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Say I've got a variety X (or a scheme locally of finite type) over an algebraically closed field k. Then closed points of X correspond to k-points of X. (correct?)

Let's define a geometric point of X as a morphism from an algebraically closed field into X. (thus for example the morphism from k[x] to the algebraic closure of the field of fractions of k[x] is a geometric point of the line)

If a (reasonable!) property P holds for all k-points of X does it then hold for all geometric points?

My question comes from moduli stuff. For example, if E is a flat family of sheaves on X parameterised by some base S, such that the fibre of E has some behaviour over all k-points of S, will this behaviour persist on geometric points?

Jacob Bell
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  • Let $k$ be an alg. closed field. Then $X(k)$ is in bijection with the closed points of $X$. This follows from Hilbert's nullstellensatz if I'm not mistaken. – Gooz Sep 30 '11 at 21:08
  • Could you give an example of your property $P$? – Gooz Sep 30 '11 at 21:09
  • well, what I'm really thinking about is a family of complexes of sheaves. For example one might request that the (derived) restriction has no negative self-extensions (e.g. paper by max lieblich). Can I check this only over k-points? – Jacob Bell Sep 30 '11 at 21:16
  • I don't quite understand your situation but I can make the following probably useless statement. The image of a geom. point is a closed point and therefore a k-point. So if you're taking stalks of sheaves (in the Zariski topology) in points lying on your variety then once you've checked your property P for k-points it should also follow for geom. points. – Gooz Sep 30 '11 at 21:23
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    wait a minute, isn't the example I gave in my question a case of a geometric point with non-closed image? – Jacob Bell Sep 30 '11 at 21:26
  • You're completely right. My apologies. Please ignore my comment above. I can make the following even more trivial statement though. A geom. point gives a k-point or a $K(X)$-point, where $K(X)$ is the function field $X$. (I assume here $X$ is irreducible for simplicity.) So if you verify it for these points you're good if your property P just requires you check it on the points of your variety. – Gooz Sep 30 '11 at 21:34
  • On one hand my naivety is in thinking of a base scheme S as something nice, after all S is allowed to be any k-scheme. On the other hand if I'm working with concrete family over some curve, say, then the question makes sense again. Thanks for your interest anyways. – Jacob Bell Sep 30 '11 at 21:43
  • You might find http://math216.wordpress.com/2011/06/13/favorite-properties-of-varieties-finite-type-k-schemes-checkable-at-closed-points/ and http://math216.wordpress.com/2011/06/10/favorite-open-or-closed-conditions-2/ helpful. – David E Speyer Sep 30 '11 at 22:26
  • thanks for the links David. – Jacob Bell Oct 04 '11 at 22:39

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