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added "geometrically" to certain properties

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edited Mar 8 2011 at 16:09

This question is somehow related to the question What properties define open loci in excellent schemes?.

Let $f:X\to S$ be a proper (or even projective) morphism between schemes (of finite type over a field or over $\mathbb{Z}$). For $t\in S$, $X_t$ is the fiber of $f$ over $t$. Let $P$ be a property of schemes. We consider the locus: $$ U_P = \{ t\in S : X_t \text{ has property } P \}. $$

For which properties $P$ is the set $U_P$ open if

$f$ is flat,

$f$ is smooth?

Examples of such $P$'s I know or suspect to be open in flat families are "being geometrically reduced", "being geometrically smooth" or "being $S_n$". In smooth families, a nice example is that of "being Frobenius split" (we assume that $S$ has characteristic $p$).

Copy-paste from the aforementioned thread:

Question 1: Do you know other interesting classes of open properties?

Question 2: Are there good heuristic reasons for why a certain property should be open? Phrased a bit more ambitiously, are there common techniques for proving openness for certain class of properties?

What properties define open loci in families?

This question is somehow related to the question What properties define open loci in excellent schemes?.

For which properties $P$ is the set $U_P$ open if

$f$ is flat,

$f$ is smooth?

Examples of such $P$'s I know or suspect to be open in flat families are "being reduced", "being smooth" or "being $S_n$". In smooth families, a nice example is that of "being Frobenius split" (we assume that $S$ has characteristic $p$).

Copy-paste from the aforementioned thread:

Question 1: Do you know other interesting classes of open properties?