[proofplan] We cover the source by the opens on which one basis element $F_i$ does not vanish. On such an open, the projective target coordinate ratios are represented by the degree-zero regular functions obtained from $F_j$ divided by $F_i$ in the localized homogeneous coordinate ring. These local affine-chart maps agree on overlaps by the transition formulas for projective affine charts, so they glue to a global morphism. The complete degree-$d$ linear system gives exactly the usual monomial coordinate formula for the Veronese map. [/proofplan]

[step:Cover $\mathbb P^n_k$ by the nonvanishing loci of the basis sections] Let \begin{align*} S:=k[x_0,\dots,x_n] \end{align*} be the homogeneous coordinate ring of $\mathbb P^n_k$. For each index $i\in\{0,\dots,r\}$, define the distinguished open subset \begin{align*} U_i:=D_+(F_i)\subseteq \mathbb P^n_k. \end{align*} By definition, a geometric point $x\in \mathbb P^n_k$ belongs to $U_i$ precisely when $F_i(x)\neq 0$. Since the linear system is base-point-free, for every geometric point $x\in \mathbb P^n_k$ at least one of the values $F_0(x),\dots,F_r(x)$ is nonzero. If the closed subset $\mathbb P^n_k\setminus\bigcup_{i=0}^r U_i$ were nonempty, then, because it is a finite type $k$-scheme, it would have a geometric point after extension to an [algebraic closure](/page/Algebraic%20Closure) of a residue field. That geometric point would be a common zero of $F_0,\dots,F_r$, contradicting base-point-freeness. Hence \begin{align*} \mathbb P^n_k=\bigcup_{i=0}^r U_i. \end{align*} [/step]

[step:Define the local map on each nonvanishing locus by affine target coordinates]Let \begin{align*} T:=k[y_0,\dots,y_r] \end{align*} be the homogeneous coordinate ring of $\mathbb P^r_k$, and for each $i\in\{0,\dots,r\}$ let \begin{align*} W_i:=D_+(y_i)\subseteq \mathbb P^r_k \end{align*} be the standard affine chart. Its coordinate ring is \begin{align*} k\left[\frac{y_0}{y_i},\dots,\widehat{\frac{y_i}{y_i}},\dots,\frac{y_r}{y_i}\right]. \end{align*} Here the hat means that the redundant coordinate $y_i/y_i=1$ is omitted. For each $i$, the standard description of projective distinguished opens gives \begin{align*} \Gamma(U_i,\mathcal O_{\mathbb P^n_k})=(S_{F_i})_0. \end{align*} Define a $k$-algebra homomorphism \begin{align*} \theta_i:k\left[\frac{y_0}{y_i},\dots,\widehat{\frac{y_i}{y_i}},\dots,\frac{y_r}{y_i}\right]\longrightarrow \Gamma(U_i,\mathcal O_{\mathbb P^n_k}) \end{align*} by \begin{align*} \theta_i\left(\frac{y_j}{y_i}\right):=F_jF_i^{-1} \end{align*} for every $j\neq i$. This is well-defined because $F_j$ and $F_i$ are homogeneous of the same degree $d$, so $F_jF_i^{-1}$ has degree zero in the localized graded ring $S_{F_i}$ and therefore is an element of $(S_{F_i})_0=\Gamma(U_i,\mathcal O_{\mathbb P^n_k})$. Since $W_i$ is affine and its coordinate ring is the displayed [polynomial ring](/page/Polynomial%20Ring) in the affine chart coordinates, the homomorphism $\theta_i$ defines a morphism \begin{align*} \varphi_i:U_i\longrightarrow W_i. \end{align*} On geometric points, this morphism is given by \begin{align*} x\longmapsto [F_0(x):\cdots:F_r(x)], \end{align*} viewed in the chart $W_i$, because $F_i(x)\neq 0$ on $U_i$ and the affine coordinates of the image are \begin{align*} F_j(x)F_i(x)^{-1} \end{align*} for $j\neq i$.[/step]

[guided]Fix an index $i\in\{0,\dots,r\}$. The open subset on which the $i$-th coordinate of the proposed target point is nonzero is \begin{align*} U_i=D_+(F_i)\subseteq \mathbb P^n_k. \end{align*} The intended image lies in the standard affine chart \begin{align*} W_i=D_+(y_i)\subseteq \mathbb P^r_k, \end{align*} because on $U_i$ the value $F_i(x)$ is nonzero. We now write the proposed map in affine coordinates on $W_i$. The chart $W_i$ has coordinate functions $y_j/y_i$ for $j\neq i$. Therefore the coordinate formula must send \begin{align*} \frac{y_j}{y_i}\longmapsto F_jF_i^{-1}. \end{align*} The point requiring verification is regularity of this element on $U_i$. The standard projective distinguished-open formula gives \begin{align*} \Gamma(D_+(F_i),\mathcal O_{\mathbb P^n_k})=(S_{F_i})_0. \end{align*} Both $F_j$ and $F_i$ are homogeneous forms of degree $d$, so $F_jF_i^{-1}$ has homogeneous degree zero after localizing at $F_i$. Hence $F_jF_i^{-1}$ belongs to $(S_{F_i})_0$ and defines a regular function on $D_+(F_i)$. Thus the assignment on coordinate rings \begin{align*} \theta_i:k\left[\frac{y_0}{y_i},\dots,\widehat{\frac{y_i}{y_i}},\dots,\frac{y_r}{y_i}\right]\longrightarrow \Gamma(U_i,\mathcal O_{\mathbb P^n_k}) \end{align*} given by \begin{align*} \theta_i\left(\frac{y_j}{y_i}\right)=F_jF_i^{-1} \end{align*} for $j\neq i$ is a well-defined $k$-algebra homomorphism. Because $W_i$ is affine with the displayed coordinate ring, this homomorphism is exactly the data of a morphism \begin{align*} \varphi_i:U_i\longrightarrow W_i. \end{align*} For a geometric point $x\in U_i$, the affine coordinates of $\varphi_i(x)$ are \begin{align*} F_j(x)F_i(x)^{-1} \end{align*} for $j\neq i$, so in homogeneous coordinates the same point is \begin{align*} [F_0(x):\cdots:F_r(x)]. \end{align*}[/guided]

[step:Check that the local morphisms agree on overlaps] Let $i,\ell\in\{0,\dots,r\}$. On the overlap \begin{align*} U_i\cap U_\ell=D_+(F_iF_\ell), \end{align*} both $F_i$ and $F_\ell$ are invertible. For every $j\in\{0,\dots,r\}$, the coordinate ratios computed from the $i$-chart and from the $\ell$-chart are compatible because \begin{align*} F_jF_\ell^{-1}=(F_jF_i^{-1})(F_iF_\ell^{-1}) \end{align*} as regular functions on $U_i\cap U_\ell$. More explicitly, on $W_i\cap W_\ell$ the affine coordinate $y_j/y_\ell$ in the $\ell$-chart is obtained from the $i$-chart coordinates by multiplying $y_j/y_i$ by $y_i/y_\ell$. Under $\theta_i$, these two factors become $F_jF_i^{-1}$ and $F_iF_\ell^{-1}$, whose product is the displayed function $F_jF_\ell^{-1}$. Thus the coordinate pullbacks for $\varphi_i$ and $\varphi_\ell$ agree after applying the projective chart transition formulas. Hence \begin{align*} \varphi_i|_{U_i\cap U_\ell}=\varphi_\ell|_{U_i\cap U_\ell}. \end{align*} The source $\mathbb P^n_k$ is covered by the open subvarieties $U_0,\dots,U_r$, and morphisms of varieties are local on the source for the Zariski topology. Since the local morphisms agree on all pairwise overlaps, they glue to a unique morphism \begin{align*} \varphi_V:\mathbb P^n_k\longrightarrow \mathbb P^r_k. \end{align*} On geometric points, the glued morphism is given by \begin{align*} x\longmapsto [F_0(x):\cdots:F_r(x)]. \end{align*} [/step]

[step:Identify the complete degree-$d$ system with the Veronese coordinate formula] Assume now that \begin{align*} V=k[x_0,\dots,x_n]_d. \end{align*} Let $\mathcal A$ be the finite set of multi-indices \begin{align*} \mathcal A:=\{\alpha=(\alpha_0,\dots,\alpha_n)\in\mathbb N_0^{n+1}:|\alpha|=d\}. \end{align*} Its cardinality is \begin{align*} |\mathcal A|=\binom{n+d}{d}=N+1. \end{align*} The ordered monomial basis consists of the forms \begin{align*} x^\alpha:=x_0^{\alpha_0}\cdots x_n^{\alpha_n} \end{align*} for $\alpha\in\mathcal A$, in the chosen order. We verify that this complete degree-$d$ linear system is base-point-free. Let $K/k$ be a [field extension](/page/Field%20Extension) and let $p=[a_0:\cdots:a_n]\in\mathbb P^n_k(K)$. By the definition of a projective point, at least one coordinate $a_b\in K$ is nonzero. For the multi-index $\beta\in\mathcal A$ defined by $\beta_b=d$ and $\beta_m=0$ for $m\neq b$, the corresponding monomial has value \begin{align*} a^\beta=a_b^d\neq 0 \end{align*} in the field $K$. Hence no geometric point is a common zero of all degree-$d$ monomials, so the construction above applies to $V=S_d$. Therefore the morphism constructed above is \begin{align*} [x_0:\cdots:x_n]\longmapsto [x^\alpha]_{|\alpha|=d}. \end{align*} This is precisely the degree-$d$ Veronese map \begin{align*} \nu_d:\mathbb P^n_k\longrightarrow \mathbb P^N_k. \end{align*} Thus the complete degree-$d$ linear system gives the Veronese map, as claimed. [/step]