On the arithmetic of rational hypersurfaces in toric varieties

2025/10/19 by Grassi, Gianluca
#12E10 #12F10 #14E08 #14M20 #Algebraic Geometry (math.AG) #FOS: Mathematics #Primary 12F20 #Secondary 14G05

paper · doi:10.48550/arxiv.2510.16773

Abstract

In the toric variety T, with Cox ring graded by °(z_2i)=(1,-1,0), °(z_2i+1)=(1,0,-1) and °(w_±)=(0,1,0),(0,0,1), we study hypersurfaces \widetildeX²ⁿ⊂\mathcal T of multidegree (2d+1,-d,-d) over a field k. These are the strict transforms of odd-degree hypersurfaces in ℙ²ⁿ⁺¹ with multiplicity d along two skew conjugate n-planes. We prove that \widetildeX²ⁿ is k-rational and birational to ℙ²ⁿ; 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.

Citations

• Multi-height analysis of rational points of toric varieties
• Rational points on even dimensional Fermat cubics
• Points of bounded height on certain subvarieties of toric varieties
• Approximating rational points on toric varieties
• Hyperbola method on toric varieties
• Manin's conjecture for toric varieties

Related

• Rational points on even dimensional Fermat cubics
• The implicit equation of a multigraded hypersurface
• Multi-height analysis of rational points of toric varieties
• Rational linear subspaces of hypersurfaces over finite fields
• Stable rationality of hypersurfaces of mock toric variety II