What is a local parameter in algebraic geometry?

Question

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 $P$."

• 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.

• The above theorem appears to me to exist only in Shafarevich and nowhere else in the mathematical literature.

• 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?

Answer 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.

Answer 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.

Answer 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.

Answer 4

In the second-to-last paragraph from Matt E's answer they interpret uniformizers of a stalk of the sheaf of algebraic functions leveraging the analytic topology. The case of the sheaf of analytic functions is similar:

Definition 3. Let $K$ be a (not necessarily commutative) field, a valuation on $K$ and $\Gamma$ the order group of $v$. $v$ is called discrete if there exists a (necessarily unique) isomorphism of the ordered group $\Gamma$ onto $\mathbb{Z}$. Let $\gamma$ be the element of $\Gamma$ corresponding to $1$ under this isomorphism; every element $u$ of $K$ such that $v(u)=\gamma$ is called a uniformizer of $v$. (...)

Let $S$ be a connected complex analytic variety¹ of dimension $1$, $K$ the field of meromorphic functions on $S$ and $z_0$ a point of $S$; the set of $f\in K$ which are holomorphic at $z_0$ is the ring of a discrete valuation $v$; for a function $f\in K$ to be uniformizing for $v$, it is necessary and sufficient that it be holomorphic and zero at $z_0$ and that there exist a neighbourhood $V$ of $z_0$ in $S$ such that the restriction of $f$ to $V$ be a homomorphism of $V$ onto a neighbourhood of the origin in $\mathbb{C}$. It is this example and other analogues which are the origin of the word “uniformizer”.

(Bourbaki, Commutative Algebra, Ch. VI, §3.6.)

One can prove Bourbaki's claim (resp., Matt E's claim) for the analytic case (resp., the algebraic case) in much the same way: since the statement is local, our complex (analytic) variety—with the analytic topology—locally looks like an open neighborhood of the origin of $\mathbb{C}$ (up to biholomorphism).² So it suffices to prove it for the stalk at the origin of $\mathbb{C}$ (of the sheaf of algebraic or analytic fucntions). In both sheaves, the valuation on the stalk is given by the order $n$ at the origin (the smallest integer $n$ for which the term of degree $n$ of the Laurent series doesn't vanish). In particular, the uniformizers (the functions of vanishing order $n=1$) are exactly the (algebraic or analytic) functions $f$ holomorphic in a neighborhood of the origin such that $f(0)=0$ but $f'(0)\neq 0$. Such a function is a local bilohomorphism at the origin by the inverse function theorem. Conversely, an (algebraic or analytic) function $f$ that is a local biholomorphism at $0$ and that preserves the origin has non-vanishing first derivative by the chain rule applied to $\operatorname{id}=f\circ f^{-1}$ (where the equation is on the level of germs).

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¹ In the French original book, “complex analytic variety” comes from variété analytique complexe. With this, I think Bourbaki means “complex manifold.” (There is no specific French word for “manifold.” Both English words “manifold” and “variety” translate to the French variété.)

² In the algebraic case, we are assuming that our variety is smooth at the point of interest (a complex algebraic curve is smooth at a point $P$ iff the stalk at $P$ is a DVR).