Shafarevich offers the following theorem-definition:
"At any nonsingular point $P$ of an irreducible algebraic curve, there exists a regular function $t$ that vanishes at $P$ and such that every rational function $u$ that is not identically $0$ on the curve can be written in the form $u = t^k v$, with $v$ regular at $P$ and $v(P) \neq 0$. A function $t$ with this property is called a local parameter on the curve at $P$."
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I've looked through six other books on algebraic geometry (The Geometry of Schemes by Eisenbud and Harris, Algebraic Curves by Fulton, Principles of Algebraic Geometry by Griffiths and Harris, The Red Book of Varieties by Mumford, and Vakil's online notes Foundations of Algebraic Geometry) and, unless I have made an error, none even contain the phrase "local parameter." Hartshorne does appear to have the phrase in a few instances, but certainly does not give any definition at all similar to the one above, and besides Hartshorne is above my level right now so I am not in a good position to decide whether his usage agrees with that above or not.
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The above theorem appears to me to exist only in Shafarevich and nowhere else in the mathematical literature.
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Wikipedia offers the following much simpler characterization: "In the geometry of complex algebraic curves, a local parameter for a curve $C$ at a smooth point $P$ is just a meromorphic function on $C$ that has a simple zero at $P$."
So my question is this: what exactly are these local parameters, and how should I think of them? How can I reconcile what Wikipedia has written with what Shafarevich writes? The name "local parameter" suggests to me there is some simple characterization of these functions which Shafarevich is keeping a mystery from me (or is Shafarevich's definition more intuitive than I am finding it?). And finally are these really present virtually nowhere in the entire mathematical literature except Shafarevich, or do equivalent ideas go under different names?
Best Answer
What Shafarevic calls a local parameter is often called a uniformizing parameter at $P$, and is also the same thing as a uniformizer of the local ring of $C$ at $P$.
The point is that if $P$ is a smooth point on a curve, then the local ring at $P$ (i.e. the ring of rational functions on $C$ which are regular at $P$) is a DVR, and hence its maximal ideal is principal; a generator of this ideal is called a uniformizer.
If $t$ is a uniformizer/local parameter/uniformizing parameter at $P$, and if $u$ is any other rational function, then if we write $u = t^k v$ where $v(P) \neq 0$ (i.e. $v$ is a unit in the local ring), then $k$ is the order of vanishing of $u$ at $P$. In particular, $u$ vanishes to order one if and only if it is equal to $t$ times a unit in the local ring, if and only if it is also a generator of the maximal ideal of the local ring at $P$, if and only if it is also a uniformizer. Thus Shafarevic and Wikipedia are reconciled.
One is supposed to think of $t$ as being a "local coordinate at $P$." In the complex analytic picture you would choose a small disk around $P$, and consider the coordinate $z$ on this disk; this a local coordinate around the smooth point $P$. This analogy is very tight: indeed, it is not hard to show (when the ground field is the complex numbers) that a rational function $t$ is a local parameter at $P$ if and only $t(P) = 0$, and if there is a small neighbourhood of $P$ (in the complex topology) which is mapped isomorphically to a disk around $0$ by $t$, i.e. if and only if $t$ restricts to a local coordinate on a neighbourhood of $P$.
Finally, this concept is ubiquitous. The fact that the local ring at a point on a smooth algebraic curve is a DVR is fundamental in the algebraic approach to the theory of algebraic curves; see e.g. section 6 of Chapter I of Hartshorne.