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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?

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    $\begingroup$ Intersections of smooth varieties have no reasons to be smooth ! $\endgroup$
    – user171326
    Apr 11 '17 at 16:04
  • $\begingroup$ @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. $\endgroup$ Apr 11 '17 at 16:09
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    $\begingroup$ 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. $\endgroup$
    – user171326
    Apr 11 '17 at 16:14
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    $\begingroup$ Thanks @N.H. - I must say I also think you did a great job of explaining this! $\endgroup$
    – Kenny Wong
    Apr 11 '17 at 16:21
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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!

However, a generic divisor in your very ample linear system is smooth. This follows from Bertini's theorem.

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