Enriques surface

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; 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.