(The blowup)
Let \(\pi : \Bl_0 \AA^2 \to \AA^2\). Every fibre over \(p \neq 0\) is a single point, and \(\pi^{-1}(0) \cong \PP^1\). Flatness over a reduced connected base forces the fibres to be equidimensional, so \(\pi\) is not flat.
(Normalisation of a node)
Let \(X\) be a nodal cubic with node \(q\) and let \(f : \tilde X \to X\) be its normalisation. If \(f\) were flat then \(f_* \OO_{\tilde X}\) would be a flat coherent sheaf, hence locally free, and of rank \(1\) hence invertible. But \(q\) has two preimages, so \((f_* \OO_{\tilde X})_q\) needs two generators over \(\OO_{X,q}\), and an invertible sheaf needs one.
The two examples are the two ways flatness fails, and they are worth separating. The blowup fails because the fibre dimension jumps; the normalisation fails because the fibre length jumps, from \(1\) to \(2\), while the dimension is constant at \(0\). So “equidimensional fibres” is necessary and not sufficient, and the invariant that flatness really keeps constant is the Hilbert polynomial.
No nontrivial normalisation is flat, for the same rank argument, and a ramified map of curves in the sense of a map that is not finite locally free is not either. The contrast to draw is with a ramified map of smooth curves, which is finite flat of constant degree: there the ramification shows up in the fibre being a fat point, not in the degree dropping.
Dependencies
D-MORFLATFlat morphismsuses