Singular curve on an abelian surface

Let $$C_2$$ be a smooth genus $$2$$ curve and $$J(C_2)$$ its Jacobian. It is well known that the blow-up of $$J(C_2)$$ at the origin $$o$$ is isomorphic to the second symmetric product $$\textrm{Sym}^2(C_2)$$, and that the blow-up morphism $$\pi \colon \textrm{Sym}^2(C_2) \to J(C_2)$$ coincides with the Abel-Jacobi map.

The exceptional divisor $$E \subset \textrm{Sym}^2(C_2)$$ corresponds to the unique $$g^1_2$$ of $$C_2$$, so it intersects transversally the diagonal $$\Delta \subset \textrm{Sym}^2(C_2)$$ at six points, corresponding to the six Weierstrass points of $$C_2$$. Hence the image $$\pi(\Delta) \subset J(C_2)$$ is a curve with an ordinary singular point of multiplicity six at $$o$$.

Finally, the curve $$\Delta$$ is the branch locus of the double cover $$C_2 \times C_2 \to \textrm{Sym}^2(C_2)$$. This implies that its class is $$2$$-divisible in the Néron-Severi group $$\textrm{NS}(\textrm{Sym}^2(C_2))$$, and so the same holds for the class of $$\pi(\Delta)$$ in $$\textrm{NS}(J(C_2))$$.

More precisely, it is possible to show that $$\pi(\Delta)$$ is algebraically equivalent to $$4 \Theta$$, where $$\Theta$$ is the theta divisor, namely the principal polarization of the Jacobian (see for instance this MO question).

In particular, we have $$\pi(\Delta) \cdot \pi(\Delta) = 16 \Theta^2 = 32.$$

I would like to know whether this is the only possible occurrence for this situation, so let me ask the following

Question. Let $$A$$ be an abelian surface and $$D$$ be an effective divisor on it such that

• $$D$$ has an ordinary point of multiplicity $$6$$ at the origin $$o \in A$$ and no other singularities;
• $$D^2=32;$$
• the class of $$D$$ is $$2$$-divisible in $$\textrm{NS}(A)$$.

Is the pair $$(A, \, D)$$ necessarily of the form $$(J(C_2), \, \pi(\Delta))$$ for some genus $$2$$ curve $$C_2$$?

• This seems very likely. A possible approach might be to first compute the geometric genus of $D$ using the first two points. Assuming one gets it to be $2$, let $C$ be the normalization of $D$ and consider the map $J(C) \to A$ induced by the map $C \to D$. Again using the first two points one should probably get that this is essentially the multiplication by $2$ map on $J(C)$. This should imply what you want (so the last condition might be superfluous).
– naf
Jun 18, 2016 at 10:23
• @ulrich: thanks for the suggestion. How do you know that $C$ is irreducible? Jun 18, 2016 at 12:01
• Sorry, I was assuming that $D$ was irreducible though you didn't say that. If this is not assumed I guess one might have to consider some cases, but first one should check that the argument does indeed work if $D$ is assumed to be irreducible.
– naf
Jun 19, 2016 at 10:00