Introduction to Algebraic
Blow up of y2=x3
In a sentence, algebraic geometry is the study of solutions to algebraic
equations. People learning it for the first time, would see a lot
of algebra, but not much geometry. But it is there. The picture above
depicts a resolution of the singular curve
y2=x3. This can be accomplished by taking
integral closures on the algebra side, or by doing a blow up.
This curve drawn in black is resolved by blowing up the plane at the
origin (replacing the yellow by blue surface) and taking the closure of
the preimage away from the origin (in red).
For a really elementary introduction, go
For a more serious introduction, you can get
my notes on
basic algebraic geometry.
This is sort of a "prequel" to Hartshorne. (It's a 340K PDF file)
"Algebraic geometry over the complex numbers" covers more: sheaf theory, cohomology and Hodge theory.
(It's a 1.3 MB PDF file).