I would like to know some global geometry of the blowup of projective spaces.


Question: Let $Z\subseteq \mathbb{P}^n$ be a subvariety. How can I embed $BL\_Z\mathbb{P}^n$ in a projective space or product of projective spaces?


Examples:


1. Let $Z\subseteq \mathbb{P}^4$ be a union of two disjoint lines, then $BL\_Z\mathbb{P}^4\subseteq \mathbb{P}^2\times\mathbb{P}^2\times\mathbb{P}^1$, which is the blowup of $\mathbb{P}^2\times\mathbb{P}^2$ along a $\mathbb{P}^1\times\mathbb{P}^1$.
2. Let $Z\subseteq \mathbb{P}^3$ be a twisted cubic curve, then $BL\_Z\mathbb{P}^3\subseteq \mathbb{P}^3\_Y\times \mathbb{P}^2\_X$ is defined by
$$X\_0Y\_0+X\_1Y\_1+X\_2Y\_2=X\_0Y\_1+X\_1Y\_2+X\_2Y\_3=0. $$


I don't know whether there is a standard way to obtain these constructions or not. Thanks very much for your answer.



Blowing up the projective space along a subvariety
Let $I\_Z$ be the ideal sheaf of $Z$. If the sheaf $I\_Z(k)$ is globally generated (i.e., if hypersurfaces of degree $k$ cut out $Z$ as a scheme) and $m := h^0(I\_Z(k)) - 1$, then
$$
\mathrm{Bl}\_Z(\mathbb{P}^n) \subset \mathbb{P}^n \times \mathbb{P}^m.
$$


