blowing up $\mathbb{A}^n$ at a point.

Question

I would like to get an informal and very visualizable explaination of the concept blowing up. I read " In blowing up $\mathbb{A}^n$ at a point $p$, the idea is to leave $\mathbb{A}^n$ unaltered except at the point $p$, which is replaced by the set of all lines through $p$, a copy of $\mathbb{P}^{n-1}$". It will be nice anyone then mathematically build up the definition from the informal discussion.and its consequenses.

Answer 1

The question is: What to do with blow-ups. Imho their main application is for resolving singularities.

If you have a singular curve, for example $x^2=y^3$ in the space $\mathbb{A}^2$, you can blow up the plane at the singular point of the curve, i.e. $(0,0)$ to resolve the singularity, which means that the curve does not have a singularity anymore. This works in more general case.

Another example: If you again blow up the space in $(0,0)$ with a $\mathbb{P}^1$, then you can "see" for a line not only if it goes to the point $(0,0)$, but also its derivative in this point. That means for example, that non-equal lines which meat in $(0,0)$ in $\mathbb{A}^2$ meet no longer in the blow up, since they have not the same derivative.

A formal definition of the blow up see Hartshorne, Algebraic Geometry, page 28ff. There is also a NICE drawing, the best one I know. You can also read the abstract definition, which requires far more algebraic geometry then the affine one.