---
tag: Y30JE2DC6Y
kind: concept
name: Enriques surface
msc: [14J28]
specializes: [KFTECAATKG]
sources:
  - url: https://en.wikipedia.org/wiki/Enriques_surface
same_as:
  wikidata: Q5379864
  wikipedia: Enriques surface
---

## Summary

An Enriques surface is a smooth projective [[algebraic surface|KFTECAATKG]] with no holomorphic $1$-forms and no holomorphic $2$-forms.
Its canonical divisor is not trivial, but twice it is trivial.
Over $\C$, every Enriques surface is the quotient of a [[K3 surface|Q3V566R6K2]] by an involution without fixed points.
Enriques surfaces form one of the four classes of minimal surfaces of Kodaira dimension $0$.
They are non-rational surfaces with $q = p_g = 0$, which shows that Castelnuovo's rationality criterion needs the second plurigenus.

## Explanations

### L1 human

Take a K3 surface with a symmetry of order two that moves every point.
Identifying each point with its image gives a new surface, half as big, which is an Enriques surface.
Its canonical bundle is not trivial, but it becomes trivial after squaring it.
So an Enriques surface is a close relative of a K3 surface that has lost its holomorphic volume form.