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Concept
Hyperelliptic curve
Summary
In algebraic geometry, a hyperelliptic curve is an algebraic curve of genus g > 1, given by an equation of the form
where f(x) is a polynomial of degree n = 2g + 1 > 4 or n = 2g + 2 > 4 with n distinct roots, and h(x) is a polynomial of degree < g + 2 (if the characteristic of the ground field is not 2, one can take h(x) = 0).
A hyperelliptic function is an element of the function field of such a curve, or of the Jacobian variety on the curve; these two concepts are identical for elliptic functions, but different for hyperelliptic functions.
The degree of the polynomial determines the genus of the curve: a polynomial of degree 2g + 1 or 2g + 2 gives a curve of genus g. When the degree is equal to 2g + 1, the curve is called an imaginary hyperelliptic curve. Meanwhile, a curve of degree 2g + 2 is termed a real hyperelliptic curve. This statement about genus remains true for g = 0 or 1, but those special cases are not called "hyperelliptic". In the case g = 1 (if one chooses a distinguished point), such a curve is called an elliptic curve.
While this model is the simplest way to describe hyperelliptic curves, such an equation will have a singular point at infinity in the projective plane. This feature is specific to the case n > 3. Therefore, in giving such an equation to specify a non-singular curve, it is almost always assumed that a non-singular model (also called a smooth completion), equivalent in the sense of birational geometry, is meant.
To be more precise, the equation defines a quadratic extension of C(x), and it is that function field that is meant. The singular point at infinity can be removed (since this is a curve) by the normalization (integral closure) process. It turns out that after doing this, there is an open cover of the curve by two affine charts: the one already given by
and another one given by
The glueing maps between the two charts are given by
and
wherever they are defined.
In fact geometric shorthand is assumed, with the curve C being defined as a ramified double cover of the projective line, the ramification occurring at the roots of f, and also for odd n at the point at infinity.
Official source
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