Computing $\pi_1$ of the complement of a non-singular plane curve

Question

The following is a well-known fact:

Theorem. The fundamental group of the complement of a non-singular curve of degree $d$ in the complex projective plane is cyclic of order $d$.

This was further generalised by Fulton, who proved that the fundamental group of a nodal curve is Abelian. (As far as I understand, Fulton used the result for non-singular curves in his proof.) I think that Zariski's idea of proof of Fulton's theorem works to prove the non-singular case, since every two non-singular curves are isotopic. (I gather this from Deligne's Séminaire Bourbaki on Fulton's theorem, but feel free to correct me!)

Another proof is using the braid monodromy technique: you prove that if a degree-$d$ curve has an order-$d$ tangent line, then the fundamental group of its complement is Abelian. (I learnt this from Cogolludo's notes on braid monodromy.) It could be that Zariski used this to prove his statement, so maybe both proofs I am aware of use braid monodromy in the end.

Here's a question I asked around a bit, googled, and thought about, but got no satisfactory answer:

Are there other (easier? more direct?) proofs of the theorem above?

If not, is there maybe a moral reason one should expect that this is hard?

Answer 1

Another approach is to use Milnor's work on isolated singularities of complex hypersurfaces. More generally, let $W$ be a smooth hypersurface in $\Bbb{CP}^n \,\,(n\geq 2)$ defined by a homogeneous polynomial $F(Z_0,\dots,Z_n)\in\Bbb{C}[Z_0,\dots,Z_n]$ of degree $d$. Let us show that $\pi_1\left(\Bbb{CP}^n-W\right)\cong\Bbb{Z}/d\Bbb{Z}$. Denote the hypersurface cut by $F(\mathbf{z})=0$ in the affine space $\Bbb{C}^{n+1}$ by $V$. Since the projective hypersurface $W$ is smooth, the origin $\mathbf{0}\in\Bbb{C}^{n+1}$ is the only singular point of the affine hypersurface $V$. For any $\epsilon>0$, the quotient map $\Bbb{C}^{n+1}-\{\mathbf{0}\}\rightarrow \Bbb{CP}^n$ restricts to a map $$\psi:S^{2n+1}_\epsilon(\mathbf{0})-V\rightarrow\Bbb{CP}^n-W$$ where $S^{2n+1}_\epsilon(\mathbf{0})$ is the real sphere of radius $\epsilon$ centered at the origin $\mathbf{0}\in\Bbb{C}^{n+1}\cong\Bbb{R}^{2n+2}$. This is clearly a fibration with fiber $S^1$. Hence, there is an exact sequence $$\pi_1(S^1)\rightarrow\pi_1\left(S^{2n+1}_\epsilon(\mathbf{0})-V\right)\rightarrow\pi_1\left(\Bbb{CP}^n-W\right)\rightarrow 0.$$ Fixing an arbitrary $(a_0,\dots,a_n)\in S^{2n+1}_\epsilon(\mathbf{0})-V$ and identifying $S^1$ with $\{\lambda\in\Bbb{C}\mid |\lambda|=1\}$, the first homomorphism is induced by $\lambda\mapsto (\lambda a_0,\dots,\lambda a_n)$.

On the other hand, a theorem of Milnor on isolated singularities implies that for $\epsilon>0$ small enough, the following map $$ \phi:S^{2n+1}_\epsilon(\mathbf{0})-V\rightarrow S^1: \mathbf{z}\mapsto\frac{F(\mathbf{z})}{|F(\mathbf{z})|} $$ is a fiber bundle with fibers homotopic equivalent to a wedge product of $S^n$'s. In particular, the fibers are simply connected and $\phi$ thus induces an isomorphism on the fundamental group level. Replacing $\pi_1\left(S^{2n+1}_\epsilon(\mathbf{0})-V\right)$ with $\pi_1(S^1)$ in the preceding exact sequence based on this fact, we arrive at the exact sequence $\pi_1(S^1)\rightarrow\pi_1(S^1)\rightarrow\pi_1\left(\Bbb{CP}^n-W\right)\rightarrow 0$ where the first homomorphism is induced by the following self-map of the unit circle $S^1\subset\Bbb{C}$: $$ \lambda\mapsto\frac{F(\lambda a_0,\dots,\lambda a_n)}{|F(\lambda a_0,\dots,\lambda a_n)|}=\frac{F(a_0,\dots,a_n)}{|F(a_0,\dots,a_n)|}\cdot\lambda^d. $$ The induced endomorphism of $\pi_1(S^1)\cong\Bbb{Z}$ is clearly multiplication by $d$. We deduce that $$\pi_1\left(\Bbb{CP}^n-W\right)\cong\Bbb{Z}/d\Bbb{Z}.$$