Question

Let $J$ be a Jacobian variety defined over a field $k$ and let $\Theta$ be a symmetric theta-divisor on $J$.

It's shown (for instance) in the book Complex Abelian Varieties by Lange and Birkenhake that the linear system of $2\Theta$ is base point free if $k=\mathbb{C}$ and that it gives an embedding of the Kummer variety of $J$ into $\mathbb{P}^{2^g-1}$, where $g$ is the dimension of $J$.

I'm looking for a reference that this holds for any algebraically closed field $k$. In fact $\mathrm{char}(k)\ne 2$ would suffice.

Answer 1

If $D$ is an ample divisor on an abelian variety, then $2D$ is base point free and $3D$ is very ample. One reference for this is Mumford's book "Abelian Varieties", II 6 and III 17.