TITLE: Are irreducible components of a flat family flat?
QUESTION [6 upvotes]: 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?

REPLY [10 votes]: 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 $Y$.