On the rational points in conics of a cubic surfac
Chunhui Liu
TL;DR
This work develops a uniform, height-sensitive bound on rational (and integral) points produced by conics on a non-ruled cubic surface, extending Salberger’s global determinant method through Arakelov geometry. Central to the approach are the height theory for varieties via arithmetic intersection and the Cayley/Chow form machinery that ties the geometry of conics to heights on Hilbert schemes; these are used to construct auxiliary hypersurfaces of controlled degree that cover the bounded-height points without containing the generic point. A key result is that conics on a non-ruled cubic surface X contribute at most C0(K)·B^{3√3/8+1}·max{log B,2} rational points of height at most B, with a parallel affine bound for integral points. The analysis blends determinant methods, Hilbert-Samuel type invariants, and Deligne pairings to relate the arithmetic of Cayley forms to the global counting problem, yielding explicit, field-dependent constants and paving the way for dimension-growth-type bounds for conics on cubic surfaces and related affine models.
Abstract
In this paper, we give a uniform upper bound on the rational points of bounded height provided by conics in a cubic surface. For this target, we give a generalized version of the global determinant method of Salberger by Arakelov geometry.