Hyperelliptic curve

In algebraic geometry a hyperelliptic curve is an algebraic curve $C$ which admits a double cover $f:C\to \mathbb {P} ^{1}$. If such a double cover exists it is unique, and it is called the "hyperelliptic double cover". The involution induced on the curve $C$ by the interchanging between the two "sheets" of the double cover is called the "hyperelliptic involution".

Weierstrass points

By the Riemann-Hurwitz formula the hyprelliptic double cover has exactly $2g+2$ branch points. For each branch point $p$ we have $h^{0}(2p)=2$.