Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

In context:

...Let's define "blow" (perhaps, I use incorrect terminology) of affine space $V=\mathbb{A}^n$ in point $P=(0,\ldots,0)$. Consider subset $$\widetilde{V}=\{(v, L)|v\in L\}\subset V\times \mathbb{P}(V),$$ and define projection $\sigma : \widetilde{V}\to V~~$ as $(v, L)\longmapsto v$. ...

So, What is $\mathbb{P}(V)$?


share|cite|improve this question
It's called a ‘blowup’. $\mathbb{P}(V)$ is the projective $(n-1)$-space obtained by considering the $1$-dimensional subspaces of $V$, considered as a vector space. – Zhen Lin Sep 29 '11 at 2:53
up vote 5 down vote accepted

It seems like you're trying to construct a blow up. Here $\mathbf P(V)$ is the projective space of lines through the origin in the vector space $V$. You can give this the structure of a variety by choosing a basis (and it seems like you already have), which will give you an isomorphism of $\mathbf{P}(V)$ with some $\mathbf{P}^{n - 1}$.

There are many things to learn about this last space. To begin with, the closed sets in $\mathbf{P}^n$ are the zero sets of collections of homogeneous polynomials. What sort of functions will be attached to these sets? How do these sets relate to varieties in $\mathbf{A}^n$? What is that product operation? This should all be explained in any textbook on basic algebraic geometry. There are some nice pictures in Harris's book, for example.

share|cite|improve this answer
Am I right that $\dim L=1$ and $L$ generated by $v$? – Aspirin Sep 29 '11 at 5:27
@Aspirin That seems right. – Dylan Moreland Sep 29 '11 at 12:29

Your Answer


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.