What is an algebraic curve?

Question

I know that this question is at least "ridiculous" if you have internet access, though maybe this is the problem as it seems (many times, at least in my head). I found many books (or notes), like Miranda's Riemann Surface and Algebraic Curves, Hartshorne's Algebraic Geometry, Fulton's Algebraic Curves and so on, each one with a different definition.

Apparently, depends on the context that you're investigating them, however, because I'm not an expert and have no clue what's the right definition, or the most general one, can you give me please the most general one that comprises all the others (I'm saying that because the dimension plays important role here, for instance in dimension one over $\mathbb{C}$ we have another definition sometimes, as it admits a "Riemann surface" structure and a couple of notes define them alternatively as in this way).

Answer 1

Classical approach.

For simplicity, let $\mathbb{K}$ be a field: an algebraic curve $X$ in $\mathbb{A}^n_{\mathbb{K}}$ (the affine $n$-dimensional space over $\mathbb{K}$) is an algebraic set $X$ (the zero locus of a finite family of polynomials with coefficients in $\mathbb{K}$) which (Krull) dimension is purely $1$;

what do I mean for "$X$ has pure (Krull) dimension $1$"? I mean that the unique closed, irreducible, proper and non-empty subsets of the irreducible components of $X$ are the points of $X$; equivalently the coordinates ring $\mathbb{K}[X]$ has Krull dimension $1$.

Scheme approach.

Let $X$ be a scheme: it is an algebraic curve if it has pure (Krull) dimension $1$;

for exact, this is equivalent to the existence of an affine open covering $\{\operatorname{Spec}R_i\}_{i\in I}$ of $X$ such that any $\operatorname{Spec}R_i$ has (Krull) dimension $1$; that is the (Krull) dimension of any $R_i$ is $1$. (See Vakil FOAG, December 29 2015 version, definition 11.1.3 and exercise 11.1.B.)

Answer 2

An algebraic plane curve over an algebraically closed field $k$ is by definition the zero set of a polynomial equation in one variable. For example $y= x^3-ax+b$ is such an example for $x\in k$. This is not a very rigorous definition but it shoulds give an idea of what an algebraic curve is. Note that many people use the words algebraic curves and Riemann surface interchangeably which is slightly incorrect. A Riemann surface (and here we are interested about compact ones) carries less information than a smooth algebraic curve since the later has additional structure about its embedding (e.g. a compact Riemann surface together with a line bundle).