Veronese subring ($S^{(d)}$)

For a nonnegatively graded ring $S={⨁}_{m\ge 0}{S}_m$ and $d\ge 1$, its $d$th Veronese subring isgraded by $(S^{(d)}{)}_m=S_{dm}$. Passing from $S$ to $S^{(d)}$ does not change its projective scheme: the standard affine charts give a canonical isomorphism $\mathrm{Proj}S\cong \mathrm{Proj}{S}^{(d)}$.

The degree-$d$ Veronese embedding sends a projective point to all monomials of degree $d$ in its homogeneous coordinates. For example,is a closed immersion with image the conic $uw-v^2=0$.