Vol. 41, No. 2, 1972

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Non-openness and non-equidimensionality in algebraic quotients

John Dustin Donald

Vol. 41 (1972), No. 2, 365–373

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.

Mathematical Subject Classification
Primary: 14C10
Received: 5 February 1971
Revised: 27 December 1971
Published: 1 May 1972
John Dustin Donald