I'm trying to solve Hartshorne Chap.III Ex.9.5.(c):
The biggest problem is to show the existence of a closed subscheme $\tilde{X}\subset \mathbb{P}_{T}^{n+1}$ such that $\tilde{X}_{t} = \operatorname{C}(X_{t})$.
I guess the projective cone $\operatorname{C}(X) \subset \mathbb{P}_{T}^{n+1}$ of $X$ works.
Note that an inclusion $\operatorname{C}(X_{t}) \subset \operatorname{C}(X)_{t}$ always holds but an equality doesn't necessarily hold in a general case.
However, one can easily see that $\operatorname{C}(X_{\eta}) = \operatorname{C}(X)_{\eta}$, where $\eta$ is the generic point of $T$.
Moreover, it is clear that the Hilbert polynomial of $\operatorname{C}(X_{t})$ is independent of $t \in T$.
Hence it suffices to show that the Hilbert polynomial of $\operatorname{C}(X)_{t}$ is independent of $t$, i.e., $\operatorname{C}(X)$ is flat over $T$.
(Then from the inclusion $\operatorname{C}(X_{t}) \subset \operatorname{C}(X)_{t}$ and the agreement of their Hilbert polynomials we conclude that $\operatorname{C}(X_{t}) = \operatorname{C}(X)_{t}$.)
Note that $\operatorname{C}(X)$ is almost everywhere an $\mathbb{A}^{1}-$bundle over $X$, so almost everywhere flat.
Therefore we only have to check the flatness of the affine cone $\operatorname{C}_{\operatorname{aff}}(X)$ of $X$.
So we have reduced the original problem to the following:
Let $A$ be a noetherian domain and write $T:= \operatorname{Spec}A$.
Let $X \subset \mathbb{P}_{T}^{n}$ be a closed subscheme which is flat over $T$.
Then, is the affine cone $\operatorname{Spec} A[x_0,\cdots x_n] / \operatorname{I}(X)$ flat over $T$?
Thank you!