Number of lines incident to a fixed line on quartic threefolds Let $X$ be a general quartic threefold over $\mathbb C$. It is known that the Fano scheme of lines $F$ is a smooth curve and every line $L\in F$ has normal bundle $N_{L|X}\cong \mathcal{O}+\mathcal{O}(-1)$. (See Collino's Lines on Quartic Threefolds.) Therefore if two lines are infinitesimally closed in $F$, they are disjoint.
Let $n_L$ be the number of lines on $X$ incident to the line $L$. I'd like to know 
(1) If $n_L$ is a constant number? 
(2) If so, how could we calculate this number?
Thanks!
 A: The answer is that $n_L=81$. The proof was originally due to Fano and the literature can be found in page 40 from Tyurin's  Five lectures on three-dimensional varieties, 1972. For convenience of others and for my own benefit, I will rewrite the proof below. The notation is the same as above. 
I. Degree of the Scroll: Let $I$ be the incidence variety of the pair $(L,x)$ such that $x\in L$
$\require{AMScd}$
\begin{CD}
I @>{\varphi}>> X\\
@V{p}VV \\
F 
\end{CD}
with $p:I\to F$ is a $\mathbb P^1$-bundle over the curve $F$, and $\varphi(I)$ is the scroll of the surface swept out by lines in quartic threefold $X$, which is singular at points where lines are incident to each other. According to a standard Schubert calculus computation in Tennison's On the quartic threefold, 1974, the degree of surface $\varphi(I)$ is $320$, or equivalently, $$\varphi(I)=80H\tag{1}\label{1}$$ in $A^1(X)$, where $H$ is a hyperplane section.
II. $n_L$ as Intersection Number: Let $\Gamma, L\subset I$ be a section and a fiber of $p:I\to F$, respectively, then among the points of the intersection $$\varphi(L)\cdot \varphi(\Gamma)\tag{2}\label{2}$$ is one point of the intersection of $L$ with $\Gamma$ and the remaining points are intersection of $L$ with other lines in the family. Therefore it suffices to find a special section such that the intersection number $(\ref{2})$ is easy to compute. 
Let $H'$ be a general hyperplane containing a line $L$ in $X$, then $S=H'\cap X$ is a smooth surface with $L$ as the unique line. So in the Chow ring $A^*(X)$, $$\varphi(I)\cdot S=L+C\tag{3}\label{3}$$ with $C$ the residue curve which does not contain a line. Therefore $\varphi^{-1}(\varphi(I)\cdot S)=L+\varphi^{-1}(C)$, and 

Proposition: $\varphi^{-1}(C)$ is a section of the bundle $p:I\to F$. It follows that 
   $$n_L=L\cdot C-1.$$

III. Final Computation: Intersect with $L$ on both side of $(\ref{3})$, and use $(\ref{1})$, we have
$$80(L+C')\cdot L=80(L^2+3)=L^2+L\cdot C$$
where $C'$ is a plane cubic intersecting $L$ transversely at $3$ points. Now use the fact that $S$ has trivial canonical bundle, so $L$ has self-intersection  $(-2)$ in $S$, which leads to $n_L=L\cdot C-1=81$. 
