Hyperelliptic curve

Remarks

If the characteristic of $k$ is not $2$, such an $X$ is the smooth projective model of an affine curve $y^2 = f(x)$. Here $f \in k[x]$ is squarefree of degree $2g + 1$ or $2g + 2$, and the degree-$2$ map is $(x, y) \mapsto x$. The degree-$2$ map is unique up to automorphisms of $\mathbb{P}^1$, and it defines the hyperelliptic involution $y \mapsto -y$. Example: $y^2 = x^5 - x$ over $\C$ has $f$ squarefree of degree $5 = 2 \cdot 2 + 1$, so it defines a hyperelliptic curve of genus $2$. In fact every curve of genus $2$ is hyperelliptic, because its canonical divisor gives a map of degree $2$ to $\mathbb{P}^1$. Non-example: a smooth plane quartic has genus $3$ but is not hyperelliptic, because its canonical map is the embedding into the plane and so has degree $1$, not $2$. Curves of genus $1$ with a point are also double covers of $\mathbb{P}^1$, but they fail clause 1 and are called elliptic curves.