# singular intersection only comes from tangent?

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.

• Of course, because the tangent space to the intersection is the intersection of tangent spaces. – Sasha Jan 20 at 18:50
• @Sasha Sorry, but I don't understand. What do you mean by "tangent space to the intersection" and why this follows my question? – User X Jan 20 at 22:36
• $X \cap H$ 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 $x$. – Sasha Jan 20 at 23:02
• @Sasha Wow... Many thanks! – User X Jan 21 at 10:34