---
tag: VSC3F53DX4
kind: concept
name: Elliptic surface
msc: [14J27]
specializes: [KFTECAATKG]
sources:
  - url: https://en.wikipedia.org/wiki/Elliptic_surface
same_as:
  wikidata: Q8519884
  wikipedia: Elliptic surface
---

## Summary

An elliptic surface is an [[algebraic surface|KFTECAATKG]] fibred over a curve so that almost every fibre is a smooth curve of genus one.
So it is a family of [[elliptic curves|427N6S6QQW]], or of genus-one curves, parametrized by a curve, degenerating over finitely many points.
Kodaira classified the possible singular fibres.
Elliptic surfaces include all minimal surfaces of Kodaira dimension $1$ and many K3 and rational surfaces.
They connect geometry with the arithmetic of elliptic curves over function fields.

## Explanations

### L1 human

Picture a curve, and above each of its points place a doughnut-shaped elliptic curve that varies with the point.
At a few special points the doughnut is pinched or broken into pieces.
The whole family forms a surface.
A simple example comes from the equation $y^2 = x^3 + t$, which for each nonzero value of the parameter $t$ is an elliptic curve and degenerates to a cusp at $t = 0$.