Given a projective surface $S$, and a smooth projective curve $C\subset S$ over $\mathbb{C}$. Furthermore we have a locally free sheaf $E$ of rank $r$ on $S$.


Then for any $l\geq 1$, the projective scheme $\operatorname{Quot}(E,l)$ classifies quotients $E\to T$, such that $\operatorname{dim}\_{\mathbb{C}}(H^0(S,T))=l$.


Now for fixed $E$ and $l$, i am interested in the subset of quotients which have their support $\operatorname{supp}(T)=\lbrace p\_1,\ldots,p\_n\rbrace$ in the given curve $C$, i.e. 


$X:=\lbrace E\rightarrow T | \operatorname{supp}(T)\subset C\rbrace \subset \operatorname{Quot}(E,l).$


Is $X$ a closed subscheme of the $\operatorname{Quot}$-scheme, i.e. is restricting the support of the quotient sheaf $T$ a closed condition? 


I always get confused, when a subset is defined by an open, closed or locally closed condition. In many text one can read something like "this is obviously an open condition". But i find it hard to see by what condition a subset is defined. Are there some "simple" rules which one can check to see this rather quick?



Is restricting the support of an Artinian sheaf a closed condition?
The equation of the curve $C$ gives a chain of maps 
$$
H^0(S,T) \stackrel{C}\to H^0(S,T(C)) \stackrel{C}\to H^0(S,T(2C)) \stackrel{C}\to \dots
$$
It is clear that $supp(T) \subset C$ iff $C^N = 0$ (as a map $H^0(S,T) \to H^0(S,T(NC))$) for $N$ sufficiently large. In fact it is enough to take $N = l$. Thus $X$ is the zero locus of a morphism of vector bundles on $Quot(E,l)$ (the first bundle has fiber $H^0(S,T)$ at $T$ and the second bundle has the fiber $H^0(S,T(lC))$ at $T$) and hence it is a closed subscheme.


