Fundamental group of the union of a quadric and line

Question

I'm trying to do the following exercise

Let be $L$ a line and $Q$ a non-singular quadric in $\mathbb{CP}^2$. How could be $\pi_1(L \cup Q)$?

My idea is: $L \cong Q \cong \mathbb{S}^2$ and they can intersect in one point or in two points, so their union could be $\cong \mathbb{S}^2 \vee \mathbb{S}^2$ or a "two-point" wedge sum of two $\mathbb{S}^2$. They seem to me "something like a sphere" and "something like a torus". So my idea is that $\pi_1(L \cup Q)$ is or $\{1\}$ or $\mathbb{Z}\oplus\mathbb{Z}$.

1. Is this correct?
2. If it is correct, how can I make it rigorous?

Thanks in advance.

Answer 1

$L$ and $G$ are both homeomorphic to $S^{2}$, and they must always intersect in one or two points by Bezout's theorem.

In the case when they intersect in two points, one may check that $\pi_{1}(L \cup Q) \cong \mathbb{Z}$. To see this note that we can add two $3$-cells to "fill in" the two spheres (because the fundamental group only depends on the $2$-skeleton), the resulting space then deformation retracts onto a circle (try to draw pictures), hence the fundamental group is $\mathbb{Z}$.

In the case when they intersect in one point then $\pi_{1}(L \cup Q) = \{1\}$ by van Kampens theorem.