What properties define open loci in families?

Question

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?

Answer 1

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).

Answer 2

Piotr,

Here are some answers for a slightly different question (although nearly a special case of your question), I'm assuming that $S$ is regular and 1 dimension (for example a curve), which one can often reduce to (especially if the original $S$ was a smooth variety). However, for what I'm about to say I don't need properness / projectiveness of the family (EDIT: although as pointed out in the comments, you do need something).

In particular, I know some answers to the question that if $R/f$ has P then $R$ has property P near the vanishing locus of $f$ assuming $f$ is a regular element. (EDIT: This is basically the affine case as mentioned in the comments.) Now, if $R$ has the property $P$, then Bertini-type theorems can often imply that the other (non-special) fibers have the desired property too (especially in characteristic zero).

• Normality is open, this is in EGA somewhere I don't remember, but it's also in the paper of Heitmann I'm about to cite.
• Seminormality is open, although this isn't trivial. See "Lifting Seminormality" by Ray Heitmann. I don't know if weak normality is open in this way.
• Frobenius split is not open in the families I mention. If the total space is $\mathbb{Q}$-Gorenstein with index not divisible by the characteristic, it is open. See the paper $F$-purity and rational singularity'' by Richard Fedder, as well as some papers by Anurag Singh on "deformation". You can jazz these examples up to projective families without much work. By working with cones, you can find a projective family of smooth varieties whose special fiber is $F$-split, but not the general fibers.
• The Frobenius split answer is quite closely related to the question of whether log canonical singularities deform, which leads you to a series of papers on ``Inversion of adjunction'', in particular culminating in a paper on that topic by Kawakita from about 5 years ago.
• It is an open question whether Du Bois singularities deform in this way, it is known that it is true in the Gorenstein case by Kawakita's inversion of adjunction.
• In characteristic $p$, Du Bois singularities are related to $F$-injective singularities (a weakening of $F$-split). It is known that Cohen-Macaulay $F$-injective singularities deform in the way I describe, but it is an open question in general.
• Rational singularities deform by a result of Elkik.

I'm sure there are many things that I'm forgetting too.

Answer 3

You find plenty of theorems of this type in EGA IV/Part 3, see especially EGA IV.9.

Let $f: X\to S$ be a morphism of finite presentation. (We do not have to make a flatness or properness assumption in what follows.) For example let $P$ be one of the following properties:

• being geometrically irreducible
• being geometrically connected
• being geometrically regular
• being geometrically normal
• being geometrically reduced
• having property R_k geometrically

Let $U$ be the set of points $s\in S$ such that $X_s/k(s)$ has property P.

Then $U$ is at least locally constructible. (cf. IV.9.7.7 and IV.9.9.4)

Hence: If $S$ is irreducible and noetherian and $U$ contains the generic point of $S$, then $U$ contains a nonempty open set.

(But in EGA IV / Part 3 there are much more results of that flavour...)

Answer 4

I wanted to add the following translation of EGA IV Theorem 12.2.4 (see page 182 here) which personally has been useful. Unfortunately, I do not read French the best, so I failed with some parts of the translation. Please feel free to edit the answer if you know what it should say.

Theorem. Let $f\colon X\rightarrow Y$ be a proper, flat, and finite presentation morphism, and let $k\ge 1$ be an integer. The following subsets of $Y$ are open:

1. The points $y\in Y$ for which $X_y$ has property $(S_k).$
2. The points $y\in Y$ for which $X_y$ is geometrically $(R_k)$, and "soit équidimensionnel en chaque point et n'ait pas de cycle premier associé immergé."
3. The points $y\in Y$ for which $X_y$ is geometrically regular (i.e. smooth over $k(y)$.
4. The points $y\in Y$ for which $X_y$ is geometrically normal.
5. The points $y\in Y$ for which $X_y$ is geometrically reduced.
6. The points $y\in Y$ for which $X_y$ is geometrically reduced and the number of geometric components of $X_y$ is equal to $k$. (should this say "at most k"
7. The points $y\in Y$ for which $X_y$ is ``géométriquement ponctuellement intègre".
8. The points $y\in Y$ for which $X_y$ is geometrically integral.
9. The points $y\in Y$ for which $X_y$ "n'ait pas de cycle premier associé immergé" and the total multiplicity of $X_y$ over $k(y)$ is $\le k$.

Answer 5

With regards to the original question, and not my original answer, one place where $F$-splitting was studied in this context is:

K. Shimomoto and Wenliang Zhang, On the localization theorem for $F$-pure rings

In a ring of characteristic $p > 0$ where the Frobenius map is a finite morphism (ie, geometric contexts) $F$-pure is equivalent to $F$-split.