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The notion of upper/lower semi-continuity is sometimes encountered in algebraic geometry.

Here by upper semi-continuity one means a function on a topological space $f:X\rightarrow S$ with value in some ordered topological space (like the field of real numbers), such that $\lim\sup_{x\rightarrow y}f(x)\leq f(y)$. Intuitively, for points $x$ that are close to a given point $y$, may the value $f(x)$ "exceeds" $f(y)$, the difference should be "small" and "vanishes" as $x$ approaches $y$. And lower semi-continuity means the opposite, namely replace "$\leq$" by "$\geq$".

Among the typical examples one thinks of the dimension function: let $k$ be a base field, and $X$ be a locally finite type $k$-scheme. The function $$x\in X \mapsto \dim_xX$$ is upper semi-continuous, where by $\dim_xX$ is understood to be the combinatory dimension of $X$ at $x$, namely the length of chains of inclusions of irreducible closed subschemes $$x\in X_1\subsetneq X_2\subsetneq \cdots\subsetneq X_d=X$$

There seems to be a lot of examples of such upper/lower semi-continuous functions in geometry counting certain "discrete invariants", especially those related to stratifications of spaces. It'll be great to have the list extended in mathoverflow.

I hope this question is well-posed...

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The dimension of the (abstract) tangent space at points of X is also semi-continuous. –  Qfwfq Oct 19 '10 at 14:55
Dim of cohomology groups with coeff's in coherent sheaves. –  Qfwfq Oct 19 '10 at 14:56
I don't think your question is quite well-posed right now: do you simply want a list of examples (in which case this should be community-wiki and the big-list tage should be added), or are you after something more (like some sort of big-picture explanation/classification, in which case the question should not be CW)? –  Thierry Zell Oct 19 '10 at 15:53
@thierry: thanks for the comments and advice. Now it is a CW question open for big list. I was rather new to Mathoverflow... –  genshin Oct 20 '10 at 8:30

5 Answers 5

The semicontinuity theorem (Hartshorne III.11) states that the ranks of cohomology groups on the fibers of a morphism is a semicontinuous function. More precisely, given a projective morphism $f:X\to Y$ of noetherian schemes and a coherent sheaf $F$ on $X$, flat over Y, then the function $$ h^i(y,F)=\dim_{k(y)}H^i(X_y,F_y) $$is upper semicontinuous as a function of $y$. Here $X_y$ denotes the fibre of $f$ over $y$. This is used widely in algebraic geometry.

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Virtually every example is this one, for the right choices of X and F. Unfortunately, that's not very helpful for a beginner. –  Alexander Woo Oct 19 '10 at 18:57
What about the Ext groups? Is the function $e^i(y, F, G) = \dim_{k(y)} Ext^i(F_y, G_y)$ upper semicontinuous under some assumptions on $F$ and $G$? Of course for $F$ locally free this is just a cohomology group, but what about more general cases? –  Piotr Achinger Oct 19 '10 at 19:43

The most basic semi-continuous function in algebraic geometry is, I think, the following: given a coherent sheaf $F$ on a scheme $X$ of (locally) finite type over $k$, the function

$$ x \mapsto \dim_{k(x)} F\mid_x = \dim_{k(x)} F \otimes k(x) $$

is upper semi-continuous. It follows from Nakayama's Lemma.

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There are a lot of examples in the representation theory of finite-dimensional algebras.

For instance, we can encode associative finite-dimensional algebras of given dimension $d$ into an algebraic variety and the map sending an algebra to its projective dimension is upper semicontinuous. Or for a fixed algebra, one can encode its modules or pairs of modules of given dimensions into an algebraic variety and functions like the dimension of the $i$th Ext group is upper semicontinuous.

It might be kind of weird because these varieties don't parametrize isomorphism classes, but rather ways of putting a module or algebra structure on a fixed vector space, but these results tend to be useful in geometric approaches to finite-dimensional algebras.

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Seshadri constants (see arXiv:0810.0728) are lower semicontinuous.

This fact is of course related to J.C. Ottem's example, but (a priori) it is not "the same". I believe that there is not a single coherent F giving the same stratification that the Seshadri constant gives (you could get it by a sequence of coherent F, or using non-coherent F, trivially.)

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Hodge numbers are (upper) semi-continuous in a family of complex manifolds and constant for Kähler manifolds. Then again, this is a special case of the above, at least for projective families. See this for links and references for related stuff.

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