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The simple-minded idea of an orbit space. 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 direct approach can be made, by means of the function field of a variety i. 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:. In Chapter 5 he isolates further the specific technical problem addressed, in a moduli problem of quite classical type — classify the big 'set' of all algebraic varieties subject only to being non-singular and a requisite condition on polarization.

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Geometric Invariant Theory for Polarized Curves | Gilberto Bini | Springer

The moduli are supposed to describe the parameter space. For example, for algebraic curves it has been known from the time of Riemann that there should be connected components of dimensions. In the coarse moduli problem Mumford considers the obstructions to be:. It is the third point that motivated the whole theory.

As Mumford puts it, if the first two difficulties are resolved. To deal with this he introduced a notion in fact three of stability. This enabled him to open up the previously treacherous area — much had been written, in particular by Francesco Severi , but the methods of the literature had limitations. The birational point of view can afford to be careless about subsets of codimension 1.

To have a moduli space as a scheme is on one side a question about characterising schemes as representable functors as the Grothendieck school would see it ; but geometrically it is more like a compactification question, as the stability criteria revealed. The restriction to non-singular varieties will not lead to a compact space in any sense as moduli space: varieties can degenerate to having singularities. On the other hand, the points that would correspond to highly singular varieties are definitely too 'bad' to include in the answer.

The correct middle ground, of points stable enough to be admitted, was isolated by Mumford's work.

The concept was not entirely new, since certain aspects of it were to be found in David Hilbert 's final ideas on invariant theory, before he moved on to other fields. The book's Preface also enunciated the Mumford conjecture , later proved by William Haboush.