What are the local properties of schemes preserved under global sections? $\newcommand{\Spec}{\mathrm{Spec}\ }$
Let $(P)$ be a property of rings. I call $(P)$ local when $(P)$ satisfy these two
conditions:


*

*if $A$ is a ring satisfying $(P)$, then the distinguished rings $A_f$ also
satisfy $(P)$;

*If $\Spec A$ is covered by distinguished open $\Spec A_i$ with the $A_i$ having
$(P)$, then $A$ satisfy $(P)$.


Now if $(P)$ is local, then it is natural to extend the property $(P)$ to
schemes by saying that a scheme $X$ has property $(P)$ iff for all open affines
$\Spec A$ of $X$, $A$ has property $(P)$.
By definition of locality, then


*

*a ring $A$ satisfy $(P)$ iff $\Spec A$ satisfy $(P)$;

*a scheme $X$ satisfy $(P)$ iff $X$ can be covered by affines open $\Spec A_i$ with
$A_i$ satisfying $(P)$.


Likewise in the relative setting, local properties of morphisms of rings
allow to define a corresponding notion for morphisms of schemes.
[I have to point out that sometimes extending a local property $(P)$ of rings
to schemes this way is called $(\mathrm{locally}\ P)$, and a scheme $X$ is said to have
property $(P)$ when $X$ is locally P and satisfy some finiteness condition.
For instance $X$ is noetherian when it is locally noetherian and
quasi-compact; a morphism is of finite presentation when it is locally of
finite presentation and quasi-compact + quasi-separated.]
Now while this is a standard construction explained in all textbooks, it is
harder to find references for what happen to the global sections of non
affine open subschemes.
Indeed, if $X$ has a local property $(P)$, then an open scheme $U$ has also
property $(P)$, but $\Spec O_X(U)$ may not have $(P)$ when $U$ is not affine.
For instance:


*

*$X$ is noetherian does not mean that each ring of sections $O_X(U)$ is
noetherian
(An example is given by the union of two disjoints plane in projective
space $P^3_k$,
see http://sma.epfl.ch/~ojangure/nichtnoethersch.pdf);

*$X$ is finitely generated (say over a field $k$) does not mean that each
ring of sections O_X(U) is finitely generated
(Same example as above, see also http://math.stanford.edu/~vakil/files/nonfg.pdf);

*$X \to \Spec A$ is flat does not mean that $O_X(X)$ is flat over $A$.
(see Global sections of flat scheme also flat?).


However there are properties that hold for sections over any open subschemes:


*

*If $X$ is reduced, then for every open subscheme $U$, $O_X(U)$ is reduced;

*If $X$ is integral, then for every open subscheme $U$, $O_X(U)$ is integral.


I am interested to what happens with other properties $(P)$. I am also
interested to what happens in the relative case: if a morphism $X \to Y$ has
property $(P)$, then does $\Spec_Y(X) \to Y$ also has $(P)$?
 A: NormalityFor an integral scheme, being normal (integrally closed in ones own fraction field) satisfies this property.  Indeed, suppose that $a, b \in A = \Gamma(X, O_X)$ are such that $a/b$ satisfy some polynomial $p(x) \in A[x]$.  Then $a|_U, b|_U$ satisfy the same polynomial after restriction to each (affine) set $U \subseteq X$.

Of course this shows up in many applications of things like Stein factorization.

Semi-normality
A reduced ring $R$ is seminormal if for any finite extension $R \subseteq S$ satisfying the following two properties is an isomorphism.


*

*The induced map on $\text{Spec}$'s is an isomorphism

*The induced residue field extensions $k(r) \subseteq k(s)$ are isomorphisms for all $s \in \text{Spec } S$ mapping to $r \in \text{Spec } R$.


The typical example of a seminormal ring is a node, the cusp $k[x^2,x^3]$ is not seminormal
Equivalently, $R$ is seminormal if and only if for any $a/b$ in the total ring of fractions of $R$, one has that if $(a/b)^2, (a/b)^3 \in R$ then $(a/b) \in R$ (see a paper by Swan, he might be assuming finitely many minimal primes, I forget the details).  It follows similarly that seminormality satisfies this property.

Weak normality
Weak normality is similar to semi-normality.  A reduced ring is called weakly normal if for any finite birational extension $R \subseteq S$ satisfying the following properties is an isomorphism:


*

*The induced map on $\text{Spec}$'s is an isomorphism

*The induced residue field extensions $k(r) \subseteq k(s)$ are purely inseparable for all $s \in \text{Spec } S$ mapping to $r \in \text{Spec } R$. 


This can also be phrased as requiring that every birational universal homeomorphism is an isomorphism.
I do NOT know if weakly normal rings satisfy the sort of property asked for.  I do not think it is in the literature (but perhaps I am wrong).  I remember I convinced myself that they did not several years ago, but never wrote down an example carefully.
