Flat but not very flat families

Question

I'm doing an exercise in Hartshorne's Algebraic Geometry, Ex 9.5 in Chapter III, whose part (a) states the following:

Given an example to show that if $\{X_t\}$ is a flat family of closed subschemes of $\mathbb{P}^n$, then the projective cone $\{C(X_t)\}$ need not be a flat family in $\mathbb{P}^n$.

Since we can determine flatness by Hilbert polynomials. I want to use the dimension formula $dim((S_t/I_t)[x])_d = \sum_{i = 0}^{d}dim(S_t/I_t)_i$, to construct a family with same Hilbert polynomial but have different dimension in lower degrees.

This turned out to be some special example of flat but not very flat families related to the remaining part of this exercise, but I failed in finding such counter examples.

Answer 1

Hint: compute the Hilbert function of three points in $\Bbb P^2$. There are two options, depending on some geometric information. Can you arrange a flat family which exploits this?

I'll put the Hilbert polynomial behind a spoiler block so you can make a go of computing it yourself before reading what it is.

The Hilbert function of three points in $\Bbb P^2$ is $$\begin{cases} 0 & n<1 \\ 2 & n=1 \\ 3 & n>1\end{cases}$$ if they're collinear, or $$\begin{cases} 0 & n<1 \\ 3 & n\geq 1\end{cases}$$ if they're not collinear.

A strategy for proving this:

Prove this by looking at the evaluation map $k[x,y,z]_d\to k^3$.

Do you see how use this to solve the problem?