I have a few questions on the construction of Hilbert scheme of hypersurfaces I found recently in Nitin Nitsure’s Construction of Hilbert and Quot Schemes (page 6 part 4)
In his paper Nitsure added a short guide how to show (4) Hilbert scheme of hypersurfaces in $\mathbb{P}^n$
is a projective space and left some details to reader. And couple of difficulties prevent me from verify step (i). Let $\Phi_d:= \binom{n+ t}{n} - \binom{n-d + t}{n} \in \mathbb{Q}[t]$ where $d \ge 1$. The exercise is to show that the Hilbert scheme $\operatorname{Hilb}^{\Phi_d, O(1)}_{\mathbb{P}^n}$ is isomorphic to $\mathbb{P}^m_{\mathbb{Z}}$ where $m= \binom{n+ d}{n} -1$. The proof guide in splited in steps (i),(ii) and (iii).
I stuck in troubles to understand some notations and how to prove to first step (i). The original claim in (i) is:
(i) Any closed subscheme $Y \subset \mathbb{P}^n_k$ with Hilbert polynomial $\Phi_d$, where $k$ is any field, is a hypersurface of degree $d$ in $\mathbb{P}^n_k$. Hint: If $Y \subset \mathbb{P}^n_k $ is a closed subscheme with Hilbert polynomial of degree $n − 1$, then show that the schematic closure $Z$ of the height $1$ primary components is a hypersurface in $\mathbb{P}^n_k$ with $\operatorname{deg}(Z) = \operatorname{deg}(Y)$.
Recall a Hilbert polynomial associates to a coherent sheaf $F$ on $X=\mathbb{P}^n_k$ together with line bundle $O_X(1)$
a polynomial
$$\Phi(t) = \sum_{i=0} ^n (-1)^i \dim_k H^i(X, F \otimes O_X(1)^{\otimes t} )= \sum_{i=0} ^n (-1)^i \dim_k H^i(X, F (t)) \in \mathbb{Q}[t] $$
(more generally $X$ can be ssumed as a finite type scheme over a field $k$ together with arbitrary line bundle $L$ on $X$). A Hilbert polynomial of a closed subscheme $Y \subset \mathbb{P}^n_k$ is the Hilbert polynomial of quotient sheaf $O_Y= O_X/I_Y$.
In Hartshorne's book Algebraic Geometry I found a proof that any hypersurface $H_d \subset \mathbb{P}^n_k$ has Hilbert polynomial $\Phi_d$. I don’t know how to use the Hint to show the converse.
The notation "height $1$ primary components" looks non standard geometric language since the topology not see the difference between primary ideals and it's prime radicals. What does Nitsure here mean?
I know only from commuatitive algebra that an ideal $I \subset R$ of a ring which is nice enough can be decomposed as an intersection of primary components $I= Q_1 \cap Q_2 \cap ... \cap Q_l$ where $Q_i \subset R$ primary ideals. It follows from definition that the radical $P_i=r(Q_i)$ is prime and therefore $V(Q_i)$ is irreducible. So I think that by a "height $1$ primary component of $Y \subset \mathbb{P}^n_k$ he simply means an irreducible component of $Y$ of codimension $1$ in $\mathbb{P}^n_k$? Is that true?
If everything what I said before is true, then: However I not understand why a closed $Y \subset \mathbb{P}^n_k$ with Hilbert polynomial $\Phi_d$ contains an irreducible component of codimension one. Also I don't know how to deduce the conclusion on $\operatorname{deg}(Z) = \operatorname{deg}(Y)$ where $Z$ is the schematic closure $Z$ of the hight $1$ primary components.