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  "body": "\n## Definition\n\nLet $k$ be a field.\nA *projective plane curve* over $k$ is the closed subscheme $C = V(F)$ of the [[projective plane|ZT9A6JK8QF]] $\\mathbb{P}^2_k$ defined by a nonconstant [[homogeneous polynomial|2C3PAQ7H50]] $F \\in k[x, y, z]$.\nIts *degree* is $\\deg F$.\nAn *affine plane curve* is the closed subscheme $V(f) \\subset \\mathbb{A}^2_k$ defined by a nonconstant polynomial $f$ in the [[polynomial ring|0SDC934R52]] $k[x, y]$.\n\n## Remarks\n\nThe polynomial is determined by $C$ up to a nonzero scalar factor.\nA plane curve is an [[algebraic curve|XJM2MT4S3Y]]: it has dimension $1$.\nIt is reduced exactly when $F$ has no repeated irreducible factor, and integral exactly when $F$ is irreducible.\nExample: $F = x^2 + y^2 - z^2$ defines a conic of degree $2$ in $\\mathbb{P}^2$, the projective closure of the circle $x^2 + y^2 = 1$.\nNon-example: the twisted cubic $\\set{[s^3 : s^2 t : s t^2 : t^3]} \\subset \\mathbb{P}^3$ is a curve in $\\mathbb{P}^3$ that lies in no plane of $\\mathbb{P}^3$, so it is not a plane curve inside $\\mathbb{P}^3$.\nAs an abstract curve it is isomorphic to $\\mathbb{P}^1$, which is a line in $\\mathbb{P}^2$; being a plane curve is a property of the embedding.\nA nonzero linear form vanishing on it would give a linear relation among $s^3, s^2 t, s t^2, t^3$, and there is none.\n",
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