)Given a closed subscheme formula_15 over a field with Hilbert polynomial formula_16, the Hilbert scheme has a universal subscheme formula_17 flat over formula_18 such that


The tangent space of the point formula_24 is given by the global sections of the normal bundle formula_33; that is,

For local complete intersections formula_35 such that formula_36, the point formula_37 is smooth.This implies every deformation of formula_35 in formula_21 is unobstructed.