Why is $\mathbb A^1$ not projective? i.e. why is it not isomorphic to a projective variety?
I'm drawing a complete blank here...
Any help would be appreciated. Thanks!
Why is $\mathbb A^1$ not projective? i.e. why is it not isomorphic to a projective variety?
I'm drawing a complete blank here...
Any help would be appreciated. Thanks!
The ring of regular functions of a projective variety is the underlying field, however the ring of regular functions of $\mathbb{A}^{1}$ is $k[t]$. In fact the only projective and affine varieties are the singletons. This can be shown using the bijection between:
$\operatorname{Hom}(X,Y) \leftrightarrow \operatorname{Hom}(A(Y),\mathcal{O}(X))$, where $X$ is any variety and $Y$ an affine variety.