Problem/Question: I not understand how concretely the mentioned techniques from the chapter on semicontinuity theorem (III, 12) was applied here to solve the reduced problem above.
  I not know what there Hiro exactly had in mind, 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:

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:

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 (from page 281).

(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).