Suppose $f:X\to Y$ is a surjective morphism bewteen algebraic varieties , does the locus of non-reduced fibers form a closed subset of $Y$?
If the condition is not good enough, one may add conditions on smoothness, projectiveness or flatness.I am not clear what are suitable for this.
Thanks for any comments!
The locus of non-reduced fibres is closed, and the locus of reduced fibres is open. (If one is not working over an alg. closed field, or is working in some more general scheme-theoretic context, then it might be better to say geometrically reduced here, and to impose some finiteness condition on the morphism, something which will be automatic for maps between varieties.)
Actually, probably what I should have said is that the locus of points where the fibre is geometrically reduced is constructible. If the map is further flat, then it is open. (See the remark at the end of section 1 in this write-up by Ravi Vakil, and also this discussion in the stacks project.)