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Let $X$ be a smooth hypersurface in $\mathbb {CP}^n$, $H$ be a hyperplane. If $X\cap H$ is singular, is it true that $H$ is a tangent plane at some point of $X$?

Note the converse is always true, if we count the multiplicities.

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  • $\begingroup$ Of course, because the tangent space to the intersection is the intersection of tangent spaces. $\endgroup$ – Sasha Jan 20 at 18:50
  • $\begingroup$ @Sasha Sorry, but I don't understand. What do you mean by "tangent space to the intersection" and why this follows my question? $\endgroup$ – User X Jan 20 at 22:36
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    $\begingroup$ $X \cap H$ has dimension $n - 2$. It is singular at point $x \in X \cap H$ iff the Zariski tangent space to $X \cap H$ has dimension greater than $n - 2$. But this space is equal to the intersection of the tangent spaces to $X$ and to $H$ at $x$. Both a hyperplanes in the Zariski tangent space to $\mathbb{P}^n$, hence $x$ is singular iff they coincide, i.e., $H$ is tangent to $X$ at $x$. $\endgroup$ – Sasha Jan 20 at 23:02
  • $\begingroup$ @Sasha Wow... Many thanks! $\endgroup$ – User X Jan 21 at 10:34

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