Let $C\subset \mathbb{P}^2$ be a reduced nodal complex plane curve of degree $d$. Let $\Sigma$ be the set of nodes of $C$, and let $I$ be the ideal of $\Sigma$. Denote with $S=\mathbb{C}[x,y,z]$ the polynomial ring in three variables. Consider a minimal free resolution $ 0 \to \oplus_{i=1}^t S(-b_i) \to \oplus_{i=1}^{t+1}S(-a_i) \to S \to S/I \to 0.$

As an exercise I tried to calculate the Hodge structure on $H^3(\mathbb{P}^2\setminus C)$. From this calculation it follows that $b_i\leq d$ for all $i$ and that the number of irreducible components of $C$ equals $\#\{i \mid b_i=d\}+1$.

My feeling is that this statement might have been known before (it sounds like a classical statement) but I could not find a reference. Moreover, I would prefer a proof for this fact that avoids the use of Hodge theory.

Does anyone knowns a reference (or a more classical proof)?