---
tag: 7H6T24G8BM
kind: definition
name: Enriques surface
msc: [14J28]
defines: Y30JE2DC6Y
uses: [KFTECAATKG, W05VYY4HBR, PQ6Q143ARJ, Q3V566R6K2]
sources:
  - url: https://en.wikipedia.org/wiki/Enriques_surface
---

## Definition

Let $X$ be a smooth projective [[algebraic surface|KFTECAATKG]] over $\C$, with structure sheaf $\mathcal{O}_X$ and [[canonical divisor|W05VYY4HBR]] $K_X$.
$X$ is an *Enriques surface* if:

1. $H^1(X, \mathcal{O}_X) = 0$, that is, the irregularity $q$ is $0$;
2. $H^0(X, \mathcal{O}_X(K_X)) = 0$, that is, the geometric genus $p_g$ is $0$; and
3. $2 K_X$ is linearly equivalent to $0$.

## Remarks

Clauses 2 and 3 together force $K_X$ to be a nonzero element of order $2$ in the Picard group, since $K_X \sim 0$ would give $p_g = 1$.
The double cover $\tilde X \to X$ defined by $K_X$ is unramified, and $\tilde X$ is a [[K3 surface|Q3V566R6K2]]; conversely, the quotient of a K3 surface by a fixed-point-free involution is an Enriques surface.
The fundamental group of an Enriques surface is $\Z/2$, and its Euler characteristic is $12$.
Example: in $\mathbb{P}^5$ with coordinates $x_0, x_1, x_2, y_0, y_1, y_2$, let $Y$ be a general intersection of three quadrics $Q_i(x) + R_i(y) = 0$, $i = 1, 2, 3$.
Then $Y$ is a K3 surface, and the involution $(x, y) \mapsto (x, -y)$ acts on $Y$ without fixed points.
The quotient of $Y$ by this involution is an Enriques surface.
Non-example: a K3 surface satisfies clauses 1 and 3, but it has $K \sim 0$ and so $p_g = 1$, which fails clause 2.
In characteristic $2$ the definition is modified, and there are three kinds of Enriques surfaces.