morphism to projective space homogeneous coordinates

Let $X\subseteq {\mathbf{P}}_k^n$ be a classical projective variety. A map $\phi :X\to {\mathbf{P}}_k^m$ is a projective morphism if there is an open cover $X={\bigcup }_{\alpha }{U}_{\alpha }$ such that, for every $\alpha$, homogeneous polynomials $F_{\alpha ,0},\dots ,F_{\alpha ,m}$ of one common degree have no common zero on $U_{\alpha }$ and $\phi (p)=[F_{\alpha ,0}(p):\cdots :F_{\alpha ,m}(p)](p\in U_{\alpha })\mathrm{.}$ On each $U_{\alpha }\cap \{F_{\alpha ,i}\mathrm{\ne }0\}$ the target-chart coordinates are the regular functions $F_{\alpha ,j}\mathrm{/}{F}_{\alpha ,i}$. The local tuples define one map precisely when, for every $p\in U_{\alpha }\cap U_{\beta }$, $F_{\alpha ,i}(p)F_{\beta ,j}(p)=F_{\alpha ,j}(p)F_{\beta ,i}(p)\text{for all }i,j;$ equivalently, they give the same projective point there. Multiplying every entry of one tuple by a common locally nonvanishing regular factor changes no projective point on its domain; the resulting representatives need not themselves be homogeneous polynomials.

This local description agrees with regular maps of classical varieties. Each target standard chart is affine space (standard projective opens are affine spaces), and the target-chart coordinates of a tuple are the regular equal-degree fractions $F_{\alpha ,j}\mathrm{/}{F}_{\alpha ,i}$ where $F_{\alpha ,i}\mathrm{\ne }0$ (regular function projective variety). The chart criterion for regular maps therefore applies (Classical algebraic prevarieties, regular maps, and varieties). Conversely, near a point mapped into the $i$th target chart, each of its finitely many coordinate functions is a regular degree-zero fraction $G_j\mathrm{/}{H}_j$ with $H_j$ nonzero at that point. Shrink to the common nonvanishing domain of the $H_j$, put $H={\prod }_{j\mathrm{\ne }i}{H}_j$, and set $F_i=H$ and $F_j=G_j{\prod }_{\mathrm{\ell }\mathrm{\ne }i,j}{H}_{\mathrm{\ell }}$. These are homogeneous polynomials of one common degree, $F_i$ is nonzero on the shrunken domain, and $F_j\mathrm{/}{F}_i=G_j\mathrm{/}{H}_j$. For $m=0$, use the one-entry tuple $(1)$. Taking all such eligible local domains forms an open cover without choosing one tuple at every point.