The even-degree subring is the second <Veronese subring>
$$
R^{(2)}=k[x^2,xy,y^2]\cong k[u,v,w]/(uw-v^2),
$$
where $u,v,w$ have degree one in the regraded ring. The canonical invariance of the <Proj construction> under passage to a Veronese subring gives
$$
\operatorname{Proj}R\xrightarrow{\sim}\operatorname{Proj}R^{(2)}.
$$
On homogeneous points this is the degree-two <Veronese embedding>
$$
[x:y]\longmapsto[x^2:xy:y^2].
$$
It is an isomorphism onto the closed subscheme
$$
V_+(uw-v^2)\subseteq\mathbb P_k^2.
$$
Thus the quotient homomorphism $k[u,v,w]\twoheadrightarrow R^{(2)}$ supplies the requested <closed immersion> of $\operatorname{Proj}R^{(2)}$ into the <projective plane>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 113 / 1 / b / Solution

{c}
{title2=$S^{(d)}$}

For a nonnegatively <graded ring> $S=\bigoplus_{m\geq0}S_m$ and $d\geq1$, its $d$th Veronese subring is
$$
S^{(d)}=\bigoplus_{m\geq0}S_{dm},
$$
graded 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 $\operatorname{Proj}S\cong\operatorname{Proj}S^{(d)}$.

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