The problem in Hartshorne essentially says that given a separated $\mathbb C$ scheme of finite type $X$, and a semilinear involution $\sigma$ we can find a $\mathbb R$ scheme $X_0$ such that $X$ is the fibre product of $X_0$ (over $\mathbb R$) with $\mathbb C$.
I want to prove part e. It says:
If $X = P^1_\mathbb C$ then either $X_0 = P^1_\mathbb R$ or is isomorphic to the conic $x^2+y^2+z^2 =0$ inside $P^1_\mathbb R$.
I am a bit confused here since in part a I had to prove that $X_0$ is unique.
How does one prove part e?
The key is that given the pair $(X,\sigma)$, the scheme $X_0$ so that $X_0\times_{\Bbb R}\Bbb C$ with the standard complex conjugation is isomorphic to $X$ with the involution $\sigma$ is unique. So the big idea is that we have to look at $\sigma$ to see which situation we're in. Throughout the following I'll give a general roadmap so that you still have some details to check yourself - if you run in to trouble, please do leave a comment.
If $\sigma$ fixes a closed point in $\Bbb P^1_{\Bbb C}$, then the complement of that closed point is a copy of $\Bbb A^1_{\Bbb C}$ which is fixed by $\sigma$. One can check that any $\sigma$ acting on $\Bbb A^1_{\Bbb C}$ must fix a closed point, so in this case $\Bbb P^1_{\Bbb C}$ is covered by two $\sigma$-invariant copies of $\Bbb A^1_{\Bbb C}$ which intersect in a $\sigma$-invariant copy of $\Bbb A^1_{\Bbb C}\setminus 0$. With an application of part (c), we see that these inclusion morphisms give rise to the standard gluing data for $\Bbb P^1_{\Bbb R}$.
If $\sigma$ does not fix a closed point, then pick a closed point $p$ and via a change of coordinates, send $p\mapsto 0$ and $\sigma(p)\mapsto \infty$. The complement of these points is fixed by $\sigma$, so we have $\sigma$ acting on $\operatorname{Spec} \Bbb C[t,t^{-1}]$ fixing no points. After some calculations (which I leave to you as to not completely spoil your enjoyment of the problem), one may find that the action of $\sigma$ on the ring side can be taken to be $\sigma(t)=-t^{-1}$ which has algebra of invariants $\Bbb R[t-t^{-1},it+it^{-1}]\cong\Bbb R[x,y]/(x^2+y^2+1)$, which gives rise to the conic $V(x^2+y^2+z^2)\subset\Bbb P^2_{\Bbb R}$.