Holomorphic differentials separate generic points

Choose a nonzero meromorphic differential $\eta$ supplied by [F2] and put $K=(\eta )$. By [F3] and [F1], $\mathrm{\ell }(K)=g$; [F2] also gives $\mathrm{\ell }(0)=1$. Applying [F2] to $D=K$ yields $g-1=\mathrm{deg}K+1-g$, so $\mathrm{deg}K=2g-2$.

Constants lie in $L(p)$, so $\mathrm{\ell }(p)\ge 1$. If $f\in L(p)$ were nonconstant, [F4] would make it a nonconstant holomorphic map $X\to \widehat{\mathbb{C}}$ whose total pole order is at most one and which is proper. By [F5] its degree is positive and at most one, hence is one. By [F6] this is a biholomorphism to the sphere, so [F7] gives $g=0$, contrary to the hypothesis. Thus $L(p)$ consists exactly of constants and $\mathrm{\ell }(p)=1$.

Let $0\le k<g$ and suppose the combined evaluation $E_k:\mathrm{\Omega }(X)\to {⨁}_{j=1}^k(K_X{)}_{a_j}$ at distinct points is surjective; for $k=0$ this is the map to the zero space. Its kernel $W$ has dimension $g-k$ by [F1]. Choose a nonzero $\omega \in W$. By [F1] its zero set is finite and contains each $a_j$, so choose $q$ outside that set; this is possible because a coordinate disk in $X$ contains infinitely many points (Riemann surfaces and holomorphic atlases). Then $q$ is distinct from the previous points and ${\mathrm{ev}}_q{\mathrm{\mid }}_W$ is nonzero, hence onto the one-dimensional fiber $(K_X{)}_q$. Given any target in ${⨁}_{j=1}^k(K_X{)}_{a_j}\oplus (K_X{)}_q$, first lift its first $k$ coordinates through $E_k$ and then adjust that lift by an element of $W$ to attain the last coordinate. Thus the combined evaluation at $a_1,\dots ,a_k,q$ is surjective, and its kernel has dimension $g-k-1$. Iterating finitely until $k=g$ gives a surjection between $g$-dimensional spaces, hence an isomorphism; starting with $k=0$ proves claim 2, and starting with any surjective family proves claim 3.

Apply [F2] to $D=K-p$. Using step 1.1 and step 1.2, $\mathrm{\ell }(K-p)-\mathrm{\ell }(p)=\mathrm{deg}(K-p)+1-g=(2g-3)+1-g=g-2$, so $\mathrm{\ell }(K-p)=g-1$. By [F3] this is the kernel dimension of ${\mathrm{ev}}_p:\mathrm{\Omega }(X)\to (K_X{)}_p$. Since [F1] gives $\dim \mathrm{\Omega }(X)=g$ and $(K_X{)}_p$ is one-dimensional, rank-nullity shows that the evaluation has rank one, hence is nonzero (indeed surjective).

Each fiber $(K_X{)}_{a_j}$ is one-dimensional. Choosing a nonzero local frame identifies it with $\mathbb{C}$, and changing frames composes the combined evaluation with an invertible diagonal map on the target. Thus surjectivity and isomorphism do not depend on those frame choices; for $k=g$, injectivity is exactly that no nonzero differential vanishes at every selected point. Step 1.3 and step 2.1 give the existence and extension claims and the one-point evaluation calculation.