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  "body": "\n## Definition\n\nLet $k$ be an [[algebraically closed field|0PF6TM8TWT]], and let $\\mathbb{P}^1$ be the [[projective line|ZT9A6JK8QF]] over $k$.\nA smooth projective [[algebraic curve|XJM2MT4S3Y]] $X$ over $k$ is *hyperelliptic* if:\n\n1. its [[genus|6W8TDV9VVN]] $g$ satisfies $g \\ge 2$, and\n2. there is a finite [[morphism|CDS1VWB7A3]] $X \\to \\mathbb{P}^1$ of degree $2$.\n\n## Remarks\n\nIf the characteristic of $k$ is not $2$, such an $X$ is the smooth projective model of an affine curve $y^2 = f(x)$.\nHere $f \\in k[x]$ is squarefree of degree $2g + 1$ or $2g + 2$, and the degree-$2$ map is $(x, y) \\mapsto x$.\nThe degree-$2$ map is unique up to automorphisms of $\\mathbb{P}^1$, and it defines the hyperelliptic involution $y \\mapsto -y$.\nExample: $y^2 = x^5 - x$ over $\\C$ has $f$ squarefree of degree $5 = 2 \\cdot 2 + 1$, so it defines a hyperelliptic curve of genus $2$.\nIn fact every curve of genus $2$ is hyperelliptic, because its canonical divisor gives a map of degree $2$ to $\\mathbb{P}^1$.\nNon-example: a smooth plane quartic has genus $3$ but is not hyperelliptic, because its canonical map is the embedding into the plane and so has degree $1$, not $2$.\nCurves of genus $1$ with a point are also double covers of $\\mathbb{P}^1$, but they fail clause 1 and are called elliptic curves.\n",
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