Take the 2-minute tour ×  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 $A\to \mathbf{P}^g_k$?  share|improve this question A
better analogue of the degree 2 map $E\to\mathbb{P}^1$ for an
elliptic curve is the degree 2 map $A\to A/\pm1$ from $A$ to the
associated Kummer variety. (Of course, you need to blow up some
points if you want a smooth quotient, and then then map is only
rational, it's not a morphism.) –  Joe Silverman Aug 1 '12 at
14:05 add comment  1 Answer  up vote 9 down vote accepted  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