The matrix F must have a single row. The output N is a matrix over the same ring as F whose columns form a basis for (a graded piece of) the normal module Hom(I,S/I), where S is the ring of F and I is ideal generated by the columns of F. Selection of graded pieces is done in the same manner as with basis. If the selected pieces are infinite dimensional, an error occurs.
For example, consider a degenerate twisted cubic curve:
i1 : S=QQ[x,y,z,w]; |
i2 : F=matrix {{x*z,y*z,z^2,x^3}} o2 = | xz yz z2 x3 | 1 4 o2 : Matrix S <--- S |
i3 : N=normalModule(0,F) o3 = {-2} | 0 0 0 0 0 0 0 0 0 zw 0 x2 xy xw 0 0 | {-2} | 0 0 0 0 0 0 0 0 0 0 zw xy y2 yw 0 x2 | {-2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 zw 0 | {-3} | zw2 y3 y2w yw2 x2y x2w xy2 xyw xw2 0 0 0 0 0 0 0 | 4 16 o3 : Matrix S <--- S |
The degree zero component of the normal module, and thus the tangent space of the Hilbert scheme, is sixteen dimensional.