Are all divisors in a very ample linear system on a smooth variety smooth?

Let $X$ be a smooth projective variety and $\mathcal{L}$ a very ample line bundle. It seems to me that in this situation any $D\in |\mathcal{L}|$ is smooth, since it is (isomorphic to) the intersection of a smooth variety with a hyperplane in $\mathbb{P}^n$, and hence smooth. Is this correct?

• Intersections of smooth varieties have no reasons to be smooth !
– user171326
Apr 11, 2017 at 16:04
• @N.H. Of course, something I shouldn't have overlooked... If you add this as an answer I'll accept it, since it adreses the flaw in my reasoning more directly then the other answer. Apr 11, 2017 at 16:09
• No problems ! You can accept the other answer :) Notice that the key word is transversality : if $X,Y$ are smooth and intersect transversally then $X \cap Y$ is smooth.
– user171326
Apr 11, 2017 at 16:14
• Thanks @N.H. - I must say I also think you did a great job of explaining this! Apr 11, 2017 at 16:21

No. For example, take $X = \mathbb P^N$ and take $\mathcal L = \mathcal O(d)$ (with $d > 1$). The divisors in your linear system are degree $d$ hypersurfaces. And it is very easy to think of singular hypersurfaces!