blow-up and embedding Let $P$ be the weighted projective space $\mathbb{P}(1,1,2,2,2)$, let $\hat{P}\to P$ be the blow-up along the singular locus $\\{x_0=x_1=0\\}$ in $P$. Using the line bundle $\mathcal{O}(2)$, one can make the embedding $P\hookrightarrow \mathbb{P}^5$ as a (singular) quadratic hypersurface. My question is if we can embed $\hat P$ into a natural space $X$ as a smooth hypersurface such that the following diagram commutes: \begin{array}{cc} \hat P&\hookrightarrow & X \\\ \downarrow & \quad & \downarrow\\\ P&\hookrightarrow & \mathbb{P}^5 \end{array} My guess is that $X$ should be the blow-up of $\mathbb{P}^5$ along $x_0=x_1=x_2=0$, is this correct? For those who are familiar with toric geometry, how can we describe the toric fan of $X$ and which divisor class in $X$ would correspond to $\hat P$? Thanks!

Indeed, the blowup of the singular quadric embeds into the blowup $X$ of the projective space, that can be alternatively described as $$ X = \mathbb{P}_{\mathbb{P}^2}(\mathcal{O} \oplus \mathcal{O} \oplus \mathcal{O} \oplus \mathcal{O}(-1)). $$