Given a complete smooth Toric surface (over $\mathbb C$), is its intersection form well-known? Or is there an algorithm to calculate it? Thanks in advance.



Intersecton form of complete smooth Toric surface
The Picard group of a toric surface is generated by the simple toric divisors $D\_1,D\_2,\dots,D\_n$, and if $v\_1,v\_2,\dots,v\_n$ are the generators of the rays of the corresponding fan, there are relations
$$
\sum f(v\_i)D\_i = 0
$$
for all linear functions $f$. The intersection product is determined by these linear relations and simple intersection formulas
$$
D\_i \cdot D\_j = 
\begin{cases} 
1, & \text{if $|i-j| = 1$},\\
0, & \text{if $|i-j| > 1$},
\end{cases}
$$
where the indicies are assumed to by ordered cyclically (and to compute $D\_i \cdot D\_i$ one should use the linear relations).


