This is a concrete question in Enumerative geometry. Let $S$ be a compact complex surface and $L\rightarrow S$ a holomorphic line bundle. Let

$$ \delta_d = \text{dim}~ \mathbb{P}(H^0(S,L^d)) $$

i.e the dimension of the space of degree $d$ curves, up to scaling. Define $N^d(m,\chi_k)$ to be the number of degree $d$ curves that pass through $\delta_d-(m+k)$ points and have $m$ distinct simple nodes and one singularity of type $\chi_k$, where $\chi_k$ is a singularity of co dimension $k$. For example you can take $\chi_k$ to be $A_k$, $D_k$ or $E_k$ singularity. My question is the following. Aside from the case of $S= \mathbb{C}\mathbb{P}^2$ and $L= \gamma^{*}$, can you give some other examples where I can make some low degree checks? Let me give a few examples to illustarte what I mean in the case of $S= \mathbb{C}\mathbb{P}^2$ and

$L= \gamma^{*}$.

Let us consider the number $N^d(0,A_1)$, the number of degree $d$ curves passing through $\delta_d-1$ points and having one simple node. I claim that the answer is $3(d-1)^2$. There are three instances where I can actually check this. For $d=1$, I am asking how many lines are there through one point that has a node. Clearly zero. For $d=2$, I am asking how many pair of lines are there through $4$ points. That is $3$. And for $d=3$, I am asking how many genus zero cubics are there through $8$ points, which can be obtained through Kontseviche's recursion formula (i.e 12). There are several other examples I can give. If it helps I can explicitly state a few more.

My question is that if someone claims a formula for $N^d(m,\chi_k)$ for some other surface and line bundle, is there a similar way to make low degree checks, atleast in some cases?

Let me just define what are $A_k$, $D_k$ and $E_k$ singularities. A curve $f(x,y)=0$ has an $A_k$ singularity at the origin if it can be expressed after a change of coordinates as $$ y^2 + x^{k+1} =0, \qquad A_k.$$ It is of type $D_k$ if it can be expressed as $$ y^2 x + x^{k-1} =0, \qquad D_k.$$ There are similar definitions for $E_6$, $E_7$ etc, which I can write down, if it helps.

Here is a related question I have asked as well