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2 added "geometrically" to certain properties

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

1. $f$ is flat,
2. $f$ 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?
1

# What properties define open loci in families?

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

1. $f$ is flat,
2. $f$ 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?

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?
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