# Dimension of the dual variety

Suppose $$X$$ is a smooth projective variety of dimension $$k$$ in $$\mathbb{P}^n$$. Let $$X^*$$ be its dual variety in $${\mathbb{P}^n}^*$$.

Question: What is the dimension of $$X^*$$ ?

• In most cases $X^*$ is a hypersurface, but there are exceptions. See L. Ein, Varieties with small dual varieties I, Invent. Math. 86 (1986), no. 1, 63-74.
– abx
Apr 5, 2022 at 3:58

The natural number $$\delta(X)=n-1-\dim X^*$$ is called the defect of $$X$$. A standard application of the incidence variety shows that $$\delta(X) \geq k$$ if and only if, for every $$x \in X$$ and for every hyperplane $$H \supset T_xX$$, we have $$\dim_x \{y \in X \; | \; H \supset T_yX \} \geq k.$$
In particular, $$X$$ is a hypersurface (namely, $$\delta(X)=0$$) if and only if, for every point $$x \in X$$ and for every hyperplane $$H \supset T_xX$$, there are only finitely many $$y \in X$$ such that $$H$$ is contained in $$T_yX$$. As explained in abx's comment, this is the usual situation, but there are exceptions.
A formula by Katz expresses the defect of $$X$$ in terms of the rank of a certain Hessian matrix, see Chapter 6 of the book Projective Dual varieties by E. Tevelev.