Canonical divisors on hyperelliptic curves

Embed $C$ projectively by [F3]. Its complex points $X$ are closed in compact projective space because its homogeneous defining equations are continuous, so $X$ is compact, Hausdorff and second countable. At each point [F4] gives a standard smooth chart with one free coordinate $z$; the implicit-function lemma supplies a holomorphic graph chart, and the transitions are holomorphic. Thus each component of $X$ is a compact Riemann surface, with finitely many components by [F10]. An algebraic morphism is holomorphic in these charts: its coordinate functions are regular fractions whose denominators are nonzero near the point, and compositions with the graph chart are holomorphic. In particular $\varphi$ induces a holomorphic map $X\to \widehat{\mathbb{C}}$.

In a standard smooth chart the invertible Jacobian minor lets [F5] eliminate all dependent coordinate differentials, leaving $dz$ as a regular algebraic frame of ${\omega }_C$ and as the analytic differential frame. The cotangent isomorphism in [F5] says $z-z(p)$ has nonzero class in the one-dimensional ${\mathfrak{m}}_p\mathrm{/}{\mathfrak{m}}_p^2$; since the local ring is a DVR by [F6], it is an algebraic uniformizer. Any rational coefficient is therefore $(z-z(p){)}^{a}u$ with $a\in \mathbb{Z}$ and $u$ a regular unit; analytically $u$ is holomorphic with nonzero value at $p$, so its algebraic and analytic orders are equal. This also proves that a nonzero rational coefficient cannot vanish identically on an analytic neighbourhood. Consequently algebraic rational differentials become nonzero meromorphic differentials with precisely the same divisor, and regular differentials become holomorphic; the restriction of their section spaces is injective. Pullbacks and line-bundle isomorphisms have the same local regular transition formulas and therefore induce the corresponding holomorphic bundle maps.

On each component $X_j$, the map $\varphi$ is nonconstant: if locally constant at a point over $t$, a local coordinate of the target vanishing at $t$ would pull back to an identically zero germ, contrary to the finite DVR order of that nonzero rational pullback in step 2.1 and [F7]. Compactness makes each restriction proper. By [F10], it has a positive integer analytic degree $d_j$. Equality of local orders in step 2.1 and the algebraic fibre formula [F7] give ${\sum }_j{d}_j=2$. If $X$ had two components, both degrees would be one. The fibre formula would then make each restriction bijective and unramified, and the local inverses in [F10] would make each component biholomorphic to the sphere. The sphere has no nonzero holomorphic differential: the meromorphic differential $dz$ has divisor $-2[\mathrm{\infty }]$, since $dz=-w^{-2}dw$ in $w=1\mathrm{/}z$. Dividing a holomorphic differential by $dz$ would give an element of $L(-2[\mathrm{\infty }])$, which is zero by [F9]. But [F8] supplies a nonzero regular algebraic differential, whose restriction is nonzero by step 2.1 and holomorphic on every component; if every component were a sphere it would vanish everywhere, contradicting this injectivity. Thus $X$ is connected and $\varphi :X\to \widehat{\mathbb{C}}$ has degree two.

Choose a nonzero regular algebraic differential by [F8], and let $K_C$ be its divisor. Step 2.1 identifies its algebraic divisor with the analytic canonical divisor $K$ on connected $X$, coefficient by coefficient. All residue degrees are one by [F7], so their degrees agree. If $h$ is the topological genus of $X$, [F8] and [F9] give $2g-2=\mathrm{deg}{K}_C=\mathrm{deg}K=2h-2$, hence $h=g$. Restriction of regular algebraic differentials is injective by step 2.1; its source has dimension $g$ by [F8] and its target has dimension $h=g$ by [F9], so it is an isomorphism. This proves the required canonical-section and genus comparison without assuming a general algebraic/analytic cohomology comparison.

A general fibre of the analytic degree-two map consists of two distinct points $P,Q$: [F10] makes the branch-value set finite. It is the same algebraic fibre by step 2.1. The pullback of the standard section of $\mathcal{O}(1)$ vanishing at $t$ has divisor $P+Q$ on $X$, so $L$ induces ${\mathcal{O}}_X(P+Q)$. The bundle isomorphism in [F2] and step 2.1 therefore give $K\sim D=(g-1)(P+Q)$. By [F9], $\mathrm{\ell }(K)=g$ and linear equivalence gives $\mathrm{\ell }(D)=g$; equivalently RR gives $\mathrm{\ell }(D)-\mathrm{\ell }(K-D)=g-1$ with $K-D\sim 0$ and $\mathrm{\ell }(0)=1$. The $g$ pulled-back monomial sections in [F2] are now a basis of both algebraic and analytic canonical spaces by step 4.1, so their coordinate map is precisely the Veronese factorization, up to a projective basis change. Since $P\mathrm{\ne }Q$ have the same image under $\varphi$, their canonical images agree; hence the canonical map is not an embedding and has degree two onto the rational normal curve. Finally [F8], [F9] and step 4.1 give all displayed RR and duality dimensions.