I will complete the argument soonCorollary 5. The basic idea now With the same hypotheses as in Proposition 3, assume further that $(X_K,\mathcal{L}|_{X_K})$ is $K$-isomorphic to use specializations$(\mathbb{P}^n_K,\mathcal{O}(1))$ for $n\geq 1$. Let $\mathcal{Y}$ be an $R$-flat Cartier divisor in $\mathcal{X}$ in the linear system of lines to prove that $\mathcal{H}$$\mathcal{L}$. Then $\mathcal{Y}$ is smooth over $\text{Spec}\ R$ (or else the multiplicity at a singular point would give an intersection number with a specialization of a line that is greater than one). Then,
Proof. Denote by an induction hypothesis,$\mathcal{Y}_{\text{sm}}$ the schememaximal open subscheme of $\mathcal{H}$$\mathcal{Y}$ that is a projective space bundlesmooth over $\text{Spec}\ R$. Finally Let $\mathcal{Y}_{\text{sing}}$ denote the closed complement of this open in $\mathcal{Y}$. The claim, fixingto be proved by contradiction, is that $\mathcal{Y}_{\text{sing}}$ is empty. These are both compatible with arbitrary base change of $\text{Spec}\ R$. Thus, if $\mathcal{Y}_{\text{sing}}$ is nonempty, then after faithfully flat base change of $\text{Spec}\ R$ assume that there exists a $k$-point $p$ of $\mathcal{Y}_{\text{sing}}$ and a $k$-point $q$ of $X_k$$X_k\setminus \mathcal{Y}_{\text{sing}}$. After further base change, assume that these are the specializations of $K$-points $p_K\in Y_K$ and $q_K \in X_K\setminus Y_K$.
Since $(X_K,\mathcal{L}|_{X_K})$ is not containedisomorphic to $(\mathbb{P}^n_K,\mathcal{O}(1))$ there exists a reduced, closed $K$-curve $C_K$ in $H_k$$X_K$ containing the $K$-points $p_K$ and $q_K$ having $\mathcal{L}$-degree equal to $1$. This curve is the image of a closed immersion of $K$-schemes, $$u_K:(\mathbb{P}^1_K,0,\infty)\mapsto (X_K,p_\eta,q_\eta),$$ with $u_K^*\mathcal{L}$ isomorphic to $\mathcal{O}(1)$.
Form the specializationsclosure of $C_K$ in $X_k$$\mathcal{X}$, and then form the normalization of linesthis closed subscheme of $\mathcal{X}$. Denote this normalization as follows, $$(\rho:\mathcal{C}\to \text{Spec}\ R,u:\mathcal{C}\to \mathcal{X}).$$ By the previous corollary, the closed fiber $C_k$ is geometrically integral. As a geometrically integral specialization of a smooth curve of genus $0$, also $C_k$ is smooth of genus $0$ (this can fail in higher genus, e.g., a smooth plane cubic can specialize to an irreducible, nodal plane cubic). The inverse image of $\mathcal{Y}$ is an effective Cartier divisor in $\mathcal{C}$ that does not contain $q$ each intersect, and thus it does not contain the entire closed fiber $H_k$$C_k$. Thus, this Cartier divisor in a unique point$\mathcal{C}$ is $R$-flat. From Since the degree of this Cartier divisor on $C_K$ equals $1$, it follows thatalso the blowing updegree on $C_k$ equals $1$. However, the intersection multiplicity at $p$ is at least as large as the multiplicity of the Cartier divisor $\mathcal{Y}$ at $p$. Thus, this Cartier divisor in the closed fiber $X_k$ alonghas multiplicity $q$$1$ at $p$, i.e., it is smooth at $p$. This contradicts that $p$ is contained in $\mathcal{Y}_{\text{sing}}$. This contradiction proves that $\mathcal{Y}$ is smooth over $\text{Spec}\ R$. QED
Let $\mathcal{Y}$ be the closure in $\mathcal{X}$ of a Cartier divisor in $X_K$ that is in the linear system of $\mathcal{L}|_{X_K}$.
Proposition 6. With hypotheses as in Corollary 5, the restriction homomorphism, $$r:H^0(\mathcal{X},\mathcal{L})\to H^0(\mathcal{Y},\mathcal{L}|_{\mathcal{Y}}),$$ is surjective, the complete linear system of $\mathcal{L}$ is globally generated, and the associated morphism of the complete linear system is an isomorphism to $\mathbb{P}^n_R$.
Proof. This is proved by induction on $n$. The base case is when $n$ equals $1$. Every smooth, proper curve of genus $0$ is geometrically isomorphic to $\mathbb{P}^1$. Thus, by way of induction, assume that $n\geq 2$ and assume that the projective bundle overresult is true for smaller values of $H_k$$n$. In particular, by Corollary 5, the pair $(\mathcal{Y},\mathcal{L}|_{\mathcal{Y}})$ satisfies the hypotheses for $n-1$. By the induction hypothesis, this pair is isomorphic to $(\mathbb{P}^{n-1}_R,\mathcal{O}(1))$. After a further finite, flat base change, assume that there exists a section $\sigma$ of $\mathcal{O}_{H_k}\oplus \mathcal{L}|_{H_k}$$\pi$ whose image is disjoint from $\mathcal{Y}$, i. Using the induction hypothesise., this equals the blowing up of $\mathbb{P}^n_k$ along a $k$-fiber is not contained in $\mathcal{Y}_k$.
Let $H_k\subset \mathcal{Y}_k$ be a Cartier divisor in the linear system of $\mathcal{L}|_{\mathcal{Y}_k}$. Since $(\mathcal{Y},\mathcal{L}|_{\mathcal{Y}})$ is $R$-isomorphic to $(\mathbb{P}^{n-1}_R,\mathcal{O}(1))$, there exists a lift of this Cartier divisor to an $R$-flat Cartier divisor $\mathcal{H}$ in $\mathcal{Y}$. In the generic fiber $X_K$, the subset $\mathcal{H}_K$ is a codimension $2$ linear subvariety, and $\sigma_K$ is a $K$-point that is disjoint. Thus, there exists a unique Cartier divisor $D_K$ in the linear system of $\mathcal{L}|_{X_K}$ that contains $\mathcal{H}_K$ and $\sigma$. By the previous corollary, the closure $\mathcal{D}$ in $\mathcal{X}$ of $D_K$ is a smooth Cartier divisor in the linear system of $\mathcal{L}$ that contains $\mathcal{H}$ and that contains $\sigma$.
If the intersection of $\mathcal{D}$ with $\mathcal{Y}_k$ is strictly larger than $\mathcal{H}_k$, then it completely contains the Cartier divisor $\mathcal{Y}_k$. Since $\mathcal{D}_k$ is irreducible, this implies that $\mathcal{D}_k$ equals $\mathcal{Y}_k$. This contradicts that $\mathcal{D}$ contains $\sigma$, since $\sigma$ is disjoint from $\mathcal{Y}$. Therefore, the restriction of $\mathcal{D}$ to $\mathcal{Y}$ equals $\mathcal{H}$.
As the divisors $\mathcal{H}$ vary over a bases for the complete linear system of $\mathcal{L}|_{\mathcal{Y}}$, the divisors $\mathcal{D}$ give a subsystem of the complete linear system of $\mathcal{L}$ that restricts isomorphically to the complete linear system of $\mathcal{L}|_{\mathcal{Y}}$. In particular, the base locus of this linear subsystem of $\mathcal{L}$ is disjoint from $\mathcal{Y}$, since the base locus of $\mathcal{L}|_{\mathcal{Y}}$ is empty. Consider the linear subsystem of $\mathcal{L}$ generated by this linear system together with the Cartier divisor $\mathcal{Y}$. This has empty base locus, and thus defines an $R$-morphism, $$\phi:\mathcal{X}\to \mathbb{P}^n_R.$$ Since $\mathcal{L}$ is ample, this morphism is finite. Moreover, since $\phi$ is an isomorphism on $K$-fibers, $\phi$ is birational. Since $\mathbb{P}^n_R$ is normal (and even regular), it follows from Zariski's Main Theorem that $\phi$ is an isomorphism. QED
Remark. If the residue field has characteristic $0$, then this result, often called "deformation-in-the-large", also holds if the geometric generic fiber is a smooth quadric hypersurface in $\mathbb{P}^n$ (the Ph.D. thesis of Jun-Muk Hwang), or if the geometric generic fiber is a projective homogeneous variety of cominuscule type (joint result of Jun-Muk Hwang and Ngaiming Mok).
MR1608587 (99b:32027)
Hwang, Jun-Muk(KR-SNU); Mok, Ngaiming(PRC-HK)
Rigidity of irreducible Hermitian symmetric spaces of the compact type under Kähler deformation.
Invent. Math. 131 (1998), no. 2, 393–418.
https://arxiv.org/abs/math/9604227
This has been extended to positive characterstic and mixed characteristic by Jan Gutt (using totally different techniques that do not reduce to Kobayashi-Ochiai as in Hwang-Mok). Precisely, for a fixed cominuscule type $(G,P)$, for a fixed integer $m$, there exists an explicit integer $p_0(G,P,m)$ depending on certain Schubert calculus computations such that for every prime $p\geq p_0(G,P,m)$, if the residue characterstic of $k$ equals $p$ (or $0$) and if $\mathcal{L}^{\otimes m}|_{\mathcal{X}_k}$ is very ample, then also $\mathcal{X}_k$ is a projective homogeneous variety of the same cominuscule type $(G,P)$.
Jan Gutt
On the extension theorem of Hwang and Mok
Journal für die reine und angewandte Mathematik (Crelles Journal),
ISSN (Online) 1435-5345, ISSN (Print) 0075-4102,
