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?
More specific questions:
- how about properties $R_n$ and normality?
- is being Frobenius split open in flat families?
- in general take a property local rings $Q$ and consider $P = $ "all local rings of $X$ satisfy $Q$". Which of the properties $Q$ listed in the cited thread give $P$'s which are open in flat families?
Best Answer
The recentish book of Görtz and Wedhorn (see http://www.algebraic-geometry.de/ ) has an Appendix E which gives a long list of properties of morphisms for which the corresponding set of the base is open or constructible (when only constructibility holds), together with references for the proofs (either to their book or to EGA/SGA).