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  "body": "\n## Definition\n\nLet $X$ be a smooth projective [[algebraic surface|KFTECAATKG]] over $\\C$, with structure sheaf $\\mathcal{O}_X$ and [[canonical divisor|W05VYY4HBR]] $K_X$.\n$X$ is an *Enriques surface* if:\n\n1. $H^1(X, \\mathcal{O}_X) = 0$, that is, the irregularity $q$ is $0$;\n2. $H^0(X, \\mathcal{O}_X(K_X)) = 0$, that is, the geometric genus $p_g$ is $0$; and\n3. $2 K_X$ is linearly equivalent to $0$.\n\n## Remarks\n\nClauses 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 $p_g = 1$.\nThe 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.\nThe fundamental group of an Enriques surface is $\\Z/2$, and its Euler characteristic is $12$.\nExample: 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 $Q_i(x) + R_i(y) = 0$, $i = 1, 2, 3$.\nThen $Y$ is a K3 surface, and the involution $(x, y) \\mapsto (x, -y)$ acts on $Y$ without fixed points.\nThe quotient of $Y$ by this involution is an Enriques surface.\nNon-example: a K3 surface satisfies clauses 1 and 3, but it has $K \\sim 0$ and so $p_g = 1$, which fails clause 2.\nIn characteristic $2$ the definition is modified, and there are three kinds of Enriques surfaces.\n",
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