Take the 2-minute tour ×  I am interested in studying moduli of
complex surfaces which arise in computing the differential on
the Heegaard Floer Homology chain complex. In particular, I am
interested in the generic case, when the holomorphic discs in
$\operatorname{Sym}^g\Sigma$ are as "bad" as possible. I have
searched online for some examples of "generic"/"complicated"
Whitney disks in the symmetric product, but most papers I see
deal with special cases (namely where one can see by inspection
and the Riemann mapping theorem that certain homotopy classes of
Whitney disks are uniquely representable by holomorphic disks).
To elaborate on what I mean by "complicated" holomorphic disks,
let's recall a certain (now standard) perspective on the Whitney
disks. If we have a holomorphic disk $\phi:\mathbb
D^2\to\operatorname{Sym}^g\Sigma$, then we can consider the
fiber product: $$\begin{matrix} S&\xrightarrow{\tilde\phi}&\Sigm
a\times\operatorname{Sym}^{g-1}\Sigma\cr \downarrow&
&\downarrow\cr \mathbb D^2&\xrightarr