I'm reading the book of Miranda (Algebraic Curves and Riemann Surfaces), and in the text he give the following definition:
Let $X$ be a compact Riemann surface, then $X$ is a algebraic curve if, $\mathcal{M}(X)$ (Meromorphic Functions on $X$) separates points on $X$ and tangents.
- Separates Tangents means that, for every $p\in X$ there is a meromorphic function $f$ such that $mult_p(f)=1$
Well, there is a classical definition of algebraic curve, that is, it's a algebraic variety of dimension one, how to show that this two definition are equivalent?