I. Shimada proved this result in Singularities of Dual Varieties in Characteristic 3, Geom. Dedicata 120 (2006), 141–177, 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. The author works over $\mathbb C$ though.)
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.
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.
Best Answer
This is the portion concerning projecting cones of my explanation as posted in math.stackexchange regarding the degree of a projective variety. I think this answers exactly what you are asking for, including references within the text listed at the end. In particular, you can find the detailed construction of projecting cones at the beginning of part B. and just before the middle of C. The degree of it and the variety it comes from is the subject of the whole discussion (my notation is $C(X_k, L_r)$ for your $C_L(X)$, with $k:=\dim X$, $r=\dim L$). Whether it intersects properly a given subvariety $Y\subset X$ is true for generically chosen center $L_r$ (disjoint with X) by iterated application of Bertini's theorem to the sections $L_{r+1}\cap Y$, i.e., for $L_r$ in a dense Zariski-open subset of the Grassmannian $\mathbb{Gr}(r,\mathbb{CP}^{n})$, cf. reference at the end of point E. below.
Projecting Cones and Degree of a Variety: All the following definitions of degree are proved to be equivalent to each other so one can pick any of them as starting definition and get the rest (and more as mentioned in the link above) as interesting theorems.
• A. If $X$ is a hypersurface of $\mathbb{CP}^n$, i.e. $\dim X=n-1$, by [Hartshorne, Exercise I.2.8] it is given by the zero locus, $X=Z(f)$, of an irreducible homogeneous polynomial $f\in S_d$ of algebraic degree $d$, i.e. any monomial summand $a_{k_0\dots k_n}x_0^{k_0}\cdots x_n^{k_n}$ in $f$ has degree $d=\sum_i k_i$, where all such monomials generate the abelian group $S_d$, all of which make the ring of polynomials a graded ring $\mathbb{C}[x_0,\dots,x_n]=\bigoplus_{d=0}^\infty S_d$. So any such $f$ has a canonical associated (algebraic) degree, so
• B. If $k:=\dim X_k< n-1$, let $L_r\cong\mathbb{CP}^r$ be a generic (i.e. in general position) linear variety (linear projective vector subspace) of $\mathbb{CP}^n$ of dimension $r\leq n-k-1$. The projecting cone, $C(X_k, L_r)$, of $X_k$ from "vertex" $L_r$ is defined to be the joint locus of the subspaces $L_{r+1}$ that join the given $L_r$ with each point of $X_k$ (this generalizes the intuitive "cone" of lines obtained by projecting from a point). By [Beltrametti et al., sec.3.4.5] the projecting cone is also a, possibly reducible, algebraic variety of dimension $r+k+1$ (which justifies the upper bound of r at the beginning). For a generic $L_{n-k-2}$ the projecting cone of $X_k$ is thus a, possibly reducible, hypersurface as in A. above (if it is irreducible it is the case A. if it is reducible then its defining zero locus polynomial is reducible but has nevertheless well defined degree). Call
where $L$ is an element of the Grasssmannian of the required dimension. The degree of a variety $X_r\subset\mathbb{CP}^n$ is thus defined to be the degree of the generic hypersurface-projecting-cone; this is proved to be well-defined as this max deg is constant for a dense Zariski-open subset of the Grassmannian, cf. [Harris, Exercise 18.2]. For example if $L_0$ is a generic point in $\mathbb{CP}^3$ and $X_1$ a spatial algebraic curve, for each point of $X_1$ there is only one line joining it with $L_0$. Moving along all such points of the curve we obtain a cone swept by the joining lines with the fixed $L_0$, cone which is an algebraic surface, thus the zero locus of an homogeneous polynomial in projective space. So we are calling the degree of the spatial curve the degree of its projecting cone generic surface polynomial. Note that a curve in projective space is generically given by the intersection of two surfaces of possibly different degrees, $X_1=Z(h_1,h_2)\subset\mathbb{CP}^3$, so it has no canonical unique polynomial degree as is the case for plane curves. It is also important to remember that any nonsingular algebraic curve (thus Riemann surface) is isomorphic to a smooth spatial curve in $\mathbb{CP}^3$ [Hartshorne, Cor.IV.3.6][Shafarevich vol. I, sec.5.4 Cor.2] and birational to a plane curve with at most node singularities [Hartshorne, Cor.IV.3.11]. Note also that the first definition A. above is necessary, since the projecting cone of a hypersurface cannot be defined due to the constraint $r\leq n-k-1$. So what we have done is defining, for any lower dimensional variety, associated hypersurfaces which have generically well-defined polynomial degree.
• C. The "vertex" $L_r$ of a generic projecting cone $C(X_k, L_r)$ of a variety $X_k$ is given by $n-r$ linearly independent linear equations: $L_r=Z(h_1,\dots,h_{n-r})$ where $h_i$ are linear forms which define hyperplanes $H_i=Z(h_i)\cong\mathbb{CP}^{n-1}$ within $\mathbb{CP}^n$, so that $L_r=\bigcap_{i=1}^{n-r} H_i$. Projecting cones take their name from the fact that they define a generalized projection of a variety to a linear subspace (e.g. projecting from a point a spatial curve into a plane): the projection [Shafarevich vol. I, sec.4.4 Ex.1], with center or vertex $L_r$, is the rational map $\pi_{L_r}(x):=[h_1(x):\dots :h_{n-r}(x)]$ which is a regular morphism on the Zariski-open set $\mathbb{CP}^n\setminus L_r$. Therefore its restriction to any variety disjoint from the vertex, $\pi_{L_r}\vert_{X_k}:X_k\rightarrow\mathbb{CP}^{n-r-1}$ is a regular map of it to a projective subspace. Take any linear variety disjoint from $L_r$ as representative, i.e. $\mathbb{CP}^{n-r-1}\cong L'_{n-r-1}\subset\mathbb{CP}^n$ such that $L_r\cap L'_{n-r-1}=\varnothing$, which is always possible by [Beltrametti et al., Th.3.3.8] (since $\dim L_r\cap L'_{n-r-1}\geq r+(n-r-1)-n=-1$ so they do not intersect necessarily). Now for every point $x\in\mathbb{CP}^n\setminus L_r$, in particular $X_k$, there is a unique $L''_{r+1}$ passing through the vertex $L_r$ and $x$ by elementary dimension counting. The locus of all these generators $L''_{r+1}$ is just the generic projecting cone $C(X_k,L_r)$ for generic center $L_r$!. Each generator intersects $L'_{n-r-1}$ in a unique point (solution of a system of $n-(r+1)+n-(n-r-1)=n$ linear equations) which corresponds to $\pi_{L_r}(x)$ through the isomorphism with $\mathbb{CP}^{n-r-1}$. Therefore, given a generic linear subspace $L_r$ we can regularly (rationally if $L_r\cap X_k\neq\varnothing$) project any variety $X_k\subseteq\mathbb{CP}^n$ to a lower dimensional generic linear subspace $L'_{n-r-1}\cong\mathbb{CP}^{n-r-1}$ by intersecting the projecting cone with it, $C(X_k,L_r)\cap L'_{n-r-1}$, and calling $\pi_{L_r}(X_k)\subset \mathbb{CP}^{n-r-1}$ the projection of $X_k$ from $L_r$ to $L'_{n-r-1}$. The case $r=0$ is the classical projection from a point into a hyperplane (like our spatial curve projected to a plane curve). Therefore for any $X_k$, projecting from a generic center $L_{n-k-2}$, we obtain a, possibly reducible, variety $\bar{X}_k$ in $\mathbb{CP}^{k+1}$ as projection; since this comes from the intersection of the hypersurface $C(X_k,L_{n-k-2})$ with a linear variety $L'_{k+1}$, by the projective dimension theorem [Hartshorne, Th.I.7.2] every of its irreducible components has dimension $\geq (n-1)+(k+1)-n=k$. In fact, if $\dim X_k\geq 2$ by repeated application of Bertini's theorem [Hartshorne, Th.II.8.18], any such intersection is generically not only smooth but connected and thus irreducible, thus any generic such projection is a hypersurface $\bar{X}_k\subset\mathbb{CP}^{k+1}$ (generically reducible for $X_1$ a curve) and so it has a defining zero locus irreducible homogeneous polynomial $q\in\mathbb{C}[x_0,...,x_{k+1}]$ with well-defined algebraic degree (if $X_1$ is a curve then each of its irreducible components after intersecting will be points solution of a reducible polynomial). This is equivalent to B. since the intersection of a generic projecting cone hypersurface with a generic linear variety has the same polynomial degree as the cone (solve as many variables as possible from the linear system defining the linear variety and substitute in the homogeneous polynomial of the cone; each of its equal-degree monomials produce new monomials in less variables but of the same degree as the original, so one gets a new homogeneous polynomial in less variables, i.e. a hypersurface in a lower-dimensional projective space). This hints a geometric construction for the theorem of the birational equivalence of any projective algebraic set of dimension $k$ with a hypersurface in $\mathbb{CP}^{k+1}$, cf. [Beltrametti et al., sec.2.6.11] and [Hartshorne, Prop.I.4.9].
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• E. Following [Beltrametti et al., Prop.3.4.8] let us go back to B. or C. above, our projection of $X_k$ into a hypersurface of $\mathbb{CP}^{k+1}$ via generic center $L_{n-k-2}\subset\mathbb{CP}^{n}$. Take a generic line $l_1\subset\mathbb{CP}^{n}$ not contained in $L_{n-k-2}$, so that the linear space $\operatorname{Join}(L_{n-k-2}, l_1)=\langle L_{n-k-2}, l_1 \rangle$ is a generic $L'_{n-k}$ because this is just a projecting cone, thus having dimension $(n-k-2)+(1)+1$ (cf. beginning of B. above). It is clear that any such generic linear $(n-k)$-space can be obtained in this way by generically decomposing it into a line and a linear $n-k-2$-subspace contained in it. Now, the intersection of a $k$-variety with a generic hyperplane has irreducible components of dimension $k-1$ [Shafarevich vol. I, sec.6.2], thus $X_k\cap L'_{n-k}$ consists generically of a finite number of points. This number of points is constant in a dense Zariski-open subset of the Grassmannian $\mathbb{Gr}(n-k,\mathbb{CP}^{n})$, cf. [Harris, Ex.18.2]. This can be readily proved by noticing that it is the number of points in the generic fiber of $\pi_{\Lambda}$ with center a generic hyperplane $\Lambda\subset L'_{n-k}$ seen in D. above. To see this, note that our generic $L'_{n-k}$ can be thought as a projecting cone with center $\mathbb{CP}^{n-k-1}\cong\Lambda\subset L'_{n-k}$, and the fiber of $\pi_{\Lambda}$, which is finite by D., is by construction the intersection of the projecting cone with the variety. Therefore $\#\, (X_k\cap L'_{n-k})=\deg \pi_{\Lambda}$, showing equivalence with all the previous notions. It is worth mentioning that many (most) classical treatments define degree as this finite number of points of a generic intersection with a linear space of dimension the codimension of the variety. An independent proof is [Mumford, Th.5.1] where it is shown that generic linear $(n-k)$-subspaces meeting transversaly our variety $X_k$, do so in a common number of points: the degree. (This relates to definitions in differential topology of intersections of submanifolds meeting properly, i.e. $T_pX_k\cap T_pL'_{n-k}={0}$ and $T_pX_k\oplus T_pL'_{n-k}=T_p\mathbb{CP}^n$).
[Beltrametti; Carletti; Gallarati; Bragadin] - Lectures on Curves, Surfaces and Projective Varieties: A Classical View of Algebraic Geometry, (EMS 2009).
[Harris] - Algebraic Geometry: A First Course, (Springer 1992).
[Hartshorne] - Algebraic Geometry, (Springer 1977).
[Mumford] - Algebraic Geometry I, Complex Projective Varieties, (Springer 1976).
[Shafarevich] - Basic Algebraic Geometry I: Varieties in Projective Space, (Springer 1994).