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Everything is over field $\mathbb C$. Let $X$ be a hypersurface of degree $d$ in $\mathbb P^n$. We know that if $X$ is smooth, then its dual $X^\vee$ is still a hypersurface in $(\mathbb P^n)^\vee$, but not necessarily smooth. I want to know that, for a general choose of $X$, can we compute the dimension of the singular locus $X^\vee_{sing}$? Where can I find a discussion of this?

For some reason, I believe in general it is of codimension $1$ In $X^\vee$, or it is empty. But I don’t know how to prove this.

Thanks in advance.

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