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Let X and Y two smooth closed subschemes of a smooth projective scheme Z over a field.

Let $W:=X\cap Y$.

I suppose that W is non empty and that the intersection of X and Y is non proper, i.e

codim(W)< codim (X) +codim(Y).

Does W necessarily have singularities?

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    $\begingroup$ Of course not. For example, you can take $X = Y$ to be a point. Then $W$ is also a point. $\endgroup$
    – Sasha
    Commented Oct 30, 2012 at 3:19
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    $\begingroup$ To have less trivial examples, you can even intersect linear spaces to have non-proper intersections, but the intersection will still be linear and hence non-singular. $\endgroup$
    – Sándor Kovács
    Commented Oct 30, 2012 at 5:12
  • $\begingroup$ In fact, if $X$ and $Y$ are sufficiently transverse and $\dim W = 0$, then $W$ is always reduced and so always a smooth point. $\endgroup$
    – Karl Schwede
    Commented Nov 1, 2012 at 12:14

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