Compute the dual graph of the fiber over 2. (Some intermediate information is printed.)
#The only non-smooth fiber is over 2
sage: X.badprimes
2
sage: X.fiberInformation(2)
Calculating ideal sheaves, 4 charts:
. . . .
Number of prime components D_1, D_2, ... over the fiber at 2 : 7
Returning a list of ideal sheaves of irreducible components of the fiber with all non-reduced structure.
The fiber over 2 is (1)*D_0 + (2)*D_1 + (1)*D_2 + (2)*D_3 + (4)*D_4 + (1)*D_5 + (1)*D_6
Returning a tuple [A,B] (of ideal sheaves (with non-reduced structure) and multiplicites):
A = ['I_{(1)*D_0}', 'I_{(2)*D_1}', 'I_{(1)*D_2}', 'I_{(2)*D_3}', 'I_{(4)*D_4}', 'I_{(1)*D_5}', 'I_{(1)*D_6}']
B = [1, 2, 1, 2, 4, 1, 1]
Computing intersection matrix: . . . . . . . . . . . . . . . . . . . . .
Intersection Matrix:
[-3 1 0 0 0 1 0]
[ 1 -3 0 0 1 1 0]
[ 0 0 -3 1 0 0 1]
[ 0 0 1 -3 1 0 1]
[ 0 1 0 1 -1 0 0]
[ 1 1 0 0 0 -3 0]
[ 0 0 1 1 0 0 -3]
Genus...
Reduced...
Ideal sheaf over CHb0 is different.
Ideal sheaf over CHb0 is different.
Ideal sheaf over CHb0 is different.
Smooth...
===============Key (Vertex/Sheaf of ideals)==================
Vertex C0
Genus of reduced normalization: Not Calculated
Reduced: True
Smooth radical: True
Multiplicity: 1
Self-intersection: -3
{'CHa2': ['1'], 'CHb3': ['a3+1', 'a1+1', 'a0+a2', 'x1', 'x0+1', '2'], 'CHb2': ['a2+1', 'a1+a3', 'a0+1', 'x1', 'x0+1', '2'], 'CHb0': ['a2+1', 'a1+a3', 'a0+1', 'x1', 'x0+1', '2']}
Vertex C1
Genus of reduced normalization: Not Calculated
Reduced: False
Smooth radical: True
Multiplicity: 2
Self-intersection: -3
{'CHa2': ['1'], 'CHb3': ['a3+1', 'x0+x1+a1', 'a2^2', 'a1*a2+a2', 'x1*a2', 'a1^2+1', 'x1*a1+x1', 'x1*a0+x1+a1+1', 'x1^2', 'a0^3*a1+a0^3+a0^2*a1+a0^2*a2+a0^2+a0*a1+a0+a1+1', '2'], 'CHb2': ['1'], 'CHb0': ['a0+1', 'x1*a3+x1+a1+a3', 'x0*a3+x1+a3', 'a2^2', 'x1*a2', 'x0*a2+a2', 'a1^2+a3^2', 'x1*a1+x1+a1+a3', 'x0*a1+x1+a1', 'x1^2', 'x0*x1+x1', 'x0^2+1', 'a1*a3^2+a3^3+a1*a3+a3^2+x0+x1+a1+a2+a3+1', 'a1*a2*a3+a2*a3^2', '2']}
Vertex C2
Genus of reduced normalization: Not Calculated
Reduced: True
Smooth radical: True
Multiplicity: 1
Self-intersection: -3
{'CHa2': ['a2+1', 'a1+1', 'x1+1', 'x0', '2'], 'CHb3': ['a3+1', 'a1+a2', 'a0+a2', 'x1+1', 'x0', '2'], 'CHb2': ['a2+1', 'a1+1', 'a0+1', 'x1+1', 'x0', '2'], 'CHb0': ['a2+1', 'a1+1', 'a0+1', 'x1+1', 'x0', '2']}
Vertex C3
Genus of reduced normalization: Not Calculated
Reduced: False
Smooth radical: True
Multiplicity: 2
Self-intersection: -3
{'CHa2': ['a2+1', 'a1^2', 'x1*a1+a1', 'x0*a1', 'x1*a0+x0+a0', 'x1^2+1', 'x0*x1+x0', 'x0^2', 'x0*a0^3+x1+a1+1', '2'], 'CHb3': ['a3+1', 'a2^2', 'a0*a2+a1*a2', 'x1*a2+a2', 'x0*a2', 'x1*a1+x1+a0+1', 'x0*a1+x1+1', 'a0^2+a1^2', 'x1*a0+x1+a1+1', 'x0*a0+x1+1', 'x1^2+1', 'x0*x1+x0', 'x0^2', 'a0*a1^2+a1^3+a1^2*a2+a0*a1+a1^2+x0+x1+a0+a1+1', '2'], 'CHb2': ['1'], 'CHb0': ['a0+1', 'x0+x1+a1', 'x1*a3+x1+a1+a3', 'a2^2', 'a1*a2+a2', 'x1*a2+a2', 'a1^2+1', 'x1*a1+x1+a1+1', 'x1^2+1', 'a1*a3^3+a1*a3^2+a3^3+a1*a3+a3^2+a1+a2+a3+1', '2']}
Vertex C4
Genus of reduced normalization: Not Calculated
Reduced: False
Smooth radical: True
Multiplicity: 4
Self-intersection: -1
{'CHa2': ['a2+1', 'a1', 'x1*a0+x0+a0', 'a0^4+1', 'x0*a0^3+x1+1', 'x0^2*a0^2+x1^2+1', 'x0^3*a0+x1^3+x1^2+x1+1', 'x0^4+x1^4+1', '2'], 'CHb3': ['a3+1', 'a2', 'x0*a1+x1+1', 'x1*a0+x1+a1+1', 'a0*a1^2+a1^3+x0*a0+a0*a1+a1^2+x0+a0+a1', 'a1^4+1', 'a0^3*a1+a0^3+a0^2*a1+a0^2+a0*a1+a0+a1+1', 'x1^3*a1+x0^3+x1^2*a1+x1*a1+a1', 'a0^4+1', 'x0^2*a0^2+x0^2+a0^2+a1^2', 'x0^3*a0+x0^3+x1^2*a1+x1*a1^2+x0^2+x1*a1+a0*a1', 'x0^4+x1^4+1', '2'], 'CHb2': ['1'], 'CHb0': ['a2', 'a0+1', 'x1*a3+x1+a1+a3', 'x0*a1+x1+a1', 'a1^2*a3+x0*a3^2+a1*a3^2+a1^2+a1*a3+a3^2+x0+1', 'a3^4+1', 'a1*a3^3+a1*a3^2+a3^3+a1*a3+a3^2+a1+a3+1', 'x0*a3^3+x0*a3^2+a1*a3^2+x0*a3+a3^2+x0+a1+1', 'x0^3*a3+x0^3+x1^2*a1+x0^2+x1*a1+x0+a1+1', 'x0^4+x1^4+1', '2']}
Vertex C5
Genus of reduced normalization: Not Calculated
Reduced: True
Smooth radical: True
Multiplicity: 1
Self-intersection: -3
{'CHa2': ['1'], 'CHb3': ['a3+1', 'a1+1', 'a0', 'x1', 'x0+1', '2'], 'CHb2': ['a2+1', 'a1+a3', 'a0', 'x1', 'x0+1', '2'], 'CHb0': ['1']}
Vertex C6
Genus of reduced normalization: Not Calculated
Reduced: True
Smooth radical: True
Multiplicity: 1
Self-intersection: -3
{'CHa2': ['1'], 'CHb3': ['a3+1', 'a1', 'a0', 'x1+1', 'x0', '2'], 'CHb2': ['a2+1', 'a1', 'a0', 'x1+1', 'x0', '2'], 'CHb0': ['1']}
===============Key (Vertex/Sheaf of ideals)==================
=============================================================
Dual graph saved to DualGraph-BVQDsfiber-2.png
=============================================================
Returning elementary divisors... [1, 1, 1, 1, 4, 4]
[
[{'CHa2': ['1'], 'CHb3': ['a3+1', 'a1+1', 'a0+a2', 'x1', 'x0+1', '2'], 'CHb2': ['a2+1', 'a1+a3', 'a0+1', 'x1', 'x0+1', '2'], 'CHb0': ['a2+1', 'a1+a3', 'a0+1', 'x1', 'x0+1', '2']}, {'CHa2': ['1'], 'CHb3': ['a3+1', 'x0+x1+a1', 'a2^2', 'a1*a2+a2', 'x1*a2', 'a1^2+1', 'x1*a1+x1', 'x1*a0+x1+a1+1', 'x1^2', 'a0^3*a1+a0^3+a0^2*a1+a0^2*a2+a0^2+a0*a1+a0+a1+1', '2'], 'CHb2': ['1'], 'CHb0': ['a0+1', 'x1*a3+x1+a1+a3', 'x0*a3+x1+a3', 'a2^2', 'x1*a2', 'x0*a2+a2', 'a1^2+a3^2', 'x1*a1+x1+a1+a3', 'x0*a1+x1+a1', 'x1^2', 'x0*x1+x1', 'x0^2+1', 'a1*a3^2+a3^3+a1*a3+a3^2+x0+x1+a1+a2+a3+1', 'a1*a2*a3+a2*a3^2', '2']}, {'CHa2': ['a2+1', 'a1+1', 'x1+1', 'x0', '2'], 'CHb3': ['a3+1', 'a1+a2', 'a0+a2', 'x1+1', 'x0', '2'], 'CHb2': ['a2+1', 'a1+1', 'a0+1', 'x1+1', 'x0', '2'], 'CHb0': ['a2+1', 'a1+1', 'a0+1', 'x1+1', 'x0', '2']}, {'CHa2': ['a2+1', 'a1^2', 'x1*a1+a1', 'x0*a1', 'x1*a0+x0+a0', 'x1^2+1', 'x0*x1+x0', 'x0^2', 'x0*a0^3+x1+a1+1', '2'], 'CHb3': ['a3+1', 'a2^2', 'a0*a2+a1*a2', 'x1*a2+a2', 'x0*a2', 'x1*a1+x1+a0+1', 'x0*a1+x1+1', 'a0^2+a1^2', 'x1*a0+x1+a1+1', 'x0*a0+x1+1', 'x1^2+1', 'x0*x1+x0', 'x0^2', 'a0*a1^2+a1^3+a1^2*a2+a0*a1+a1^2+x0+x1+a0+a1+1', '2'], 'CHb2': ['1'], 'CHb0': ['a0+1', 'x0+x1+a1', 'x1*a3+x1+a1+a3', 'a2^2', 'a1*a2+a2', 'x1*a2+a2', 'a1^2+1', 'x1*a1+x1+a1+1', 'x1^2+1', 'a1*a3^3+a1*a3^2+a3^3+a1*a3+a3^2+a1+a2+a3+1', '2']}, {'CHa2': ['a2+1', 'a1', 'x1*a0+x0+a0', 'a0^4+1', 'x0*a0^3+x1+1', 'x0^2*a0^2+x1^2+1', 'x0^3*a0+x1^3+x1^2+x1+1', 'x0^4+x1^4+1', '2'], 'CHb3': ['a3+1', 'a2', 'x0*a1+x1+1', 'x1*a0+x1+a1+1', 'a0*a1^2+a1^3+x0*a0+a0*a1+a1^2+x0+a0+a1', 'a1^4+1', 'a0^3*a1+a0^3+a0^2*a1+a0^2+a0*a1+a0+a1+1', 'x1^3*a1+x0^3+x1^2*a1+x1*a1+a1', 'a0^4+1', 'x0^2*a0^2+x0^2+a0^2+a1^2', 'x0^3*a0+x0^3+x1^2*a1+x1*a1^2+x0^2+x1*a1+a0*a1', 'x0^4+x1^4+1', '2'], 'CHb2': ['1'], 'CHb0': ['a2', 'a0+1', 'x1*a3+x1+a1+a3', 'x0*a1+x1+a1', 'a1^2*a3+x0*a3^2+a1*a3^2+a1^2+a1*a3+a3^2+x0+1', 'a3^4+1', 'a1*a3^3+a1*a3^2+a3^3+a1*a3+a3^2+a1+a3+1', 'x0*a3^3+x0*a3^2+a1*a3^2+x0*a3+a3^2+x0+a1+1', 'x0^3*a3+x0^3+x1^2*a1+x0^2+x1*a1+x0+a1+1', 'x0^4+x1^4+1', '2']}, {'CHa2': ['1'], 'CHb3': ['a3+1', 'a1+1', 'a0', 'x1', 'x0+1', '2'], 'CHb2': ['a2+1', 'a1+a3', 'a0', 'x1', 'x0+1', '2'], 'CHb0': ['1']}, {'CHa2': ['1'], 'CHb3': ['a3+1', 'a1', 'a0', 'x1+1', 'x0', '2'], 'CHb2': ['a2+1', 'a1', 'a0', 'x1+1', 'x0', '2'], 'CHb0': ['1']}],
[-3 1 0 0 0 1 0]
[ 1 -3 0 0 1 1 0]
[ 0 0 -3 1 0 0 1]
[ 0 0 1 -3 1 0 1]
[ 0 1 0 1 -1 0 0]
[ 1 1 0 0 0 -3 0]
[1, 2, 1, 2, 4, 1, 1], [ 0 0 1 1 0 0 -3],
, [1, 1, 1, 1, 4, 4]
]
