I've read the wikipedia article but I don't know what an affine plane is and the definition/example did not seem clear. What I know is that in the 1880s mathematicians like Hilbert, Kronecker, Lasky and Macauley were responsible for the development of the algebraic variety concept. This is for a history of mathematics paper that I am working on.
[Math] Could someone give me an example of an algebraic variety and explain what it is
abstract-algebraalgebraic-geometrymath-history
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Victor Katz is renowned for his writing and research in the History of Mathematics. I read an earlier edition of the text History of Mathematics while taking an undergraduate course in the History of Math during a Spring term (it was the required text for the class.) It is an excellent book. We couldn't cover the entire text over one semester, so I persisted in reading it to completion over the summer which followed.
You can go as far back as you'd like (it goes very far back in history!), or pick up where your interest is piqued.
If you can't take the course, for credit, or as an "auditor", I'd recommend this book for your library. It is a good complement to "doing" hard-core math. That's not to say that it's necessarily "easy", because it invites you to engage in mathematics using only the tools available at a given point in history and in a given culture. At any rate, I found the text to be very engaging, it helped me appreciate the field of mathematics more than I ever thought I could, and it has served me well as a reference, too.
I just noticed that there is a "brief" version of Katz's History of Mathematics which might not be as overwhelming, and likely highlights the best and the biggest breakthroughs in mathematics, as they developed over time.
You might also want to peruse the following list: Resources: History of Mathematics, to find some helpful recommendations.
In short: "varieties are nice schemes over fields". In particular, the study of varieties is a subscase of the study of schemes. A variety (depending on who you ask) over a field $k$ is a finite type separated (usually reduced, sometimes geometrically integral!) scheme over $\text{spec}(k)$.
The reason that the two fields look so different, is purely in language. When one studies varieties on their own, it's likely that one is using a source that uses the classical (read "pre-Grothendieck") language. This language suffices to talk about such well-behaved schemes, but fails to distinguish between the more sophisticated properties of general schemes (e.g. a point $\text{spec}(k)$ and a 'fuzzy point' $\text{spec}(k[x]/(x^m))$).
That said, much of general scheme theory actually can be reduced to the study of varieties. This is because, for a sufficiently nice schemes (most that we encounter) we can think about them as being 'fibered' over varieties. By this, I mean that if $X$ is a scheme, with an equipped map of schemes $X\to S$ which satisfies some mild properties, then all of the fibers $X_s$ for $s\in S$ are varieties over $k(s)$. This allows one to think about general schemes as being 'families of varieties indexed by other schemes'.
It is a mistake to confuse 'variety land' with 'commutative algebra land'. What is more likely happening, is that when you are looking at texts on varieties, they happen to be focusing (perhaps at the beginning) on affine varieties. This is why the following correct identification can be confused with the above incorrect one: 'affine scheme land' IS 'commutative algebra land'. This holds true for schemes over arbitrary bases, and is made precise by an (anti)equivalence of categories between affine schemes (over an affine base) and algebras over that affine base.
In terms of learning it, historically varieties came before fields. They are also the most tenable examples of schemes (besides, perhaps, number rings), and so are useful to have in your back pocket. The classical language, what you would most likely learn varieties in, has the advantage of being simpler, and perhaps easier to see the geometry. Unfortunately, it is often times sloppy, and hard to analyze the fine structure results that you'd need.
In this way, I think that a cursory reading of a book on varieties, and almost more helpful a book on complex manifolds, is a helpful first step to studying schemes. That said, there will be no technical loss (only a loss of intuition) by starting directly with schemes.
Good luck, and feel free to ask a follow up question.
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I'm not sure why you mention the word "affine plane" since it is not used in the Wikipedia article about algebraic varieties.
In any case, it's not as hard as it might seem. I have not studied algebraic geometry yet but I have come across one type of variety in one of my other courses. As you can see in the Wikipedia article, there are two types of algebraic varieties, affine and projective. Let's ignore projective varieties for now and first understand what an affine variety is:
You start with a ring of polynomials in $n$ variables $\mathbb F [x_1, \dots, x_n]$ where $\mathbb F$ is any algebraically closed field. Let $S \subset \mathbb F [x_1, \dots, x_n]$ be any set of polynomials. Now you can define the affine variety associated to $S$ as $$ V(S) = \{ \vec{x} \in \mathbb F^n \mid p(\vec{x}) = 0 \text{ for all } p \in S \}$$
So, an affine algebraic variety is a special subset of $\mathbb F^n$, namely, the set of all points that make all polynomials in $S$ vanish.
If $F= \mathbb C$ and $S = \{ p_n(z) = z^n + 1 \mid n \in \mathbb N \}$ then $V(S)$ would be all the roots of unit on the circle in the complex plane.