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 $g^{2}\in O_{U}(B)$.

The plane model

A plane model for a hyperelliptic curve of genus 3. Two Weierstrass points have non-real coordinates, and one is at infinity

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 system

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 $g-1$.

Level 2 structure

If $S$ is a set of at most $g+1$ Weierstrass points of $C$ such that $\#S-(g+1)$ is even, then $D_{S}:=[S]+{\frac {\#S-(g+1)}{2}}H_{C}$ is a theta characteritic of $C$; i.e. $2D_{S}\sim K_{C}$ in the Picard group of $C$. Moreover, it can be shown that $h^{0}(D_{S})=1+{\frac {\#S-(g+1)}{2}}$, and if there are two such sets $S\neq S'$, then either $D_{S}\neq D_{S}'$ or $S'\cup S$ is the set of all Weirstrass points on $C$.

If we count each set $S$ together with its complementary set in the set of Weierstrass points (and then divide by 2) then the combinatorial description above tells us that any parition of the set of Weierstrass points to two sets such that the difference between the cardinalities is divisible by 4 induces a theta characteritic. Hence we constructed ${\frac {1}{2}}\sum _{4|2g+2-2k}binom(2g+2,k)$. This combinatorial sum is $2^{2g}$ which is the number of theta characteritic on a curve of genus $g$. Hence our description exhausts all the theta characteritics.

a family of bitangents to cannonical curves of genus 3 degenerate to a line connecting images of two Weierstrass points of a cannonical image of a hyperelliptic curve of genus 3

• In genus 2, there are 6 Weiertrass points. Each of them is an odd theta characteritics. There are $binom(6,3)=20$ three-tuples of distinct Weierstrass points, and hence there are $20/2=10$ odd theta characteritic, as expected. It is interesting to understand the geometrical meaning of pairs of Weierstrass points: if $p_{1},p_{2}$ are two distinct Weierstrass points on $C$ then $p_{1}+p_{2}-H_{C}$ is a 2-torsion point on the Jacobian of $C$, in this way we can express all the $2^{4}-1=15=binom(6,2)$ non trivial 2-torsion points on this Jacobian. Moreover, if $p_{3}+p_{4}-H_{C}$ is another (and different) two torsion point, then the $Weilpairing$ between the two 2-torsion points is given by $\{p_{1},p_{2}\}\cap \{p_{3},p_{4}\}$.
• in genus 3, there are 8 Weierstrass points. The even theta characteritics are given by the empty set, and by paritions of these 8 points to two distinct sets - hence we get $1+binom(8,4)/2=36$ even theta characteritics. the odd theta characteritics are given by the $binom(8,2)=28$ pairs of Weierstrass points. The cannonical map in thithis case maps the curve $C$ to a double cover of plane conic. Consider a family of cannonicaly embedded genus 3 curves with parmeter $t$ given by $Nulls(Q^{2}+tF)$, where we identify the double conic $Q^{2}$ with the cannonical image of $C$, where the Weierstarss points the intersection points of $Nulls(Q)\cap Nulls(F)$. Then it can be shown that the "limit" of the bitangents of the curves in the family (which are the odd theta characteritics) are the lines connecting pairs of images of Weierstrass points on $C$.

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 $2g-1$.