# What is an algebraic curve?

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).

• see here for an answer en.wikipedia.org/wiki/Algebraic_curve – Dr. Sonnhard Graubner Dec 12 '16 at 11:22
• Thano you for your answer Graubner, though, as you do (I guess :)) wikipedia was the first thing I've checked. Because it is very vague in my head, can you please give me a better guidance? – user321268 Dec 12 '16 at 11:26
• @mayer_vietoris I'm confused by your question, since what happens usually is that what the author means by an algebraic curve depends heavily on context. Armando j18eos is right that the most general thing that someone might call an algebraic curve is a scheme of Krull dimension one, but your references would probably define an algebraic curve as a "proper integral scheme of Krull dimension one over an algebraically closed field $k$," and Miranda in particular would probably demand that $k = \mathbf{C}$. – Takumi Murayama Dec 12 '16 at 19:15
• @TakumiMurayama thank you for your comment! I'm confused though by it. If you read the comment below his answer I think that you will understand why he edited the answer. I've written that I do understand the relation between the context and the definition, I'm just seeking for the general definition. And in Hartshorne's book seems that is exactly the one you mentioned. Armando's answer justifies more or less what you written. That's the story! – user321268 Dec 12 '16 at 22:10
• @TakumiMurayama Also you are right!, for example: a compact Riemann surface can be view as an integral, proper, separated scheme of finite type over $\operatorname{Spec}\mathbb{C}$ of (Krull) dimension $1$. But because mayer_vietoris cites Miranda, Hartshorne and Fulton's books: I supposed that he looked for a general definition of algebraic curve. – Armando j18eos Dec 13 '16 at 9:58

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 $\mathbb{K}$) of pure (Krull) dimension $1$;

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

Scheme approach.

Let $X$ be a scheme: it is an algebraic curve if it is of 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.)

• Do you know anything about the scheme-theoritic approach? How does this goes through the above definition that you gave me? – user321268 Dec 12 '16 at 11:27
• Ah!, I did not undestand your necessity of scheme-theoretic approach: no problem; I modify my answer. ;) – Armando j18eos Dec 12 '16 at 11:32
• just out of curiosity, why the downvote here? – user321268 Dec 12 '16 at 12:10
• By the way, thanks for editing your answer! – user321268 Dec 12 '16 at 12:11
• I don't know: I can't vote my answers\questions; if my answer is useful: can you upvote it? – Armando j18eos Dec 12 '16 at 12:18

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).