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Aug 2, 2011 at 8:24 comment added naf Yes. The argument works in any dimension and the singularities need not be isolated.
Aug 1, 2011 at 15:59 comment added Maharana Does a similar reason hold for $Y$ a normal threefold with isolated singularities ($X$ is again the singular locus) ?
Jul 31, 2011 at 17:38 comment added Francesco Polizzi Right. I was just thiking of a line in the quadric cone in $\mathbb{P}^3$. There is no way to move it in its linear equivalence class in order to avoid the vertex
Jul 31, 2011 at 17:32 history answered naf CC BY-SA 3.0