Let $X, Y$ be irreducible projective schemes over $\mathbb{C}$, $W \subset X \times Y$ a closed irreducible subscheme. Suppose that the natural projection map $pr_2:W \to Y$ is surjective on the underlying topological spaces. Note that for all $x \in W$, there is a natural map of tangent spaces $\phi_x:T_{W,x} \to T_{Y,pr_2(x)}$. If there is a point $y \in Y$ such that the $\mathbb{C}$-vector space generated by the union $\cup_{x \in pr_2^{-1}(y)} \phi_x(T_{W,x})$ does not coincide with $T_{Y,y}$, does this mean that $Y$ is nonreduced at the point $y$?
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No. Take for $X$ and $Y$ smooth projective curves (say), $W\subset X\times Y$ a smooth curve such that $pr_2$ has degree 2. Any branch point $y$ of $pr_2$ satisfies your requirement.
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$\begingroup$ I am a bit confused. I thought that a (schematic) branch point i.e., a point counted with multiplicity more than one is non-reduced. $\endgroup$– KaliCommented Aug 12, 2014 at 14:01
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$\begingroup$ No! The fiber is non-reduced. But as a point in $W$ (or $Y$), it is smooth. Think of $\mathbb{A}^1\rightarrow \mathbb{A}^1$, $t\mapsto t^2$. $\endgroup$– abxCommented Aug 12, 2014 at 14:03
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$\begingroup$ Am I correct to think that this means the tangent space to the scheme theoretic image is NOT the one generated by the union of the images of the tangent spaces, as above? $\endgroup$– KaliCommented Aug 12, 2014 at 14:14
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$\begingroup$ Thanks once again. Could you please suggest some reference that could help me understand when this does hold true (i.e., the tangent space to the scheme theoretic image is the same as the one generated by the union of the images of the tangent spaces as above). $\endgroup$– KaliCommented Aug 12, 2014 at 15:35