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.

  • 6
    $\begingroup$ Do you know Jacobian criterion ? $\endgroup$
    – Mohan
    Aug 12 '21 at 0:11
  • 4
    $\begingroup$ 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. $\endgroup$ 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.

(Ref: "The topology of complex projective varieties after S. Lefschetz" by Klaus Lamotke.)


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