The Hilbert Scheme

Many important moduli spaces can be constructed as quotients of the Hilbert scheme by a group action. For example, to construct the moduli space of smooth curves of genus g ≥ 2, we can first embed all smooth curves of genus g in Pn(2g−2)−g by a sufficiently large multiple of their canonical bundle K C . Any automorphism of a variety preserves the canonical bundle. Hence, two n-canonically embedded curves are isomorphic if and only if they are projectively equivalent. If we had a parameter space for n-canonically embedded curves, then the moduli space of curves would be a quotient of this parameter space by the projective linear group. The Hilbert scheme parameterizes subschemes of projective space with a fixed Hilbert polynomial, thus provides the starting point for all such constructions.