# Singular locus of the discriminant variety

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?

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.)