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Let $k$ be an algebraically closed field, $C$ a non-singular projective curve over $k$ of genus at least $2$ and $\mathcal{F}$ a locally free sheaf on $C$. Let $r,d$ be two integers satisfying $\mathrm{gcd}(r,d)=1$ and $r>0$. Denote by $Q$ the Quot-scheme parametrizing quotients of $\mathcal{F}$ of rank $r$ and degree $d$. Is every irreducible component of $Q$ (generically) reduced i.e., for a general point $x$ of the irreducible component of $Q$, the local ring $\mathcal{O}_{Q,x}$ is reduced?

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  • $\begingroup$ The Quot schemes can be disconnected. For $g$ equals $4$, for $C$ general, for $\mathcal{F}$ equals $\mathcal{O}_C^{\oplus 2}$, for $r$ equals $1$, and for $d$ equals $3$, the Quot scheme has two connected components. This corresponds to the fact that there are two $g^1_3$s on $C$. $\endgroup$ – Jason Starr Oct 22 '15 at 12:38
  • $\begingroup$ Sorry, I misread the question. If you specialize the curve $C$ from my previous comment to one whose canonical image lies on a singular quadric cone, then I believe that the Quot scheme is nonreduced. Certainly the corresponding scheme $G^1_3$ is nonreduced. $\endgroup$ – Jason Starr Oct 22 '15 at 13:05
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This is false, at least for special $C$ (perhaps it is true for $C$ that are sufficiently general in moduli). The simplest counterexample I know of is a genus $4$ curve $C$ that is non-hyperelliptic and whose canonical image is contained in a singular quadric hypersurface in $\mathbb{P}^3$. In this case, the scheme $G^1_3$ equals $W^1_3$ inside $\text{Pic}^3_C$, and this is one nonreduced point. For the invertible sheaf $\mathcal{L}$ on $C$ parameterized by this point, $\mathcal{L}$ is globally generated. Thus for $\mathcal{F}$ defined to be $H^0(C,\mathcal{L})\otimes_k \mathcal{O}_C$, the natural homomorphism, $$\phi:\mathcal{F} \to \mathcal{L},$$ is surjective.

Since being invertible is an open condition on flat families of coherent $\mathcal{O}_C$-modules, every small deformation of the quotient $\mathcal{L}$ is an invertible $\mathcal{O}_C$-module of degree $3$ that has a $2$-dimensional vector space of global sections, and conversely. Thus, there is a smooth morphism from a Zariski open neighborhood $U$ of $[\phi]$ in the Quot scheme to $W^1_3$. Since $W^1_3$ is nonreduced with nilradical a rank $1$ vector space over the residue field, also $U$ is nonreduced with nilradical an invertible sheaf on the reduced scheme of $U$ (which is smooth).

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