Examples of curves

A curve is an absolutely irreducible projective curve defined over a field field''. Such a curve is determined by a polynomial of degree which is irreducible over . The projective curve is composed of three affine components''

where

The projective version of is

We shall sometimes abuse notation and confuse and .

The Klein quartic is given by . The projective version is . (You can compute , , and above by setting , , and , respectively, in the projective version.)

The Fermat curve (of degree ) is given by . The projective version is .

The Hermitian curve is a special case of the Fermat curve. It is defined over and is given by .

Exercise 6.4.1 (a)   Compute , , and for the Klein quartic.

(b) Compute , , and for the Fermat curve.

(c) Compute , , and for the Hermitian curve.

Subsections

david joyner 2008-04-20