  Sheaves on non-reduced curves can appear in moduli space of 1-dimensional
semistable sheaves over a surface, and moduli space of Higgs bundles as well.
We estimate the dimension of the stack $\mathbf{M}_{X}(nC,\chi)$ of pure
sheaves supported at the non-reduced curve $nC~(n\geq 2)$ with $C$ an integral
curve on $X$. We prove that the Hilbert-Chow morphism
$h_{L,\chi}:\mathcal{M}^H_X(L,\chi)\rightarrow |L|$ sending each semistable
1-dimensional sheaf to its support have all its fibers of the same dimension
for $X$ Fano or with trivial canonical line bundle and $|L|$ contains integral
curves.
