Let us fix the notation. Let $V$ be a 4-dimensional complex vector space and let $$\mathbb P(V)=\operatorname{Proj}(\operatorname{Sym}V)=\operatorname{Proj}(\mathbb C[x,y,z,w]).$$ Consider the $\mathbb C^*$-action on $\mathbb P(V)$ given by $$t \cdot [x:y:z:w]=[t^{-1}x:t^{-1}y:tz:tw].$$ Since $\mathbb C^*$ is a reductive algebraic group, the ring of invariant polynomials on $V$ is finitely generated and it holds $$\mathbb C[V]^{\mathbb C^*}=(xz,xw,yz,yw) \subset \mathbb C[V]=\mathbb C[x,y,z,w].$$ Let us consider the quotient $\mathbb C[V]/(xz,xw,yz,yw)$: there is a surjective map $\mathbb C[V] \to \mathbb C[V]/(xz,xw,yz,yw)$, hence there is an inclusion $$A:=\operatorname{Proj}\left(\frac{\mathbb C[V]}{(xz,xw,yz,yw)}\right) \subset \mathbb P(V).$$ Let us define $V_- \subset V$ and $V_+ \subset V$ to be the invariant subspaces of $V$, that is such that $$\mathbb P(V_-)=\operatorname{Proj}(\mathbb C[x,y]), \qquad \mathbb P(V_+)=\operatorname{Proj}(\mathbb C[z,w]).$$
Question: Does it hold that $A=\mathbb P(V_-) \sqcup \mathbb P(V_+)$?