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Every (smooth, projective, geometrically integral) hyperelliptic curve $X$ over $\Q$ of genus $g$ can be defined by an integral Weierstrass equation $$y^2+h(x)y=f(x),$$ where $h(x)$ and $f(x)$ are integral polynomials of degree at most $g+1$ and $2g+2$, respectively. Each such equation has a discriminant $\Delta$. A minimal equation is one for which $|\Delta|$ is minimal among all integral Weierstrass equations for the same curve. Over $\Q$, every hyperelliptic curve has a minimal equation. The prime divisors of $\Delta$ are the primes of bad reduction for $X$.

The equation $y^2+h(x)y=f(x)$ uniquely determines a homogeneous equation of weighted degree 6 in variables $x,y,z$, where $y$ has weight $g+1$, while $x$ and $z$ both have weight 1: one homogenizes $h(x)$ to obtain a homogeneous polynomial $h(x,z)$ of degree $g+1$ and homogenizes $f(x)$ to obtain a homogeneous polynomial $f(x,z)$ of degree $2g+2$. This yields a smooth projective model $y^2+h(x,z)y=f(x,z)$ for the curve $X$.

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  • Review status: reviewed
  • Last edited by John Cremona on 2018-05-24 16:32:36
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