bit of a computation question here.
Let $C$ be a (smooth) curve in $\mathbb{C}$P$^2$ (or more generally $\mathbb{C}$P$^N$) of degree $d$. Then the homology class $[C]$ is $d \cdot [\mathbb{C}\text{P}^1]$, where $[\mathbb{C}\text{P}^1]$ is the class of the canonical copy of $\mathbb{C}\text{P}^1$ in $\mathbb{C}\text{P}^2$. I know (or at least, I think I know) how to show this by using Poincare duality and number of points of intersection, etc. However, I was trying to prove this another way by showing that the integral of the Fubini-Study form $\omega_{FS}$ of $\mathbb{C}\text{P}^2$ over $C$ was $d$.
Is this a valid approach? I thought so, but actually doing the computation does not yield $d$. For example, let $C$ be the degree two curve given by
$[X_0 : X_1] \rightarrow [X_0^2 : X_0X_1 : X_1^2]$
which in affine charts is just $z \rightarrow (z, z^2)$. Pulling back $\omega_{FS}$ under this map gives me the form
$\dfrac{3|z|^2}{(1 + |z|^2 + |z|^4)^2} dzd\bar{z}$
whose integral, according to Wolfram Alpha, is certainly not 2 (or 2 times whatever multiple of $\pi$). Could someone verify this calculation/tell me what I am doing wrong?
Thanks!