I am new here, but hopefully you can help me with a concrete problem I have. I try to compute a Self-Intersection Number of a constructed curve in an analytic surface. I know the answer by some references, but have no idea how it has been found.
First my surface:
Let $N\in \mathbb N$. We take $M_N:=\coprod\limits_{n\in\mathbb Z} (\mathbb C \times \mathbb C) / \sim $, where $"\sim"$ is the following equivalence relation:
- $\forall a,b \in\mathbb C \setminus \{0\} \ \forall n,k\in \mathbb Z : (a,b)_n \sim (a(ab)^{-k},b(ab)^k)_{n+k}$ , where $(a,b)_n$ denotes the pair $(a,b)$ in the $n$th-copie of $\mathbb C \times \mathbb C$.
- $\forall b \in\mathbb C \setminus \{0\} \ \forall n,k\in \mathbb Z: (0,b)_n \sim (b^{-1},0)_{n+1}$
- $\forall a,b \in\mathbb C \ \forall n\in \mathbb Z : (a,b)_n \sim (a,b)_{n+N}$
The elements of $M_N$ are equivalence-classes of Elements $(a,b)_n$ under $\sim$ and shall be denoted by $[a,b]_n$.
The curve:
For $n\in\mathbb Z$ let $S_n := \{ [0,b]_{n-1}\} \cup \{ [0,0]_n\}$. Immedeatly we see: 1. $\forall n \in \mathbb Z : S_n \cup S_{n+1} = \{[0,0]_n\}$ 2. $\forall n \in \mathbb Z : S_n \simeq \mathbb{P^1(C)}$
I am searching for the Self-Intersection Number $S_n.S_n$. I know it is -2. But why is it? I have no experience in computing Intersection Numbers. The usual way seems to intersect a sligthly moved copy of the curve with the original curve. But this won't work, if the result is negative.
The idea must lie in the fact, that for $b\not =0$ the identity $[a,b]_{n-1} = [b^{-1},ab^2]_n$ ist given. But how to use this?
I am thankful for any hint. Shall I blow up anywhere?