Let $k$ be a field, let $n\geq 0$, let $d\geq 1$, and let $S:=k[x_0,\dots,x_n]$ with degree-$d$ homogeneous part $S_d$. Let $V\subseteq S_d$ be a finite-dimensional $k$-linear subspace with $\dim_k V=r+1$, and let $F_0,\dots,F_r$ be a $k$-basis of $V$. Assume that $V$ is base-point-free on $\mathbb P^n_k$ after extension to geometric points: for every [field extension](/page/Field%20Extension) $K/k$ and every point $x\in \mathbb P^n_k(K)$, at least one value $F_i(x)$ is nonzero. Equivalently, there is no geometric point $x\in \mathbb P^n_k$ such that $F_0(x)=\cdots=F_r(x)=0$. Then the rule on geometric points $x\mapsto [F_0(x):\cdots:F_r(x)]$ defines a morphism of projective $k$-varieties $\varphi_V:\mathbb P^n_k\to \mathbb P^r_k$.

In particular, if $V=S_d$ and $F_0,\dots,F_N$ is the ordered monomial basis of all degree-$d$ monomials, where $N=\binom{n+d}{d}-1$, then $\varphi_V$ is the degree-$d$ Veronese map $\nu_d:\mathbb P^n_k\to \mathbb P^N_k$.