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. Stability If a reductive group acts linearly on a vector space , then a non-zero point of is called unstable if 0 is in the closure of its orbit, semi-stable if 0 is not in the closure of its orbit, stable if its orbit is closed, and its stabilizer is finite. There are equivalent ways to state these (this criterion is known as the Hilbert–Mumford criterion): A non-zero point is unstable if and only if there is a 1-parameter subgroup of all of whose weights with respect to are positive. A non-zero point is unstable if and only if every invariant polynomial has the same value on 0 and . A non-zero point is semistable if and only if there is no 1-parameter subgroup of all of whose weights with respect to are positive. A non-zero point is semistable if and only if some invariant polynomial has different values on 0 and . A non-zero point is stable if and only if every 1-parameter subgroup of has positive (and negative) weights with respect to . A non-zero point is stable if and only if for every not in the orbit of there is some invariant polynomial that has different values on and , and the ring of invariant polynomials has transcendence degree . A point of the corresponding projective space of is called