Consider a holomoprhic map from a Riemann surface $$ f: \Sigma_g \to \mathbb{CP}^n. $$ This is given by some homogeneous polynomials in some variables.
How can we show that the homogeneous degree $d$ of these polynomials coincides with the definition of degree from fundamental homology class, namely $d = f_* ([\Sigma_g]) \in H_2(\mathbb{CP}^n;\mathbb{Z})$?
Let me rename $C=\Sigma_g$, as the genus is unimportant here. To give a map from a smooth complex projective curve $C$ to $\mathbb P^n$ is the same as to give a line bundle $L$ on $C$. This line bundle can be recovered up to isomorphism as $$L=f^\ast(\mathscr O_{\mathbb P^n}(1))=g^\ast(\mathscr O_{\mathbb P^n}(1)|_D),$$ where I have denoted by $g$ the same map $f$, but restricted to the image $D=f(C)$; in other words we factor $f$ as follows: $$C\overset{g}{\longrightarrow}D\hookrightarrow\mathbb P^n.$$ On $\mathbb P^n$, the one-cycle $f_\ast[C]$ has degree $$\deg f_\ast[C]=\deg g_\ast[C]=(\deg g)\cdot(\deg D).$$ On the other hand, the degree of the polynomials defining $f$ is the degree of $L$, which is $$\deg L=(\deg g)\cdot \deg \mathscr O_{\mathbb P^n}(1)|_D=(\deg g)\cdot(\deg D).$$