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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?

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2 Answers 2

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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.

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This looks interesting. Two questions remain: i see how to get the maps, but the condition $supp(T)\subset C$ iff $C^l=0$ is not quite clear. I think $C^l=0$ implies something like: every element of $T$ is annihilated by some power of the equation of $C$, is this equivalent with the support condition? 2nd question: can we write down a vector bundles with the given properties, i.e. having fiber exactly $H^0(S,T)$ over $T$? –  TonyS Aug 31 '11 at 14:25
    
The first answer is yes, this is analogous to the fact that $T$ is supported at $(0,0) \in A^2$ iff both $x$ and $y$ act nilpotently on $H^0(T)$. The bundle can be constructed as follows. Take the universal quotient sheaf $F$ on $S\times Quot(E,l)$. Then its pushforward to $Quot(E,l)$ is the required bundle. –  Sasha Aug 31 '11 at 19:50
    
Thanks a lot for your answer and the further explanations. –  TonyS Aug 31 '11 at 21:04

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.

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Thanks, +1. That's a nice way to see the type of this condition and other conditions in general. –  TonyS Aug 31 '11 at 21:03

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