Fibres non r\'eduites d'un sch\'ema arithm\'etique
Algebraic Geometry
2021-01-19 v5 Number 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.
Keywords
Cite
@article{arxiv.1609.08979,
title = {Fibres non r\'eduites d'un sch\'ema arithm\'etique},
author = {Chunhui Liu},
journal= {arXiv preprint arXiv:1609.08979},
year = {2021}
}
Comments
30 pages, in French. Some details are revised