Degree of dual variety of complete intersection Let $X$ be a complete intersection in $\mathbb P^n$, does anyone know how to calculate the degree of a dual variety of $X$? 
Some preliminaries can be found in 3264 and all that, chapter 10. For instance, such dual varieties are always codimension one (except when $X$ is linear); and when $X$ is a hypersurface, the answer is well know.
In particular, I'd like to know the answer when $X$ is the curve as a complete intersection by a cubic and quadric in $\mathbb P^3$.
 A: Let $X$ be a smooth complete intersection of type $(d_1,\dots,d_k)$ in $\mathbb{P}(V)$. Consider the bundle
$$
E := \bigoplus \mathcal{O}_X(1 - d_i) \hookrightarrow V^\vee \otimes \mathcal{O}_X,
$$
where the morphism is given by the derivatives of the equations of $X$. Then $\mathbb{P}(E)$ is the universal tangent hyperplane to $X$, hence the dual variety is the image of the map
$$
\mathbb{P}_X(E) \to \mathbb{P}(V^\vee)
$$
induced by the embedding $E \to V^\vee \otimes \mathcal{O}_X$. Therefore, the degree of the dual variety is 
$$
\deg(X^\vee) = s_{n}(E),
$$ 
where $n = \dim(X)$ and $s_n$ is the $n$-th Segre class (this is true under the assumption that the map $\mathbb{P}_X(E) \to X^\vee$ is birational, otherwise you need to divide by its degree). This class is easy to compute: this is the coefficient of $h^n$ in
$$
\left(\prod_{i=1}^k(1 - (d_i - 1)h)^{-1}\right)\prod_{i=1}^k d_i.
$$
In the case of a $(2,3)$ intersection in $\mathbb{P}^3$ this gives $3 \cdot 2 \cdot 3 = 18$.
A: I think the $h^n$ in the explanation should be $h^{n-k}$. E.g. in the case $k=n$ the expression in the parentheses must be equal to $1$.
