9.1 Hyperelliptic curves

Loosely speaking, hyperelliptic curves represent the solutions of the equations of the form

y² + a(x)y + b(x) = 0

where a and b are polynomials in x. To put this in the language of schemes developed earlier, we first restrict our attention to schemes over Spec($\mathbb {F}$) where $\mathbb {F}$ is a finite field called the ground field. Next we consider the d - tuple Veronese embedding of $\mathbb {P}$¹ in $\mathbb {P}$^d; also known as the ``rational normal curve of degree d''; this is given as the locus of (1 : x : x<sup>2</sup> : <sup>...</sup> : x<sup>d</sup>) as (1 : x) varies over $\mathbb {P}$<sup>1</sup>. Alternatively, it is described by the system of equations X<sub>p</sub>X<sub>q</sub> = X<sub>r</sub>X<sub>s</sub> for all p, q, r, s such that p + q = r + s. Let us consider $\mathbb {P}$<sup>d + 1</sup> with (X<sub>0</sub> : <sup>...</sup> : X<sub>d</sub> : Y) as its co-ordinates so that $\mathbb {P}$<sup>d</sup> is obtained by projecting from the point (vertex) v = (0 : <sup>...</sup> : 0 : 1). Let S<sub>d</sub> denote the ``cone'' over the rational normal curve of degree d; it is the subvariety of $\mathbb {P}$^d defined by the same set of equations as above (in other words the variable Y is ``free''). Now suppose that a(x) = $\sum_{i}^{}$a_ixⁱ is a polynomial of degree at most d and b(x) = $\sum_{i}^{}$b_ixⁱ is a polynomial of degree at most 2d. We consider the linear forms

$$\sum_{i=0}^{d}$$ A(X) =

$$\sum_{i=0}^{d}$$ B(X) =

$$\sum_{i=1}^{d}$$ C(X) =

and the quadratic equation Y² + A(X)Y + B(X)X₀ + C(X)X_d = 0. The addition of this equation to the equations for S defines a subvariety T of S. It is clear that the vertex v does not lie on T so that projection gives a morphism on T which lands in the rational normal curve of degree d. Thus, we have a morphism T$\to$$\mathbb {P}$¹. There is an involution on $\mathbb {P}$^{d + 1} which fixes the X's and sends Y to A(X) - Y. Clearly this involution $\iota$ sends T to itself and pairs of points that are involutes of each other are sent to the same point in $\mathbb {P}$¹. The variety T is called a hyperelliptic curve, the involution is called the hyperelliptic involution and the morphism T$\to$$\mathbb {P}$¹ is called the canonical morphism.

Now it is clear that a solution (x, y) of the equation y² + a(X)y + b(x) = 0 gives rise to the solution (1 : x : ^... : x^d : y) of the above system. Conversely, if we have a solution (X₀ : X₁ : ^... : X_d : Y) of the system of equations with X₀ a unit, then we can put (x, y) = (X₁/X₀, Y/X₀) to obtain a solution of the two variable equation. Similarly, if (X₀ : ^... : X_d : Y) is a solution of the system of equations and X_d is a unit then consider the pair (u, v) = (X_{d - 1}/X_d, Y/X_d); this pair satisfies a two variable equation v² + a'(u)v + b'(u) = 0, where a'(u) = u^da(1/u) and b'(u) = u^2db(1/u). One sees from the above system that either X₀ or X_d must be a unit so we have covered all cases. The Jacobian criterion for regularity can be used to show that the curve defined by y² + a(x)y + b(x) = 0 is regular when either,

1. the discriminant a(x)² - 4b(x) has distinct roots, or
2. the field $\mathbb {F}$ has characteristic 2, a(x) has distinct roots and for each point (x₀, y₀) where x₀ is a root of a(x), the polynomial b(x) - b(x₀) - y₀a(x) vanishes with multiplicity one at 0.

To apply this to the equation v² + a'(u)v + b'(u) = 0, we note that

a'(u)² - 4b'(u) = u^2d(a(1/u)² - 4b(1/u))

Thus, if a(x)² - 4b(x) has distinct roots, then the only multiple root of a'(u)² - 4(b'(u) can be at u = 0; moreover, this happens only if a(x) has degree less than d - 1 and b(x) has degree less than 2d - 1. From now one we will assume the T is regular or non-singular; in fact we will assume that b(x) has degree equal to 2d - 1. The point (0 : ^... : 0 : 1 : 0) is a point on the curve T is called the ``point at infinity'' and denoted $\infty$. The number g = d - 1 is called the genus of the hyperelliptic curve. The points on T where a(x)² - 4b(x) vanishes and the point at infinity are called the Weierstrass points of the hyperelliptic curve; these are precisely the fixed points of the Weierstrass involution.