Some questions about complex curves in $\mathbb CP^2$

Question

I would like to ask for some clarifications in the following questions about complex curves.

1. My first question is if I correctly understand what the complex curve in $\mathbb CP^2$ is. Is it only a set which is locally given as the nullset of some analytic function with non-zero gradient at each zero? If this is correct, what is the complex curve with boundary?
2. If I understand the definition of complex curve correctly, then the complex curve needn't to be a Riemann surface in general, it may have finite number of points in which it fails to be a smooth manifold. With these points excluded it is a Riemann surface. Right?
3. I know the definition of the normalization of a complex curve (A holomorphic map from a Riemann surface to this complex curve such that ...), but in one article the authors used in this definition meromorphic map instead of holomorphic one. What is the meromorphic map in this context? Does one call by a meromorphic map from a Riemann surface to $\mathbb C^2$ a holomorphic map from this Riemann surface to $\mathbb CP^2$?

Answer 1

1. no your understanding is a little bit off, while not being off. If we define a complex curve locally as the nullset of some analytic function with non-zero gradient by the implicit function theorem we will get that we can locally express one variable as analytic function of the other: $(z,w(z))$ on some open subset of the space. We can now define a complex atlas on our curve via the projection on the first variable, or saying that our curve is locally the graph on an analytic function and so it is a Riemann surface. To define a Riemann surface with boundary we use the same definition used for smooth manifolds: we ask for the atlas to contain homeomorphisms with the half-disk: $D_0^+=\{z\in\mathbb{C}\,:\,|z|<1\, ,\, \Im(z)\geq 0\}$.
2. At this point here you understand what you get. Note that we can define Riemann surfaces in many ways and in such ways we may have to be careful at some points as you mentioned, but there are many ways to eliminate such problems such as plugging holes.
3. And here when we talk of meromorphic maps between Riemann Surfaces what we mean is the underlying map in the local coordinates to be meromorphic. You need to talk about meromorphic maps or functions every time you deal with compact surfaces because of the Riemann-Roch Theorem.

Hope it was clear enough!