I have been having trouble understanding some statements regarding flatness in Hartshorne - in particular relating to some of the examples in the text. Any help would be appreciated!

Here is the issue:

In example III.9.8.4 Hartshorne discusses an example of a family of twisted cubics arising from the projection of $\mathbb{P}^3$ to $\mathbb{P}^2$ from a point. The result is that the flat limit of the twisted cubic is not only singular, but it has an embedded point at the singular point.

In example III.9.10.1 he explains why if one takes the reduced induced structure on the fibers, then the family is not flat.

My question is: If one takes the flat family $Y\to\mathbb{A}^1$ from the first bullet, and uses the canonical map $Y_{red}\to Y$, then by composing one gets a family $Y_{red}\to \mathbb{A}^1$, which should be flat by proposition III.9.7 (which states that you have flatness over a smooth curve if every associated point of $Y$ maps to the generic point of the curve). Now, I expected $Y_{red}$ to be the family in the second bullet, but since it is not flat, this cannot be the case. What is going on?

Thanks!

fibersby their nilreductions. There are lots of maps of reduced schemes with non-reduced fibers. Maybe try to write down a simple example. – Ramsey Jun 20 '11 at 22:28