## Some obvious singularities

Wei Yue asked a question about an assertion that came up in lecture today. It was that if a curve in has defining equation , where for two non-constant homogeneous polynomials and , then is necessarily singular. The reason is the equation

So if is a point where and , then . But the zero sets of and must meet (by Bezout’s theorem) and hence, the curve is singular.

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

and

.

Since , we get

But there must be a point , and this is a singular point of . If you visualize as the union of two curves, you can imagine that these isn’t a single tangent line to 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 two lines *must* meet.)

Another question was about the factorization of such an . That is, doesn’t 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 of large degree are *irreducible*. In fact, what we showed above is that if is reducible (in that case, we also say is reducible), then is necessarily singular.

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