2
$\begingroup$

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?

$\endgroup$

1 Answer 1

6
$\begingroup$

(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.)

$\endgroup$
3
  • $\begingroup$ It has been a long time since I thought about this, but is it possible the discriminant locus is also singular at a hypersurface with just one singularity at which the Milnor number is > 1? $\endgroup$
    – roy smith
    Mar 26 at 22:12
  • $\begingroup$ Here is a partial argument. The singular locus of D should be closed, so a hypersurface with one singularity arising from 2 distinct sings which come together, should be a singular point of the discriminant, such as a conic with a tangent line, on the discriminant of plane cubics. $\endgroup$
    – roy smith
    Mar 26 at 22:34
  • $\begingroup$ It is a bit confusing, since at a hypersurface with a finite number of singularities, the tangent cone to D is a finite union of hyperplanes, one for each singularity. But if the singularity is the limit of 2 nearby singularities, then the hyperplane is also a limit of 2 hyperplanes, so counts twice. And just as a hyperplane is dual to a point, at a general singular hypersurface X, the set theoretic tangent cone to D is the dual variety of a certain projective model of the singular locus of X.jstor.org/stable/2001192?origin=JSTOR-pdf $\endgroup$
    – roy smith
    Mar 26 at 23:03

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.