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  "body": "\n## Summary\n\nA hyperelliptic curve is a smooth projective [[algebraic curve|XJM2MT4S3Y]] of genus at least $2$ that is a double cover of the projective line.\nAway from characteristic $2$, it has an affine model $y^2 = f(x)$ with $f$ a squarefree polynomial.\nHyperelliptic curves are the simplest curves of higher genus.\nThey are the exceptions in many results, such as [[Clifford's theorem|RSW29AVKGM]] and the embedding by the canonical divisor.\n\n## Explanations\n\n### L1 human\n\nTake a polynomial $f(x)$ with distinct roots and look at all solutions of $y^2 = f(x)$.\nFor most values of $x$ there are two values of $y$, one the negative of the other, and they come together where $f(x) = 0$.\nSo the curve is two copies of the line glued along cuts between the roots.\nFor $y^2 = x^5 - x$ there are six branch points, counting the point at infinity, and the curve has genus $2$.\n",
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