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I am trying to understand a particular case of this question.

Let $T$ be an affine scheme of finite type over the ground field C. Let $X \to T$ be a morphism. I'll assume this to be the base change of a smooth and projective variety over C. In particular $X \to T$ is flat and proper.

Let $E$ be a coherent sheaf on $X$, which I do not assume to be flat over $T$ (and this is where the problems come in). Fix a number $r$. If $t \in T$ is a (geometric) point of $T$, let $E_t$ be the pullback to the fibre $X_t$. I would like to understand what the locus of points in $T$ such that $E_t$ is supported in dimension at most $r$.

In other words, I want to characterise the subfunctor of $T$, taking an affine $U$ to the set of morphisms $U \stackrel{g}{\to} T$, such that the pullback sheaf $g^*E$ is fiberwise supported in dimension at most $r$.

The condition can be phrased with chern characters (or hilbert polynomials I guess), as if $r$ corresponds to codimension $k$ in $X_t$ then I'm looking at the locus where $E_t$ has chern character with zeros in degrees lower than $k$. If $E$ were flat over $T$ then this locus would be open in $T$.

I suspect this condition for general $E$ should be closed or locally closed, but I can't seem to work it out. One thought would be to use the flattening stratification for $E$ and then impose the condition on chern characters, but that isn't really what we want. The other thought would be to take some locus where $E$ itself has support of dimension at most $r + \dim T$. While it is true that, for any $d$, one can split $E$ in $0 \to E_{\leq d} \to E \to F \to 0$, with $E$ support in dimension at most $d$, I'm not sure taking, say, the schematic image of the (schematic) support of $E_{\leq r + \dim T}$ is the right thing.

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up vote 2 down vote accepted

Maybe I misunderstand something, but by Nakayama, the support of $E_t$ is just the support of $E$ intersected with $X_t$, and the support of $E$ is closed in $X$. Let $Z$ be the suppor of $E$. By Chevalley's semi-continuity theorem (EGA, IV.13.1.3), the set of $x\in X$ such that $\dim_x Z_t\le r$ ($x\mapsto t$) is open, so the set of $t\in T$ such that $\dim Z_t\le r$ is constructible in $T$.

This is true for any morphism of finite type $X\to T$ over a noetherian scheme $T$.

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thanks for the answer! (and for the reference), I'll think about it more. Can one use your idea to describe what the scheme parameterising these things looks like? (It's the comment I made before the horizontal line) In other words this constructible subset of T comes with a moduli structure, is it representable? – Jacob Bell Feb 25 '13 at 11:42
As the fiber dimension doesn't change by base change, if there is a scheme which represents your functor, it will be reduced with underlying space equal to the set of $t\in T$ such that $\dim Z_t\le r$. If the latter is locally closed, then endow it with the structure of a reduced subscheme of $T$ and you get a scheme representing your functor. Otherwise I don't know, probably your functor is then not representable. – Qing Liu Feb 25 '13 at 12:13
Could you please clarify how you deduce reducedness from the fiber dimension? – Jacob Bell Feb 25 '13 at 13:33
sorry for the barrage of comments. Your reference got me reading the wonderful book by Goertz-Wedhorn again, and Corollary 14.113 applied to Z = supp E --> T, seems to suggest that the locus of where the fibre $Z_t$ is of dimension $\geq r+1$ is closed. My foggy brain right now concludes that then the locus I'm interested in is in fact open. – Jacob Bell Feb 25 '13 at 14:51
Yes because in your special case $X\to T$ is proper. As for the reducedness (in the general case), apply your functor to all $U$ affine open in the (moduli scheme)$_{\mathrm{red}}$. – Qing Liu Feb 25 '13 at 17:20

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