What is a local parameter in algebraic geometry?

Solution 1:

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.

Solution 2:

A synonym for this term is "uniformizing parameter" or "uniformizer," and this term does appear in other books. I believe it is supposed to be the algebraic analogue of a chart; in other words, it is a "local coordinate" at the point. Perhaps to understand the geometry behind the term you should look at textbooks on Riemann surfaces first.

Solution 3:

I think page 146 of this seems like a clear explanation of local parameters. This is an article about algebraic geometry applied to coding theory. It also uses "uniformizing parameter" as a synonym as Qiaochu mentioned.