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Every (smooth, projective, geometrically integral) curve of genus 2 can be defined by a Weierstrass equation of the form $$y^2+h(x)y=f(x)$$ with nonzero discriminant and $\deg h \le 3$ and $\deg f \le 6$; in order to have genus 2 we must have $\deg h = 3$ or $\deg f =5,6$. Over a field whose characteristic is not 2 one can complete the square to make $h(x)$ zero, but this will yield a model with bad reduction at 2 that is typically not a minimal equation for the curve.

This equation can be viewed as defining the function field of the curve, or as a smooth model of the curve in the weighted projective plane. Every curve of genus 2 admits a degree 2 cover of the projective line (consider the function $x$) and is therefore a hyperelliptic curve.

Knowl status:
  • Review status: reviewed
  • Last edited by Andrew Sutherland on 2020-10-25 13:56:16
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