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 $$\pi \colon \textrm{Sym}^2(C_2) \to J(C_2)$$ be 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 $o$, and 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 $$\pi(\Delta) \cdot \pi(\Delta) = 16 \Theta^2 = 32.$$

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 $\textrm{NS}(J(C_2))$.

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 $C_2$?