= Homological degree of a smooth complex plane curve
{title2=$[C]=d[\mathbb{CP}^1]$}
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.
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