Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
    
How is what you quoted not visualisable? Take, for example, the affine plane, and blow it up at the origin to obtain the Möbius strip. –  Zhen Lin May 24 '12 at 18:16

1 Answer 1

up vote 1 down vote accepted

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.

share|improve this answer
    
thank you dear sir. –  Une Femme Douce May 25 '12 at 11:10
    
You're welcome. –  sebigu May 26 '12 at 17:29

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.