Hyperelliptic curves and hyperelliptic maps

Choice premise. The definitions of hyperelliptic and geometrically hyperelliptic are unconditional. Assume AC for all proved assertions below: the map-finiteness and function-to-map, global-sections, projective-space, Čech-comparison, long-exact-sequence, Riemann--Roch, Serre-duality, and birational-curve suppliers used here explicitly require it (The Axiom of Choice, A nonconstant rational function defines a finite map to the projective line, Nonconstant morphisms of proper curves are finite and surjective, Functions on a proper curve, Maps to projective space equal generating line-bundle data, Cohomology of O(d) on projective space, Cech cohomology computes quasi-coherent cohomology on a separated scheme, Long exact sequence of sheaf cohomology, The full Riemann-Roch theorem for divisors on a smooth proper curve, Serre duality for line bundles on a smooth proper curve, and the residue realization, The canonical divisor has degree 2g - 2, The canonical bundle has exactly g independent sections, Birational smooth proper curves are isomorphic).

Let $k$ be a field and let $C$ be a smooth proper geometrically integral curve over $k$ (Curves over a field) of genus $g$. A $k$-morphism $\varphi :C⟶{\mathbf{P}}_k^1$ is a hyperelliptic map on $C$ when it is nonconstant of degree two, $\mathrm{deg}\varphi =2$ in the sense of Degree of a nonconstant morphism of curves, with target the projective line ${\mathbf{P}}_k^1$ in its standard charts (Relative projective space from standard charts). Such a morphism is finite. The curve $C$ is hyperelliptic over $k$ when it admits such a map; this means that the degree-two target is the split projective line over $k$. The curve is geometrically hyperelliptic when $C_{\bar{k}}=C{\times }_k\bar{k}$ admits a degree-two map to ${\mathbf{P}}_{\bar{k}}^1$, where $\bar{k}$ is an algebraic closure. These definitions impose no restriction on the genus or characteristic. A geometrically hyperelliptic curve need not be hyperelliptic over $k$: its degree-two quotient can descend to a nonsplit genus-zero curve rather than to ${\mathbf{P}}_k^1$.

For every genus, a nonconstant rational function $f\in k(C{)}^{\times }$ gives a finite map ${\varphi }_f:C\to {\mathbf{P}}_k^1$ of degree $[k(C):k(f)]$, with pole divisor the fibre over infinity (A nonconstant rational function defines a finite map to the projective line). Thus $C$ is hyperelliptic over $k$ if and only if some nonconstant $f\in k(C{)}^{\times }$ has $[k(C):k(f)]=2$; every hyperelliptic map is obtained this way, up to an automorphism of ${\mathbf{P}}_k^1$.

If $g\ge 1$, then hyperellipticity over $k$ is equivalent to $\mathrm{gon}(C)=2$, where gonality is the minimum degree of a nonconstant $k$-map to ${\mathbf{P}}_k^1$ (Gonality). Indeed, a degree-one map between smooth proper curves is an isomorphism, so would force $C\cong {\mathbf{P}}_k^1$ and $g=0$; conversely, a gonality of two is attained by a map of degree two. This equivalence is not asserted in genus zero: ${\mathbf{P}}_k^1$ itself has degree-two self-maps and gonality one.

For every field $k$ and every $g\ge 1$, the following is also equivalent to hyperellipticity over $k$: there is an invertible sheaf $L$ on $C$ (Invertible sheaves) of degree two with $h^0(C,L)=2$ whose complete linear system is base-point-free. A hyperelliptic map gives $L={\varphi }^{\ast }{\mathcal{O}}_{{\mathbf{P}}^1}(1)$; the section count is proved below. Conversely, a basis of the two sections of such an $L$ gives a nonconstant map $C\to {\mathbf{P}}_k^1$, and the degree of its fibre over a rational point is $\mathrm{deg}L=2$, so it is a hyperelliptic map. The genus qualification is necessary: on ${\mathbf{P}}^1$ the pullback of $\mathcal{O}(1)$ under a degree-two map is $\mathcal{O}(2)$, with three sections.

For $g\ge 2$, any two hyperelliptic maps over $k$ are related by a $k$-automorphism of ${\mathbf{P}}_k^1$. The proof below is over the given field; it does not infer a split-target map from geometric hyperellipticity.

Let $\varphi :C\to {\mathbf{P}}_k^1$ be a degree-two map and put $L={\varphi }^{\ast }{\mathcal{O}}_{{\mathbf{P}}^1}(1)$. The map is finite. On a local ring of ${\mathbf{P}}_k^1$, a discrete valuation ring, the finite algebra ${\varphi }_{\ast }{\mathcal{O}}_C$ is a finitely generated torsion-free module of generic rank two, hence free of rank two by Every finitely generated torsion-free module over a PID is free. (Torsion-freeness follows from the injection of the target function field into $k(C)$.) Write $E={\varphi }_{\ast }{\mathcal{O}}_C$. The unit map ${\mathcal{O}}_{{\mathbf{P}}^1}\to E$ is a subbundle. Indeed, in a local basis $e_1,e_2$ of the finite free rank-two algebra, write $1=u_1{e}_1+u_2{e}_2$. Its reduction in the fiber algebra is nonzero, so at least one of $u_1,u_2$ is a unit in the local ring. If $u_1$ is a unit, $\{1,e_2\}$ is a basis; if $u_2$ is a unit, $\{e_1,1\}$ is a basis. Thus the unit spans a direct summand locally, and its quotient $Q$ is a line bundle.

On each standard affine chart of ${\mathbf{P}}^1$, the module of sections of $Q$ is a finitely generated rank-one projective module, hence torsion-free; since $k[t]$ and $k[t^{-1}]$ are PIDs, it is free by Every finitely generated torsion-free module over a PID is free. The transition on their intersection is a unit of $k[t,t^{-1}]$, hence is $c{t}^a$ for some $c\in k^{\times }$ and $a\in \mathbb{Z}$. Rescaling a frame absorbs $c$; thus $Q\cong \mathcal{O}(a)$ for an integer $a$. The projective-line cohomology calculation Cohomology of O(d) on projective space gives $\chi (\mathcal{O}(a))=a+1$, $h^0(\mathcal{O}(-g))=0$ for $g\ge 1$, $h^0(\mathcal{O}(1))=2$, and $h^1(\mathcal{O}(1))=0$. For the cohomology of $E$, cover ${\mathbf{P}}^1$ by its two standard affines and $C$ by their inverse images. These inverse images and their intersection are affine by Finite is affine and local on its target. The Čech complexes for $E$ on ${\mathbf{P}}^1$ and ${\mathcal{O}}_C$ on $C$ have identical terms by the definition $E={\varphi }_{\ast }{\mathcal{O}}_C$. Since both schemes are quasi-compact and separated and the sheaves are quasi-coherent, the published Čech comparison theorem Cech cohomology computes quasi-coherent cohomology on a separated scheme identifies their sheaf cohomology. Hence $\chi (E)=\chi (C,{\mathcal{O}}_C)=1-g$. Additivity of Euler characteristic in the exact sequence below follows from the long exact sequence of sheaf cohomology Long exact sequence of sheaf cohomology. Therefore the exact sequence $0⟶{\mathcal{O}}_{{\mathbf{P}}^1}⟶E⟶\mathcal{O}(a)⟶0$ has Euler characteristic $1+(a+1)=a+2=1-g$, so $a=-g-1$.

Twist this sequence by $\mathcal{O}(1)$. For $g\ge 1$ it becomes $0⟶\mathcal{O}(1)⟶E(1)⟶\mathcal{O}(-g)⟶0.$ The displayed Cech calculations imply $H^0(E(1))\cong H^0(\mathcal{O}(1))$, of dimension two. On the two standard charts, trivializing $\mathcal{O}(1)$ identifies the sections of $E(1)$ with the sections of ${\varphi }^{\ast }\mathcal{O}(1)$; hence $h^0(C,L)=2$. The pullback of $\mathcal{O}(1)$ is generated, and its degree is two: the pullback of a rational point is the fiber of the finite flat rank-two map, an effective divisor of degree two.

Conversely, if $L$ has degree two, has two independent sections and is base-point-free, those sections define a nonconstant map to ${\mathbf{P}}_k^1$ with pullback of $\mathcal{O}(1)$ equal to $L$. The map is finite; its finite flat rank is the degree of a fiber over a rational point, which is $\mathrm{deg}L=2$. This proves the sheaf criterion over any field in the stated positive-genus range.

It remains to prove uniqueness when $g\ge 2$. The $g$ monomials in $H^0({\mathbf{P}}^1,\mathcal{O}(g-1))$ pull back along $\varphi$ to independent sections of $L^{\otimes (g-1)}$: pullback is injective because $\varphi$ is dominant. Since $\mathrm{deg}L=2$, this tensor power has degree $2g-2$. Riemann--Roch and Serre duality give $h^0(L^{\otimes (g-1)})-h^0({\omega }_C\otimes L^{-\otimes (g-1)})=g-1.$ The first term is at least $g$, so the second term is positive. The line bundle in that second term has degree zero; a nonzero section has an effective zero divisor of degree zero and is nowhere vanishing. Consequently $L^{\otimes (g-1)}\cong {\omega }_C$. The pulled-back monomials are now a basis of $H^0(C,{\omega }_C)$, whose dimension is $g$. Thus the canonical map factors through the degree-$(g-1)$ Veronese closed immersion of ${\mathbf{P}}_k^1$, followed by $\varphi$, up to the projective coordinate change from this basis to any fixed basis of canonical sections.

For a second hyperelliptic map $\psi$, the same argument factors the same canonical map through another Veronese closed immersion. Each Veronese image is the canonical map's image, since $\varphi$ and $\psi$ are finite and surjective. The two closed immersions therefore identify their copies of ${\mathbf{P}}_k^1$ with the same canonical image. Composing one identification with the inverse of the other gives a $k$-automorphism $\alpha$ of ${\mathbf{P}}_k^1$ and the equality $\psi =\alpha \circ \varphi$. This proves the stated uniqueness, including in characteristic two.