 Grothendieck constructed the Hilbert scheme formula_9 of -dimensional projective space over a Noetherian scheme as a subscheme of a Grassmannian defined by the vanishing of various determinants.Its fundamental property is that for a scheme over , it represents the functor whose -valued points are the closed subschemes of formula_10 that are flat over .If is a subscheme of -dimensional projective space, then corresponds to a graded ideal formula_11 of the polynomial ring in formula_12 variables, with graded pieces formula_13.