Let $X$ be a smooth projective variety over a field $K$ of characteristic zero such that $$ X_{\bar{K}} \cong \mathrm{Gr}(k,n)_{\bar{K}} $$ and $X(K) \ne \varnothing$.

First, consider the case $n \ne 2k$. Let $r = \gcd(k,n)$. Then there is an $r$-torsion Brauer class $\beta \in \mathrm{Br}(K)$ and a Severi--Brauer variety $Y$ of dimension $n - 1$ with class $\beta$ such that $$ X \cong \mathrm{Hilb}_{\mathrm{P}^{k-1}}(Y) $$ is the component of the Hilbert scheme that parameterizes subschemes with the same Hilbert polynomial as $\mathrm{P}^{k-1}$. Let us call such $X$ a Severi--Brauer Grassmannian of class $\beta$.

Now consider the case $n = 2k$. Then either $X$ is a Severi--Brauer Grassmannian as before, or there is a quadratic extension $K'/K$ such that $X_{K'}$ is a Severi--Brauer Grassmannian.

Let $X$ be a smooth projective variety over a field $K$ of characteristic zero such that $$ X_{\bar{K}} \cong \mathrm{Gr}(k,n)_{\bar{K}} $$ and $X(K) \ne \varnothing$.

First, consider the case $n \ne 2k$. Let $r = \gcd(k,n)$. Then there is an $r$-torsion Brauer class $\beta \in \mathrm{Br}(K)$ representable by a Severi--Brauer variety of dimension $k - 1$ and a Severi--Brauer variety $Y$ of dimension $n - 1$ with class $\beta$ such that $$ X \cong \mathrm{Hilb}_{\mathrm{P}^{k-1}}(Y) $$ is the component of the Hilbert scheme that parameterizes subschemes with the same Hilbert polynomial as $\mathrm{P}^{k-1}$. Let us call such $X$ a Severi--Brauer Grassmannian of class $\beta$.

Now consider the case $n = 2k$. Then either $X$ is a Severi--Brauer Grassmannian as before, or there is a quadratic extension $K'/K$ such that $X_{K'}$ is a Severi--Brauer Grassmannian.