On the arithmetic of rational hypersurfaces in toric varieties

Abstract

In the toric variety $\mathcal{T}$, with Cox ring graded by $°(z_{2i})=(1,-1,0)$, $°(z_{2i+1})=(1,0,-1)$ and $°(w_{\pm })=(0,1,0),(0,0,1)$, we study hypersurfaces ${\tilde{X}}^{2n}\subset \mathcal{T}$ 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 ${\tilde{X}}^{2n}$ 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.