Canonical maps of general hypersurfaces in Abelian varieties

Abstract

<p style='text-indent:20px;'>The main theorem of this paper is that, for a general pair <inline-formula><tex-math id="M1">\begin{document}$ (A,X) $\end{document}</tex-math></inline-formula> of an (ample) hypersurface <inline-formula><tex-math id="M2">\begin{document}$ X $\end{document}</tex-math></inline-formula> in an Abelian Variety <inline-formula><tex-math id="M3">\begin{document}$ A $\end{document}</tex-math></inline-formula>, the canonical map <inline-formula><tex-math id="M4">\begin{document}$ \Phi_X $\end{document}</tex-math></inline-formula> of <inline-formula><tex-math id="M5">\begin{document}$ X $\end{document}</tex-math></inline-formula> is birational onto its image if the polarization given by <inline-formula><tex-math id="M6">\begin{document}$ X $\end{document}</tex-math></inline-formula> is not principal (i.e., its Pfaffian <inline-formula><tex-math id="M7">\begin{document}$ d $\end{document}</tex-math></inline-formula> is not equal to <inline-formula><tex-math id="M8">\begin{document}$ 1 $\end{document}</tex-math></inline-formula>).</p><p style='text-indent:20px;'>We also easily show that, setting <inline-formula><tex-math id="M9">\begin{document}$ g = dim (A) $\end{document}</tex-math></inline-formula>, and letting <inline-formula><tex-math id="M10">\begin{document}$ d $\end{document}</tex-math></inline-formula> be the Pfaffian of the polarization given by <inline-formula><tex-math id="M11">\begin{document}$ X $\end{document}</tex-math></inline-formula>, then if <inline-formula><tex-math id="M12">\begin{document}$ X $\end{document}</tex-math></inline-formula> is smooth and</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \Phi_X : X {\rightarrow } {\mathbb{P}}^{N: = g+d-2} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>is an embedding, then necessarily we have the inequality <inline-formula><tex-math id="M13">\begin{document}$ d \geq g + 1 $\end{document}</tex-math></inline-formula>, equivalent to <inline-formula><tex-math id="M14">\begin{document}$ N : = g+d-2 \geq 2 \ dim(X) + 1. $\end{document}</tex-math></inline-formula></p><p style='text-indent:20px;'>Hence we formulate the following interesting conjecture, motivated by work of the second author: if <inline-formula><tex-math id="M15">\begin{document}$ d \geq g + 1, $\end{document}</tex-math></inline-formula> then, for a general pair <inline-formula><tex-math id="M16">\begin{document}$ (A,X) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M17">\begin{document}$ \Phi_X $\end{document}</tex-math></inline-formula> is an embedding.</p>