The Hilbert polynomial of a coherent sheaf $F$ on smooth normal toric variety $X$ is the Euler characteristic of $F ** OO_X(i_0,i_1,...,i_r)$ where $r$ is the rank of the Picard group of $X$ and $i_0,i_1,...,i_r$ are formal variables. The Hilbert polynomial agrees with the Hilbert function when evaluated at any point sufficiently far in the interior of in the nef cone.
For a graded module over the total coordinate ring of $X$, this method computes the Hilbert polynomial of the corresponding coherent sheaf. Given an ideal $I$ in the total coordinate ring, it computes the Hilbert polynomial of the coherent sheaf associated to $S^1/I$.
The cotangent bundle on a smooth surface provides simple examples.
i1 : PP2 = toricProjectiveSpace 2; |
i2 : OmegaPP2 = cotangentSheaf PP2
o2 = cokernel {2} | x_2 |
{2} | x_0 |
{2} | -x_1 |
3
o2 : coherent sheaf on PP2, quotient of OO (-2)
PP2
|
i3 : h0 = hilbertPolynomial (PP2, OmegaPP2)
2
o3 = i - 1
0
o3 : QQ[i ]
0
|
i4 : for i to 10 list hilbertFunction(i, module OmegaPP2)
o4 = {0, 0, 3, 8, 15, 24, 35, 48, 63, 80, 99}
o4 : List
|
i5 : R = ring h0; |
i6 : for i to 10 list sub(h0, R_0 => i)
o6 = {-1, 0, 3, 8, 15, 24, 35, 48, 63, 80, 99}
o6 : List
|
i7 : FF3 = hirzebruchSurface 3; |
i8 : OmegaFF3 = cotangentSheaf FF3
o8 = cokernel {2, 0} | 3x_1x_3 |
{-2, 2} | x_0 |
{-2, 2} | -x_2 |
1 2
o8 : coherent sheaf on FF3, quotient of OO (-2, 0) ++ OO (2, -2)
FF3 FF3
|
i9 : h1 = hilbertPolynomial (FF3, OmegaFF3)
2
o9 = 2i i + 3i - 2
0 1 1
o9 : QQ[i ..i ]
0 1
|
i10 : matrix table(5,5, (i,j) -> hilbertFunction({j,4-i}, module OmegaFF3))
o10 = | 47 54 62 70 78 |
| 26 31 37 43 49 |
| 11 14 18 22 26 |
| 2 3 5 7 9 |
| 0 0 1 2 3 |
5 5
o10 : Matrix ZZ <--- ZZ
|
i11 : R = ring h1; |
i12 : matrix table(5,5, (i,j) -> sub(h1, {R_0 => j, R_1 => 4-i}))
o12 = | 46 54 62 70 78 |
| 25 31 37 43 49 |
| 10 14 18 22 26 |
| 1 3 5 7 9 |
| -2 -2 -2 -2 -2 |
5 5
o12 : Matrix QQ <--- QQ
|
Example 2.16 in Diane Maclagan and Gregory G. Smith's Uniform bounds on multigraded regularity, J. Algebraic Geom. 14 (2005), 137-164 shows that the Hilbert function for a set of points agrees with the Hilbert polynomial precisedly on the multigraded regularity.
i13 : X = toricProjectiveSpace (1) ** toricProjectiveSpace (1); |
i14 : S = ring X; |
i15 : I = intersect (ideal (S_1-S_0, S_3-S_2), ideal (S_1-S_0, S_3-2*S_2), ideal (S_1-2*S_0, S_3-S_2), ideal (S_1-2*S_0, S_3-2*S_2), ideal (S_1-3*S_0, S_3-3*S_2), ideal (S_1-4*S_0, S_3-4*S_2))
4 3 2 2 3 4 3 2 2
o15 = ideal (24x - 50x x + 35x x - 10x x + x , 2x x - 2x x x - 3x x x
2 2 3 2 3 2 3 3 1 2 0 2 3 1 2 3
-----------------------------------------------------------------------
2 2 3 3 2 2 2
+ 3x x x + x x x - x x , 24x x - 50x x x + 2x x x + 33x x x -
0 2 3 1 2 3 0 3 0 2 0 2 3 1 2 3 0 2 3
-----------------------------------------------------------------------
2 3 3 2 2 2 2 2 2 2
3x x x - 7x x + x x , 2x x - 3x x x - 2x x + 3x x x , 2x x x -
1 2 3 0 3 1 3 1 2 1 2 3 0 3 0 1 3 0 1 2
-----------------------------------------------------------------------
2 2 2 2 2 2 2
2x x x - x x x + x x x , 24x x - 50x x x + 21x x x x - 7x x x +
0 2 3 1 2 3 0 1 3 0 2 0 2 3 0 1 2 3 1 2 3
-----------------------------------------------------------------------
2 2 2 2 2 2 2 3 3 2
14x x - 3x x x + x x , 2x x x - 3x x x + x x - 2x x + 3x x x -
0 3 0 1 3 1 3 0 1 2 0 1 2 1 2 0 3 0 1 3
-----------------------------------------------------------------------
2 3 2 3 3 2 2
x x x , 24x x - 42x x x + 18x x - 50x x + 77x x x - 28x x x +
0 1 3 0 2 0 1 2 1 2 0 3 0 1 3 0 1 3
-----------------------------------------------------------------------
3 4 3 2 2 3 4
x x , 24x - 50x x + 35x x - 10x x + x )
1 3 0 0 1 0 1 0 1 1
o15 : Ideal of S
|
i16 : assert (I == saturate (I, ideal X)) |
i17 : hilbertPolynomial (X, I)
o17 = 6
o17 : QQ[i ..i ]
0 1
|
i18 : matrix table(5,5, (i,j) -> hilbertFunction({j,4-i}, I))
o18 = | 4 6 6 6 6 |
| 4 6 6 6 6 |
| 3 6 6 6 6 |
| 2 4 6 6 6 |
| 1 2 3 4 4 |
5 5
o18 : Matrix ZZ <--- ZZ
|