Fibres non r\'eduites d'un sch\'ema arithm\'etique

TL;DR

This paper provides an explicit upper bound on the product of norms of places where fibers of a reduced projective scheme over the integers of a number field are not reduced, using height generalizations and Chow varieties.

Contribution

It introduces a generalized notion of height over the adelic ring and reduces the problem to hypersurfaces using Chow varieties and resultants.

Findings

Explicit upper bound for product of norms of non-reduced fibers

Reduction of general scheme case to hypersurface case

Application of resultants and height theory

Abstract

For a reduced projective scheme over the ring of integers of a number field, the set of places over which the fibres of the scheme are not reduced is a finite set. We give an explicit upper bound for the product of the norms of places in this set. For this purpose, we introduce a generalization of the notion of height over the adelic ring. We reduce the general case of a scheme of pure dimension to the case of a hypersurface by using the theory of Chow varieties. The case of a hypersurface is then treated with the help of the resultant of the equation of the hypersurface with some partial derivatives of the equation.