[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 $j\neq i$.
[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]
[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]
