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My name is Chemy (Przemysław). I am a PhD student at UvA (Amsterdam), and I work on projects related to foliations in algebraic geometry in positive characteristic. Therefore I am avidely reading two papers from 1987 by Ekedahl, and Miyaoka, accordingly "Foliations and Inseparable Morphisms", and "Deformations of a morphisms along a foliation and applications". The papers define classes of "inseparable morphisms" called n-foliations, study them, and use them, but they do not define "inseparable morphisms"... And I am trying to understand the papers more by putting them into a big picture about "inseparable morphisms", but I do not know much about them, what probably should be expected from most of us, because we have a (over 10 years old) quote of Ravi Vakil:

This seems to me to not be a notion that absolutely everyone should see in a first serious schemes course.

Here are my questions:

  1. "Inseparable morphism" is (probably) the best formalized by the notion of a radicial morphism. Is it true? Are there any other candidates? What are good references about them?
  2. Is there any (general) classification of inseparable morphisms? I say "general", because there is a theorem that between curves, there are only compositions of the Frobenius morphism, the theorem,
  3. and there is a classification of "exponent 1 inseparable extensions of fields" using derivatives by Jacobson, Jacobson–Bourbaki theorem, and its generalizations. What is the state of art with extending it beyond the exponent 1 and beyond fields?

It is hard to find anything about the topic; there is not much about it? Consequently, there are not many threads about it here. The best I have found is Who uses radicial morphisms?, but it doesn't have what I am looking for, namely, an exposition of the ideas, examples, uses, and problems around the notion, and what we know about them today. I don't believe that precisely such thing exists right now(if so, that would be a great surprise to me), but I believe that it is possible to find a covering of such a thing consisting of some few papers, books. Also, there is a paper by Liedtke from about 2010 on surfaces in positive characteristic. There is something in Liu's book as well. Finally, I have found that there is a recently published book called "Algebraic Surfaces In Positive Characteristics: Purely Inseparable Phenomena in Curves and Surfaces", The book, but I do not have any access to it right now. I am waiting for my library to get it.

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    $\begingroup$ I recommend that you ask some of these questions to Prof. Mingmin Shen at UvA. He is an expert on purely inseparable morphisms, foliations, and the articles by Ekedahl and Miyaoka from the Bowdoin volumes. $\endgroup$
    – Jason Starr
    Commented Mar 16, 2022 at 17:17
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    $\begingroup$ One application of these techniques is Ekedahl's vanishing theorem / semipositivity theorem for minimal surfaces in all characterisic. This is a major component in Koll'ar's proof of projectivity over $\text{Spec}\ \mathbb{Z}$ of the (coarse) moduli space of Deligne-Mumford stable curves, cf. "Projectivity of complete moduli." $\endgroup$
    – Jason Starr
    Commented Mar 16, 2022 at 17:21
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    $\begingroup$ @JasonStarr Thank you, he is one of my supervisors. I am interested in what I can get from others. What other stories are out there. I want a bigger picture. This is vague, but I don't want it to be more concrete for now. I just want to listen. Probably, at some level, I am also trying to relive some anxiety that I could be missing something important in a narrative about such morphisms. Normal human thing. :) "Projectivity of complete moduli.", this is interesting. Thank you for sharing that with me. I have seen the Ekedahl's paper, but I didn't know about this application. $\endgroup$
    – P. Grabowski
    Commented Mar 16, 2022 at 21:20
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    $\begingroup$ Using inseparable morphisms on projective homogeneous spaces, i.e., $G/P$, Haboush-Lauritzen and, later, Totaro found examples of Fano manifolds in positive characteristic where Kodaira vanishing fails. $\endgroup$
    – Jason Starr
    Commented Mar 16, 2022 at 23:35
  • $\begingroup$ I have got the book! $\endgroup$
    – P. Grabowski
    Commented Mar 17, 2022 at 22:10

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