2024/12/20 by Roberto Gualdi, Martín Sombra, Martı́n Sombra +2
Mathematics · Computer Science · #Commutative Algebra and Its Applications #Polynomial and algebraic computation #Algebraic Geometry and Number Theory
paper · pdf · doi:10.48550/arxiv.2412.16308
The height of a toric variety and that of its hypersurfaces can be expressed in convex-analytic terms as an adelic sum of mixed integrals of their roof functions and duals of their Ronkin functions. Here we extend these results to the 2-codimensional situation by presenting a limit formula predicting the typical height of the intersection of two hypersurfaces on a toric variety. More precisely, we prove that the height of the intersection cycle of two effective divisors translated by a strict sequence of torsion points converges to an adelic sum of mixed integrals of roof and duals of Ronkin functions. This partially confirms a previous conjecture of the authors about the average height of families of complete intersections in toric varieties.