Recall that a flag variety over a field $k$ is a smooth projective variety over $k$, which is a homogeneous space for some linear algebraic group.


My question concerns specialisations of flag varieties over discrete valuation rings. Namely, let $R$ be a discrete valuation ring with field of fractions $K$ and residue field $k$. Assume for simplicity that $K$ and $k$ are algebraically closed.



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> Let $X \to \textrm{Spec} R$ be a smooth projective scheme over $R$, and assume that the generic fibre is a flag variety. Is the special fibre also a flag variety?
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Some remarks:


* If the generic fibre is isomorphic to a projective space, then the answer to my question is yes. This is Theorem 2 of [1].
* The converse to my question has a positive answer. Namely, if the special fibre is a flag variety, then the generic fibre is a flag variety. This is Proposition 5 of [2].


[1] Goren - Characterization and algebraic deformations of projective space.


[2] Démazure - Automorphismes et déformations des variétés de Borel.



Specialisations of flag varieties
There is a counter-example (with $k=\mathbb{C}$) due to Pasquier and Perrin, Math. Zeitschrift 265 (2010), 589-600. The generic fiber is an orthogonal grassmannian $\mathrm{Gr}\_q(2,7)$, while the special fiber is a smooth
projective non-homogeneous ("horospherical") variety with an action of $G\_2$. 


