---
abstract: 'Let $X$ be an irreducible smooth projective curve, of genus at least two, defined over an algebraically closed field of characteristic different from two.  If $X$ admits a nontrivial automorphism $\sigma$ that fixes pointwise all the order two points of $\text{Pic}^0(X)$, then we prove that $X$ is hyperelliptic with $\sigma$ being the unique hyperelliptic involution.  As a corollary, if a nontrivial automorphisms $\sigma''$ of $X$ fixes pointwise all the theta characteristics on $X$, then $X$ is hyperelliptic with $\sigma''$ being its hyperelliptic involution.' 
address:
- 'School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India'
- 'School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India'
author:
- Indranil Biswas
- 'A. J. Parameswaran'
title: Automorphisms of curves fixing the order two points of the Jacobian
---

Introduction
============

Let $Y$ be a compact connected Riemann surface of genus at least two.  Assume that there is a nontrivial holomorphic automorphism $$\sigma_0\, :\, Y\, \longrightarrow\, Y$$ satisfying the condition that for each holomorphic line bundle $\xi$ over $Y$ with $\xi^{\otimes 2}$ trivializable, the pull back $\sigma^*_0\xi$ is holomorphically isomorphic to $\xi$. In [@BGP] it was shown that $Y$ must be hyperelliptic and $\sigma_0$ is the unique hyperelliptic involution (see [@BGP p. 494, Theorem 1.1]). 

We recall that a theta characteristic on $Y$ is a holomorphic line bundle $\theta$ such that $\theta^{\otimes 2}$ is holomorphically isomorphic to the homomorphic cotangent bundle $K_Y$. The group of order two line bundles on $Y$ acts freely transitively on the set of all theta characteristics on $Y$. From this it follows immediately that if an automorphism of $Y$ fixes pointwise all the theta characteristics, then it also fixes pointwise all the order two line bundles on $Y$. Therefore, if $Y$ admits a nontrivial automorphism $\sigma'_0$ that fixes pointwise all the theta characteristics on $Y$, then $Y$ is hyperelliptic and $\sigma'_0$ is its unique hyperelliptic involution. 

The proof of Theorem 1.1 in [@BGP] is topological.  Here we investigate the corresponding algebraic geometric set–up, where the topological proof of Theorem 1.1 in [@BGP] is no longer valid. 

Let $X$ be an irreducible smooth projective curve defined over an algebraically closed field $k$. We will assume that $\text{genus}(X)\, >\, 1$ and $\text{char}(k)\, \not=\, 2$. We prove the following:

\[thm1\] Let $$\sigma\, :\, X\, \longrightarrow\, X$$ be a nontrivial automorphism that fixes pointwise all the theta characteristics on $X$. Then $X$ is hyperelliptic with $\sigma$ being its unique hyperelliptic involution. 

This theorem is proved by showing that if $$\sigma'\, :\, X\, \longrightarrow\, X$$ is a nontrivial automorphism of $X$ that fixes pointwise all the order two points in ${\rm Pic}^0(X)$, then $X$ is hyperelliptic with $\sigma'$ being its unique hyperelliptic involution. ( See Lemma \[lem1\].) 

It should be pointed out that Theorem \[thm1\] is not valid if the assumption that the field $k$ is algebraically closed is removed.  There exists a geometrically irreducible smooth projective real algebraic curve $Y$ of genus $g\, \geq\, 2$ which admits a nontrivial involution $\sigma$ that fixes pointwise all the real points $\xi\, \in\, \text{Pic}^{g-1}(Y)$ with $\xi^{\otimes 2}\,=\,
K_Y$, and $\text{genus}(Y/\langle \sigma\rangle)\,
\not=\,0$. (The details are in [@BG].) 

Automorphisms of polarized abelian varieties
============================================

Let $k$ be an algebraically closed field whose characteristic is different from two.  Let $A$ be an abelian variety defined over $k$ and $L$ an ample line bundle over $A$.  For any positive integer $n$, let $$\label{ker.} 
A_n\, \subset\, A$$ be the scheme–theoretic kernel of the endomorphism $A\, \longrightarrow\, A$ defined by $x\, \longmapsto\, nx$.

\[prop1\] Let $$\tau\, :\, A\, \longrightarrow\, A$$ be a nontrivial automorphism such that $\tau^*L \, =\, L\bigotimes L_0$ for some $L_0\, \in\, {\rm Pic}^0(A)$, and the restriction of $\tau$ to the subscheme $A_{n_0}$ (see Eq. )  is the identity map for some $n_0\, \geq\, 2$. Define the two endomorphisms $$f_{\pm}\,:=\, {\rm Id}_A\pm \tau\, :\, A\, \longrightarrow\, A\, .$$ Let $A_+$ (respectively, $A_-$) be the image of $f_+$ (respectively, $f_-$).  Then

1.   $n_0\,=\, 2$.

2.   $\tau^2\, =\, \tau\circ\tau$ is the identity automorphism of $A$. 

3.   The natural homomorphism $$\label{inc.} 
    \beta\, :\, A_+ \times A_-\, \longrightarrow\, A$$ defined by the inclusions of $A_+$ and $A_-$ in $A$ is an isomorphism. 

4.   The pull back $\beta^*L$ is of the form $p^*_+L_+
    \bigotimes p^*_-L_-$, where $p_+$ (respectively, $p_-$) is the projection of $A_+ \times A_-$ to $A_+$ (respectively, $A_-$). 

A proof of statement (1) is given in [@Mu p. 207, Thoerem 5].  See [@LB p. 120, Corollary 1.10] for a proof under the assumption that $k$ is the field of complex numbers. 

To prove statement (2), we will show that the restriction of $\tau^2$ to $A_4$ is the identity map.  Take any point $x\,\in\, A_4$. Then $\tau(2x)\,=\,2x$ because $2x\,\in\, A_2$. Hence $\tau(x)\,=\, x' + x$ for some $x'\,\in\, A_2$. Thus $$\tau(\tau(x))\,=\,\tau(x' + x)\,=\,\tau(x') +\tau(x) \,=\, x' + 
(x' +x) \,=\, x\, .$$ Consequently, the restriction of $\tau^2$ to $A_4$ is the identity map.  Now statement (2) follows from statement (1). 

To prove statement (3), consider the composition homomorphism $$A\,\stackrel{f_+\times f_-}{\longrightarrow}\,
A_+ \times A_- \, \stackrel{\beta}{\longrightarrow}\,  A\, ,$$ where $\beta$ is the homomorphism in Eq. .  It coincides with the endomorphism of $A$ defined by $x\, \longmapsto\, 2x$. We also note that $A_2\, \subset\, \text{kernel}(f_+\times f_-)$. Hence $$\label{kerl.} 
\text{kernel}(\beta\circ (f_+\times f_-)) \, \subset\,
\text{kernel}(f_+\times f_-)\, .$$

Since $\tau^2\, =\, \text{Id}_A$, the composition $f_+\circ
f_-$ is the zero homomorphism.  Hence $\dim (A_+ \times A_{-})\,
\leq\,\dim A$.  Now From Eq.  it follows that $\beta$ is an isomorphism. 

To prove statement (4), let $$\phi_{\beta^*L}\, :\, A_+\times A_-\, \longrightarrow\,
\text{Pic}^0(A_+\times A_-) \, =\, \text{Pic}^0(A_+)\times
\text{Pic}^0(A_-)$$ be the homomorphism that sends any $k$–rational point $x\, \in\,
A_+\times A_-$ to the line bundle $(t_x^*\beta^*L)\bigotimes \beta^*L^*$, where $t_x$ is the translation map of $A_+\times A_-$ defined by $y\, \longmapsto\,
y+x$; see [@Mu p. 131, Corollary 5] for a precise definition of the morphism $\phi_{\beta^*L}$. Let $$\tau'\, :=\, \text{Id}_{A_+}\times (-\text{Id}_{A_-})$$ be the automorphism of $A_+\times A_-$. We note that the isomorphism $\beta$ in Eq.  takes $\tau$ to $\tau'$.

Let $$\widehat{\tau}'\, :=\, \text{Id}_{\text{Pic}^0(A_+)}\times 
(-\text{Id}_{\text{Pic}^0(A_-)})$$ be the automorphism of $\text{Pic}^0(A_+)\times \text{Pic}^0(A_-)\, =\, \text{Pic}^0(A_+
\times A_-)$. Since $\tau^*L \, =\, L\bigotimes L_0$ for some $L_0
\,\in\, {\rm Pic}^0(A)$, the following diagram is commutative $$\begin{matrix}
A_+\times A_- & \stackrel{\phi_{\beta^*L}}{\longrightarrow} &
\text{Pic}^0(A_+)\times \text{Pic}^0(A_-)\\
~\,\Big\downarrow \tau' && ~\,\Big\downarrow \widehat{\tau}'\\
A_+\times A_- & \stackrel{\phi_{\beta^*L}}{\longrightarrow} &
\text{Pic}^0(A_+)\times \text{Pic}^0(A_-)
\end{matrix}$$ Therefore, the homomorphism $\phi_{\beta^*L}$ takes the subgroup $A_+$ (respectively, $A_-$) of $A_+\times A_-$ to the subgroup $\text{Pic}^0(A_+)$ (respectively, $\text{Pic}^0(A_-)$) of $\text{Pic}^0(A_+)\times 
\text{Pic}^0(A_-)$. Now from the injectivity of the homomorphism $$\text{NS}(A_+\times A_-)\, \longrightarrow\,
\text{Hom}(A_+\times A_-\, ,\text{Pic}^0(A_+)
\times \text{Pic}^0(A_-))$$ defined by $\xi\, \longmapsto\,\phi_\xi$ it follows immediately that the Néron–Severi class of $\beta^*L$ coincides with that of some line bundle of the form $p^*_+L_+ \bigotimes p^*_-L_-$ (see [@Mu p. 178] for the injectivity of the above homomorphism).  Therefore, statement (4) follows using the fact that $\text{Pic}^0(A_+)\times \text{Pic}^0(A_-)\, =\, \text{Pic}^0(A_+
\times A_-)$. This completes the proof of the proposition. 

Automorphisms and theta characteristics
=======================================

Let $X$ be an irreducible smooth projective curve, of genus at least two, defined over the field $k$.

\[lem1\] Let $$\sigma\, :\, X\, \longrightarrow\, X$$ be a nontrivial automorphism of $X$ that fixes pointwise all the order two points ${\rm Pic}^0(X)_2\, \subset\,
{\rm Pic}^0(X)$. then $X$ is hyperelliptic with $\sigma$ being its unique hyperelliptic involution. 

Let $\text{Pic}^d(X)$ denote the moduli space of line bundles over $X$ of degree $d$. Let $g$ denote the genus of $X$. On $\text{Pic}^{g-1}(X)$, we have the theta divisor $\Theta$ given by the locus of the line bundles admitting nontrivial sections.  Fix a $k$–rational point $x_0\, \in\, X$. Let $L$ be the pull back of the line bundle ${\mathcal O}_{\text{Pic}^{g-1}(X)}
(\Theta)$ by the morphism $\text{Pic}^0(X)\, \longrightarrow\, 
\text{Pic}^{g-1}(X)$ that sends any $\zeta$ to $\zeta\bigotimes
{\mathcal O}_X((g-1)x_0)$.

Let $\tau\, :\, \text{Pic}^0(X)\, \longrightarrow\,
\text{Pic}^0(X)$ be the automorphism defined by $\zeta\,
\longmapsto\, \sigma^*\zeta$. This $\tau$ satisfies the conditions in Proposition \[prop1\].  Hence $\tau$ is an involution (see Proposition \[prop1\](2)).  This implies that $\sigma$ is an involution. 

A hyperelliptic smooth projective curve $Y$ of genus at least two admits a unique involution $\sigma_Y$ such that $\text{genus}(Y/\langle\sigma_Y\rangle)\, =\, 0$. Therefore, to complete the proof of the lemma it suffices to show that $\text{genus}(X/\langle\sigma\rangle)\, =\, 0$. We note that the theta divisor $\Theta$ on $\text{Pic}^{g-1}(X)$ is irreducible.  Indeed, it is the image of $\text{Sym}^{g-1}(X)$ by the obvious map.  Also, $h^0({\mathcal O}_{\text{Pic}^{g-1}(X)}(\Theta))\, =\,1$ because $\Theta$ defines a principal polarization. 

On the other hand, any ample hypersurface of the form $(A_+\times D_{-})\bigcup (D_+\times A_{-})$ on $A_+\times A_{-}$ is never irreducible unless at least one of $A_+$ and $A_{-}$ is a point; here $D_+$ (respectively, $D_{-}$) is a hypersurface on $A_+$ (respectively, $A_{-}$).  Therefore, from statement (4) of Proposition \[prop1\] and the irreducibility of $\Theta$ we conclude that either $\dim A_+ \, =\, 0$ or $\dim A_{-} \, =\, 0$. But $\dim A_{-} \, =\, \text{genus}(X)-
\text{genus}(X/\langle\sigma\rangle)$, and $\dim A_{+} \, =\, \text{genus}(X/\langle\sigma\rangle)$. Since $\text{genus}(X)\, >\, \text{genus}(X/\langle\sigma\rangle)$, we now conclude that $\text{genus}(X/\langle\sigma\rangle)
\,=\, 0$. This completes the proof of the lemma. 

A line bundle $\theta$ is called a *theta characteristic* of $X$ if $\theta^{\otimes 2}$ is isomorphic to the canonical line bundle $K_X$ of $X$. The space of theta characteristics on $X$ is a principal homogeneous space for ${\rm Pic}^0(X)_2$. Therefore, if an automorphism $\sigma$ of $X$ fixes pointwise all the theta characteristics on $X$, then $\sigma$ fixes ${\rm Pic}^0(X)_2$ pointwise.  Consequently, the following theorem is deduced from Lemma \[lem1\]. 

\[thm2\] Let $\sigma\, :\, X\, \longrightarrow\, X$ be a nontrivial automorphism that fixes pointwise all the theta characteristics on $X$. Then $X$ is hyperelliptic with $\sigma$ being its unique hyperelliptic involution. 

[AAAA]{}

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