Veronese map (${\nu }_d$)

For an integer $d\ge 1$, the degree-$d$ Veronese map sends a projective point to all degree-$d$ monomials in its homogeneous coordinates:Multiplying the original coordinates by $\lambda \ne 0$ 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 $X\ne 0$, the ratio $Y/X$ recovers $t/s$; the chart $Z\ne 0$ recovers $s/t$. The degree-$d$ construction similarly makes homogeneous degree-$d$ expressions into linear coordinates in a larger projective space.