Hyperelliptic curve

A hyperelliptic curve of genus at least two admits a degree-two map to ${\mathbb{P}}^1$ and can be represented in characteristic other than two by an equation $y^2=f(x)$ with $f$ square-free.

If $y^2=f(x)$ with $\mathrm{deg}f=2n$, the coordinatesgive the second chartThere are two points $P_{+},P_{-}$ above infinity.

The affine germ surface over the plane with the eight roots of unity removed is compactified by adding the eight simple branch points above those roots and two unbranched points above infinity. The degree-two map to the Riemann sphere has total ramification eight, so the Riemann-Hurwitz formula gives genus three.

For the smooth curve $y^2=f(x)$ with square-free $f$ of degree $2n$, the differential $dx/y$ has divisorThus the canonical degree is $2n-4$, the genus is $n-1$, and $L(K)$ has basis $1,x,\dots ,x^{n-2}$.

The canonical map of an even-degree hyperelliptic curve isso it identifies the two points $(x,y)$ and $(x,-y)$ of a general fibre and is not an embedding.