correspondencePolynomial -- computes the Hilbert polynomial of a correspondence scroll

Synopsis

Usage:

H = correspondencePolynomial(M,L)

H = correspondencePolynomial(M,L,VariableName =>h)
Inputs:
- M, a module,
- L, a list, of ZZ
- VariableName, a string,
Optional inputs:
- VariableName => ..., default value "s"
Outputs:
- H, a ring element, in QQ[h]

Description

Let M be a module over a polynomial ring P = kk[x_{0,0}..x_{0,a_0}..x_{n-1,0}..x_{n-1,a_{n-1}}] graded with degree x_{i,j} = e_i, the i-th unit vector, and let b = {b_0..b_{n-1}} be a list of integers. The code computes the multigraded Hilbert polynomial mH(h_0,..,h_{n-1}) and returns H(h) = mH(b_0*h_0, .., b_{n-1}*h_{n-1}).

i1 : P = productOfProjectiveSpaces {1,1}

o1 = P

o1 : PolynomialRing

i2 : Delta = smallDiagonal P

o2 = ideal(- x   x    + x   x   )
              0,1 1,0    0,0 1,1

o2 : Ideal of P

i3 : M = P^1/(Delta^2)

o3 = cokernel | x_(0,1)^2x_(1,0)^2-2x_(0,0)x_(0,1)x_(1,0)x_(1,1)+x_(0,0)^2x_(1,1)^2 |

                            1
o3 : P-module, quotient of P

i4 : correspondencePolynomial (M,{1,1})

o4 = 4s

o4 : QQ[s]

i5 : correspondencePolynomial (M,{2,2})

o5 = 8s

o5 : QQ[s]

Ways to use correspondencePolynomial :

correspondencePolynomial(Module,List)

For the programmer

The object correspondencePolynomial is a method function with options.

correspondencePolynomial -- computes the Hilbert polynomial of a correspondence scroll

Synopsis

Description

See also

Ways to use correspondencePolynomial :

For the programmer