I would like to clarify an important aspect from the discussion in this question. The OP discussed an obstacle to solve part (c) from Exercise 9.5 from Hartshorne's Algebraic Geometry Chap. III page 266. The exercise is also displayed in the linked thread.
The main obstacle there was to find a closed subscheme
$\tilde{X}\subset \mathbb{P}_{T}^{n+1}$ ($T$ noetherian integral base) such that
$\tilde{X}_{t} = \operatorname{C}(X_{t})$ holds for every $t \in T$.
The natural guess is to choose as candidate
the projective cone $\operatorname{C}(X) \subset \mathbb{P}_{T}^{n+1}$
and to check that it does the job. Using the observations that $\operatorname{C}(X_{t}) \subset \operatorname{C}(X)_{t}$
and $\operatorname{C}(X_{\eta}) = \operatorname{C}(X)_{\eta}$ for generic point $\eta$ of
base scheme $T$ hold, the OP reduced to original problem to verification of
the Hilbert polynomial of $\operatorname{C}(X_{t})$ is independent of $t \in T$. Or equivalently according to Thm 9.9 to check if the cone $C(X)$ is flat over $T$.
By locality of the problem it suffice to check that the following:
Let $A$ be noetherian domain and write $T:= \operatorname{Spec}A$ and let $X \subset \mathbb{P}_{T}^{n}$ be a closed subscheme which is very flat over $T$.
Is the affine cone $\operatorname{Spec} A[x_0,\cdots x_n] / \operatorname{I}(X)$ flat over $T$?
ATTENTION: This statement above is not the original statement from the linked page, but due to explanations by Jason Starr the original statement was wrong, and $X$ must be assumed to be very flat and not just flat.
(Very flat means for a flat famaily $X \to T$ for every degree $d$ the dimension of the $d$-th piece $\dim_{k(t)} (S_t/I_t)_d$ is independent of $t$. Here $S_t/I_t$ is the homogeneous coordinate ring associated to fiber $X_t$. )
Following the discussion below KReiser's answer Hiro Wat remarked that meanwhile he already solved the problem using a technique of the semicontinuity theorem., Chap. III, 12 (starts from page 281).
Problem/Question: I not understand how concretely the mentioned techniques from the chapter on semicontinuity theorem (III, 12) was applied here to solve the explaned problem. Here I'm also not sure if the methods/results were applied directly to show that the projective cone $C(X) \subset \mathbb{P}_{T}^{n+1}$ does the job or to the reduced problem in the grey box.
I not know what there Hiro exactly had in mind there, but I conjecture that probably he made use of one of following methods from the chapter 12 to relate to attack the original problem. My guesses:
Scenario I: Corollary 12.6 (page 287): The goal might be to relate $T^0(A)$ to $\operatorname{Spec} A[x_0,\cdots x_n] / \operatorname{I}(X)$ induced by the the $T^i$-functors defined in the book. Since for any $A$-module $M$ the module $T^i(M)$ and a fixed coherent sheaf $F$ flat over $A$ defined as
$$ T^i(M) = H^i(X, F \otimes_A M) $$
it seems not to be clear as what $F$ here should be choosen. As the only involved sheaf that is flat over $A$ is the structure sheaf $O_X$, it seems to be the only canonical choice for $F$, but then $T^0$ cannot be related to $\operatorname{Spec} A[x_0,\cdots x_n] / \operatorname{I}(X)$.
And another problem if we even succeed to find such a flat $F$ with $T^0(A)= \operatorname{Spec} A[x_0,\cdots x_n] / \operatorname{I}(X)$, why it should be that $T^i$ is exact there? (indeed, according to Corollary 12.6 exactness of $T^i$ is equivalent to flatness of $T^i(A)$)
Scenario II: Theorem 12.8 (Semicontinuity) + Proposition 9.3 (compatibility of Base Change
with cohomology).
Problem: To use this we have to assure that $C(X_t)$ is the pullback of
$i: \{t\} \to T$ and $C(X) \to T$. But that's also not clear at all, see: For a projective scheme over a base
Therefore I doubt if I really understand how the idea of the mentioned approach via thes techniques of the semicontinuity theorem would lead to a solution of original problem and would like to discuss in this question how the probably used argument should work here.
Some remarks: I posted an identical question several weeks ago in MSE.
I also would like to add that the problem might be not really have reseach level, but I think it could also uncover an interesting facet I haven't found anywhere before: In standard literature like Hartshorne's book one learns that the techniques/philosophy around standard formulation of Semicontinuity theorem and it's applications in algebraic geometry used flatness as assumpion to study behaviour of certain functions $f: X \to \mathbb{N}$ from the studied scheme to natural numbers (in most cases the dimension of a group like cohomology) while "scanning" $X$ along it's points.
But here it seems that we can also use tools from same toolbox to work "reversally", namely to answer when a certain scheme is flat, anstead to use flatness as technical assumption. That's my motivation behind this question.
