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A hyperelliptic curve $X$ over a field $k$ is a smooth projective algebraic curve of genus $g\ge 2$ that admits a 2-to-1 map $X\to \mathbb{P}^1$ defined over the algebraic closure $\bar k$.

If $X$ is a hyperelliptic curve over $k$, then the canonical map $X \to \mathbb{P}^{g-1}$ is a 2-to-1 map onto a smooth genus 0 curve $Y$. The curve $Y$ is isomorphic to $\mathbb{P}^1$ if and only if $Y$ has a $k$-rational point.

If $X$ admits a 2-to-1 map to $\mathbb{P}^1$ that is defined over $k$, then $X$ has a Weierstrass model of the form $y^2+h(x)y=f(x)$; when the characteristic of $k$ is not $2$ one can complete the square to put this model in the form $y^2=f(x)$.

In general, there is always a model for $X$ in $\mathbb{P}^3$ of the form \[ h(x,y,z)=0\qquad w^2=f(x,y,z) \] where $h(x,y,z)$ is a homogeneous polynomial of degree $2$ (a conic) and $f(x,y,z)$ is a homogeneous polynomial of degree $g+1$.

Knowl status:
  • Review status: reviewed
  • Last edited by Bjorn Poonen on 2022-03-24 16:23:46
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