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 $S^2(C_2)$; let $$\pi \colon S^2(C_2) \to J(C_2)$$ be the blow-up map.

The exceptional divisor $E \subset S^2(C_2)$ corresponds to the unique $g^1_2$ of $C_2$, so it intersects transversally the diagonal $\Delta \subset S^2(C_2)$ at $6$ 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 sextuple point 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 $$\pi(\Delta) \cdot \pi(\Delta) = 16 \Theta^2 = 32.$$

I would like to know whether this is the only possible occurrence of this situation, in other words 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;
• $D^2=32.$

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