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.