i.e. the quotient space of "X" by the group action, runs into difficulties in algebraic geometry, for reasons that are explicable in abstract terms. There is in fact no general reason why equivalence relations should interact well with the (rather rigid) regular functions (polynomial functions), which are at the heart of algebraic geometry. The functions on the orbit space "G"\"X" that should be considered are those on "X" that are invariant under the action of "G". The direct approach can be made, by means of the function field of a variety (i.e. rational functions): take the "G"-invariant rational functions on it, as the function field of the quotient variety. Unfortunately this — the point of view of birational geometry — can only give a first approximation to the answer. As Mumford put it in the Preface to the book: