So I recently started teaching myself about weighted projective spaces from Harris' Algebraic geometry. It was going well until I came across this exercise, which has me stumped:
"Show that any weighted projective space of the form $\mathbb{P}(1, \dots, 1, k, \dots, k)$ is isomorphic to a cone over a Veronese variety $v_k(\mathbb{P})^l$ (and in particular $\mathbb{P}(1, k, \dots, k) \cong \mathbb{P}^n$)."
I feel like I understand the discussion leading up to this exercise (on p. 128), but I don't see how to prove the statement. Help would be appreciated
Let's take coordinates $x_1, \dots, x_r, y_1, y_s$ where $x_i$ have degree $1$ and $y_j$ degree $k$. The homogenous functions of degree $k$ are given by multi-monomials $X^I$ where $X^I = x_1^{i_1} \dots x_r^{i_r}$ and $i_1 + \dots + i_r = k$. They are $s$ more, namely $y_1, \dots, y_s$.
It follows that the map $\Bbb P(1, \dots, k) \to \Bbb P^N, (x_1, \dots, y_s) \mapsto (x_1^k, \dots, x_r^k, y_1, y_2, \dots, y_s) $ is an embedding. But this map is exactly the Veronese embedding $v_k : \Bbb P^r \to \Bbb P^M$ on the first components. Moreover, the image is stable by scalar multiplication in the $y$-coordinates, i.e as expected the image is the cone over the $k$-Veronese embedding.
For surfaces, this is particulary nice as the blow-up at the vertex of the cone gives the well-known Hirzerbruch surface $F_n \cong \Bbb P(\mathcal O_{P^1}(k)\oplus \mathcal O_{P^1})$. This can be seen using toric geometry.