Genus of a curve My question is:

How to show that for all $g>0$ there exists an algebraic curve of genus $g$?
The intuition of this is simple, but I don't know how to show it.


Here is a perspective from working over the field of complex numbers. I'll assume you want to talk about nonsingular curves, in which case the genus unambiguously means the geometric genus $h^0(X,\Omega^1)$, i.e. dimension of the global sections of the holomorphic cotangent bundle. Let $X$ denote a Riemann surface. It is well known that the geometric genus of $X$ defined as above equals the topological genus of the underlying real surface.
Next, using Kodaira's Embedding Theorem (or simply any other way to prove that a Riemann surface holomorphically embeds in $\Bbb{P}^n$), you can holomorphically embed $X$ as an analytic subvariety of some $\Bbb{P}^n$ and by Chow's Theorem this realizes the image of $X$ as an algebraic variety in $\Bbb{P}^n$. The genus of this variety is then $g$.
Of course, a better construction in some sense is given by using hyperelliptic curves as mentioned in Somos' answer.
