On objects of study of Classical Algebraic Geometry

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A subject of Algebraic Geometry (AG) is one of central branches of Mathematics, yet I have almost no knowledge of objects and spaces that algebraic geometers study. The mere look at the name suggests some kind of relationship between algebra and geometry. So does algebra lends its hand to solve geometric problems, or is it other way around. I did not know. Realising my ignorance of the matter, I decided to rectify situation I decided to read some Wikipedia articles along with some introductory books on the subject matter. In this post I intend to share fruits of my explorations.

 How AG was born?
AG was born out of desire to solve systems of polynomial equations. Suppose we have a system
\[
f_i(X_1,\ldots,X_n) = 0, \quad i=1,\ldots, m
\]
where each $f_i$ is a polynomial of $n$ variables with coefficients drawn from some field $k$, which can succinctly be written as $f_i\in k[X_1,\ldots,X_n]$.

What to do next?
Mostly it is pretty difficult to solve such systems. However, instead of of exact solutions, we could look at collection of all solutions. In general the collection of solutions to above system forms some (continuous?) structure, which is called algebraic set. If an algebraic set can not be written as a union of two smaller algebraic sets it is called irreducible, or algebraic variety. These are central objects of study for classical AG. One could also look at maps between algebraic sets. Let $A_1\subset k^n$ and $A_2\subset k^m$ be algebraic sets and $f_1,\ldots, f_m$ be polynomials. One then could form a function $f = (f_1,\ldots,f_m) : A_1\to k^m$. If range of $f$ happens to be within $A_2$, then we have polynomial or regular map between two algebraic sets. One definition of classical AG can be given as study of algebraic sets and regular maps between them.

 Where it led?
In the middle of XX century there was a shift of perspective on objects that AG studies. People wanted to study algebraic sets intrinsically, rather than as parts of some external space. This required some machinery such as sheaves and schemes. These days AG became quite abstract study of new interpretations of algebraic sets.

What is AG all about at the end?
There are might be many opinions. For now, I think of it as abstract study of zeroes of polynomial equations. 

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