Segre embedding and its line bundle

The generating sections. On the affine product of charts $U_i{\times }_S{V}_j$ the pullbacks ${\mathrm{p}\mathrm{r}}_1^{\ast }{x}_i$ and ${\mathrm{p}\mathrm{r}}_2^{\ast }{y}_j$ are frames of ${\mathrm{p}\mathrm{r}}_1^{\ast }\mathcal{O}(1)$ and ${\mathrm{p}\mathrm{r}}_2^{\ast }\mathcal{O}(1)$, so their tensor product $z_{ij}$ is a frame of $L$ on $U_i{\times }_S{V}_j$ by [F2]. Since the products $U_i{\times }_S{V}_j$ cover $P$ by [F1], the sections $z_{ij}$ generate $L$, and $L$ is invertible; this is claim (1).

The chartwise map. Fix $i,j$ and an affine open $T=\mathrm{Spec}A\subseteq S$. By [F1] the source over $T$ has chart $U_i^T{\times }_T{V}_j^T=\mathrm{Spec}A[u_a,v_b]$ and the target has chart $W_{ij}^T=\mathrm{Spec}A[w_{ab}]$, with $u_i=v_j=w_{ij}=1$. The section $z_{ij}$ is a frame of $L$ exactly on $U_i{\times }_S{V}_j$, so [F3] gives ${\sigma }^{-1}(W_{ij}^T)=U_i^T{\times }_T{V}_j^T$ and the induced $A$-algebra map is $A[w_{ab}:(a,b)\mathrm{\ne }(i,j)]⟶A[u_a:a\mathrm{\ne }i,v_b:b\mathrm{\ne }j],w_{ab}⟼u_a{v}_b\mathrm{.}$ It is surjective because $w_{aj}\mapsto u_a$ and $w_{ib}\mapsto v_b$. Every dehomogenised $2\times 2$ minor $w_{ab}{w}_{cd}-w_{ad}{w}_{cb}$ maps to zero. Conversely, in the quotient $Q_{ij}$ by those minors, the relation involving rows $a,i$ and columns $b,j$ gives $w_{ab}=w_{aj}{w}_{ib}$; hence $u_a\mapsto w_{aj}$ and $v_b\mapsto w_{ib}$ define an inverse $A$-algebra map $A[u_a,v_b]\to Q_{ij}$. Thus the kernel is generated by the dehomogenised minors and the source chart is isomorphic to their closed subscheme in $W_{ij}^T$. This calculation holds for every commutative base ring $A$.

The morphism. By [F3] applied to the invertible sheaf $L$ and its generating sections $z_{ij}$ there is a unique $S$-morphism $\sigma :P\to {\mathbb{P}}_S^N$ with ${\sigma }^{\ast }\mathcal{O}(1)\cong L$, the index set being $((i,j))$ with $N+1=(m+1)(n+1)$ elements; this is the Segre embedding and gives claim (2), including the pullback identity.

The image is the minors subscheme. On every affine base open $T=\mathrm{Spec}A\subseteq S$, the homogeneous minors define a closed subscheme of ${\mathbb{P}}_T^N$ by [F4]. Their chart ideals are exactly the kernels calculated in step 1.2. The minors have integral coefficients, so these local closed subschemes agree under restriction to overlaps of base opens and glue to a closed subscheme $Z\hookrightarrow {\mathbb{P}}_S^N$. The chart isomorphisms of step 1.2 are compatible because each is induced by the same morphism $\sigma$ and the same minor equations; they show that $P\to Z$ is an isomorphism on the open cover $\{U_i^T{\times }_T{V}_j^T\}$, hence globally. Therefore $\sigma$ is a closed immersion and its scheme-theoretic image is exactly $Z$, the subscheme defined by the rank-one $2\times 2$ minors.

Conclusion. Step 1.1 establishes the generation of $L$ by the products $z_{ij}$, step 2.1 produces the morphism with ${\sigma }^{\ast }\mathcal{O}(1)\cong L$, and steps 1.2 and 2.2 identify the image with the minors subscheme, so $\sigma$ is the Segre embedding. The construction is uniform in $m,n\ge 0$, including $m=0$ or $n=0$: then one family of coordinates is empty, the products are just the other coordinates, and the same chart computation gives an isomorphism onto a linear subspace. The equalities use no field hypothesis and no choice beyond [A1], inherited from the projective-space constructions. [A1, step 1.1, step 1.2, step 2.1, step 2.2, cases: m=0 or n=0] \qed