How to construct the "best" resolution for a multi projective space?

Apologies for a slightly vague question. Consider the involution of $\mathbb{CP}^n$, given by $[z_{0}: \ldots : z_{n}] \mapsto [-z_{0}: \ldots : z_{n}]$.

The quotient $X$ is a projective variety with one isolated, normal singularity. Hence some resolution should exist by Hironaka's theorem.

Question:1. Is there a "simplest resolution" of X with some nice properties?

Solution 1:

Question: "The quotient $X$ is a projective variety with one isolated, normal singularity. Hence some resolution should exist by Hironaka's theorem. Question:1. Is there a "simplest resolution" of X with some nice properties?"

Answer: This is the weighted projective space $\mathbb{P}^n/(\mathbb{Z}/(2)) :=\mathbb{P}(1,..,1,2)$ associated to the integers $1,1,..,1,2$, and this is a much studied variety. You may find some references in Harris "Algebraic geometry - a first course". Any such is isomorphic to a cone $\tilde{X}$ over a Veronese variety $X:=v_2(\mathbb{P}^n)$ (Harris, Ex10.28). The blow up of $\tilde{X}$ at $0$ is

$$Bl_0(\tilde{X}) \cong \mathbb{V}(\mathcal{O}_X(-1)).$$

You must check if this is a resolution. If not I suspect this is a much studied topic and that something is known.