Suppose given an equivalence
relation R on an algebraic variety V and the associated fibering of V by a family of
subvarieties. This paper treats the question of the existence of a quotient structure
for this situation when the fibering is non-equidimensional. For this purpose a general
definition of quotient variety for algebraic equivalence relations is used which contains
no topological requirements.
The results are of two types. In §1 it is shown that certain maps into nonsingular
varieties are quotient maps for the induced equivalence relation whenever the union
of the excessive orbits has codimension ≧ 2. This theorem yields many examples of
non-equidimensional quotients. Section 2 contains a converse showing that
no excessive orbit containing a normal hypersurface can be fitted into a
quotient. This theorem depends on a stronger and less conceptual fieldtheoretic
result which fails without the normality hypothesis. Section 3 contains a
counterexample.
