Let $(A,L)$ be a polarized abelian variety of dimension $g$, with an indecomposable polarization of degree $\chi(L)=d$. There is a theorem of Debarre and Hacon about the singularities of pairs:
Theorem: Let $D \in |mL|$ for some $m\ge 2$ be a divisor, and suppose $d=2$. Then the pair $(A,(1/m)D)$ is log-canonical.
In his notes (https://www.math.ens.fr/~debarre/DivVarAb.pdf) Debarre says that, for $d>2$ and $m\ge d-1$, there are pairs $(A,(1/m)D)$ which are not log-canonical anymore, and gives an example:
Let $(A_1,L_1)$ be a general polarized abelian variety of type $(d-1)$ and let $E$ be an elliptic curve. Pick an isomorphism $\psi\colon K(L_1) \to E[d-1]$ and consider the quotient $A$ of $A_1 \times E$ by the subgroup $\{(x,\psi(x))~|~x \in K(L_1)\}$. There is a divisor $\Theta$ on $A$ that defines a principal polarization and $\mathcal{O}_A(\Theta)$ restricts to $L_1$ on $A_1$. The line bundle $L:=\mathcal{O}_A(\Theta+A_1)$ defines an indecomposable polarization of degree $d$ on $A$ that is indecomposable if $d\ge3$.
The linear system $|(d-1)\Theta-A_1|$ is nonempty, hence so is the linear system $|m\Theta-(m'-m)A_1|$ for $\frac{d}{d-1}m\ge m' > m\ge d-1$. If $D'$ is in that linear system, $D=D'+m'A_1$ is in $|mL|$ and the pair $(A,(1/m)D)$ is not log-canonical since it has a component of multiplicity $>1$.
I have some problem understanding this example, in particular:
- Why does such a $\Theta$ on $A$ exists?
- Why is $L$ an ample line bundle and why is its degree as a (indecomposable?) polarization equal to $d$?
- Why is the linear system nonempty?
