 Elliptic fibration

An elliptic fibration is a proper surjective morphism $\pi :X\to B$ with connected fibres whose generic fibre is a smooth genus one curve. Some conventions additionally require a section of a morphism, which chooses an origin and makes that generic fibre an elliptic curve; the presence of a section should be stated explicitly. Over special points of $B$, fibres may be singular.

For a fixed elliptic curve $E$, projection $E\times {\mathbb{P}}^1\to {\mathbb{P}}^1$ is an elliptic fibration with constant smooth fibres and a section from the origin of $E$. A nonconstant family can be described on a chart by a Weierstrass equation of an elliptic curve $y^2=x^3+a(t)x+b(t)$. In characteristic different from two and three, zeros of $4a(t{)}^3+27b(t{)}^2$ mark singular cubic fibres, when the displayed family defines the relevant model.

The fibration viewpoint separates the geometry of individual curves from how they vary and degenerate across the base. In dimension two it is the structure underlying an elliptic surface.