Hyperelliptic curve

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In algebraic geometry and in the theory of Riemann surfaces a hyperelliptic curve is an algebraic curve C of genus greater than 1, which admits a double cover \scriptstyle f:C\to\mathbb{P}^1. These curves are among the simplest algebraic curves: they are all birationally equivalent to curves of the form y2f(x) in the affine plane, where f(x) is a polynomial in x, and the degree of f(x) is either twice the genus of the curve plus 2, or twice the genus of the curve plus one.

If a double cover f:C\to\mathbb{P}^1 exists, then it is the unique double cover C\to\mathbb{P}^1, and it is called the "hyperelliptic double cover". The involution induced on the curve C by 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".


[edit] Weierstrass points

By the Riemann-Hurwitz formula the hyperelliptic double cover has exactly 2g + 2 branch points. For each branch point p we have h0(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(g2 − 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 unique double cover of C\to\mathbb{P}^1 whose branch divisor is the set B. From an algebro-geometric point of view one 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).

[edit] The plane model

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

If the 2g + 2 branch points of a hyperelliptic double cover C\to\mathbb{P}^1 are w_1,\ldots w_{2g+2} then C is birational to the plane curve \mbox{Nulls}(y^2=(x-w_1)(x-w_2)\cdots(x-w_{2g+1})(x-w_{2g+2}))\subset\mathbb{A}^2, where if one of the branch points is infinity, we omit the corresponding term in the product. The closure of this curve in the projective plane has a singularity on the line \mathbb{P}^2\setminus\mathbb{A}^2.

[edit] Curves of genus 2

If the genus of C is 2, then the degree of the canonical class KC is 2, and h0(KC) = 2. Hence the canonical map is a double cover. The Jacobian variety of such a curve is an Abelian surface.

[edit] 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 KC. 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.

[edit] 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 characteristic of C; i.e. 2DSKC is 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 Weierstrass points on C.

Every theta characteristics may be obtained in this way: Indeed 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 partition of the set of Weierstrass points into two sets such that the difference between the cardinalities is divisible by 4, induces a theta characteristic. In this way we count a total of \frac{1}{2}\sum_{4|2g+2-2k}\mbox{binom}(2g+2,k) distinct theta characteristics . This combinatorial sum is 22g which is the number of theta characteristics on a curve of genus g. Hence our description exhausts all the theta characteristics.

File:Family bitangents.png
a family of bitangents to canonical curves of genus 3 degenerate to a line connecting images of two Weierstrass points of a canonical image of a hyperelliptic curve of genus 3

[edit] Moduli of hyperelliptic curves

Since for any set B of 2g + 2 points on \mathbb{P}^1 there is a unique double cover C\to\mathbb{P}^1 with branch divisor B, the coarse 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 than three points in B, there is a finite non-empty subset of \mbox{Aut}(\mathbb{P}^1)=\mbox{PGL}_2 that sends three of the points in B to 0,1,\infty. Thus, the moduli of space which parametrizes 2g + 2 distinct points on \mathbb{P}^1 up to projective transformations, is a finite quotient of the space of distinct 2g − 1 points on \mathbb{P}^1\setminus\{0,1,\infty\}. Specifically this space is an irreducible affine scheme of dimension 2g − 1.

[edit] Compactifications of the moduli

There are two standard "natural" compactifications of the moduli of n distinct points on \mathbb{P}^1. One is the binary forms compactification \overline{\mathcal{B}}_n: the GIT quotient of homogeneous polynomials of degree n in two variables by the group PGL2 acting on the linear span of the variables. The second compactification is the Deligne-Mumford compactification of the moduli of pointed curves \overline{\mathcal{M}}_{0,n}. For the case n = 2g + 2 Avritzer and Lange constructed a surjective morphism \overline{\mathcal{M}}_{0,2g+2}\to\overline{\mathcal{B}}_{2g+2}, thus relating these compactifications in the hyperelliptic case. Finally, the closure of the locus of hyperelliptic curves in the moduli stack \overline{\mathcal{M}}_g of curves of genus g, is a degree 1 / 2 cover of \overline{\mathcal{M}}_{0,2g+2}.

[edit] References

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