= Veronese map
{c}
{title2=$\nu_d$}
For an <integer> $d\geq1$, the degree-$d$ <Veronese map> sends a <projective point> to all degree-$d$ <monomials> in its <homogeneous coordinates>:
$$
\nu_d:\mathbb P^n\longrightarrow\mathbb P^{\binom{n+d}{d}-1}.
$$
Multiplying the original <coordinates> by $\lambda\ne0$ multiplies every output by $\lambda^d$, so the <map> is well-defined on <projective points>. It is a <closed immersion>, given by the complete <linear system> of $\mathcal O(d)$.
For $n=1,d=2$, the formula is $[s:t]\mapsto[s^2:st:t^2]$. Its image in the <projective plane> satisfies $XZ=Y^2$, a <smooth plane conic>. On the chart $X\ne0$, the ratio $Y/X$ recovers $t/s$; the chart $Z\ne0$ recovers $s/t$. The degree-$d$ construction similarly makes homogeneous degree-$d$ expressions into linear <coordinates> in a larger <projective space>.
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