, and progress in resolving this more delicate question was rather modest. The invariants had classically been described only in a restricted range of situations, and the complexity of this description beyond the first few cases held out little hope for full understanding of the algebras of invariants in general. Furthermore, it may happen that any polynomial invariant takes the same value on a given pair of points and in , yet these points are in different orbits of the -action. A simple example is provided by the multiplicative group of non-zero complex numbers that acts on an -dimensional complex vector space by scalar multiplication. In this case, every polynomial invariant is a constant, but there are many different orbits of the action. The zero vector forms an orbit by itself, and the non-zero multiples of any non-zero vector form an orbit, so that non-zero orbits are parametrized by the points of the complex projective space . If this happens (different orbits having the same function values), one says that "invariants do not separate the orbits", and the algebra reflects the topological quotient space rather imperfectly. Indeed, the latter space, with the quotient topology, is frequently non-separated (non-Hausdorff). (This is the case in our example – the null orbit is not open because any neighborhood of the null vector contains points in all other orbits, so in the quotient topology any neighborhood of the null orbit contains all other orbits.) In 1893 Hilbert formulated and proved a criterion for determining those orbits which are not separated from the zero orbit by invariant polynomials. Rather remarkably, unlike his earlier work in invariant theory, which led to the rapid development of abstract algebra, this result of Hilbert remained little known and little used for the next 70 years. Much of the development of invariant theory in the first half of the twentieth century concerned explicit computations with invariants, and at any rate, followed the logic of algebra rather than geometry. Mumford's book Geometric invariant theory was founded and developed by Mumford in a monograph, first published in 1965, that applied ideas of nineteenth century invariant theory, including some results of Hilbert, to modern algebraic geometry questions. (The book was greatly expanded in two later editions, with extra appendices by Fogarty and Mumford, and a chapter on symplectic quotients by Kirwan.) The book uses both scheme theory and computational techniques available in examples.