Canonical maps of general hypersurfaces in Abelian varieties

The main theorem of this paper is that, for a general pair \begin{document}$ (A,X) $\end{document} of an (ample) hypersurface \begin{document}$ X $\end{document} in an Abelian Variety \begin{document}$ A $\end{document} , the canonical map \begin{document}$ \Phi_X $\end{document} of \begin{document}$ X $\end{document} is birational onto its image if the polarization given by \begin{document}$ X $\end{document} is not principal (i.e., its Pfaffian \begin{document}$ d $\end{document} is not equal to \begin{document}$ 1 $\end{document} ). We also easily show that, setting \begin{document}$ g = dim (A) $\end{document} , and letting \begin{document}$ d $\end{document} be the Pfaffian of the polarization given by \begin{document}$ X $\end{document} , then if \begin{document}$ X $\end{document} is smooth and\begin{document}$ \Phi_X : X {\rightarrow } {\mathbb{P}}^{N: = g+d-2} $\end{document}is an embedding, then necessarily we have the inequality \begin{document}$ d \geq g + 1 $\end{document} , equivalent to \begin{document}$ N : = g+d-2 \geq 2 \ dim(X) + 1. $\end{document} Hence we formulate the following interesting conjecture, motivated by work of the second author: if \begin{document}$ d \geq g + 1, $\end{document} then, for a general pair \begin{document}$ (A,X) $\end{document} , \begin{document}$ \Phi_X $\end{document} is an embedding.