Let $k$ be a field of characteristic zero. Given integers $2 \leq s \leq r < n$, define the variety $X_n$ in $P_K^n$ with coordinates $y_0, \cdots, y_n$ and $K=k(u_0, \cdots, u_n)$ ( where $u_i$'s are independent transcendental vairables) by the folloing $n-r$ equatuions:
$$ f_{i-r}(y_0, y_1, \cdots, y_n)= \begin{vmatrix} 1 & 1 & \cdots & 1 & 1 \\ u_0& u_1& \cdots & u_r & u_i\\ \vdots & \vdots & \cdots & \vdots & \vdots \\ u_0^r & u_1^r & \cdots & u_r^r& u_i^r\\ y_0^s & y_1^s & \cdots & y_r^s & y_i^s \end{vmatrix} =0, \ (r+1 \leq i \leq n). $$
It is easy to see that for fixed $r$, $s$, and $n > (sr + 1)/(s−1)$, $X_n$ is a smooth complete intersection variety of general type with $\dim(X_n)=r$. For a set of pairwise distinct elements $b=\{b_0, b_1, \cdots, b_n\}$ in $k$, letting $u_i=b_i$, one can get $ X_{b, n}$ a smooth variety defined over $k$.
Question: Is it possible to show that $X_{b, n}$ is a Geometric Mordellic variety, i.e., $X_{b, n}$ does not contain subvarieties which are not of general type over $\bar{k}$ the algebraic closure of $k$?
