A famous result of Lefschetz says that if $L$ is an ample line bundle on an abelian variety then $L^{\otimes 3}$ is very ample.
In Mumford's book on abelian varieities, this is mentioned in Chapter 17.
First one needs to show that the associated map to $L^3$ is an embedding. i.e. sections separates points. I will outline how this is proven in Mumford's book.
Let $0\neq x_1$ be points where any sections $s\in H^0(L^3)$ do no separate the two points.
Using the divisor language, this is to say that for every effective divisor $D'\in |L^3|$, $0\in Supp(D')\iff x_1\in Supp(D')$.
By the theorem of the square, $D':=t_x^*D+t_y^*D+t_{-x-y}^*D\in |3L|$ where $D$ is some effective divisor such that $L=\mathcal{O}_X(D)$. If $0\in Supp(t_x^*D)$ then, $0\in Supp(D')$ and so $x_1\in Supp(t_x^*D)\cup Supp(t_y^*D)\cup Supp(t_{-x-y}^*D)$. Varying $y$ appropriately so that $x_1$ doesn't belong to $Supp(t_y^*D)\cup Supp(t_{-x-y}^*D)$, we obtain that $x_1\in Supp(t_x^*D)$. In particular, this means that $x\in D-x_1$ and $x\in D$. In particular, $t_{x_1}^*D=D$ so $x_1\in K(L)$. Since $L$ is a nondegenerate line bundle (it's ample), $K(L)$ is a finite group. Call the subgroup generated by $x_1$, $F$ and call the canonical projection map $\pi: X\to X/F$. In this case, I can pushforward $D$ to get a divisor $D_1=\pi(D)$ on $X/F$. Also by pulling back, $\pi^*(D_1)=D$.
Denote $L_1=\mathcal{O}_{X/F}(D_1)$
Using that $H^0(L)=\chi(L)$ for ample line bundles and using Riemann-Roch (the second equality can be obtained from a Riemann-Roch statement),
$h^0(X,L)=\chi(L)=\deg(\pi)\chi(L_1)=\deg(\pi)h^0(X/F,L_1)>h^0(X/F,L_1)$.
This equation is telling us that there are only finitely many divisor class (up to linear equivalence) on $X/F$ that give line bundles that pullback to $L\in Pic(X)$ (namely, $\deg \pi=\# F$ many divisor classes).
Mumford then says, "all sections $s\in \Gamma(L)$ either define multiple divisors, or lie in one of a finite number of lower-dimensional subspaces $\pi^*\Gamma(\mathcal{O}_{X/F}(D_1))$".
What I don't understand is the contradiction that arises from this. I understand where the second statement is coming from but not why this is a contradiction. I am also not entirely sure what is meant by a regular section of a line bundle defining multiple divisors.
(I am omitting for brevity the part where we have to check separation of tangent vectors. However, the contradiction obtained is a similar remark to the first part above.)