Calculate the intersection numbers by a plane section

Question

This question is from the chapter A of Reid's note: Chapters on algebraic surfaces

Let X = X$_d$ $\subset$ P$^3$ be a nonsingular surface of degree d and suppose that X has a plane section P that decomposes as a union of two curves P$\cap$X = A + B of degrees a and b. How to calculate the intersection numbers A$^2$, B$^2$ and AB?

If possible, please do not use genus formula, because later there is a question ask me check it again with genus formula.

An added question: If the plane section has more than two parts, what can we say?

Answer 1

Let me sketch an answer here.

1. Since $A$ and $B$ are curves in the plane $P$, we get $A \cdot B = ab$ by Bézout.

2. Now since $A$ is contained in $X$, we have $A \cdot (A+B) = A \cdot P$; to calculate this we can replace $P$ with a plane that meets $A$ properly, so we get $A \cdot P = a$.

3. Similarly for $B \cdot (A+B)$.

Combining the above gives the numbers $A^2$ and $B^2$.