Let $X$ be an integral (and singular) curve over the complex field and let $A$ be a torsion free sheaf (not necessarily locally free) of rank $1$ on $X$. We denote by $\mathrm{Quot}^n\_A$ the Quot scheme parametrizing quotient $q:A\to Q$ of length $n$. Is this Quot scheme always isomorphic to the Hilbert scheme $\mathrm{Hilb}^n(X)$? If not, can you give me a counterexample?



Quot schemes and Hilbert schemes on a singular curve
That is definitely not true. Let $X$ be an integral curve that has a single ordinary double point $p$ (and no other singular points). Let $\nu:\widetilde{X}\to X$ be the normalization. The fiber of $\nu$ over $p$ consists of two closed points, $p\_1$ and $p\_2$.


Let $A$ be $\nu\_\*\mathcal{O}\_{\widetilde{X}}$. Let $n$ be $1$. Then $\text{Quot}^1\_A$ admits a "fundamental cycle" morphism, $$\text{FC}:\text{Quot}^1\_A \to X.$$  

But this is not an isomorphism. In fact $\text{Quot}^1\_A$ has two irreducible components, one of which is $\widetilde{X}$, and the second of which is the projective line associated to the two-dimensional vector space $\nu\_\*\mathcal{O}\_{\widetilde{X},p}/\mathfrak{m}\_p\nu\_\*\mathcal{O}\_{\widetilde{X},p}$, which is canonically isomorphic to $$
(\mathcal{O}\_{\widetilde{X},p\_1}/\mathfrak{m}\_{p\_1} \mathcal{O}\_{\widetilde{X},p\_1})
\oplus
(\mathcal{O}\_{\widetilde{X},p\_2}/\mathfrak{m}\_{p\_2} \mathcal{O}\_{\widetilde{X},p\_2}).
$$
This $\mathbb{P}^1$ is attached to $\widetilde{X}$ at the two points $p\_1$ and $p\_2$, with $p\_i$ being attached to the point of $\mathbb{P}^1$ determined by the corresponding summand of the two-dimensional vector space. Also $\text{FC}$ is constant on the $\mathbb{P}^1$, mapping it to $p$. All-in-all, $\text{Quot}^1\_A$ is quite different from $X$.


