Let $R$ be an excellent Noetherian ring. A property $P$ is said to be *open* if the set $\{q \in \operatorname{Spec}(R) \ | \ R_q \ \text{satisfies} \ (P)\}$ is Zariski open. Examples of open properties include:

Regular (well-known), complete intersections, Gorenstein, Cohen-Macaulay, Serre's condition $(S_n)$ (Matsumura's book). These imply openness of other properties, for example normality, which means $(R_1)$ and $(S_2)$.

Factorial (for $R$ of characteristic $0$, since the proof uses resolution of singularities). UPDATE: in a recent very interesting preprint, the factorial and $\mathbb Q$-factorial property are proved to be open for varieties over any algebraically closed field.

$\mathbb Q$-Gorenstein, i.e. the canonical module is torsion in the class group (I don't know a convenient reference, please provide if you happen to know one).

My questions are:

**Question 1**: Do you know other interesting class 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?

Some comments: The excellent condition is quite mild but crucial. Question 2 was motivated by another question of mine.

`springerlink.com`

is broken. Perhaps you could take a look, whenever possible... $\endgroup$