Let $k$ be an [algebraically closed field](/page/Algebraically%20Closed%20Field), let $n\geq 0$, and let $d\geq 1$. All projective and affine varieties are understood in the classical sense over the algebraically closed field $k$. Set
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\begin{align*}
N:=\binom{n+d}{d}-1.
\end{align*}
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For each $i\in\{0,\dots,n\}$, let $e_i\in\mathbb N_0^{n+1}$ denote the $i$-th standard basis vector. Let
Give $\mathbb P_k^N$ homogeneous coordinates $Y_\alpha$ indexed by $\alpha\in M_d$, and for homogeneous coordinates $[x_0:\cdots:x_n]$ on $\mathbb P_k^n$ write
is a well-defined morphism of projective varieties: for every nonzero representative $(x_0,\dots,x_n)$ at least one coordinate $x^{d e_i}=x_i^d$ is nonzero, and rescaling the representative by $\lambda\in k^\times$ rescales all target coordinates by the common factor $\lambda^d$. This morphism is injective, has image a closed projective algebraic subset of $\mathbb P_k^N$, and is an isomorphism of projective varieties from $\mathbb P_k^n$ onto its image.