Local parameter

In the geometry of algebraic curves, a local parameter for a curve C at a (smooth) point P is a rational function that has a simple zero at P. They are used mainly to count properly (in a local way). This has an algebraic resemblance with the concept of a uniformizing parameter (or just uniformizer) found in the context of discrete valuation rings in commutative algebra; a uniformizing parameter for the DVR (R, m) is just a generator of the maximal ideal m. The link comes from the fact that the local ring at a smooth point of an algebraic curve is always a discrete valuation ring. If this is the case of $P\in C$, then the maximal ideal of ${\mathcal {O}}_{C,P}$ consists of all those regular functions defined around P which vanishes at P, and finally a local parameter at P will be a uniformizing parameter for the DVR (${\mathcal {O}}_{C,P}$, $m_{P}$) (where the valuation function is given by ${\text{ord}}_{P}(f)={\text{max}}\{d=0,1,2,...:f\in m_{P}^{d}\}$).

Definition

Let C be an algebraic curve (defined over an algebraically closed field K). The valuation on K(C) (the field of rational functions of C) corresponding to a smooth point $P\in C$ is defined as ${\text{ord}}_{P}(f/g)={\text{ord}}_{P}(f)-{\text{ord}}_{P}(g)$. A local parameter for C at P is a function $t\in K(C)$ such that ${\text{ord}}_{P}(t)=1$.

See also

• Discrete valuation ring