A general fact about the singular locus $Sing(X)$ of a variety $X$ (analytic or projective) is that they form a subvariety of the oringinal variety $X$. And we know that the boundary of a manifold have no boundary itself. My simple question is that
Is $Sing(X)$ (as a variety itself) a smooth variety ?
Intuitively, I can't imagine a picture such that the answer is no.
If the answer is no, another question is that does the singular locus $Sing(X)$ necessarily have dimension less than that of $X$ ?
What is true is that the singular locus of a variety has lower dimension. It can be singular, and its singular locus singular, and the singular locus of the latter singular, and so on.