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.

  • 1
    $\begingroup$ There is plenty of ample, irreducible, singular curves in abelian surfaces. $\endgroup$ – Francesco Polizzi Aug 20 '17 at 12:44

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.

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.