To see how this might work, consider the problem of parametrizing smooth curves of the genus "g" > 2. A smooth curve together with a complete linear system of degree "d" > 2"g" is equivalent to a closed one dimensional subscheme of the projective space P"d−g". Consequently, the moduli space of smooth curves and linear systems (satisfying certain criteria) may be embedded in the Hilbert scheme of a sufficiently high-dimensional projective space. This locus "H" in the Hilbert scheme has an action of PGL("n") which mixes the elements of the linear system; consequently, the moduli space of smooth curves is then recovered as the quotient of "H" by the projective general linear group.