---
tag: 5NK10NGY13
kind: concept
name: Ruled surface
msc: [14J26]
specializes: [KFTECAATKG]
sources:
  - url: https://en.wikipedia.org/wiki/Ruled_surface
---

## Summary

In algebraic geometry, a ruled surface is a smooth projective [[algebraic surface|KFTECAATKG]] that maps onto a curve with every fibre a projective line.
So it is a family of lines parametrized by a curve, the algebraic analogue of a surface swept out by a moving line.
Ruled surfaces, together with the projective plane, are the minimal surfaces of Kodaira dimension $-\infty$.
They play a basic role in the classification of surfaces.

## Explanations

### L1 human

Imagine a curve, and attach to each of its points a copy of the projective line, varying smoothly.
The result is a surface ruled by these lines.
The simplest example is the product of a curve with a line.
The Hirzebruch surfaces are ruled surfaces over the projective line that are twisted versions of $\mathbb{P}^1 \times \mathbb{P}^1$.