Morphisms preserving weak normality

Question

I would like to find a class of morphisms for which weakly normality descends. The notion of seminormlaity is very close to the one of weakly normality and for seminormal schemes one has Theorem 5.8 of "On seminormal schemes" (Greco and Traverso) which says the following

"Let $f:A\rightarrow B$ be a reduced ring homomorphism, and assume $A$ is seminormal. Then $B$ is seminormal iff $B$ is Mori and the generic fibers of $f$ are seminormal"

This is a useful criterion to deduce the seminormality of the source space. For example one can deduce from this that the product of seminormal schemes over a perfect field is still seminormal.

We know that the notion of weak normality is different from seminormality only in characteristic $p$.

I have 2 questions:

1) does there exists a criterion as before for weak normality?

2) "more easy": is a finite product of weakly normal schemes, say over an algebraically closed field, weakly normal?

Answer 1

Manaresi, Mirella. Some properties of weakly normal varieties. Nagoya Math. J. 77 (1980), 61–74. DOI: 10.1017/s0027763000018663. MR: 556308.

contains the following (which is not quite the same as the result in Greco–Traverso.):

Proposition III.3. Let $A$ be a Mori ring, and let $f \colon A \to B$ be a reduced ring homomorphism with normal generic fibers. If $A$ is weakly normal, then $B$ is weakly normal, and the converse holds if $f$ is faithfully flat.