Intersection plane and non-singular cubic surface $\mathbb{P}^3$

Question

I found in the paper of Reid "Undergraduate Algebraic Geometry" the following statement:

Every plane $\pi \subset \mathbb{P}^3$ intersects S (a nonsingular cubic surface) in one of the following:

1) an irreducible cubic; or

2)a conic plus a line; or

3) 3 distinct lines

The proof claims : "I have to prove that a multiple line is impossible." I can't find out why this is what we need to prove. I guess it has something to do with dimensions, but can't make an argument. And if I would accept this I don't have any intuition why this are the three cases.

Thank you for your help!

Answer 1

For simplicity assume that your plane $\pi$ is given by $x_3 = 0$. If $f(x_0,x_1,x_2,x_3) = 0$ was the equation of the cubic, you are looking by the curve given by the equation $x_3 = 0$ and $f(x_0,x_1,x_2,0) = 0$ which you can consider as polynomial in $x_0,x_1,x_2$, and this is of course a polynomial of degree $3$.

(Indeed, if this is identically zero, this would means that $f$ was divisible by $x_3$, i.e that the plane $\pi$ was contained in my cubic which is not possible as we assumed $S$ was smooth.)

So we have to think how looks like a plane cubic curve $C \subset \pi$, but there are not much possibility : if $C$ is not irreducible, it can contains a conic and then it is the union of a conic and a line. If it does not contains a conic, then it is the union of $3$ lines, but possibly some lines can have multiplicities, e.g the cubic $x_0^2x_1 = 0$ or worst $x_0^3 = 0$. So indeed if the line have no multiplicites then the proposition by Reid follows.

Answer 2

The intersection of a smooth degree $3$ surface and a plane (which is a smooth degree $1$ surface) in $\mathbb{P}^3$ is a degree $3$ curve $C$, i.e. a cubic curve.

If $C$ is irreducible, then it is an irreducible cubic curve.

Suppose now that $C$ is reducible. The degree of a reducible curve is the sum of degrees of the irreducible components. As the degree of a curve is a positive integer, this leaves us with two possibilities: two components, one of degree two and the other of degree one, or three degree one components.

If $C$ has two irreducible components, then one has degree two (i.e. an irreducible conic) and the other has degree one (i.e. a line).

If $C$ has three irreducible components, then $C$ is the union of three lines. In this case two (or even three) of the lines could be the same line. As $S$ is nonsingular, all three lines are distinct and we see that the three options you listed are the only possibilities.