Let k be a field, and L/k a finitely generated field extensions. I would like to know if one can classify intermediate extensions L/K/k such that K/k has transcendence degree one.

This question comes from geometry, where given a birational class of a variety X over k (with function field L), one considers pencils on X parametrized by a curve C over k (with function field K).

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    $\begingroup$ Looks like a tall order to me. What kind of classification were you thinking of? $\endgroup$ – Lubin Aug 28 '17 at 18:30
  • $\begingroup$ Anything, even interesting example would be nice. Looking at the generic fiber, this gives an equivalence relation for varieties over global fields of characteristic p. My motivation was that for transcendence degree 2, the Tate-Shafarevich group of the Jacobian of the generic fiber of one is finite if and only if the Tate-Shafarevich group of the other is. $\endgroup$ – Thomas Geisser Aug 28 '17 at 19:27

If $X$ is fibered over $C$ and $C$ has positive genus then, as $C$ has non-zero holomorphic differentials, you can pull back those to $X$, which is already a non-trivial condition on $X$. Moreover, these pullbacks have the property that their wedge squares are zero. I seem to recall a theorem (Catanese?) that states that the converse is true in char. 0. I am sure that this can be (with a bit of pain) stated in terms of function fields.

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