Homogeneous curves and automorphisms of P^1

In (a), geometric genus is $0$ because $C_{k^{\mathrm{a}}}\cong {\mathbf{P}}^1$. A genus-zero smooth complete curve is a smooth conic, and a conic with a rational point is isomorphic to ${\mathbf{P}}^1$, for example by projection from that point. Hence $C\cong {\mathbf{P}}^1$.

For (b), base change to an algebraic closure. The action remains transitive on the positive-dimensional curve and is therefore nontrivial; [F2] gives $C_{k^{\mathrm{a}}}\cong {\mathbf{P}}^1$. The assumed $k$-point and step 1.1 then give $C\cong {\mathbf{P}}^1$ over $k$. The group ${\mathrm{G}\mathrm{L}}_2$ is smooth affine, being the determinant-open subscheme of ${\mathbf{A}}^4$. Thus [F2] represents its fppf quotient by the closed central scalar subgroup ${\mathbf{G}}_m$; multiplication and inversion descend to this quotient, denoted ${\mathrm{PGL}}_2$, and its action on lines descends because scalars act trivially.

Let $f$ be an automorphism of ${\mathbf{P}}_R^1$ for any $k$-algebra $R$. Locally on $\mathrm{Spec}R$, choose generators $v_{\mathrm{\infty }},v_0$ of the lines $f(\mathrm{\infty }),f(0)$. Their fibrewise distinctness makes these columns a basis of $R^2$. In this basis a generator of $f(1)$ has two unit coordinates $a,b$, since it is distinct from both other lines in every fibre. The matrix with columns $a{v}_{\mathrm{\infty }},b{v}_0$ therefore carries $(\mathrm{\infty },0,1)$ to $(f(\mathrm{\infty }),f(0),f(1))$. An automorphism $h$ fixing these three sections is the identity: on the two affine charts it has coordinate polynomials $P(t),Q(t^{-1})$ with zero constant terms and unit linear coefficients (their polynomial inverses force those coefficients to be units), and $P(t)Q(t^{-1})=1$ on the overlap. If $P$ had highest nonzero degree $N>1$, the coefficient of $t^{N-1}$ in this product would be its leading coefficient times the unit linear coefficient of $Q$, a contradiction. Hence $P(t)=ct$, and $h(1)=1$ gives $c=1$; the overlap then gives $Q(t^{-1})=t^{-1}$. Finally, a matrix fixing the three lines is scalar: the first two force it to be diagonal and the third makes its diagonal entries equal. Thus the local matrices inducing $f$ are unique up to scalar and glue to a unique point of the fppf quotient. This proves ${\mathrm{PGL}}_2\cong \mathrm{Aut}({\mathbf{P}}^1)$ as functors, hence as group schemes, including over nonreduced test algebras.

For (c), a line is fixed by $T$ exactly when it is a one-dimensional subrepresentation, hence lies in a single weight space. Thus the fixed locus is the disjoint union of the projective spaces $\mathbf{P}(V_{\chi })$. When $T={\mathbf{G}}_m$ and every occurring weight space, including weight zero, has dimension one, these are finitely many $k$-rational points, giving a finite constant fixed scheme. For a non-fixed point $[v]$, write $v={\sum }_n{v}_n$ according to integer weights, with least and greatest occurring weights $r<s$. The orbit map extends to ${\mathbf{P}}^1$, with endpoints $[v_r]$ and $[v_s]$; on the two affine charts this follows by factoring $t^r$ at zero and $t^s$ at infinity. This extension is surjective onto the orbit closure because its image is closed and contains the dense orbit. A point of the image of ${\mathbf{G}}_m$ is non-fixed, so the only fixed points of the closure are precisely the two endpoints.