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.