# Dual varieties and nodal sections

Let $$X$$ be a(n even dimensional) smooth complex projective variety in $$\mathbb{P}^N$$, and let $$X^{\vee}$$ be its dual variety; up to an higher degree Veronese embedding of $$X$$, I assume that $$X^{\vee}$$ is a hypersurface (that is, $$X$$ has no defect).

By construction, for any $$H\in X^{\vee}$$, the correspondig hyperplane cuts $$X$$ in a singular section $$X_H=X\cap H$$.

By Theorem of Reflexivity, for any $$(P,H)\in X\times X^{\vee}_{sm}$$ such that $$H$$ is tangent to $$X$$ at $$P$$ is equivalent to say the hyperplane corresponding to $$P$$ in $$\overset{\vee}{\mathbb{P}^N}$$ is tangent to $$X^{\vee}$$ at $$H$$.

Question: assuming that the singular points of $$X_H$$ are only double ordinary (for simplicity, all singularity are nodes, so $$X_H$$ is a nodal section), what can I say about $$H\in X^{\vee}$$?

• What kind of property are you asking for? $H$ is a smooth point if and only if $X_H$ has exactly one ordinary double point. What else? – abx May 16 at 10:26
• 1) I am not assuming that $X_H$ has exactly only one node. 2) How can I see the equivalence between the smoothness of $H$ and the "nodality" of $X_H$? – Armando j18eos May 16 at 10:33
• One reference is SGA 7, Exp. XVII, Proposition 3.5. If you have $n$ nodes, $X^{\vee}$ has $n$ branches locally around $H$, each branch being smooth. – abx May 16 at 12:06
• Perfect! This is very enlightening. – Armando j18eos May 16 at 12:54
• Also see Tevelev's Projectively Dual Varieties. – AG learner May 16 at 15:58