---
tag: FMHA649HJ9
kind: concept
name: Fano variety
msc: [14J45]
specializes: [A4H4CZ2V35]
sources:
  - url: https://en.wikipedia.org/wiki/Fano_variety
same_as:
  wikidata: Q846881
  wikipedia: Fano variety
---

## Summary

A Fano variety is a smooth projective variety whose anticanonical bundle is ample.
Fano varieties are the positively curved end of the classification of varieties, opposite to varieties of general type.
They appear as the general fibres of Mori fibre spaces in the [[minimal model program|Q8CTDCZH01]].
In dimension $2$ they are the del Pezzo surfaces.

## Explanations

### L1 human

The canonical bundle measures how a variety curves, and a Fano variety curves positively in every direction, like a sphere.
Projective space is the main example.
A smooth hypersurface of degree $d$ in $\mathbb{P}^n$ is Fano exactly when $d \le n$: a smooth conic or a smooth cubic surface is Fano, but a smooth plane cubic is not.
Fano varieties have many rational curves, which makes them close to rational varieties.