Question

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?

Answer 1

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.

Answer 2

There should be a regular map

$$\mathrm{Quot}(E,l)\to \mathrm{Chow}^l(S)$$ $$T \mapsto \sum_{p\in S} h^0(T_p)\cdot p$$

where the latter space is the Chow variety of zero-cycles of length $l$. This Chow variety has $\mathrm{Chow}^l(C)$ as a closed subvariety, and the locus you describe is the preimage of this subvariety.

In general, the main tools for determining if something is open or closed is just to look at maps and/or incidence correspondences, and reduce the problem to known spaces. Joe Harris' "First course" book has several examples of this sort, albeit in lower-tech situations.