Ag.Algenic Geometry – Definition of Low Normality


Let $ X $ to be a variety on a field $ F $ of characteristic $ p> 0 $.

The usual (not 100% sure) definition of low normality should be

$ X $ is weakly normal if each time $ f: Y to X $ is a morphism of varieties
satisfied

1) $ f $ is a finite birational bijective morphism, and

2) for each $ p in
> X, q in Y $
, $ f (q) = p $, the extension of the field k (p) to k (q) $ is purely
inseparable.

Then, $ f $ is an isomorphism.

In the book of Brion and Kumar Frobenius fractionation methods in geometry and representation theory. The definition (1.2.3) of low normality in the book is different, condition 2) is abandoned. Moreover, in the proof of the proposition (1.2.5) that each divided Frobenius variety is weakly normal, I think that the authors somehow use condition 2) in their evidence.

I would like to know if condition 2) is redundant in the definition of weak normality and I am happy to see a reference for a counterexample of the proposition (1.2.5) if we do not impose condition 2) in the definition. Thank you in advance.