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What is the difference between the word local coordinate system and unifomization ?

I know the notion of local coordinate sytem, well. However, I cannot understand the unifomization.

Example.

Let $C$ be a non-singular curve and denote by $t$ the local unifomization varibale

In this context $C$ is not a disc.

Example

Let $\Delta$ be a curve . We denote by $t$ the local unifomization variable on $\Delta$.

In this context $C$ is a disc.

Why the above two examples use the word unifomization instead of local coordinate?

Jean Billie
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1 Answers1

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It is the difference between local and global. The sentence

Let $C$ be a non-singular curve and denote by $t\in \Delta$ the local unifomization variable

is very sloppy and one should refrain from using this language. It should be something like:

Let $C$ be a connected Riemann surface of hyperbolic type (e.g. a compact Riemann surface of genus $\ge 2$). Let $f: \Delta\to C$ be the holomorphic universal covering map, where $\Delta=\{t\in {\mathbb C}: |t|<1\}$ is the unit disk. By abusing the terminology, we regard $t$ as the uniformization variable on $C$.

See this Wikipedia article for more details.

Edit. the answer also depends on the context. For instance, in Algebraic Geometry, local uniformizer is used for a generator of the local ring of an algebraic curve at the given point, see this question. From the view point of complex analysis or differential geometry, such local uniformizer is the same thing as a local chart once restricted to a suitable open subset of the Riemann surface. In many cases, the word "local" is dropped.

Moishe Kohan
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  • I would like to point out that In algebraic geometry, the first is phrasing is not "very sloppy" - the local ring of any curve at a smooth closed point $x$ is a DVR and thus has a uniformizer $\pi$. Locally, translates of this function $\pi-\pi(x_0)$ for $x_0$ another closed point become uniformizers for the local rings at those points. This language is relatively standard, see for instance here for more discussion. – KReiser Nov 20 '19 at 23:35
  • @KReiser: Oh, you are right, somehow I did not think about the AG aspect. I will modify my answer. – Moishe Kohan Nov 21 '19 at 17:00