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I am interested in examples of hypersurfaces $X, Y$ defined by polynomials $F(x_1, \cdots, x_n), G(x_1, \cdots, x_n)$ respectively, so that the intersection $X \cap Y$ has a singular locus of high dimension. I am interested in both cases where $F,G \in \mathbb{C}[x_1, \cdots, x_n]$, so that $X,Y$ are affine hypersurfaces in $\mathbb{A}^n(\mathbb{C})$, and when $F,G$ are homogeneous polynomials, so that $X,Y$ are hypersurfaces in $\mathbb{P}^{n-1}(\mathbb{C})$. In particular, I am interested to know if when say $F$ is non-singular, is it possible for the intersection $X \cap Y$ to have a singular locus of codimension 2 or less? Or is the hypothesis that one of the hypersurfaces $X,Y$ being defined by a non-singular polynomial sufficient to show that the singular locus of the intersection must have a low dimension?

Thanks for any help or reference.

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The intersection of two smooth hypersurfaces $X,Y$ is singular at a point $p$ if and only if $X$ and $Y$ fail to be transverse at $p$. It should be easy to show that the singular locus of the intersection can have any codimension you want. As you should interpret the intersection scheme-theoretically, it is even possibly for the intersection to be everywhere non-reduced, in which case you should view it as being singular everywhere. (For example, this is the case for the two hypersurfaces $z=0$ and $z=x^2$ in 3-space, whose intersection should be viewed as a double line in the plane $z=0$). –  Jack Huizenga May 21 '13 at 7:42
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