In the geometry of complex algebraic curves, a local parameter for a curve C at a smooth point P is a meromorphic function on C that has a simple zero at P. This concept can be generalized to curves defined over fields other than
C
Local parameters, as its name indicates, are used mainly to properly count multiplicities in a local way.
If C is a complex algebraic curve, count multiplicities of zeroes and poles of meromorphic functions defined on it.[2] However, when discussing curves defined over fields other than
C
f
P\inC
f\inmP\subsetl{O}C,P
The valuation function on
l{O}C,P
\operatorname{ord}P(f)=max\{d=0,1,2,\ldots:f\in
d | |
m | |
P\}; |
l{O}C,P
f\inK(C)
d | |
m | |
P |
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 a local parameter at P will be a uniformizing parameter for the DVR (
l{O}C,P
mP
Let C be an algebraic curve defined over an algebraically closed field K, and let K(C) be the field of rational functions of C. The valuation on K(C) corresponding to a smooth point
P\inC
\operatorname{ord}P(f/g)=\operatorname{ord}P(f)-\operatorname{ord}P(g)
\operatorname{ord}P
l{O}C,P
mP
t\inK(C)
\operatorname{ord}P(t)=1