\documentclass[11pt]{article} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{lmodern} \usepackage{amsmath,amssymb,amsthm} \usepackage{mathtools} \usepackage{amscd} \usepackage[all]{xy} \usepackage{enumitem} \usepackage[colorlinks=true,linkcolor=black,citecolor=black,urlcolor=black]{hyperref} \usepackage[margin=1.15in]{geometry} % ---------- Macros ---------- \newcommand{\CC}{\mathbb{C}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\F}{\mathbb{F}} \newcommand{\calO}{\mathcal{O}} \DeclareMathOperator{\Pic}{Pic} \DeclareMathOperator{\NS}{NS} \DeclareMathOperator{\End}{End} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\rk}{rk} \DeclareMathOperator{\degop}{deg} \DeclareMathOperator{\Nm}{Nm} \DeclareMathOperator{\Gal}{Gal} \DeclareMathOperator{\Spec}{\mathbf{Spec}} \renewcommand{\deg}{\degop} \newcommand{\isom}{\simeq} \newcommand{\into}{\hookrightarrow} \newcommand{\onto}{\twoheadrightarrow} \newcommand{\ee}{\mathrm{e}} \newcommand{\wA}{\widehat{A}} \newcommand{\wT}{\widehat{T}} \newcommand{\wt}{\widehat{\tau}} \newcommand{\cM}{\mathcal{M}} \newcommand{\cL}{\mathcal{L}} \newcommand{\cN}{\mathcal{N}} \newcommand{\cO}{\mathcal{O}} \newcommand{\cP}{\mathcal{P}} \newcommand{\cR}{\mathcal{R}} \newcommand{\cQ}{\mathcal{Q}} % ---------- Theorem environments ---------- \theoremstyle{plain} \newtheorem{theorem}{Theorem}[section] \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newtheorem{warning}[theorem]{Warning} \numberwithin{equation}{section} \title{\textbf{A Counterexample to Beauville's\\ Stability Conjecture}\\[4pt]} \author{Yi Gu \and Lifan Guan} \begin{document} \maketitle \begin{abstract} We construct a morphism $f: X\to Y$ from a genus-two curve to an elliptic curve such that $f_*\mathcal L$ is unstable for every $\mathcal L\in\Pic^3(X)$. This gives a counterexample to a conjecture of Beauville. \end{abstract} \paragraph{Keywords.} Direct images, stability, Frey--Kani construction. \paragraph{2020 Mathematics Subject Classification.} 14H60, 14H40, 14K02. %===================================================================== \section{Introduction}\label{sec:intro} Let $f\colon X\to Y$ be a finite morphism of smooth projective curves. The stability of vector bundles under pullback and direct image has been studied extensively; see \cite{BiswasDasParameswaran,BiswasParameswaran,Gieseker,JRXU,Langer,SunDirect,SunSurvey,Weissmann}. In particular, direct images need not preserve semistability. In \cite{Beauville}, Beauville proposed a conjecture concerning the stability of $f_*\mathcal E$ for a \textit{general} vector bundle $\mathcal E$ on $X$ and reduced it to the line-bundle case. Namely: \begin{conjecture}[Beauville \cite{Beauville}]\label{conj:beauville} Let $f\colon X\to Y$ be a finite morphism of smooth projective connected complex curves. For every $d\in\mathbb Z$ and a general line bundle $\mathcal L\in\Pic^d(X)$, the direct image $f_*\mathcal L$ is stable if $g(Y)\geq2$ and semistable if $g(Y)=1$. \end{conjecture} Since being stable/semistable is an open condition, it suffices to prove there exists one such $\mathcal{L}\in \Pic^d(X)$. Beauville proved stability/semi-stability in a range controlled by the Euler characteristic, together with several further cases \cite[\S~3--4]{Beauville}. The conjecture has recently been established when the cover itself is general: Coskun, Larson, and Vogt proved it for $f$ that is \textit{general} in any component of the relevant Hurwitz space \cite{CLV}. On the other side, we will give a concrete counterexample to Conjecture~\ref{conj:beauville} in this paper. \begin{theorem}\label{thm:main} There exist a smooth projective curve $X$ of genus $2$, an elliptic curve $Y$, and a ramified finite morphism $f\colon X\to Y$ of degree $7$ such that $f_*\mathcal L$ is unstable for every $\mathcal L\in\Pic^3(X)$. \end{theorem} This counterexample is a $(2,7)$-elliptic curve of genus $2$, in the sense studied by Franciosi, Pardini, and Rollenske \cite{FPR}. The construction is essentially motivated by the Frey--Kani construction of genus-two curves from isogenies and anti-symplectic torsion gluings \cite{FreyKani}. The curve $X$ is equipped with another morphism $b\colon X\to B$ to an elliptic curve $B$ of degree $2$. These curves are fitted into the following commutative diagram of fibre products, where the vertical maps $v$, $p$, and $q$ are \'{e}tale covers of degree $3$. $$\xymatrix{ B' \ar[d]_q && X'\ar[ll]_{b'} \ar[rr]^{f'}\ar[d]^v && Y'\ar[d]^p \\ B && X \ar[ll]_b \ar[rr]^f && Y} $$ We will show in \S~\ref{subsec:commoncover} that for every $\mathcal L\in\Pic^3(X)$, there are $\mathcal N\in\Pic^1(Y')$ and $\mathcal M\in\Pic^1(B')$ such that \[ v^*\mathcal L=f'^*\mathcal N\otimes b'^*\mathcal M. \] Then we will prove in \S~\ref{sec:proof} that $\mathcal A_{\mathcal L}:=p_*\mathcal N$ admits a nonzero morphism to $f_*\mathcal L$. By \cite{Atiyah, Beauville}, $\mathcal A_{\mathcal L}=p_*\mathcal N$ is stable. A simple calculation gives: \[ \mu(\mathcal A_{\mathcal L})=\frac13>\frac27=\mu(f_*\mathcal L). \] Therefore, the saturated image of $\mathcal A_{\mathcal L}$ destabilizes $f_*\mathcal L$. Although the example above disproves Conjecture~\ref{conj:beauville}, we also prove a complementary positive result for Galois covers. The precise statement is the following. \begin{theorem}\label{thm:galois-generic} Let $h\colon X\to Y$ be a finite ramified \emph{Galois} morphism between smooth projective connected complex curves. Assume that $g(Y)\geq 1$, then there is a nonempty Zariski open dense subset $U_d\subseteq\Pic^d(X)$ for every integer $d$ such that $h_*\mathcal L$ is semistable for $\mathcal L\in U_d$. \end{theorem} If $h$ is unramified, then the semistability is already obtained by \cite[\S~4]{Beauville}. This paper is organized as follows: \S~\ref{Sec: Preliminaries} provides necessary preliminaries, \S~\ref{sec:counterexample} gives the degree-$7$ counterexample and \S~\ref{sec:galois} proves the Galois-cover theorem. Experts can skip \S~\ref{Sec: Preliminaries} at first. %\textbf{AI disclosure}: \section{Preliminaries}\label{Sec: Preliminaries} We work over $\CC$. All curves are smooth, projective and connected unless otherwise stated. \subsection{Stability of vector bundles on curves} For a nonzero vector bundle $\mathcal V$ on a smooth projective curve, its \emph{slope} is $\mu(\mathcal V):=\deg(\mathcal V)/\rk(\mathcal V)$. \begin{definition} A vector bundle $\mathcal V$ is \emph{stable} (respectively, \emph{semistable}) if \[ \mu(\mathcal V')>\mu(\mathcal V), \,\,\,\,\,\,\,\,\,(\text{respectively,} \,\,\,\,\,\,\,\,\,\mu(\mathcal V')\geq\mu(\mathcal V)) \] for every proper nonzero quotient bundle $\mathcal V\twoheadrightarrow \mathcal V'$. We say $\mathcal V$ is \emph{unstable}, if it is not semistable. \end{definition} The following results on stability are well known. \begin{proposition}\label{Prop: stable} (1) Let $f: C\to D$ be a separable morphism of curves, $\mathcal E$ be a vector bundle on $D$, then $\mathcal{E}$ is semistable if and only if $f^*\mathcal{E}$ is so. In particular, $\mathcal E$ is unstable if $f^*\mathcal{E}$ is unstable. (2) (see \cite[\S~4]{Beauville}) Let $f: C\to D$ be an \'etale morphism of curves, then for any semistable vector bundle $\mathcal E$, $f_*\mathcal{E}$ is semistable. \end{proposition} \subsection{On Abelian varieties} Let $A$ be an Abelian variety, $\widehat{A}$ be the dual Abelian variety. Denote by $e_n: A[n]\times \wA[n]\to \mu_n$ the Weil pair. \subsubsection{On theta group of Abelian varieties} We recall the theory of theta group on Abelian varieties. The standard reference is \cite[Chapter~IV]{Mumford}. Given an ample line bundle $\mathcal M$ on $A$, translation defines the polarization \[ \lambda_{\mathcal M}\colon A\longrightarrow\widehat A=\Pic^0(A), \qquad a\longmapsto t_a^*\mathcal M\otimes\mathcal M^{-1}, \] and we write $K(\mathcal M)=\ker\lambda_{\mathcal M}$ the translation stabilizer. The theta group $\mathcal{G}(\mathcal{M})$ of $\mathcal M$ is the central extension \[ 1\longrightarrow\CC^*\longrightarrow\mathcal G(\mathcal M) \longrightarrow K(\mathcal M)\longrightarrow 1, \] classifying the pair $(a,\tau)$, where $a\in K(\mathcal M)$ and $\tau: T_a^* \mathcal M \simeq \mathcal{M}$ is an isomorphism. Its commutators define the nondegenerated bilinear alternating pairing, known as the \textit{theta pairing} \[ e^{\mathcal M}\colon K(\mathcal M)\times K(\mathcal M)\longrightarrow\CC^*,\,\,\,\,\,\,\,\, (a,b)\mapsto \overline{\mathstrut a}\cdot\overline{\mathstrut b}\cdot \overline{\mathstrut a}^{-1}\cdot \overline{\mathstrut b}^{-1}, \] for any lifting $\overline{\mathstrut a},\overline{\mathstrut b}\in \mathcal{G}(\mathcal{M})$ of $a,b\in K(\mathcal{M})$. A finite subgroup $H\subseteq K(\mathcal M)$ is \emph{isotropic} if $e^{\mathcal M}|_{H\times H}=1$. \begin{theorem}[Chapter~IV, \S23 \cite{Mumford}]\label{Thm: descend} (1) If $H\subseteq K(\mathcal M)$ is isotropic, then the line bundle $\mathcal M$ descends to the quotient $\pi\colon A\to A/H$. If $H$ is furthermore maximal isotropic, the descended polarization is principal. (2) We have $$e_n(a,\lambda_{\mathcal{M}}(b))=e^{\mathcal{M}^{\otimes n}}(a,b),$$ for any $a\in A[n], b\in [n]^{-1}(K(\mathcal{M}))$. \end{theorem} \begin{remark}\label{rem: numerical equiv} It is well known that the $\lambda_\cM, K(\mathcal{M}), \mathcal{G}(\mathcal{M})$ and $e^{\mathcal{M}}$ all depend only on the numerical class of $\mathcal{M}$. In particular, in the elliptic curve case, all these depend only on $\deg\mathcal{M}$. \end{remark} \subsection{On elliptic curves} Given an elliptic curve $(E,o)$, denote by $\lambda_E$ the canonical principal polarization associated to the line bundle $\cM=\cO_E(o)$: \[ \lambda_E=\lambda_{\cM}: E\simeq \widehat{E}, \,\,\,\,\, x\mapsto t_x^*\cM\otimes \cM^{-1}=\cO_E(x-o). \] We denote by $e_{n,E}$ the theta pairing on $E[n]$ giving by any line bundle of degree $n$ (see Remark~\ref{rem: numerical equiv}). Following from Theorem~\ref{Thm: descend}(2), $$e_{n,E}(a,b)=e_n(a,\lambda_E(b)), \,\,\,\,\,\,\, \forall a,b\in E[n].$$ By abuse of language, we also call this $e_n$ the Weil pairing. \subsection{On the decomposition of cyclic cover of Elliptic curves} Let $\pi_n=[n]: A \longrightarrow A$ be the multiplication-by-$n$ map. Then $\pi_n$ is a Galois cover with Galois group $A[n]$. Following \cite[Example~11.18]{vdGM}, we have the following decomposition: $$ (\pi_n)_*\mathcal{O}_A=\bigoplus\limits_{\chi} \mathcal{L}_\chi, $$ where \begin{itemize} \item $\chi$ runs through all multiplicative character of $A[n]$; \item $\mathcal{L}_\chi$ is the line subbundle of $(\pi_n)_*\mathcal{O}_A$ so that the translation operator $$t_a^*|_{\mathcal{L}_\chi}=\chi(a)\cdot \mathrm{id}, \,\,\,\,\,\,\,\forall a\in A[n].$$ \item $\cL_\chi\in \wA[n]$ is such that $e_n(a,\mathcal{L}_\chi)=\chi(a), \,\,\, \forall a\in E[n]$. \end{itemize} Let $u: F\to E$ be a finite homomorphism of elliptic curves of degree $n$. Then the multiplication-by-$n$ map of $E$ factors through $u$ as: $$\pi_n: E \stackrel{v}{\longrightarrow} F \stackrel{u}{\longrightarrow} E.$$ And thus $v$ is a Galois cover with Galois group $G:=\mathrm{Ker}(v)$ and $u$ with Galois group $\mathrm{Ker}(u)=E[n]/G$. The above decomposition of $(\pi_n)_*\cO_E$ then gives: $$u_*\mathcal{O}_F=\bigoplus\limits_{\chi,\chi(G)=1}\mathcal{L}_\chi=\left(\bigoplus\limits_{\chi}\mathcal{L}_\chi\right)^G\subseteq \bigoplus\limits_{\chi}\mathcal{L}_\chi=(\pi_n)_*\mathcal{O}_E.$$ It is classical that $v$ is dual to $u$ in the sense of canonical principal polarizations: $$v=\lambda_F^{-1}\cdot \widehat{u} \cdot \lambda_E.$$ In other words, given $a\in E[n]$, $\cM:=\lambda_E(a)=\cO_E(a-o_E)$, then \[u^*\cM= \cO_F(v(a)-o_F)=\lambda_F(v(a)).\] In summary, we have \begin{proposition}\label{prop: decomposition} Given any $\cM\in \widehat{E}[n]$, and $b\in \mathrm{Ker}(u)$ then $u^*\cM= \lambda_F(b)$ if and only if $$\chi(b)=e_{n,\widehat{E}}(\cL_\chi,\cM)$$ for all $\chi$ with $\chi|_G\equiv 1$. \end{proposition} \begin{proof} We can write $\cM=\cO_E(a-o_E)= \lambda_E(a)$ for some $a\in E[n]$, then $u^*\cM= \lambda_F(b)$ if and only if $b=v(a)$. That is equivalent to \[\chi(b)= \chi(a)=e_n(\cL_\chi,a)=e_{n,\widehat{E}}(\cL_\chi, \lambda_E(a))=e_{n,\widehat{E}}(\cL_\chi,\cM),\] for all $\chi$ with $\chi_G\equiv 1$. \end{proof} \section{The counterexample to Conjecture~\ref{conj:beauville}}\label{sec:counterexample} %===================================================================== \subsection{A genus-two Jacobian with two elliptic directions}\label{sec:genus2} %===================================================================== We now construct the geometric input to Theorem~\ref{thm:main}. The construction is a specialization of the Frey--Kani construction as organized in \cite[Example~3.9]{FPR}. We include the intersection and torsion calculations because they are essential for the counterexample. Choose an elliptic curve $F$ with $\End(F) = \Z$, and choose a cyclic subgroup $\Gamma_5 \subset F$ of order $5$. Put \[ E:= F, \qquad E':= F/\Gamma_5, \] and consider \[ \varphi = [3] \colon F \longrightarrow E, \qquad \varphi' \colon F \longrightarrow E' = F/\Gamma_5. \] Thus \[ \deg \varphi = 9, \qquad \deg \varphi' = 5, \qquad \ker \varphi \cap \ker \varphi' = F[3] \cap \Gamma_5 = \{0\}. \] The homomorphism \[ i = (\varphi, \varphi') \colon F \into E \times E' \] is therefore an embedding. Set \[ D_1: = E \times \{0\}, \qquad D_2: = \{0\} \times E', \qquad H: = i(F[2]) \subset (E \times E')[2]. \] and $\cL_{2,2}:=\cO(2D_1+2D_2)$ on $E\times E'$. \begin{lemma}\label{lem: H is isotropic} We have $K(\cL_{2,2})=(E\times E')[2]$ and the subgroup $H$ is maximal isotropic for the theta pairing $e^{\cL_{2,2}}$. \end{lemma} \begin{proof} The line bundle $\cL_{2,2}$ is of type $(2,2)$ on $E\times E'$, so $K(\cL_{2,2})=(E\times E')[2]$ and $$ e^{\cL_{2,2}}(i(a),i(b))=e_{2,E}(\varphi(a),\varphi(b))\cdot e_{2,E'}(\varphi'(a),\varphi'(b)). $$ for any $a,b \in F[2]$. On the other hand, following the functoriality of theta pairing, we have $$ e_{2,E}(\varphi(a),\varphi(b))=e_{2\cdot \deg\varphi, F}(a,b)=e_{2,F}(a,b)^{\deg \varphi}=e_{2,F}(a,b)^9=e_{2,F}(a,b), $$$$ e_{2,E'}(\varphi'(a),\varphi'(b))=e_{2\cdot \deg\varphi', F}(a,b)=e_{2,F}(a,b)^{\deg \varphi'}=e_{2,F}(a,b)^5=e_{2,F}(a,b). $$ Thus $e^{\cL_{2,2}}(i(a),i(b))=e_{2,F}(a,b)^2\equiv 1.$ Namely, $H$ is isotropic. It is maximal since $H$ has half $\Z/2\Z$ dimension of $K(\cL_{2,2})$. \end{proof} Denote by \[ \pi: E\times E' \to A:=E\times E'/H \] the quotient Abelian variety by $H$. Following Theorem~\ref{Thm: descend} and Lemma~\ref{lem: H is isotropic}, $\cL_{2,2}$ descends to a principal polarization $\Theta$ on $A$. This is the degree-$2$ Frey--Kani construction \cite[Proposition~2.3]{FPR}. \begin{lemma}\label{lem:indec} The principal polarization $\Theta$ is indecomposable. This theta divisor is a smooth curve $X$ of genus $2$, and $(A, \Theta) \isom (J(X), \Theta_X)$. \end{lemma} \begin{proof} The standard classification of principally polarized abelian surfaces gives two possibilities: either the wanted case: (1) $\Theta$ is indecomposable, and $(A,\Theta)\simeq(J(X),\Theta_X)$ for a smooth genus-two curve $X$; or the case to be excluded: (2) $\Theta=C_1+C_2$ were reducible, the $C_i$ would be elliptic curves with $C_1\cdot C_2=1$. Let $Z_i=\pi(D_i), i=1,2$. By construction, the map $\pi: D_i\to Z_i$ is an isomorphism, hence $\pi^*Z_i\equiv 4D_i$ as $\deg\pi=4$. So \[ Z_1\cdot Z_2=\frac{\pi^*Z_1 \cdot \pi^* Z_2}{4}=4D_1\cdot D_2=4, \] and \[2\Theta\equiv Z_1+Z_2, \,\,\,\,\,\text{since} \,\,\,\,\, 2\pi^*\Theta=4(D_1+2D_2)\equiv \pi^*(Z_1+Z_2). \] If $\Theta=C_1+C_2$ were reducible, intersecting with $2\Theta\equiv Z_1+Z_2$ so that $(C_1+C_2)\cdot (Z_1+Z_2)=4$ forces either $C_1\cdot Z_1=C_1\cdot Z_2=1$ or $C_1\cdot Z_1=0, C_1\cdot Z_2=2$. In the first case, quotient by $C_1$ gives isomorphisms $A/C_1\simeq Z_i$ for both $i=1,2$. A contradiction to the fact $Z_1\simeq E$ and $Z_2\simeq E'$ are not isomorphic, see the Lemma~\ref{lemma: non-isomorphic} in the following. In the latter case, we have $C_1\equiv Z_1$. A contradiction to that $Z_1\cdot Z_2=4, C_1\cdot Z_2=2$. We are done. \end{proof} \begin{lemma}\label{lemma: non-isomorphic} The elliptic curves $E$ and $E'$ are not isomorphic. \end{lemma} \begin{proof} Otherwise, an isomorphism \(E\simeq E'\) would produce an endomorphism of \(F\) of degree \(5\), impossible because \(\End(F)=\mathbb Z\). \end{proof} From now on, we identify $X=\Theta$ and $J(X)=A$. \subsection{Elliptic directions of degrees \texorpdfstring{$2$}{2} and \texorpdfstring{$7$}{7}}\label{subsec:directions} Next we construct the two morphisms from $X$ to two elliptic curves. Given an elliptic curve $T$, and a non-zero homomorphism $\tau: T \to J(X)$, it dual homomorphism $\wt: J(X)\to \wT$ induces a finite map $j: X\hookrightarrow J(X) \stackrel{\wt}{\longrightarrow} \wT$ onto an elliptic curve. \begin{proposition}\label{prop: degree of dual map} We have identity $\deg j=\deg \tau^*\Theta$. \end{proposition} \begin{proof} The map $\wt$ can be identified with $\mathrm{Alb}(j)$ while $\tau$ is identified with $j^*: T=\Pic^0(\wT) \to J(X)=\Pic^0(X)$. In particular, it is classical that $\wt\cdot \lambda_\Theta \cdot \tau=[\deg j]\cdot \lambda_T$, see \cite[{\S}3, proof of Theorem~3.7, pp.~161--162]{BeauvillePrym}. On the other hand, following from the functoriality of the $\lambda$ map, we have $\wt\cdot \lambda_\Theta \cdot \tau=\lambda_{j^*\Theta}$. $$ \xymatrix{ T \ar[rr]^\tau \ar[d]_{\lambda_{\tau^*\Theta}=[j]\cdot \lambda_T} && J(X)\ar[d]^{\lambda_{\Theta}} \\ \widehat{T} && \ar[ll]_{\widehat{\tau}} \widehat{J}(X) } $$ Counting degrees then gives: \[ (\deg j)^2[\deg j]\cdot \lambda_T=\lambda_{j^*\Theta}=(\deg j^*\Theta)^2. \] We are done. \end{proof} Inside $A = J(X)$, we have two elliptic subcurves: \[ S_2 = \pi(D_2), \qquad S_7 = \pi(i(F)). \] The restriction of $\pi$ to $D_2$ is injective, while its restriction to $i(F)$ has kernel $i(F[2])$ and degree $4$. The projection formula gives \[ \Theta \cdot S_2 = \pi^*\Theta \cdot D_2 = 2, \] and \[ 4(\Theta \cdot S_7) = \pi^*\Theta \cdot i(F) = 2(D_1 + D_2) \cdot i(F) = 2(\deg \varphi' + \deg \varphi) = 2(5 + 9) = 28. \] Thus \[ \Theta \cdot S_2 = 2, \qquad \Theta \cdot S_7 = 7. \] By the beginning of this subsection, we therefore obtain elliptic curves $B=\widehat{S}_2$, $Y=\widehat{S}_7$ and finite morphisms \[ b \colon X \longrightarrow B, \quad \deg b = 2, \qquad f \colon X \longrightarrow Y, \quad \deg f = 7, \] chosen so that we have the following commutative diagram: $$ \xymatrix{ \Pic^0(B)\ar[d]^{\simeq }\ar[rr]^{b^*} && \mathrm{Pic}^0(X)\ar@{=}[d] &&\Pic^0(Y)\ar[d]^{\simeq }\ar[ll]_{f^*}\\ S_2\ar@{^{(}->}[rr] && J(X) && S_7\ar@{_{(}->}[ll] } \label{eq:directions} $$ Moreover, these two elliptic curves $S_2$ and $S_7$ share their $3$-torsion. \begin{proposition}\label{prop:3torsion} The two elliptic directions satisfy \[ S_2 \cap S_7 = \pi(i(F[3])) = S_2[3] = S_7[3]. \] In particular, there is a unique isomorphism \[ \alpha \colon \Pic^0(Y)[3]\stackrel{f^*}{\simeq} S_7[3]=S_2[3]\stackrel{{b^*}^{-1}}{\simeq} \Pic^0(B)[3] \] such that \begin{equation}\label{eq:alpha} f^*a = b^*\alpha(a) \qquad \text{for every } a \in \Pic^0(Y)[3]. \end{equation} This isomorphism is anti-symplectic with respect to the pairing $e_{3,S_7}$ and $e_{3,S_2}$. \end{proposition} \begin{proof} For $x\in F$, $i(x)=(3x,\overline{x})\in E\times E'$. By construction $\pi(i(x))\in S_7$ belongs to $S_2$ if $\pi(i(x))=\pi(0,\overline{y})$ for some $\overline{y}\in E'$. Namely, $$(3x,\overline{x}-\overline{y})= i(x')=(3x',\overline{x}')$$ for some $x'\in F[2]$. Namely $x$ is a $3$-torsion after modulo $F[2]$. In other words, $S_2\cap S_7=S_7[3]$. The other equality $S_2\cap S_7=S_2[3]$ follows immediately then. Note that the two natural projections $F \to S_i, i=2,7$ has degree $5,4$ coprime to $3$, so the natural map $\pi\cdot i: F[3]\to S_2[3]=S_7[3]$ is an isomorphism. For any $(a,b)\in S_2[3]=S_7[3]$, choose $a',b' \in F[3]$ so that $a=\pi(a'),b=\pi(i(b'))$. Then the functoriality of theta pairing gives: \begin{align*} e_{3,S_7}(a,b)&=e_{12,F}(a',b')=e_{3,F}(a',b')^4=e_{3,F}(a',b'),\\ e_{3,S_2}(a,b)&=e_{15,F}(a',b')=e_{3,F}(a',b')^5=e_{3,F}(a',b')^{-1}. \end{align*} Hence the isomorphism $\alpha$ is anti-symplectic. \end{proof} \subsection{The common cover}\label{subsec:commoncover} Fixing any $\eta_Y\in \Pic^0(Y)[3]$, set $\eta_B:=\alpha(\eta_Y)\in \Pic^0(B)[3]$, then $\eta:=f^*\eta_Y=b^*\eta_B$ in $\Pic^0(X)[3]$. In particular, the \'etale cyclic covers $p: Y'\to Y$ and $q: B'\to B$ defined by \begin{align*} \cO_{Y'}=\cO_Y\oplus \eta_Y \oplus \eta_Y^{\otimes 2}\\ \cO_{B'}=\cO_B\oplus \eta_B \oplus \eta_B^{\otimes 2} \end{align*} pull back to a common \'etale cover on $X$. Namely, we have the following commutative cartesian diagram $$\xymatrix{ B' \ar[d]_q && X'\ar[ll]_{b'} \ar[rr]^{f'}\ar[d]^v && Y'\ar[d]^p \\ B && X \ar[ll]_b \ar[rr]^f && Y}. $$ We then fix a generator $\sigma$ (resp. $\sigma_B, \sigma_Y$) of the cyclic Galois group of the \'etale covers $v$ (resp. $q, p$) such that it acts on $\eta$ (resp. $\eta_B, \eta_Y$) by a fixed primitive cubic root $\mu_3$. Given elliptic curve structures on $Y'$ and $B'$ so that $p,q$ are homomorphisms, then the Galois automorphism $\sigma_Y, \sigma_B$ are realized by translations $\sigma_Y=t^*_{\overline{y}}$ and $\sigma_B=t^*_{\overline{b}}$ for unique $\overline{y}\in \mathrm{Ker}(p)$ and $\overline{y}\in \mathrm{Ker}(q)$. Thus for any line bundle $\cN_1$ of degree $1$ on $Y'$, we have \[ \sigma_Y^*\cN_1\otimes \cN_1^{-1}=t_{\overline{y}}^*\cN_1\otimes \cN_1^{-1}=\cO_{Y'}(\overline{y}-o_{Y'})=\lambda_Y(\overline{y}). \] \begin{lemma} \label{lem:deck} For $\cP\in \Pic^0(Y)[3]$, $p^*\cP\simeq \sigma^*\cN_1\otimes \cN_1^{-1}$ if and only if $e_{3,\widehat{Y}}(\eta_Y,\cP)=\mu_3$. \end{lemma} \begin{proof} By construction, \[t^*_{\overline{y}}|_{\eta_Y}=\mu_3\cdot \mathrm{id}, \quad t^*_{\overline{y}}|_{\eta^{\otimes 2}_Y}=\mu^2_3\cdot \mathrm{id}.\] It follows from Proposition~\ref{Sec: Preliminaries}, $p^*\cP\simeq \sigma^*\cN_1\otimes \cN_1^{-1}=\lambda_{Y}(\overline{y})$ if and only if \[ e_{3,\widehat{Y}}(\eta_Y,\cP)=\mu_3, \quad e_{3,\widehat{Y}}(\eta^{\otimes 2}_Y,\cP)=\mu^2_3. \]We are done. \end{proof} Now fixing one such $\cP$ and take $\cQ=\alpha(\cP)\in \Pic^0(B)[3]$. Then for any $\cM_1$ of degree one on $B'$ we have $$q^*\cQ^{-1}\simeq \sigma_B^*\cM_1\otimes \cM_1^{-1}.$$ In fact, it follows from the fact $\alpha$ is anit-symplectic that $e_{3,\widehat{B}}(\eta_B,\cQ^{-1})=e_{3,\widehat{Y}}(\eta_Y,\cP)=\mu_3$. \begin{lemma} Take $\cR:=f'^*\cN_1\otimes b'^*\cM_1$, with $\cN_1,\cM_1$ line bundles on $Y',B'$ of degree one as above, then $\sigma^*\cR\simeq \cR$. \end{lemma} \begin{proof} We have: \begin{align*} \sigma^*\cR\otimes \cR^{-1}&\simeq f'^*(\sigma_Y^*\cN_1\otimes \cN_1^{-1})\otimes b'^*(\sigma_B^*\cM_1\otimes \cM_1^{-1})\\ &\simeq f'^*p^*\cP\otimes b'^*q^* \cQ^{-1}\\ &\simeq v^*(f^*\cP\otimes b^*\cQ^{-1})\\ &\simeq v^*\cO_X\\ &\simeq \cO_{X'}. \end{align*} \end{proof} Since $v: X'\to X$ is a cyclic \'etale cover with generator $\sigma$, this $\cR$ can descend to a line bundle $\cL_1$ on $X=X'/<\sigma>$. Computation gives its degree: $$ \deg \cL_1=\dfrac{\deg(\cR)}{3}=\dfrac{2+7}{3}=3. $$ \begin{corollary}~\label{cor:factor} For any line bundle $\cL\in \mathrm{Pic}^3(X)$ of degree $3$, there are $\cN\in \mathrm{Pic}^1(Y')$ and $\cM\in\mathrm{Pic}^1(B')$ such that $v^*\cL\simeq f'^*\cN\otimes b'^*\cM$. \end{corollary} \begin{proof} By the previous lemma, $v^*\cL_1=f'^*\cN_1\otimes b'^*\cM_1$. Note $\cL$ differs from $\cL_1$ by an line bundle in $\Pic^0(X)$ and the homomorphism \[ \Phi=(f^*,b^*) \colon \Pic^0(Y) \times \Pic^0(B) \longrightarrow \Pic^0(X) \] is an isogeny, in particular surjective. The corollary then follows. \end{proof} %===================================================================== \subsection{Instability of the direct image}\label{sec:proof} %===================================================================== We can now prove the instability \begin{proof}[Proof of Theorem~\ref{thm:main}] Fix $\cL\in\Pic^3(X)$ and choose $\cN,\cM$ as in Corollary~\ref{cor:factor}. A simple calculation gives that $f_*\cL$ has degree $2$ and rank $7$. Set $\mathcal A_{\cL}:=p_*\cN$. It is a semistable bundle (see \cite[\S~4]{Beauville}) of degree $1$ and rank $3$. Noticing a degree-one line bundle on an elliptic curve has a section, so $H^0(X',b'^*\cM)\ne0$. Recall in finite \'etale case, $p^*$ is also the right adjoint of $p_*$, we have: \[ \begin{aligned} \operatorname{Hom}_Y(\mathcal A_\cL,f_*\cL)&=\operatorname{Hom}_Y(p_*\cN,f_*\cL)\\ &\simeq \operatorname{Hom}_{Y'}(\cN,p^*f_*\cL)\\ &\simeq \operatorname{Hom}_{Y'}(\cN,f'_*v^*\cL)\\ &\simeq \operatorname{Hom}_{X'}(f'^*\cN,v^*\cL)\\ &=H^0(X',v^*\cL\otimes f'^*\cN^{-1})\\ &=H^0(X',b'^*\cM)\ne0. \end{aligned} \] But \[ \mu(\mathcal A_{\cL})=\frac{1}{3}>\frac{2}{7}=\mu(f_*\cL). \] So $f_*\cL$ is not semistable. \end{proof} \begin{remark}\label{rem:basechange} One can also argue as: $p^*f_*\cL=f'_*(v^*\cL)=\cN\otimes f'_*b'^*\cM$. As $\cM$ is effective, there is an embedding $\cN\to p^*f_*\cL$. As $$\mu(\cN)=1>\mu(p^*f_*\cL)=6/7,$$ $p^*f_*\cL$ is unstable. So is $f_*\cL$, see Proposition~\ref{Prop: stable}. \end{remark} %===================================================================== \section{Generic semistability for Galois covers}\label{sec:galois} The counterexample above concerns a fixed non-Galois cover. We now prove Theorem~\ref{thm:galois-generic}, beginning with two auxiliary lemmas. The argument shows that the instability constructed here depends on the special cover and on the chosen degree component of its Picard variety. \begin{definition} A finite morphism $h\colon X\to Y$ is \emph{genuinely ramified} if it does not factor as $X\to Y'\to Y$, where $Y'\to Y$ is a connected finite \'{e}tale morphism of degree greater than one. \end{definition} By definition, any $h: X\to Y$ factors as \[ X\stackrel{h'}{\longrightarrow} Y'\stackrel{\pi}{\longrightarrow} Y \] where $h'$ is genuinely ramified and $\pi$ is \'etale. In case $h$ is Galois, $h'$ is automatically Galois while $\pi$ may not. \begin{lemma}\label{lem:galois-injective} If $h\colon X\to Y$ is genuinely ramified, then the pullback \[ h^*\colon \Pic^0(Y)\longrightarrow \Pic^0(X) \] is injective. \end{lemma} \begin{proof} If $h^*\cM \simeq\mathcal O_X$, then $\cM^{\deg h}=Nm_h(h^*\cM)=Nm_h(\cO_X)=\cO_Y$ by the norm identity. If $n:=\mathrm{ord}(\cM)$ in $\Pic^0(Y)$ is greater than one, the cyclic cover \[ Y'=\operatorname{Spec}_Y\left(\bigoplus_{i=0}^{n-1}\cM^{i}\right) \longrightarrow Y \] is finite \'{e}tale and connected and trivialized over $h:X\to Y$. In particular, $h$ factors through $Y'$, a contradicting genuine ramification. Thus $\alpha=\mathcal O_Y$. \end{proof} \begin{lemma}\label{lem:galois-norm} Let $h\colon X\to Y$ be a finite Galois morphism with group $G$. For every $\cL\in J(X)$, \[ h^*\operatorname{Nm}_h(\cL)\simeq \bigotimes_{\gamma\in G}\gamma^*\cL. \] \end{lemma} \begin{proof} For a divisor $D$ representing $\cL$, the Galois identity $h^*h_*D=\sum_{\gamma\in G}\gamma D$ gives \[ h^*\operatorname{Nm}_h(B)\simeq \bigotimes_{\gamma\in G}\gamma^*B. \] \end{proof} We first prove the generic semistability for genuinely ramified covers. \begin{theorem}\label{thm: genuinely ramified} Let $h: X\to Y$ be a genuinely ramified Galois cover, then for any $d\in \Z$, there is an open dense subset $U_d\subseteq \Pic^d(X)$ such that $h_*\cL$ is semistable for all $\cL\in U_d$. \end{theorem} \begin{proof} Fix any $d\in \Z$ and set $G=\mathrm{Gal}(X/Y)$ with $g:=|G|=\deg h$. The pushforward $(f\times \mathrm{id})_*\cP$ of the Poincar\'{e} bundle on $X\times \Pic^d(X)$ to $Y\times \Pic^d(X)$ gives a flat family of vector bundles on $Y$ parametrized by $\Pic^d(X)$ whose fiber at $\cL\in \Pic^d(X)$ is $h_*\cL$. Suppose $f_*L$ is unstable for any $\cL$, By the relative Harder--Narasimhan theorem of Shatz \cite{Shatz}, the Harder--Narasimhan type is constant on a nonempty open subset $U\subseteq \Pic^d(X)$. Take $e$ and $r$ to be the degree and rank of the first Harder--Narasimhan term over $U$. In particular, the first term of the relative filtration is a subbundle of $(f\times \mathrm{id})_*\cP$ on $Y\times U$. Take $e$ and $r$ to be the degree and rank of the first Harder--Narasimhan term over $U$. Taking determinant over $U$ then define a morphism $\delta: U \to \Pic^e(Y)$. Following the celebrate Weil's trick, the map $\delta$ actually extends to a genuine morphism $\delta: \Pic^d(X)\to \Pic^e(Y)$ because its target is Abelian. The projection formula gives \[ \delta(\cL\otimes h^*\cN)=\delta(\cL)\otimes \cN^{\otimes r} \qquad (\cN\in \Pic^0(Y)). \] and the natural identity $h_*(\gamma^*\cL)=h_*\cL$ gives $\delta\circ \gamma^*=\delta$ for any $\gamma\in G$. Choose $\cL_0\in \Pic^d(Y)$ and define \[ \psi: \Pic^0(X)\to \Pic^0(Y), \quad \psi(\cM)=\delta(\cL_0\otimes \cM)\otimes\delta(\cL_0)^{-1}. \] Then $\psi\colon \mathrm{Pic}^0(X)\to \Pic^0(Y)$ is a homomorphism satisfying $\psi\circ h^*=[r]$. Take $\cN_\gamma:=(\gamma^{-1})^*\cL_0\otimes \cL_0^{-1}\in \mathrm{Pic}^0(X)$, then for any $\cM\in \mathrm{Pic}^0(X)$, \[ \begin{aligned} \psi(\gamma^*\cM)&=\delta(\cL_0\otimes \gamma^*\cM)\otimes \delta(\cL)^{-1}\\ &=\delta((\gamma^{-1})^*(\cL_0\otimes \gamma^*\cM))\otimes \delta(\cL)^{-1}\\ &=\delta(\cL_0\otimes \cN\otimes \cM) \otimes \delta(\cL)^{-1}\\ &=\psi(\cN\otimes \cM)\\ &=\psi(\cN)\otimes \psi(\cM). \end{aligned} \] The special case $\cM=\cO_X$ gives $\psi(\cN)=\cO_Y$ and thus $\psi\circ \gamma^*=\psi$. The pullback--norm identity then gives \[ h^*\circ\operatorname{Nm}_h = \sum_{\gamma\in G}\gamma^* \] as homomorphisms from $J(X)$ to $J(X)$. Since $h_*\gamma^*\mathcal L\simeq h_*\mathcal L$ for every $\gamma\in G$, the determinant of the maximal destabilizing subbundle is invariant under $\gamma^*$. After identifying $\Pic^d(X)$ with $J(X)$, this gives \[ \psi\circ\gamma^*=\psi. \] Consequently, \[ \begin{aligned} [r]\circ\operatorname{Nm}_h &=\psi\circ h^*\circ\operatorname{Nm}_h\\ &=\psi\circ\sum_{\gamma\in G}\gamma^*\\ &=\sum_{\gamma\in G}\psi\circ\gamma^*\\ &=[g]\circ\psi. \end{aligned} \] Write $g=g_0\cdot N$, $r=r_0\cdot N$ with $\gcd(g_0,r_0)=1$. Since $\operatorname{Hom}(\Pic^0(X),\Pic^0(Y))$ is torsion-free, there is an $\varrho \in \operatorname{Hom}(\Pic^0(X),\Pic^0(Y))$ so that \[ \operatorname{Nm}_h=[g_0]\circ\varrho, \quad \psi=[r_0]\circ \varrho. \] As $r1$. It is classical that $\operatorname{Nm}_h$ can be identified with the induced map $\mathrm{Alb}_h$ on Albanese varieties through the canonical principal polarizations, and $h^*$ is then its dual map, see \cite[{\S}3, proof of Theorem~3.7, pp.~161--162]{BeauvillePrym}. Following $\operatorname{Nm}_h=[g_0]\circ\varrho$, $h^*$ factor through the multiplicative-by-$g_0$ map, too. A contradiction to Lemma~\ref{lem:galois-injective}. We are done. \end{proof} \begin{proof}[Proof of Theorem~\ref{thm:galois-generic}] In general, if $h:X\to Y$ is Galois, then it factor through $h=\pi\circ h'$ with $h'$ genuinely ramified Galois and $\pi$ \'etale. For any $d$, it is proved in Theorem~\ref{thm: genuinely ramified}, $h'_*\cL$ is semistable for $\cL$ belonging to an open dense subset $U_d\in \mathrm{Pic}^d(X)$. It follows from \cite[\S~4]{Beauville}, that $\pi_*$ preserves semistability. Thus, for $\cL\in U_d$, $h_*\cL=\pi_*h'_*\cL$ is semistable. We are done. \end{proof} \begin{remark}\label{rem:galois-scope} The theorem proves generic semistability and makes no assertion of generic stability. The Galois hypothesis enters through the pullback--norm identity and invariance under deck transformations. The degree-$7$ map constructed in this paper is not Galois: by Riemann--Hurwitz its total ramification degree is $2$, whereas a ramified Galois cover of prime degree $7$ has total ramification degree divisible by $6$. Thus Theorem~\ref{thm:galois-generic} does not conflict with Theorem~\ref{thm:main}. \end{remark} %===================================================================== \begin{thebibliography}{18} %===================================================================== \bibitem{Atiyah} M.~F. Atiyah, \emph{Vector bundles over an elliptic curve}, Proc. London Math. Soc. (3) \textbf{7} (1957), 414--452. \bibitem{BeauvillePrym} A.~Beauville, \emph{Prym varieties and the Schottky problem}, Invent. Math. \textbf{41} (1977), no.~2, 149--196. \bibitem{Beauville} A. Beauville, \emph{On the stability of the direct image of a generic vector bundle}, preprint, 2000. \url{https://math.univ-cotedazur.fr/~beauvill/pubs/imdir.pdf}. \bibitem{BiswasDasParameswaran} I. Biswas, S. Das, and A. J. Parameswaran, \emph{Genuinely ramified maps and stable vector bundles}, Int. J. Math. \textbf{33} (2022), no.~5, 2250039. \bibitem{BiswasParameswaran} I. Biswas and A. J. Parameswaran, \emph{Ramified covering maps and stability of pulled-back bundles}, Int. Math. Res. Not. IMRN (2022), no.~17, 12821--12851. \bibitem{CLV} I. Coskun, E. Larson, and I. Vogt, \emph{Generic Beauville's conjecture}, Forum Math. Sigma \textbf{12} (2024), e51. \bibitem{vdGM} G. Van der Geer, P. Moonen, \emph{Abelian Varieties}, preprint, \url{https://www.math.ru.nl/~bmoonen/research.html}. \bibitem{FPR} M. Franciosi, R. Pardini, and S. Rollenske, \emph{$(d, d')$-elliptic curves of genus two}, Ark. Mat. \textbf{56} (2018), no.~2, 299--317. \bibitem{FreyKani} G. Frey and E. Kani, \emph{Curves of genus $2$ covering elliptic curves and an arithmetical application}, in \emph{Arithmetic Algebraic Geometry} (Texel, 1989), Progress in Mathematics \textbf{89}, Birkh\"auser, Boston, 1991, 153--176. \bibitem{Gieseker} D. Gieseker, \emph{Stable vector bundles and the Frobenius morphism}, Ann. Sci. \'Ecole Norm. Sup. (4) \textbf{6} (1973), no.~1, 95--101. \bibitem{JRXU} K.~Joshi, S.~Ramanan, E.~Z. Xia, and J.-K. Yu, \emph{On vector bundles destabilized by Frobenius pull-back}, Compositio Math. \textbf{142} (2006), no.~3, 616--630. \bibitem{Langer} A. Langer, \emph{Semistable sheaves in positive characteristic}, Ann. of Math. (2) \textbf{159} (2004), no.~1, 251--276. \bibitem{Mumford} D. Mumford, \emph{Abelian Varieties}, 2nd ed., Tata Institute of Fundamental Research Studies in Mathematics \textbf{5}, Oxford University Press, 1974. \bibitem{Shatz} S.~S. Shatz, \emph{The decomposition and specialization of algebraic families of vector bundles}, Compositio Math. \textbf{35} (1977), no.~2, 163--187. \bibitem{SunDirect} X.~Sun, \emph{Direct images of bundles under Frobenius morphism}, Invent. Math. \textbf{173} (2008), no.~2, 427--447. \bibitem{SunSurvey} X.~Sun, \emph{Frobenius morphism and semi-stable bundles}, in \emph{Algebraic Geometry in East Asia--Seoul 2008}, Adv. Stud. Pure Math. \textbf{60}, Math. Soc. Japan, 2010, 161--182. \bibitem{Weissmann} D.~Wei{\ss}mann, \emph{A functorial approach to the stability of vector bundles}, Math. Z. \textbf{308} (2024), Article 63. \end{thebibliography} \vspace{2cm} \noindent \begin{minipage}[t]{0.48\linewidth} \textbf{Yi Gu}\\ School of Mathematical Sciences,\\ Soochow University,\\ Suzhou 215006, P. R. of China.\\ Email: sudaguyi2017@suda.edu.cn \end{minipage}% \hfill \begin{minipage}[t]{0.48\linewidth} \textbf{Lifan Guan}\\ Institute for Theoretical Sciences, School of Science, \\ Westlake University, \\ 600 Dunyu Road, Hangzhou 310030, China. \\ Email: guanlifan@westlake.edu.cn \end{minipage} \end{document}