In the geometry of complex algebraic curves, a local parameter for a curve C at a smooth pointP is just a meromorphic function on C that has a simple zero at P. This concept can be generalized to curves defined over fields other than , because the local ring at a smooth point P of an algebraic curveC is always a discrete valuation ring. This valuation will endow us with a way to count the order of rational functions having a zero or a pole at P. Local parameters, as its name indicates, are used mainly to properly count multiplicities in a local way.
Introduction
When C is a complex algebraic curve, we know how to count multiplicities of zeroes and poles of meromorphic functions defined on it. However, when discussing curves defined over fields other than, we do not have access to the power of the complex analysis, and a replacement must be found in order to define multiplicities of zeroes and poles of rational functions defined on such curves. In this last case, we say that the germ of the regular function vanishes at if. This is in complete analogy with the complex case, in which the maximal ideal of the local ring at a point P is actually conformed by the germs of holomorphic functions vanishing at P. Now, the valuation function on is given by this valuation can naturally be extended to K because it is the field of fractions of. Hencethe idea of having a simple zero at a point P is now complete: it will be a rational function such that its germ falls into, with d at most 1. This has an algebraic resemblance with the concept of a uniformizing parameter found in the context of discrete valuation rings in commutative algebra; a uniformizing parameter for the DVR is just a generator of the maximal ideal m. The link comes from the fact that a local parameter at P will be a uniformizing parameter for the DVR, whence the name.
Definition
Let C be an algebraic curve defined over an algebraically closed fieldK, and let K be the field of rational functions of C. The valuation on K corresponding to a smooth point is defined as , where is the usual valuation on the local ring. A local parameter for C at P is a function such that.
