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 $y$?



Nonreducedness of schemes and projective morphisms
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.


