  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.

Algebraic Geometry;Number Theory