hilbertFunction(List,ProjectiveVariety) -- compute the Hilbert function of a projective variety

Synopsis

Usage:
hilbertFunction(d,V)
Function: hilbertFunction
Inputs:
- d, a list, a multidegree, or if V is singly graded, then one may write just ZZ
- V, a projective variety
Outputs:
- an integer, the value of the Hilbert function of the projective variety

Description

i1 : R =ZZ/101[x_0..x_2,Degrees=>{3:{1,1}}];

i2 : V = Proj R;

i3 : hilbertFunction({3,3}, V)

o3 = 10

i4 : basis({3,3}, ring V)

o4 = | x_0^3 x_0^2x_1 x_0^2x_2 x_0x_1^2 x_0x_1x_2 x_0x_2^2 x_1^3 x_1^2x_2
     ------------------------------------------------------------------------
     x_1x_2^2 x_2^3 |

             1       10
o4 : Matrix R  <--- R

In the case where the ring is singly graded, then instead of having the input be a list of length 1 containing the degree, it is sufficient to write an integer.

i5 : R =ZZ/101[x_0..x_2];

i6 : V = Proj R;

i7 : hilbertFunction(3, V)

o7 = 10

i8 : basis(3, ring V)

o8 = | x_0^3 x_0^2x_1 x_0^2x_2 x_0x_1^2 x_0x_1x_2 x_0x_2^2 x_1^3 x_1^2x_2
     ------------------------------------------------------------------------
     x_1x_2^2 x_2^3 |

             1       10
o8 : Matrix R  <--- R