Formulation and Choice of Model
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 > 4. 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 a cover of the curve with two affine pieces: the one already given by
and another one given by
- .
The glueing maps between the two pieces 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. In this way the cases n = 2g + 1 and 2g + 2 can be unified, since we might as well use an automorphism of the projective line to move any ramification point away from infinity.
Read more about this topic: Hyperelliptic Curve
Famous quotes containing the words formulation, choice and/or model:
“You do not mean by mystery what a Catholic does. You mean an interesting uncertainty: the uncertainty ceasing interest ceases also.... But a Catholic by mystery means an incomprehensible certainty: without certainty, without formulation there is no interest;... the clearer the formulation the greater the interest.”
—Gerard Manley Hopkins (18441889)
“...there are important considerations in the world beyond plain sewing and teaching dull little boys the alphabet. Any woman who has brains and willing hands finds twenty remunerative occupations open to her where formerly she would have found merely the inevitable twoplain sewing, or the dull little boys. All she had to do is to make her choice and then buckle on her armor of perseverance, while the world applauds.”
—Clara (Marquise)
“She represents the unavowed aspiration of the male human being, his potential infidelityand infidelity of a very special kind, which would lead him to the opposite of his wife, to the woman of wax whom he could model at will, make and unmake in any way he wished, even unto death.”
—Marguerite Duras (b. 1914)