# Are irreducible components of a flat family flat?

Let $f:X\rightarrow Y$ be a flat morphism of schemes of finite type over a field $k$, and assume $Y$ is irreducible. Let $X_1, \dots, X_n$ be the scheme-theoretic irreducible components of $X$ (i.e., including embedded components).

• Is it true that each $X_i$ is flat over $Y$?
• If there are counterexamples to flatness of the $X_i$, is it true at least that each of them has equidimensional fibers?
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If you take a plane and a line intersecting at the origin, isn't that a flat map of finite type over your ground field with components of different dimensions? – Jacob Bell Nov 22 '12 at 23:22
By each component having equidimensional fibers I mean that $dim (X_i)_y$ is independent of $y$. Of course these dimensions will depend on $i$. – quim Nov 23 '12 at 9:27

No to the first question. Let $Y$ be a nodal cubic curve and let $X$ be its connected two-sheeted covering space. Each of the two components of $X$ is the normalization of $Y$.