Hyperelliptic curve

In algebraic geometry a hyperelliptic curve is an algebraic curve $C$ fo genus geate then 1, 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". The divisor class of a fiber of the hyperelliptic double cover is called the "hyperelliptic class".

Weierstrass points

By the Riemann-Hurwitz formula the hyperelliptic double cover has exactly $2g+2$ branch points. For each branch point $p$ we have $h^{0}(2p)=2$. Hence these points are all Weierstrass points. Moreover, we see that for each of these points $h^{0}((2(k+1))p)\geq h^{0}(2kp)+1$, and thus the Weierstrass weight of each of these points is at least $\sum _{k=1}^{g}(2k-k)=g(g-1)/2$. However, by the second part of the Weierstrass gap theorem, the total weight of Weierstrass points is $g(g^{2}-1)$, and thus the Weierstrass points of $C$ are exactly the branch points of the hyperelliptic double cover.

Given a set $B$ of $2g+2$ distinct points on $\mathbb {P} ^{1}$, there is a uniqe double cover of $C\to \mathbb {P} ^{1}$ whose branch divisor is the set $B$. From an algebro-geometric point of view this on can construct the curve $C$ by taking the $Proj$ of the sheaf whose sections $g$ over an open subset $U\subset \mathbb {P} ^{1}$ satisfy $g^{2}\in O_{U}(B)$.

Curves of genus 2

If the genus of $C$ is 2, then the degree of the canonical class $K_{C}$ is 2, and $h^{0}(K_{C})=2$. Hence the canonical map is a double cover.

The canonical embedding

If $p$ is a rational point on a hyperelliptic curve, then for all $k$ we have $h^{0}((2(k+1))p)\geq h^{0}(2kp)+1$. Hence we must have $h^{0}((2g-2)p)\geq g$. However, by Riemann-Roch this implies that the divisor $(2g-2)p$ is rationally equivalent to the canonical class $K_{C}$. Hence the canonical class of $C$ is $g-1$ times the hyperelliptic class of $C$, and the canonical image of $C$ is a rational curve of degree $g-1$.

Moduli of hyperelliptic curves

Since for any set $B$ of $2g+2$ on $\mathbb {P} ^{1}$ there is a unique double cover $C\to \mathbb {P} ^{1}$ branch divisor $B$, the course moduli space of hyperelliptic curves of genus $g$ is isomorphic to the moduli of $2g+2$ points on $\mathbb {P} ^{1}$, up to projective transformations. However, as there are more then three points in $B$, there is a finite non-empty subset of $Aut(\mathbb {P} ^{1})=PGL_{2}$ that send three of the points in $B$ to $0,1,\infty$. Thus, the moduli of $2g+2$ distinct points on $\mathbb {P} ^{1}$ up to projective transformations is a finite quotient of the space of distincit $2g-1$ on $\mathbb {P} ^{1}\setminus \{0,1,\infty \}$. Specifically this space is an affine space of dimension $2g-1$.