Galois descent for the gonality of curves (Q1727881)

By a curve we mean a smooth, projective, geometrically connected curve over a field $k$. The gonality of $C$ over $k$, denoted by $\gamma(C)$, is defined to be an integer $\gamma\geq 1$ such that the number $\gamma$ turns to be the degree of a rational non-constant map $f: C\rightarrow \mathbb{P}^1$ which is defined over $k$ and there is no rational map from $C$ to $\mathbb{P}^1$ of degree less than $\deg(f)$. The gonality over an algebraic closure $\bar{k}$ (resp. a separable closure $k^{\mathrm{sep}}$), denoted by $\bar{\gamma}(C)$ (resp. $\gamma(C(k^{\mathrm{sep}}))$), is called the geometric gonality (resp. separable gonality). Replacing $\mathbb{P}^1$ by an arbitrary genus zero curve $D$, one obtains the conic gonality $\gamma^{\mathrm{con}}(C)$. One has clearly \[\bar{\gamma}(C)\leq \gamma(C(k^{\mathrm{sep}}))\leq \gamma^{\mathrm{con}}(C)\leq \gamma(C).\] \textit{J. Roé} and \textit{X. Xarles} [Math. Res. Lett. 25, No. 5, 1567--1589 (2018; Zbl 07027232)] studied conditions under which the stated inequalities within the invariants $\gamma(C(k^{\mathrm{sep}}))$, $\gamma^{\mathrm{con}}(C)$ and $\gamma(C)$ turn to be equalities. Their results, interestingly, generalize known existing results specifically the result attributed to \textit{J.-F. Mestre} [Prog. Math. 94, 313--334 (1991; Zbl 0752.14027)] stating that a curve of even genus $g\geq 2$ which has conic gonality $\gamma^{\mathrm{con}}=2$ has to be a $2$-gonal curve. They mean a gonal map $f: C\rightarrow \mathbb{P}^1$ unique if there exists a unique subfield $F$ of $k(C)$ which is isomorphic to $k(\mathbb{P}^1)$ and with $[k(C):F]=\gamma$. With these pre-assumptions, as their main result, they prove: \par {Theorem 1}. Let $C$ be a curve with genus $g$ and separable gonality $\gamma^{\mathrm{sep}}$. Suppose that the gonal map $f_{k^{\mathrm{sep}}}$ over $k^{\mathrm{sep}}$ is unique. Then \begin{itemize} \item[(1)] $\gamma^{\mathrm{con}}=\gamma^{\mathrm{sep}}$, \item[(2)] If the curve $C$ has a $k$-rational divisor of odd degree, then $\gamma=\gamma^{\mathrm{con}}=\gamma^{\mathrm{sep}}$. \item[(3)] There exists some degree $2$ extension $L/k$ such that $\gamma(C_L)=\gamma^{\mathrm{con}}=\gamma^{\mathrm{sep}}$. \item[(4)] If $\gamma^{\mathrm{sep}}\equiv g (\mathrm{mod} 2)$, then $\gamma=\gamma^{\mathrm{sep}}$. \end{itemize} \par Using the facts (I) and (II) as \begin{itemize} \item[Fact I:] If $k$ is algebraically closed and $C$ is a curve on $k$ admitting a simple pencil $g^1_d$ giving a morphism $f: C\rightarrow \mathbb{P}^1_k$ and if $g>(d-1)^2$, then the given series is the unique simple pencil $g^1_d$ on $C$. \item[Fact II:] If $k$ is an arbitrary field, $C$ is a genus $g$ curve on $k$ and $f:C\rightarrow D$ is a $k$-morphism on projective curves, then if $f_{k^{sep}}$ is simple then so is $f_{\bar{k}}$, \end{itemize} they obtain a uniqueness result as \par {Theorem 2}. Let $C$ be a curve with genus $g$ and separable gonality $\gamma^{\mathrm{sep}}$. The gonal map over $k^{\mathrm{sep}}$ is unique if it is simple and $(\gamma^{\mathrm{sep}}-1)^2< g$. \par While the {Fact I} was known to Riemann for curves on the field of complex numbers, for arbitrary algebraically closed fields this is a special case of the authors' {Theorem 8}. The {Fact II} is also a case of their {Lemma 10}. \par The Brill-Noether theory gives various evidences to the existence and uniqueness of the gonality maps, but {Theorem 2} of their paper gives a useful criteria to determine the uniqueness of the gonal maps in the situations that the Brill-Noether theory fails to be efficient. \par The authors prove {Theorem 1} in two different ways, each of them with enlightening nature. The first one relies on the theory of Brauer-Severi varieties, which naturally leads to analogous results for maps from $C$ to $\mathbb{P}^r$. Particularly, they obtain their {Theorem 3} which is a generalization of {Theorem 1} to maps from curves to $\mathbb{P}^2$. \par Their second proof in {Section 5}, it seems there exists a typo in their addressing because they address the second proof to {Section 4}, is based on the study of Galois descent for rational normal scrolls.