Let $A$ be an abelian variety over a field $k$ of dimension $g\geq 2$.


There exists a finite morphism $A\to \mathbf{P}^g\_k$. Here's the question. 


Does there exist a finite morphism $A\to \mathbf{P}^g\_k$ of degree two?


Can we say something about the minimal degree of a finite morphism $A\to \mathbf{P}^g\_k$?



Are abelian varieties degree two covers of some projective space
For a very general, principally polarized Abelian variety $(A,\Theta)$ of dimension $g$ over $\mathbb{C}$, every Cartier divisor $D$ on $A$ is numerically equivalent to $m\Theta$ for some integer $m$. In particular, the intersection number $D^g$ is $m^g \Theta^g$. So the minimal degree of an effective, nonzero divisor is $g!$, not $2$. 


