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  "body": "\n## Definition\n\nLet $k$ be a field and let $X$ be a smooth [[projective variety|A4H4CZ2V35]] over $k$ with [[canonical bundle|W05VYY4HBR]] $\\omega_X$.\nThe variety $X$ is a *Fano variety* if the anticanonical bundle $\\omega_X^{-1}$ is [[ample|RRK8P6TXG2]].\n\n## Remarks\n\nIn divisor notation, $X$ is Fano when $-K_X$ is ample.\nExample: on $\\mathbb{P}^n$ one has $\\omega = \\mathcal{O}(-n-1)$, so $\\omega^{-1} = \\mathcal{O}(n+1)$ is ample and $\\mathbb{P}^n$ is Fano.\nMore generally, a smooth hypersurface $X \\subset \\mathbb{P}^n$ of degree $d \\ge 1$, with $n \\ge 2$, has $\\omega_X = \\mathcal{O}_X(d - n - 1)$ by the adjunction formula, so it is Fano exactly when $d \\le n$.\nNon-example: a smooth plane cubic curve has $\\omega = \\mathcal{O}(3 - 3) = \\mathcal{O}$, and the trivial bundle on a curve is not ample.\nMany authors also allow mild singularities, such as terminal or log terminal ones, and then ask that $-K_X$ be an ample $\\Q$-Cartier divisor.\nOver $\\C$, smooth Fano varieties are simply connected and rationally connected, and in each dimension they form finitely many deformation families.\n",
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