec(k)$, $E=O_{k}^r$, $L=O_k$, $P=d<r$, although obviously I would like something a bit less narrow.

Perhaps a better question of more general interest: what are some stronger, sufficient conditions from which reducedness follows? These may be easier to check on the functor level.



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