Math 480, Spring 2014, Assignment 13

Carefully define the following terms, then give one example and one non-example of each:

1. Intersection multiplicity (of two curves at a given point). Do not try to define this rigorously (unless you happen to know some local algebra) but do give a few examples. (If you really insist on knowing a rigorous definition, see Fulton's lovely little book, freely available here.)

Carefully state the following theorems (you need not prove them):

1. Bezout's Theorem.
2. Mystic Hexagon Theorem.

Solve the following problems:

1. Prove Pappus' Theorem. (Hint: the union of two lines in $\mathbb{P}^2$ is a reducible curve of degree two. Modify the proof of the Mystic Hexagon Theorem, allowing for the fact that the "conic" is now reducible.)
2. Section 8.7, problem 16.
3. Suppose that $C$ is an irreducible plane projective curve of degree $d$. Show that the number of singular points of $C$ (see part (b) of the text problem above) is at most $d(d-1)$. (Hint: suppose $C=\mathbb{V}(f)$. Then a singular point of $C$ necessarily lies on the intersection of $C$ with $\mathbb{V}(\frac{\partial f}{\partial x})$. Now apply Bezout's Theorem.) Note that a stronger result turns out to be true: the number of singular points is at most $(d-1)(d-2)/2$, and this bound is sharp -- but these facts are harder to prove. (See Fulton's book if interested.)