Segre embedding of $\mathbb{P}^n\times \mathbb{P}^m\times\mathbb{P}^q$ How could I define the segre embedding of three projective spaces? Usually in references only the Segre embedding of two projective spaces are defined:
$\mathbb{P}^n \times \mathbb{P}^m \rightarrow \mathbb{P}^N$, $N=(n+1) (m+1)-1$ where the coordinates of $\mathbb{P}^N$ are $[u_{00},u_{01},...,u_{n+1,m+1}]=[x_0y_0,x_0y_1,...,x_{n+1}y_{m+1}]$ and the $u_{ij}$ satisfies $u_{ij}u_{kl}-u_{il}u_{kj}=0$.
Thanks in advance.
 A: The general Segre embedding discussed in EGA I$_{\text{new}}$, 9.8 (which I quite like because it is coordinate-free and transparent) can be generalized without effort as follows: Let $S$ be a scheme, and consider quasi-coherent sheaves $\mathcal{E}_1,\dotsc,\mathcal{E}_n$ on $S$. Then there is a closed immersion of $S$-schemes
$$\mathbb{P}(\mathcal{E}_1) \times_S \dotsc \times_S \mathbb{P}(\mathcal{E}_n) \hookrightarrow \mathbb{P}(\mathcal{E}_1 \otimes_{\mathcal{O}_S} \dotsc \otimes_{\mathcal{O}_S} \mathcal{E}_n),$$
given on $X$-valued points by $((\mathcal{L}_1,s_1),\dotsc,(\mathcal{L}_n,s_n)) \mapsto (\mathcal{L}_1 \otimes \dotsc \otimes \mathcal{L}_n,s_1 \otimes \dotsc \otimes s_n)$ (where $\mathcal{L}_i$ is invertible on $X$, $p : X \to S$ is the structural morphism and $s_i : p^* \mathcal{E}_i \to \mathcal{L}_i$ is an epimorphism). In particular, for all $k_1,\dotsc,k_n \in \mathbb{N}$ there is a closed immersion
$$\mathbb{P}^{k_1}_S \times_S \dotsc \times_S \mathbb{P}^{k_n}_S \hookrightarrow \mathbb{P}^{(k_1+1) \cdot \dotsc \cdot (k_n+1)-1}_S.$$
It corresponds to the graded quasi-coherent ideal $\subseteq \mathcal{O}_S[(x_{i_1,\dotsc,i_n})_{0 \leq i_p \leq k_p}]$ generated by the relations $x_{i_1,\dotsc,i_{p-1},a,i_{p+1},\dotsc,i_n} \cdot x_{i_1,\dotsc,i_{p-1},b,i_{p+1},\dotsc,i_n} = x_{i_1,\dotsc,i_{p-1},b,i_{p+1},\dotsc,i_n} \cdot x_{i_1,\dotsc,i_{p-1},a,i_{p+1},\dotsc,i_n}$.
A: You could do the usual Segre embedding twice. That is, first embed $\mathbb{P}^n \times \mathbb{P}^m  \times \mathbb{P}^q \to \mathbb{P}^{N} \times \mathbb{P}^q$, with $N=(n+1)(m+1)-1$ and then embed $\mathbb{P}^n \times \mathbb{P}^q \to \mathbb{P}^M$, with $M=(N+1)(q+1)-1=(n+1)(m+1)(q+1)-1$. 
