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

algebraic-geometrysurfaces

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 $k(C)$.

Best Answer

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 $\pi(\eta)$.

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