elliptic surface

Definition 1.

Let $k$ be a field and let $C\mathrm{/}k$ be a smooth projective curve defined over the field $k$ and has genus $g$. The function field of $C\mathrm{/}k$ will be denoted by $K\mathrm{=}k\mathit{}\mathrm{(}C\mathrm{)}$. An elliptic surface $\mathrm{E}$ over the curve $C$ is, by definition, a two-dimensional projective variety together with:

1. A morphism $\pi :ℰ\to C$ such that for all but finitely many points $t\in C(\overline{k})$, the fiber ${ℰ}_t={\pi }^{-1}(t)$ is a non-singular curve of genus $1$,

2. A section to $\pi$ (the zero section) ${\sigma }_0:C\to ℰ$.

With this definition, $\mathrm{E}\mathrm{/}K$ may be regarded as an elliptic curve over the field $K$.

Example 1.

The surface $y^2=x^3+t$ is an elliptic surface over the curve ${\mathbb{P}}^1(\mathbb{Q})$. It may be regarded as an elliptic curve over the function field $\mathbb{Q}(t)$.

Title	elliptic surface
Canonical name	EllipticSurface
Date of creation	2013-03-22 15:34:16
Last modified on	2013-03-22 15:34:16
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	5
Author	alozano (2414)
Entry type	Definition
Classification	msc 14J27
Related topic	EllipticCurve