hilbertPolynomial(CoherentSheaf) -- compute the Hilbert polynomial of the coherent sheaf

Synopsis

• Function: hilbertPolynomial

• Usage:

  hilbertPolynomial S

• Inputs:
  • S, a coherent sheaf
• Optional inputs:
  • Projective => ..., default value true, choose how to display the Hilbert polynomial
• Outputs:
  • a projective Hilbert polynomial, unless the option Projective is false

Description

i1 : R = ZZ/101[x_0..x_2];

i2 : V = Proj R;

i3 : S = sheaf(image matrix {{x_0^3+x_1^3+x_2^3}})

o3 = image | x_0^3+x_1^3+x_2^3 |

                                         1
o3 : coherent sheaf on V, subsheaf of OO
                                        V

i4 : h = hilbertPolynomial S

o4 = 3*P  - 3*P  + P
        0      1    2

o4 : ProjectiveHilbertPolynomial

i5 : hilbertPolynomial(S, Projective=>false)

     1 2   3
o5 = -i  - -i + 1
     2     2

o5 : QQ[i]