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 $H\in X^{\vee}$?

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 corresponding 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 singularities are nodes, so $X_H$ is a nodal section), what can I say about $H\in X^{\vee}$?

Let $X$ be a  (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 hyperplan 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 $H\in X^{\vee}$?

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 $H\in X^{\vee}$?