Projective embedding of a compact Riemann surface

Choose a canonical divisor $K$ by [F2]. Duality at $0$ gives $\mathrm{\ell }(K)=g$ and at $K$ gives $i(K)=\mathrm{\ell }(0)=1$, so Riemann–Roch at $K$ yields $\mathrm{deg}K=2g-2$. For $A=D,D-[p],D-[p]-[q]$ or $D-2[p]$, the degree is at least $2g-1>2g-2$. Thus [F3] gives $\mathrm{\ell }(K-A)=0$, and [F2] gives $\mathrm{\ell }(A)=\mathrm{deg}A+1-g$. This proves every displayed dimension drop and $\mathrm{\ell }(D)\ge g+2$.

By [F4], sections vanishing at $p$ correspond exactly to $L(D-[p])$: in a local frame their zero order is ${\mathrm{ord}}_p(h)+D(p)$. Step 1.1 makes this a proper codimension-one subspace, so some section is nonzero at each $p$. Hence $H^0(X,E)$ is base-point-free and [F6] supplies ${\phi }_D$. At distinct $p,q$, the sections vanishing at both form $L(D-[p]-[q])$, a proper subspace of $L(D-[p])$ by step 1.1; choose a section vanishing at $p$ but not at $q$, and symmetrically one vanishing at $q$ but not at $p$. If ${\phi }_D(p)={\phi }_D(q)$, their nonzero evaluation functionals would be proportional and have the same kernel, contrary to those sections. Thus ${\phi }_D$ is injective.

Fix $p$. By step 1.1 choose a section $s\in H^0(E(-[p]))\setminus H^0(E(-2[p]))$, and by step 2.1 choose $t\in H^0(E)$ with $t(p)\mathrm{\ne }0$. In a local coordinate $z$ centred at $p$ and a holomorphic frame $e$, write $s=a(z)e$, $t=b(z)e$; [F4] and [F5] give $a(z)=zu(z)$ with $u(0)\mathrm{\ne }0$ and $b(0)\mathrm{\ne }0$. Since $s,t$ are independent, extend them to a basis of $H^0(E)$. In the target affine chart corresponding to $t$, a coordinate of ${\phi }_D$ is $a\mathrm{/}b$, whose derivative at $p$ is $u(0)\mathrm{/}b(0)\mathrm{\ne }0$. Basis changes are holomorphic projective automorphisms by [F6], so the differential is nonzero for every basis. It is a nonzero complex-linear map from a one-dimensional complex tangent space, hence injective as a real-linear map. Thus [F7] and [F8] make ${\phi }_D$ a smooth and holomorphic immersion. This argument uses the regular coefficients $h{f}_i$, even when the representing meromorphic functions have poles.

The map is injective by step 2.1 and immersive by step 3.1; $X$ is compact and projective space is Hausdorff by [F7]. Hence [F8] makes it a homeomorphism onto its image and a smooth embedding. Its local expressions are holomorphic by [F6], so it is the asserted holomorphic embedding. The intrinsic target and basis covariance are those in [F6].