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Favorite semicontinuous functions?

Posted by ravivakil under

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[22] Comments
Semicontinuous functions are a useful tool. Because I want to make sure I don’t miss any important ones, I took an inventory of the ones I use, and found surprisingly few — they just get used a lot.

What are your favorite semicontinuous functions of a scheme (with reasonable hypotheses)?

For simplicity, please put one per comment, so people can respond. I’ve posted my favorite five (ranked).

June 8, 2011 at 9:28 am

Rank of a finite type quasicoherent sheaf (Important Exercise 14.7.I in the May 13 version).

Hence: (a) dimension of Zariski tangent space at closed points under good circumstances, and (b) degree of a finite morphism at a point of .

June 8, 2011 at 9:29 am

Fiber dimension of a morphism is upper semicontinuous on the source.

Hence: fiber dimension of proper morphisms is upper semicontinuous on the target.

July 20, 2011 at 8:46 am

Further remark: I cheaply (modulo Krull’s Principal Ideal Theorem) prove that fiber dimension of projective morphisms (of locally Noetherian schemes) is upper semicontinuous on the target. I feel like there should be some trick to use this to show that fiber dimension of locally finite type morphisms of locally Noetherian schemes is upper semicontinuous on the source, but I haven’t yet succeeded. (Perhaps this is reminiscent of how I tried to prove Chevalley’s Theorem using elimination theory, and managed to tie myself into knots.) Does anyone know of any argument, or see a straightforward argument?

(Then upper semicontinuity of fiber dimension on the target of *proper* morphisms of locally Noetherian schemes would then follow. And of course locally Noetherian hypotheses can be traded in for finite presentation hypotheses.)

The current proof I know goes through Grothendieck’s generic freeness lemma (see for example Eisenbud’s Commutative Algebra Theorem 14.4, which is used for uppersemicontinuity on the source in Thm 14.8), which I currently don’t have need for.

August 17, 2011 at 1:21 pm

Update: I’ll now add a short proof without generic freeness, in the version to appear around the end of August.

June 8, 2011 at 9:25 am

The rank of a matrix of functions is upper semicontinuous.

May 20, 2012 at 7:36 am

This is lower semicontinuous: a determinant remains nonzero in a open.

December 25, 2015 at 6:03 pm

Whoops! That’s what I meant!

June 8, 2011 at 9:28 am

cohomology groups in proper flat families

June 8, 2011 at 9:30 am

For a proper flat morphisms, the number of geometric components of the fiber is lower semicontinuous. (I’m not sure offhand what hypotheses are necessary — perhaps locally Noetherian target?)

June 8, 2011 at 5:25 pm

Isn’t this coming from Zariski’s Main Theorem, so requiring a normal target? Consider the normalization of a nodal curve.

June 9, 2011 at 7:20 am

That example isn’t flat. But your comment still makes me nervous, so I need to figure out why I jotted this down — I can’t remember offhand!

June 9, 2011 at 4:00 pm

Why isn’t it flat? Certainly it’s torsion free.

June 10, 2011 at 8:13 am

The fastest way for me to see it is to note that for finite morphisms (of locally Noetherian schemes) , if $f$ is flat then the fiber degree is locally constant.

(This is a special case of the fact that if is a proper morphism of locally Noetherian schemes, and is a coherent sheaf on , flat over , then the euler characteristic of the fibers of is a locally constant function on . The case where is projective is Theorem 25.5.1, although the full proof isn’t in the last posting.)

December 4, 2015 at 12:31 pm

I do not think the number of geometrically irreducible components of the fiber will be lower semicontinuous, except under very stringent hypotheses. See http://mathoverflow.net/questions/217488/is-the-locus-of-points-which-have-irreducible-fibers-constructible

As a counterexample, the universal curve over the Hilbert scheme of conics in P^2 will have irreducible fibers on a set which is constructible, but not even locally closed.

December 7, 2015 at 5:26 am

Very good example (irreducible conic degenerates to union of two lines degenerates to double line). The very stringent hypothesis I like is that the fibers are all reduced.

December 7, 2015 at 8:53 am

Ah, that’s very interesting! Do you have a reference for this fact (the number of irreducible components is lower semicontinuous on the target for a flat projective map with reduced fibers)? Or, if not, would you be able to sketch a proof? It is exactly what I need for a project I’ve been working on, but I’ve been stuck on how to prove it.

December 7, 2015 at 11:50 am

I have to think… We think a lot about this in my paper with Alexeev on “branchvarieties”. Check out the section on forests.

December 7, 2015 at 5:57 pm

Thank you for pointing out that reference. Also, I meant to say “upper semicontinuous” not “lower semicontinuous” in my previous comment. I spent 30 minutes browsing through your paper, but I don’t see upper semicontinuity of the number of irreducible components there. Perhaps I’m missing something obvious though. (Local constancy of the number of connected components is proven there, and I think that is also in EGA IV.) Perhaps I’ll ask this question on mathoverflow.

June 8, 2011 at 12:43 pm

Under reasonable hypotheses, when an algebraic group acts on a variety X, the orbit dimension is lower semicontinuous on X.

June 9, 2011 at 9:35 am

This is a nice application of comment 4 above: if acts on , then let , which comes with a map to . (This is easier to say with a commutative diagram, which I don’t know how to do on wordpress.) Then can be interpreted informally as an ordered pair such that ; the map to is the “obvious one”. Then by comment 4, the fiber dimension of is upper semicontinuous; consider this along the identity section . This is the statement that the dimension of the stabilizer subgroup is upper semicontinuous. By the algebro-geometric version of the “orbit-stabilizer theorem”, the dimension of the orbit of (which is the dimension of minus the dimension of the stabilizer of a point ) is lower semicontinuous.

Aside: this is my first time using latex in wordpress, and it is surprisingly easy: just put your latex between single dollar signs as usual, and write “latex” right after the first dollar sign.

June 19, 2011 at 12:30 pm

Dear Ravi, I have found the following pithy slogan/theorem surprisingly useful:

The normalization of a (non-normal) scheme is NEVER flat over the scheme.

July 21, 2011 at 10:31 am

[…] 1. I’ve proved uppersemicontinuity of fiber dimension on the target (for a projective morphism). But I haven’t proved uppersemicontinuity of fiber dimension on the source (for locally finite type morphisms to locally Noetherian schemes; or if you really care, for locally finitely presented morphisms in general, but that’s just an easy generalization once you’ve got the hard part). I don’t know an easy proof (i.e. short given what is already done in the notes). Does anyone know one (or have a reference)? It seems to be surprisingly hard work. (I also asked for a trick solution here.) […]