Some obvious singularities

Wei Yue asked a question about an assertion that came up in lecture today. It was that if a curve C in {\bf P}^2 has defining equation F=0, where F=GH for two non-constant homogeneous polynomials G and H, then C is necessarily singular. The reason is the equation

\nabla F=(\nabla G)H+G(\nabla H).

So if a is a point where G(a)=0 and H(a)=0, then \nabla F(a)=0. But the zero sets of G and H must meet (by Bezout’s theorem) and hence, the curve C is singular.

Another way of thinking about this is to explicitly consider the zero sets

D: G=0


E: H=0.

Since F=GH, we get

C=D\cup E.

But there must be a point a\in D\cap E, and this is a singular point of C. If you visualize C as the union of two curves, you can imagine that these isn’t a single tangent line to C along their intersection (and it *looks* singular there). For the easiest example, consider the union of two lines that meet at a point. (By the way, in {\bf P}^2 two lines *must* meet.)

Another question was about the factorization of such an F. That is, doesn’t F factorize into linear factors anyways? The answer is no in general. The factorization we discussed in the lecture was for homogeneous polynomials in two variables. In three variables, many F of large degree are *irreducible*. In fact, what we showed above is that if F is reducible (in that case, we also say C is reducible), then C is necessarily singular.


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