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.



Dimension of the set of singular hypersurfaces
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.


(*Ref*: ["The topology of complex projective varieties after S. Lefschetz" by Klaus Lamotke](https://www.sciencedirect.com/science/article/pii/0040938381900136).)


