# Nonreducedness of schemes and projective morphisms

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 $y$?

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.
• No! The fiber is non-reduced. But as a point in $W$ (or $Y$), it is smooth. Think of $\mathbb{A}^1\rightarrow \mathbb{A}^1$, $t\mapsto t^2$. – abx Aug 12 '14 at 14:03