# Spectrum of closure of function field?

Take $$V$$ and $$W$$ to be projective varieties over $$\mathbb{C}$$ of dimension $$n$$ and $$m$$, respectively. Let $$f : V \to W$$ be a fibre space, i.e., $$f$$ is a surjective map whose generic fiber is connected. In a 1983 paper of Viehweg (Weak Positivity and the Additivity of the Kodaira dimension for Certain Fibre Spaces), Viehweg introduces the notion of variation:

Definition: The variation, denoted $$\text{var}(f)$$, is the smallest number $$k$$ such that there is a subfield $$L$$ of $$\overline{\mathbb{C}(W)}$$ of transcendence degree $$k$$ over $$\mathbb{C}$$ and a variety $$F$$ over $$L$$ with $$F \times_{\text{Spec}(L)} \text{Spec}(\overline{\mathbb{C}(W)})$$ birational to the fibre $$V_w = f^{-1}(w)$$.

Aim: Understand the notion of variation geometrically.

• Which paper, where? It might be easier to give you intuition if we know some more of the surrounding context. In general, what's going on here is that you're just base-extending a variety to an algebraically closed field - are you familiar with examples of this, independent of this paper of Viehweg you're reading? – KReiser Sep 26 at 22:24
• @KReiser I will give more details to my question :) – ABBC Sep 26 at 22:31
• @ABitTooCurious I think you might find it useful to first think about families of curves of genus at least two $V\to W$. Let's say $\dim W=1$ to begin. To say that the family $f:V\to W$ is isotrivial is equivalent to $V_{\bar{K(W)}}$ being trivial (this is not a trivial statement). In other words, to see whether $V\to W$ is isotrivial or not it suffices to look at the "geometric generic fibre" of $V\to W$. Note that being non-isotrivial is the same as the image of the moduli map $W\to M_g$ being one-dimensional (or: the moduli map is non-constant). If $\dim W>1$.... – Ariyan Javanpeykar Sep 27 at 8:08
• ...things are more complicated. The moduli map $W\to M_g$ can be a point (i.e., the family is isotrivial), but it could also be positive-dimensional. However, the dimension of the image of $W\to M_g$ is of course at most the dimension of $W$. Intuitively speaking, the family $V\to W$ has maximal variation in moduli if this dimension is as large as possible, i.e., equal to $\dim W$. (With this definition: if $\dim W > 3g-3$, then the family $V\to W$ can't have maximal variation in moduli.) – Ariyan Javanpeykar Sep 27 at 8:10
• I'd also recommend you simply think of "maximal variation in moduli" as meaning that the moduli map is generically finite onto its image. For example, if the moduli map is quasi-finite (think about what this means for each fibre...), then your family $V\to W$ has maximal variation in moduli. Hope this helps. – Ariyan Javanpeykar Sep 27 at 8:12