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

The even-degree subring is the second Veronese subringwhere $u,v,w$ have degree one in the regraded ring. The canonical invariance of the Proj construction under passage to a Veronese subring givesOn homogeneous points this is the degree-two Veronese embeddingIt is an isomorphism onto the closed subschemeThus the quotient homomorphism $k[u,v,w]\twoheadrightarrow R^{(2)}$ supplies the requested closed immersion of $\mathrm{Proj}{R}^{(2)}$ into the projective plane.