I am taking an introductive course in (real) algebraic geometry and I got stuck at some basic exercises.
They regard affine and (real) projective varieties, as follows:
Prove that the punctured projective space, $\mathbb{P}^n - \{x\}$ is neither projective, nor quasi-affine, when $n \geq 2$.
Prove that $\mathbb{P}^1 \times \mathbb{P}^1$ and $\mathbb{P}^2$ are birationally equivalent, but not isomorphic.
Now, to clarify some things. We study more or less based on I. R. Shafarevich - Basic Algebraic Geometry (or something like that).
Our definitions of the notions involved are:
X is quasi-affine if it is a Zariski open set in an affine variety
$\mathbb{P}^n$ is supposed to mean $\mathbb{P}^n(k)$, for an algebraically closed field $k$ and is the space of "directions" in $\mathbb{A}^{n+1}-\{0\}$. Specifically, $\mathbb{P}^n=\mathbb{A}^n/\sim$, where $x\sim y \Leftrightarrow \exists \lambda \in k, \ s.t. x=\lambda\cdot y$.
birational equivalence means that there exist rational functions from either to the other, whose composite is the identity (either way), but that these need not be defined everywhere
isomorphism is usually treated in terms of isomorphic fields under the isomorphism induced by the initial morphism.
Note that the course is absolutely basic, without (co)homology, schemes, sheaves etc. Just the basics that I listed, along with Krull dimension.
Thank you.