The complex curve $X^n + Y^m = 1$ is the Milnor fiber (at the origin) of the weighted-homogeneous polynomial $f(X,Y)=X^n + Y^m$. This Milnor fiber deform-retracts onto a (minimal) Seifert fiber for the singularity link, which is an $(n,m)$-torus link. This Seifert fiber consists of $n$ stacked disks, each one joined to the one above by $m$ once-twisted bands. It is now a simple exercise to see that the genus of this surface is $(n-1)(m-1)/2$.

The complex curve $X^n + Y^m = 1$ is the Milnor fiber (at the origin) of the weighted-homogeneous polynomial $f(X,Y)=X^n + Y^m$. Suppose $\gcd(n,m)=1$. Then the Milnor fiber deform-retracts onto a (minimal) Seifert fiber for the singularity link, which is an $(n,m)$-torus link. This Seifert fiber consists of $n$ stacked disks, each one joined to the one above by $m$ once-twisted bands. It is now a simple exercise to see that the genus of this surface (equal to the Milnor number of $f$) is $(n-1)(m-1)/2$.

More generally, if $f=f(z_1,\dots,z_m)$ is a weighted-homogeneous polynomial with weights $(w_1,\dots, w_m)$, then the Milnor fiber $f=1$ has the homotopy type of a wedge of $(m-1)$-dimensional spheres, and the (Milnor) number of these spheres is given by $\mu=(w_1−1)(w_2−1)\cdots (w_m−1)$, according to John Milnor and Peter Orlik, Isolated singularities defined by weighted homogeneous polynomials, Topology 9 (1970), 385-393.