Homological degree of a smooth complex plane curve ($[C]=d[{\mathbb{CP}}^1]$)

A smooth compact Riemann surface embedded in Complex projective space ${\mathbb{CP}}^2$ determines an integral homology class $d[{\mathbb{CP}}^1]$. Since a projective line has self-intersection one, $d$ is the intersection number with a transverse projective line. Every transverse intersection of complex curves has positive sign, because a complex-linear change of tangent basis has positive real determinant. For the Fermat curve $x^d+y^d=z^d$, the line $y=0$ meets it transversely in the $d$ points $[\zeta :0:1]$, ${\zeta }^d=1$, directly verifying its homological degree.