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

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 $\mathbb{P}(V)$?

Thanks.

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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

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.
Am I right that $\dim L=1$ and $L$ generated by $v$? – Aspirin Sep 29 '11 at 5:27