I am currently reading Fulton's expository lectures "Introduction to intersection theory in algebraic geometry".
On pg. 4, Fulton sketches an argument of George Salmon which I don't understand. Let $S_1,S_2$ and $S_3$ be surfaces of degree $m,n,p$ in $\mathbb{P}^3$ and suppose that $C$ is a double line that lies on $S_1$ and suppose further that $C$ is a component of $S_1 \cap S_2 \cap S_3.$ Then Fulton sketches how Salmon argued that the contribution of the double line to the total intersection equals $m+2n+2p-4.$.
If now $C$ is a double line on a surface $S$ of degree $n,$ and $Q_1$ and $Q_2$ are distinct points lying on a general line $L,$ let $S_{Q_1}$ and $S_{Q_2}$ be the polar surfaces associated to $Q_1$ and $Q_2.$ Fulton mentions that one would then expect, based on Salmon's analysis, that the contribution of the line $C$ to the intersection $S \cap S_{Q_1} \cap S_{Q_1}$ would be $5n-8.$ However, as Salmon showed, the pinch points contribute further to the intersection.
I don't understand why the pinch points should be viewed as contributing further to the intersection: is this because I should somehow view the pinch points as giving some extra multiplicity of the intersection? Could someone please help me gain intuition for why the pinch points should contribute further to the intersection? I am more interested in intuition than proofs for now.
