Let $X$ be $\mathbb{P}^2$ blownup at one point and $\beta := d L -2E \in H_2(X, \mathbb{Z})$, where $L$ and $E$ denote the class of a line and the exceptional divisor respectively. Let $\mathcal{L}_{\beta} \longrightarrow X $ be the divisor bundle corresponding to the class $\beta$. Let $\chi$ be some specific singularity. For instance you can take $\chi$ to be an $A_k$-singularity ($y^2+x^{k+1}=0$). Let me define two numbers:
$N_1$ is the number of curves in the linear system $H^0(X, \mathcal{L}_{\beta})$ through the right number of generic points and having one singularity of type $\chi$.
$N_2$ is the number of degree $d$ curves in $\mathbb{P}^2$ through the right number of generic points having a singularity of type $\chi$ and having a simple node ($y^2 + x^2 =0$) at a fixed point (the point at which we blowup and get $X$).
$\textbf{Question:}$ Is it always true that $N_1 = N_2$? Assume that $d$ is sufficiently large, so that all necessary transversality assumptions hold.
$\textbf{Remark:}$ I think naively that there should be some sort of a correspondence between the curves counted in $N_1$ and those counted in $N_2$. More generally, if $\beta := dL- mE$, then this would be equivalent to counting degree $d$ curves in $\mathbb{P}^2$ with an $m$-fold point ($y^m + x^m =0$).
But I am not sure if this is always true (i.e. maybe it depends on what that singularity is).
