On the arithmetic of rational hypersurfaces in toric varieties
Gianluca Grassi
TL;DR
This work studies the arithmetic of rational hypersurfaces arising as strict transforms of odd-degree hypersurfaces in projective space within a toric framework. By constructing a concrete toric realisation via a $ abla Z^3$-graded Cox ring and employing a rationality construction based on restriction of scalars from a quadratic extension, the authors prove that the hypersurface $ ilde{X}^{2n}$ is $k$-rational and birational to $ ext{P}^{2n}$, with explicit parametrizations and a single multidegree equation in the Cox model. They extend the analysis to characteristic two with a direct parametrization and connect the two frameworks through a quadro-cubic Cremona transformation, yielding exact finite-field point-count identities and height-bounds for rational points. The paper also provides precise formulas for point counts over finite fields under congruence and gcd conditions, including special cases for $n=1$, and establishes height-growth estimates for rational points via the birational parametrizations. Overall, it delivers a concrete, computable toric construction that links birational geometry, finite-field arithmetic, and height theory for a family of higher-dimensional Fermat-type hypersurfaces.
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 $\widetilde{X}^{2n}\subset\mathcal T$ 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.