To make my question precise, suppose you have a complex curve locally given by $$f(x,y) =0 $$
and $f$ has singularity of type $\chi_k$ at the origin. The codimension of this singularity is $k$. Let $g$ be the contribution of the singularity to the genus of that curve. For instance if it was a simple node ($A_1$) then $g=1$. If it was a tacnode ($A_3$) then $g$ is $2$ and so on. Is it true that
$$ g \leq k-2 $$
provided, $k \geq 5$ ? This does seem to be true if $k =5, 6,7$ because in these dimensions we know that the only type of singularities are A, D and E. We just need to verify for each of them. But beyond seven dimensions there is no classification of singularities. I do not wan to make any additional assumption on the singularity, such as being simple or anything else. Basically my question is can we give an upper bound for the genus (hopefully the one I am suggesting), without even having a classification? Similarly is there a lower bound for the genus? Intuitively, the meaning of $g$ is how many nodes you get after you change the singularity a bit. So if you deform a tacnode a bit, you get two nodes.