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  "body": "\n## Definition\n\nLet $X$ be a smooth projective [[variety|HV6YBS0YDH]] of [[dimension|36X2WTHYCB]] $n$ over a field $k$, and let $\\omega_X$ be its [[canonical bundle|W05VYY4HBR]].\nFor $m \\ge 1$, the *$m$-th plurigenus* is $P_m(X) = \\dim_k H^0(X, \\omega_X^{\\otimes m})$.\nThe *Kodaira dimension* $\\kappa(X)$ is defined as follows:\n\n1. If $P_m(X) = 0$ for all $m \\ge 1$, then $\\kappa(X) = -\\infty$.\n2. Otherwise, $\\kappa(X)$ is the least integer $\\kappa \\ge 0$ such that $P_m(X) / m^{\\kappa}$ is bounded as $m \\to \\infty$.\n\n## Remarks\n\nAlways $\\kappa(X) \\in \\set{-\\infty, 0, 1, \\dots, n}$.\nEquivalently, when some $P_m(X) \u003e 0$, $\\kappa(X)$ is the largest dimension of the image of the rational map $X \\dashrightarrow \\mathbb{P}(H^0(X, \\omega_X^{\\otimes m}))$ over all $m$ with $P_m(X) \u003e 0$.\nThe variety $X$ is of *general type* if $\\kappa(X) = n$.\n\nExample: for $X = \\mathbb{P}^n_k$ with $n \\ge 1$, $\\omega_X \\cong \\mathcal{O}(-n-1)$, so $\\omega_X^{\\otimes m} \\cong \\mathcal{O}(-m(n+1))$ has no nonzero sections and $\\kappa(X) = -\\infty$.\nFor a smooth projective curve of genus $g \\ge 2$, $\\deg \\omega_X^{\\otimes m} = m(2g-2)$ and Riemann–Roch gives $P_m = (2m-1)(g-1)$ for $m \\ge 2$, which grows like $m^1$, so $\\kappa(X) = 1$.\n\nNon-example of clause 2 with $\\kappa = 0$: the curve of genus $2$ has $P_m / m^0 = P_m$ unbounded, so its Kodaira dimension is not $0$.\n",
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