Singular locus of the discriminant variety

Question

I asked this question on MSE some time ago but didn't get a response.

Everything can be assumed in $ \mathbb{C} $, or atleast in characteristic $ 0 $.

Consider degree $ d $ hypersurfaces in projective space $ \mathbb{P}^n $. These correspond, upto nonzero scalars, to homogeneous polynomials $ h $ of degree $ d $ in $ \mathbb{C} [ x_0, ... , x_n ] $. If $ M_d = \{ x^{\alpha} \} $, ($ \alpha $ a vector of $n+1 $ non-negative integers summing to $ d $) is the set of monomials of degree $ d $, the set $ M_d $ forms a basis for $ \mathbb{C} [ x_0, ... , x_n ]_d $, so write $ h $ as $ \sum c_{\alpha} x^{\alpha} $. Then there is a homogeneous polynomial $D(d,n) $ in the coefficients $ c_{\alpha} $, called the discriminant which satisfies the property that the hypersurface $ V(h) $ is singular iff $ D(d,n) $ vanishes. So $ D(d,n) $ itself defines a hypersurface in $ \mathbb{P}^{ \binom{d+n}{n} - 1 } $.

Question (1) : Is the hypersurface $ V (D(d,n) )$ singular always?

Here is an example, taking $ d=n=2 $, so we're looking at conics in $ \mathbb{P}^2 $. Write $$ h = ax^2 + by^2 + cz^2 + dxy + eyz + fzx $$ Then one can compute $$ D(2,2) = 8abc + 2def - 2ae^2 - 2bf^2 - 2cd^2 $$ which defines a singular cubic fourfold in $ \mathbb{P}^5 $.

The singular locus is (a slight modification of) the $ 2 $-uple Veronese embedding given by $$ [u,v,w] \rightarrow [u^2, v^2, w^2, 2uv, 2vw, 2wu] $$

We see that the singular locus corresponds precisely to the non-reduced conics, of the type $ h=(ux+vy+wz)^2 $.(As every conic, upto an invertible change of coordinates is either $ xy - z^2 $ which is isomorphic to a $ \mathbb{P}^1 $ or $ xy $ which is two intersecting lines or $ x^2 $ which is a nonreduced line.) This leads to my next question-

Question (2) : Does the singular locus of $ V(D(d,n)) $ correspond to nonreduced degree $ d $ hypersurfaces always?

Answer 1

(Details of what follows can be found in any exposition of dual varieties such as Lamotke's paper.)

Given a smooth projective variety $X\subset\mathbb{P}^M$ we can look at the subvariety $D(X)\subset\mathbb{P}^M\times\mathbb{P}^{M*}$ which is the locus of pairs $(p,H)$ where $H\in\mathbb{P}^{M*}$ is a hyperplane in $\mathbb{P}^M$ containing $p$ such that $T_pX$ and $T_pH$ are not transversal at $p$. One can show that $D(X)$ is the projective bundle over $X$ of a suitable vector bundle on $X$ of rank $M-\dim(X)$. It follows that $D(X)$ is a smooth variety of dimension $M-1$. The image $X^{*}$ of $D(X)$ in $\mathbb{P}^{M*}$ is called the dual variety of $X$ and the morphism $D(X)\to X^{*}$ is usually birational. Moreover, one can show that the hyperplane $L(p)$ in $\mathbb{P}^{M*}$ corresponding to $p$ is tangent to $X^{*}$ at a smooth point of $X^{*}$.

The relevance of the above to your questions is that one can take $X$ to be the $d$-tuple Veronese embedding of $\mathbb{P}^n$ in $\mathbb{P}^M$ for $M=\binom{n+d}{d}-1$. In that case, the map $D(X)\to X^{*}$ is birational and $X^{*}$ is what you have called $D(d,n)$. Note that a hypersurface $Y$ in $\mathbb{P}^n$ of degree $d$ is of the form $H\cap X$ for a suitable hyperplane $H$ in $\mathbb{P}^M$.

When $d>1$ there is a hypersurface $Y$ in $\mathbb{P}^n$ which is singular at (at least) $2$ distinct points. The point in $X^{*}$ corresponding to $Y$ is a singular point. The converse is also true. This answers both your questions. (The answer to the second question is in the negative when $d>2$ or $n\geq 3$, when one can find reduced hypersurfaces which have more than one singular point.)