On the arithmetic of rational hypersurfaces in toric varieties

In the toric variety $\mathcal{T}$, with Cox ring graded by $\deg(z_{2i})=(1,-1,0)$, $\deg(z_{2i+1})=(1,0,-1)$ and $\deg(w_\pm)=(0,1,0),(0,0,1)$, we study hypersurfaces $\widetilde{X}^{2n}\subset\mathcal T$ of multidegree $(2d+1,-d,-d)$ over a field $k$. These are the strict transforms of odd-degree hypersurfaces in $\mathbb{P}^{2n+1}$ with multiplicity $d$ along two skew conjugate $n$-planes. We prove that $\widetilde{X}^{2n}$ is $k$-rational and birational to $\mathbb{P}^{2n}$; and derive result on the distribution of its rational points over numbers and finite field. The case $d=1$ recovers the even-dimensional Fermat cubic.