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

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

$X$ is weakly normal if whenever $f:Y\to X$ is a morphism of varieties satisfy

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

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

Then, $f$ is an isomorphism.

In Brion and Kumar's book Frobenius splitting methods in Geometry and Representation Theory. The definition (1.2.3) of weak normality in the book is different, the condition 2) is dropped. Also, in the proof of the proposition (1.2.5) that every Frobenius split variety is weakly normal, I think the authors somehow use condition 2) in their proof.

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

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  • $\begingroup$ Welcome new contributor. If the field $F$ is separably closed, then condition (2) should be implied by condition (1). However, if $F$ is not closed, there are counterexamples. In fact, consider the case that $F$ equals $\mathbb{R}$, that $Y$ is the affine line $\mathbb{A}^1_{\mathbb{R}}$, and that $X$ is a nodal curve obtained from $Y$ by identifying a complex conjugate pair of complex points to one real node. Then $f$ is a bijective, birational morphism, yet it fails condition (2). $\endgroup$ – Jason Starr Jan 29 at 10:44
  • $\begingroup$ @JasonStarr, Thank you for your comment, can you direct me to a reference about (1)$\implies$ (2) in the case $F$ is separably closed? $\endgroup$ – chan kifung Jan 29 at 11:32
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    $\begingroup$ I do not know a reference (possibly EGA IV_2, somewhere around Section 4 . . .). If $F$ is separably closed, then the algebraic closure is purely inseparable over $F$. Thus, the induced field extensions of $\kappa(p)$ and $\kappa(q)$ after base change from $F$ to the algebraic closure are purely inseparable, and thus do not change (2). Thus, assume that $F$ is algebraically closed. Then there exists a finite stratification of $X$ into locally closed subvarieties that are normal. By Zariski's Main Theorem, the inverse image in $Y$ of each stratum is purely inseparable over the stratum. $\endgroup$ – Jason Starr Jan 29 at 11:50
  • $\begingroup$ @JasonStarr, I think Zariski's Main theorem says that a bijective birational map to a normal variety has to be an isomorphism. However, if $Z$ is a normal subvariety of $X$, the map $f:f^{-1}(Z)\to Z$ may not be birational, am I correct? $\endgroup$ – chan kifung Jan 30 at 5:29
  • $\begingroup$ The hypothesis in (1) is that $f$ is bijective. Thus, the restriction of $f$ over the subset $Z$ is also bijective. $\endgroup$ – Jason Starr Jan 30 at 11:54

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