Through reading Hartshorne's book I coarsely understand the flat family of varieties as a flat flat morphism between two varieties and the important objects are those fibers. But if that is the definition then the exercise 9.5 really confuses me.
In the part a of this exercise we need to find a flat family of projective varieties $\{X_t\}_t$ such that their projective cones $\{C(X_t)\}_t$ is not a flat family.
OK, $X_t$s is a flat family means we have a flat morphism $f:X\to Y$(here $X\subset \mathbb P^n$ being projective), and $X_t=f^{-1}(t)$. But then how to understand $C(X_t)$s as a family? I know that if we set $P=(0,\cdots,0,1)\in \mathbb P^{n+1}$, then we have a canonical morphism $C(X)\setminus P\to X$, and we can compose it with $f$ to get a morphism $g:C(X)\setminus P\to Y$. Does he mean $C(X_t)=g^{-1}(t)$? So I need to find an example such that $g$ is not flat?
I wonder if these understandings are right. I'm pleased to accept corrections if I have misunderstood something. Thanks.
Best Answer
What you're asking about is a key part of later portions of the problem: given a flat projective family $X\to T$ with fibers $X_t$, how does one construct a flat projective family $X'\to T$ with fibers $C(X_t)$? You'd like to say that the cone on $X$ works, but proving that you can commute the operations of cones and fibers is difficult. Your attempt is not really on the right track.
It's considerably easier to show that some collection of varieties cannot be the members of a flat family, though: recall theorem III.9.9 which states that a projective family is flat iff the Hilbert polynomials for each member $X_t$ are the same. So you need to be on the lookout for a projective family where all the members $X_t$ have the same Hilbert polynomial, but the Hilbert polynomials for $C(X_t)$ are not all the same. (Just to be clear, when I say $C(X_t)$, I mean that $X_t\subset\Bbb P^n_t$ is a projective scheme over a field, so take the cone in that setting.) To do this, you'll need to start by investigating the relationship between the Hilbert polynomials of $X_t$ and $C(X_t)$: once you do, you should notice something interesting which will tell you what to look for next.