In several algebraic geometry texts, I have seen projective space $\mathbb P^n$ (over a field $k$) be defined in two ways for $n\ge0$:
- The set of one-dimensional subspaces of $\mathbb A^{n+1}$.
- The quotient set $\left(\mathbb A^{n+1}\setminus\{0\}\right)/\sim$, where two points $p,q\in A^{n+1}\setminus\{0\}$ are considered to be equivalent if one is a scalar multiple of the other.
Although many authors assert that these definitions of $\mathbb P^n$ are equivalent, it seems to me that technically they are not. The elements in $\mathbb P^{n}$ in (1) contain the origin $0$, whereas the elements in $\mathbb P^{n}$ in (2) do not.
Am I correct in thinking this? If so, does it matter which definition we adopt in practice?
The identification from definition $2$ to definition $1$ sends $x\in (\mathbb{A}^{n+1}\setminus \{0\})/\sim$ to the vector space $\langle{x}\rangle\subset \mathbb{A}^{n+1}$ it generates; the identification in the other direction sends a $1$-dimensional subspace $V\subset \mathbb{A}^{n+1}$ to a basis element of it, which is well-defined modulo $\sim$. (It may be more intuitive over a field like $\mathbb{R}$ or $\mathbb{C}$, where you can also require $|x| = 1$.) The asymmetry comes from the fact that $0$ does not generate the line it's on. But that's fine: we're treating $\mathbb{P}^n$ as an abstract space of its own, not trying to embed it in $\mathbb{A}^{n+1}$.