Why is the generic fiber of an elliptic fibration $X\to C$ an elliptic curve over $k(C)$?

Question

Let all varieties be over a number field $k$.

Let $\pi:X\to C$ be an elliptic surface in the sense of Schütt & Shioda, i.e., having a section $\sigma:C\to X$ and being relatively minimal.

I'm trying to understand formally how the generic fibre of $\pi$ can be seen as an elliptic curve over $k(C)$.

I was confused about this at first because a generic point presupposes a scheme, but all books and articles I've read about ellipitc surfaces refer to varieties, so I don't know what's going on.

Anyway, assuming we're dealing with schemes, let $\eta\in C$ be the generic point of $C$. In this case the local ring $\mathcal{O}_{C,\eta}$ of $C$ at $\eta$ is a field, so I suppose we can say $k(C)=\mathcal{O}_{C,\eta}$. But I still can't see how this allows me to see $\pi^{-1}(\eta)$ as an elliptic curve over $k(C)$.

Answer 1

Let's recall the definition of an elliptic surface from Schuett & Schioda:

An elliptic surface $S$ over $C$ is a smooth projective surface $S$ with an elliptic fibration over $C$, i.e. a surjective morphism $f:S\to C$ so that

1. almost all fibers are smooth curves of genus 1;
2. no fiber contains an exceptional curve of the first kind.

By generic smoothness for morphisms in characteristic zero, we have that there exists $C'$ open in $C$ so that $f^{-1}(C')\to C'$ is smooth, and in particular, flat. By invariance of the Hilbert polynomial for a flat family (see Hartshorne theorem III.9.9, for instance), we have that the arithmetic genus and dimension of the fibers of $S$ over $C'$ are constant. As the generic point of $C$ is contained in $C'$, this gives that $S_\eta$ is a smooth curve over $k(\eta)$ of genus one, so all that's left to do to show it's an elliptic curve is to find a $k(\eta)$-rational point. This is exactly what their additional assumption of the existence of a section $\pi:C\to S$ does by considering $\pi(\eta)$.

Answer 2

The word curve is being used in a relative sense to mean a variety/scheme which is one-dimensional over some field. Classically, a curve is one-dimensional over a "small" field such as $\mathbb C$, but the generic fiber is a curve defined over a "larger" field, the function field $k(C)$. Since this is itself one-dimensional over $k$, $k(S)$ is two-dimensional over $k$, but the word curve is being used to describe the extension $k(S)/k(C)$.

Another way to think about this might be: an elliptic curve is (the usual projective compactification of) a variety/scheme defined by $y^2 = x^3 + ax + b$, whether $a$ and $b$ are rational numbers, complex numbers, rational functions in many variables...

Answer 3

Although this isn't necessary, I've decided to post another answer to my own question, this time in my own words. I hope this can be helpful to someone as it was to me.

If $\eta$ is the generic point of $C$, then $k(\eta)=\mathcal{O}_{C,\eta}\simeq k(C)$. In particular $\eta\simeq\text{Spec }k(C)$.

Now if $E:=\pi^{-1}(\eta)$ is the generic fiber, we have a natural morphism $$\pi|_E:E\to \eta=\text{Spec }k(C).$$

This makes $E$ a $k(C)$-scheme. Moreover, the section $\sigma:C\to X$ induces $$\sigma|_\eta:\eta=\text{Spec }k(C)\to E$$

whose composition with $\pi|_E$ is the identity, which means $E$ has a $k(C)$-rational point.

Since the general fiber of $\pi:X\to C$ is a smooth, genus 1 curve, we conclude that $E$ is an elliptic curve over $k(C)$.

N.B.: I thank Tabes Bridges and KReiser, whose answers helped me to light the way.