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 $\mathbb{P}^2$.



singularities of the dual variety of a surface
I. Shimada proved this result in [*Singularities of Dual Varieties in Characteristic 3*, Geom. Dedicata 120 (2006), 141–177](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.74.8099&rep=rep1&type=pdf), by assuming the linear series is *"sufficiently ample"*. (**Edited:** This result seems to be classical. Here is an earlier reference: Proposition 6.1 in [C. Simpson, *Some families of local systems over smooth projective varieties*, Annals of Math 138 (1993), no. 2, 337-425](https://www.jstor.org/stable/2946614?seq=1#metadata_info_tab_contents). The author works over $\mathbb C$ though.)



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> **Theorem**: *(Shimada) 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.*
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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](https://mathoverflow.net/users/4096/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.


