Canonical Maps of general Hypersurfaces in Abelian Varieties

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