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  "body": "\n## Definition\n\nLet $k$ be a field.\nAn *algebraic curve* over $k$ is an [[algebraic variety|HV6YBS0YDH]] over $k$ of [[dimension|36X2WTHYCB]] $1$.\n\n## Remarks\n\nConventions differ on whether a curve must be irreducible, reduced, projective or smooth; each text states its choice.\nFor an integral variety $X$ over $k$, the dimension equals the transcendence degree of its function field over $k$.\nSo $X$ is a curve exactly when $k(X)$ has transcendence degree $1$.\nExample: the plane curve $X \\colon y^2 = x^3 - x$ in $\\mathbb{A}^2_k$ is a curve.\nIts function field $k(x)[y]/(y^2 - x^3 + x)$ is a finite extension of $k(x)$, so it has transcendence degree $1$.\nNon-example: the affine plane $\\mathbb{A}^2$ has dimension $2$, and a finite set of points has dimension $0$, so neither is a curve.\nLet $k$ be algebraically closed.\nThen each finitely generated field extension of $k$ of transcendence degree $1$ is the function field of exactly one smooth projective curve, up to isomorphism.\n",
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