I am looking for a proof/reference of the following simple fact, which I think it holds true.

Let $S\subset \mathbb{P}^n$ be a surface embedded by a very ample linear system. Then I know that the dual variety $S^*$ is an irreducible hypersurface. Is it true that the generic plane section of $S^*$ is an irreducible curve, smooth except for at most ordinary double points and cusps?

I guess one can translate this in something concerning the generic projection of $S$ onto a $\mathbb{P}^2$.

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    $\begingroup$ Are you in char 0? If so, you can use Bertini applied to the normalization of $S^*$, as described in Jouanolou's book. $\endgroup$ – Jason Starr Nov 6 '13 at 22:47
  • $\begingroup$ at.algebraic-topology? $\endgroup$ – Fernando Muro Nov 6 '13 at 23:08
  • $\begingroup$ In particular I think Jason is saying that the fact that $S^*$ is the dual of a surface is irrelevant. But I'm confused by your first claim, since e.g. the Segre embedding of $P^1 \times P^1$ is self-dual. $\endgroup$ – Allen Knutson Nov 7 '13 at 4:54
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    $\begingroup$ @Jason: yes I am working over $\mathbb{C}$. It is the first time that I check Jouanolou's book: what is exactly the argument you are mentioning? $\endgroup$ – IMeasy Nov 7 '13 at 7:56
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    $\begingroup$ Oops, embarassingly bad example. How about the 2nd Veronese of $P^2$, in $P^5$? I think that's self-dual too (and hope I haven't made as stupid a mistake this time). $\endgroup$ – Allen Knutson Nov 7 '13 at 8:07

Shimada proved this result in Singularities of Dual Varieties in Characteristic 3 in 2006, by assuming the linear series is "sufficiently ample":

Theorem: Let $X$ be a smooth projective variety of dimension $n>0$, over an algebraically closed field $k$ with $\text{char} \ k>3$ or $\text{char}\ k=0$. Take a sufficiently ample embedding $|\mathcal{L}|:X\hookrightarrow\mathbb P^N$, then a general plane in $(\mathbb P^N)^{\vee}$ cuts the dual variety along a curve with only ordinary cusps as its unibranched singularities.

Note that by picking the plane general, the "multibranched" point should have exactly two branches, and such singularities are just ordinary nodes.

Here, the "sufficient ampleness" of a linear system $|\mathcal{L}|$ is the surjectivity of the evaluation map

$$v_p:\mathcal{L}\to\mathcal{L}_p/\mathfrak{m}_p^4\mathcal{L}_p\cong \mathcal{O}_p/\mathfrak{m}_p^4 \tag{1}$$

for all $p\in X$. This condition gurantees that up to the third order tangent cone is reachable by restricting linear functionals from the ambient projective space to a neighborhood at each point.

To get some understanding, we put $\mathcal{C}:=\{(p,H)\in X\times(\mathbb P^N)^{\vee}|\ p\in X\cap H \ \text{is singular}\}$. The image of the second projection

$$\pi:\mathcal{C}\to (\mathbb P^N)^{\vee}\tag{2}$$

is called the dual variety and denoted as $X^{\vee}$, which has expected dimension $N-1$. Therefore, a general plane cuts $X^{\vee}$ along a curve $\Sigma$ whose singularities are general in $(X^{\vee})^{sing}$. We know that a smooth point of $\Sigma$ corresponds to a hyperplane section with a single ordinary node. But assuming $|\mathcal{L}|$ being sufficiently ample $(1)$, if $q$ is a unibranched singularity of $\Sigma$, then it corresponds to a hyperplane section with a single $A_2$ singularity (analytically equivalent to $x_1^2+...+x_{n-2}^2+x_{n-1}^3=0$ working over $\mathbb C$). More precisely, if we put $$\mathcal{E}^{A_2}:=\{(p,H)\in X\times(\mathbb P^N)^{\vee}|\ p\in X\cap H \ \text{is singular of type $A_2$}\},$$

the author showed in Prop. 4.9 that $\mathcal{E}^{A_2}$ is Zariski dense in $\mathcal{E}$, which has codimension one in $\mathcal{C}$ and is defined as the singular locus of $(2)$. Finally, in Theorem 5.2, the author showed that such $q$ is an ordinary cusp.

This gives a sufficient condition to solve @IMeasy's question, but I'm not sure if we can remove "sufficient ampleness" condition in $\text{char}\ k=0$ or even over $\mathbb C$. Perhaps, one should study examples which violate the conclusion of Prop. 4.9.


The answer is yes, the claim is true. Moreover, as long as the dual variety is an irreducible hypersurface (i.e. most cases), then the claim on the singularity of the plane section holds true for any smooth complex projective variety embedded in the projective space. This is done in a paper by Dolgachev-Libgober from the 80s.

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    $\begingroup$ In the paper I cite above there's only a claim of the fact. I would be extremely happy to see a proof. $\endgroup$ – IMeasy Nov 12 '13 at 22:14

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