# Topology of plane curve complements after blow-ups

Let $C$ be a (singular, reducible) curve in the complex projective plane $\mathbb{P}^2$. An old problem is to study the topology of the complement $\mathbb{P}^2-C$. A famous result in this direction was originally stated by Zariski, but not proven until 1980 by Fulton. It states that if $C$ has only nodes for singularities (that is, double points with distinct tangents), then the fundamental group $\pi_1(\mathbb{P}^2-C)$ is abelian.

My problem is as follows: again, let $C$ be a (singular, reducible) curve in $\mathbb{P}^2$. Now, blow up the plane at one of the singular points of $C$. Denote the resulting surface $S_1$, the exceptional divisor by $E_1$, and the proper transform of $C$ by $\hat{C}_1$. Then blow up $S_1$ at a singular point of $\hat{C}_1$, and repeat this procedure to get a surface $S=S_n$, exceptional divisors $E_1,\ldots,E_n\subset S$, and the proper transform $\hat{C}\subset S$ of the original curve. I am interested in the topology of the complement $X=S-(\hat{C} \; \cup E_{i_1} \cup \ldots \cup E_{i_k})$. In other words, $X$ is obtained from $S$ by deleting $\hat{C}$ as well as some (but not all) of the exceptional divisors obtained from blowing up. In particular, I would like to prove that the fundamental group $\pi_1(X)$ is abelian.

If the $C$ has only nodes for singularities, then $\pi_1(\mathbb{P}^2-C)$ is already abelian, and it is not hard to see that $\pi_1(X)$ will also be abelian. However, I am more interested in some cases where $C$ has slightly more complicated singularities. For example, if $C$ has a triple point with distinct tangents, $\pi_1(\mathbb{P}^2-C)$ is non-abelian; but I suspect that if I blow up the plane at the triple point, then $\pi_1(S-\hat{C})$ will be abelian. Does anyone know of any techniques or references for dealing with this situation?

(Note: It is certainly not the case that $\pi_1(X)$ will always be abelian. I am interested in techniques that will allow me to prove it is abelian for certain specific curves and configurations of exceptional divisors.)

Corrected according to the comment of Scott.

Let me show that in the case you blow up $\mathbb P^2$ in one point an throw away preimages of three lines trough the point you get something with non-abelian fundamental group. The blow up of $\mathbb P^2$ at one point is a $\mathbb P^1$ bundle over $\mathbb P^1$, so when we throw away $3$ fibers we get a fibration $\mathbb P^1\to S\to \ C$, where $C$ is $\mathbb P^1$ punctured in $3$ points. The long exact sequence of homothopy groups reads as follows:

$...\to \pi_1(\mathbb P^1)\to \pi_1(S)\to \pi_1(C)\to \pi_0 \mathbb (\mathbb P^1)\to ...$

$...\to 0 \to \pi_1(S)\to \pi_1(C)\to 0$

So $\pi_1(S)=F_2$, where $F_2$ is a free group on two generators.

I can not say anything about the general situation...

• If I'm not mistaken, the long exact sequence of homotopy groups should decrease in degree (but I think this makes your proof even easier). Mar 3, 2011 at 12:05
• Scott, huge thanks, I was a bit sleepy while writing this :) Mar 3, 2011 at 12:47
• Thanks for the counterexample--obviously the situation is even more complicated than I thought. Mar 4, 2011 at 4:22

The following Theorem by Nori (Proposition 3.27 of this paper) is in the spirit of what you are looking for.

Theorem. Let $$D$$ and $$E$$ be curves on a smooth projective surface $$X$$. Assume that

1. the only singularities of $$D$$ are nodes;
2. $$D$$ and $$E$$ intersect transversally;
3. every irreducible component $$C$$ of $$D$$ satisfies $$C^2 > 2 r(C)$$ where $$r(C)$$ is the number of singularities of $$C$$.

Then the kernel of the natural morphism $$\pi_1(X-D-E) > \longrightarrow \pi_1(X-E)$$ is abelian.

In particular if you take an irreducible plane curve $$C$$ having a triple point with distinct tangents as its only singularity then the fundamental group of the complement of its strict transform is abelian (apply Theorem above to $$D = \hat C$$ and $$E = \emptyset$$ ). As Dmitri pointed out, if you drop irreducibility this is no longer true.

You might want to take a look at this survey. There you will find Nori's Theorem in Section 2.3.

1. If $$C \subset \mathbb P^2$$ is an irreducible curve with only one singularity having smooth branches intersecting pairwise transversely then the complement of $$C$$ in $$\mathbb P^2$$ and well as the complement of its strict transform in the blow-up of $$\mathbb P^2$$ are abelian ($$D = \hat C$$, $$E =$$ exceptional divisor in the first case; and $$D = \hat C$$, $$E= \emptyset$$ in the second).
2. If you have a reduced connected curve $$C = E_1 + \ldots + E_k$$ with rational irreducible components, the intersection matrix $$(E_i\cdot E_j)$$ is negative definite, and the dual graph is a tree then the fundamental group of the complement of a neighborhood of $$C$$ has been determined by Mumford in this paper.
• Thanks for the references. Obviously this is a very relevant theorem to my situation. Unfortunately it does not completely solve my problem since I am blowing up reducible plane curves, and typically all components will have negative self-intersection. Any ideas of what I can do if $C^2<0$? Mar 4, 2011 at 4:25