# Dimension of the set of singular hypersurfaces

Let $$N$$ be the number of degree $$d$$ monomials in $$n$$ variables. We can then view each non-zero point in $$\mathbb{A}^N_k$$ as a degree $$d$$ homogeneous form, $$k$$ an algebraically closed field. Let $$X$$ be the union of $$\mathbf{0}$$ and the set of points where the corresponding form is not smooth. I have heard that smoothness is a generic condition. This implies that $$X$$ is a Zariski closed set, I have not been able to find a proof of this fact. Any reference or explanation how to prove this along with what $$\dim X$$ is would be highly appreciated.

• Do you know Jacobian criterion ? Aug 12 '21 at 0:11
• You should look up the notion of multidimensional resultant and discriminant. Here $X$ is a hypersurface given by the vanishing of the discriminant which is a polynomial of degree $n(d-1)^{n-1}$. See the book by Gelfand, Kapranov and Zelevinsky. Aug 12 '21 at 20:11

Given a smooth projective variety $$X\subset\mathbb{P}^{N-1}$$, let $$I(X)\subset X\times(\mathbb{P}^{N-1})^*$$ denote the locus of pairs $$(x,H)$$ where $$x\in H$$ and $$T_x(X)\subset T_x(H)$$; here we are thinking of $$(\mathbb{P}^{N-1})^{*}$$ as the collection of hyperplanes in $$\mathbb{P}^{N-1}$$.
One can show (using the Euler sequence which is the "global" version of the Jacobian criterion mentioned by Mohan) that $$I(X)\to X$$ is the projective bundle $$\mathbb{P}_X(K_{X/\mathbb{P}^{N-1}})$$, where $$K_{X/\mathbb{P}^{N-1}}$$ is the co-normal bundle of $$X$$ in $$\mathbb{P}^{N-1}$$. This can be used to show that $$I(X)$$ is smooth of dimension $$N-2$$ in $$(\mathbb{P}^{n})^{*}$$.
Since $$X\times(\mathbb{P}^{N-1})^{*}\to(\mathbb{P}^{N-1})^{*}$$ is proper, the image $$D(X)$$ of $$I(X)$$ is a closed subvariety of dimension $$\leq (N-2)$$.
This is the closed subvariety the question is looking for when $$X\subset\mathbb{P}^{N-1}$$ is the embedding of $$X=\mathbb{P}^{n-1}$$ given by the monomials mentioned in the question. (In other words, $$X$$ is the Veronese embedding of $$\mathbb{P}^{n-1}$$ in $$\mathbb{P}^{N-1}$$.)
There are examples where $$D(X)$$ is not a of dimension $$N-2$$. However, "in general" one finds that $$D(X)$$ is a hypersurface called the dual hypersurface of $$X$$. It appears to be rather difficult to write the equation of $$D(X)$$ in general.