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I am new to abelian varieties and became interested in the singular set of a subvariety in a complex torus.

The space of ample divisors (non necessarily smooth) in a complex torus $T^n$ seems quite large, as we can always move it, however, I don't have an example of a global hypersurface $M$ in $T^n$ with codim 1 singular set.

We may consider the theta divisor in a ppav. For example, for a generic ppav the theta divisor is smooth. For a particular ppav, a result I am aware of (the book of Birkenhake-Lange) is that if $C$ is a curve of genus $g \geq 4$ and $J(C)$ its Jacobian, the singular set of Theta divisor is a closed set on $J(C)$ with dim $Sing \Theta=g-4$ if $C$ is not hyperelliptic and dim $Sing \Theta=g-3$ if $C$ is hyperelliptic.

Now my question is: is there any example of an ample divisors (or subvariety) in a complex torus whose singular set has codimension 1. In general I am interested in any reference on estimates on the singular set.

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    $\begingroup$ There is plenty of ample, irreducible, singular curves in abelian surfaces. $\endgroup$ – Francesco Polizzi Aug 20 '17 at 12:44
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Let $C$ be a smooth genus $2$ curve and let $J(C)$ be its Jacobian. Now, take a symmetric theta divisor $\Theta \subset J(C)$ and its translate $\Theta + x$, where $x$ is a $2$-torsion point. Finally, consider the isogeny of degree 2 $$f \colon J(C) \to A,$$ where $A$ is the quotient of $J(C)$ by the subgroup $\langle x \rangle$.

Therefore $A$ is a $(1, \, 2)$-polarized abelian surface, and the image via $f$ of the reducible divisor $\Theta + (\Theta +x)$ is an ample curve in $A$ having an ordinary double point as its unique singularity.

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  • $\begingroup$ Many thanks! Originally I would prefer an 2-dim example and see what kind of curves the singular locus looks like. Does a similar construction in 2-dim work? $\endgroup$ – Bo_Y Aug 22 '17 at 4:56
  • $\begingroup$ Take an abelian surface $A$ containing a nodal curve $D$ as in my answer, and consider the product $$B = A \times E,$$ where $E$ is an elliptic curve. Then $B$ is an abelian threefold containing the irreducible surface $D \times E$, which is singular along a curve. $\endgroup$ – Francesco Polizzi Aug 22 '17 at 7:43

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