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  "body": "\n## Definition\n\nLet $X$ be a smooth projective [[algebraic surface|KFTECAATKG]] over an algebraically closed field.\n$X$ is a *surface of general type* if its [[Kodaira dimension|ZW00Q0SAGR]] is $2$.\n\n## Remarks\n\nEquivalently, with $K_X$ the [[canonical divisor|W05VYY4HBR]], there is a constant $c \u003e 0$ with $h^0(X, nK_X) \\ge c n^2$ for all large divisible $n$; that is, $K_X$ is big.\nBeing of general type is a birational property.\nA minimal surface of general type satisfies $K_X^2 \\ge 1$ and $\\chi(\\mathcal{O}_X) \\ge 1$.\nExample: a smooth surface $X \\subset \\mathbb{P}^3$ of degree $d \\ge 5$ has $K_X = (d - 4) H$ by adjunction, with $H$ a plane section.\nSince $d - 4 \\ge 1$, $K_X$ is ample and $h^0(nK_X)$ grows like $n^2$, so $X$ is of general type.\nNon-example: a smooth quartic surface in $\\mathbb{P}^3$ has $K_X = 0$, so $h^0(nK_X) = 1$ for all $n \\ge 1$ and its Kodaira dimension is $0$.\nExample: the product $C_1 \\times C_2$ of two curves of genus at least $2$ is of general type.\n",
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