Check if a Weil divisor is Cartier. For example, the following divisor is not Cartier
i1 : R = QQ[x, y, u, v] / ideal(x * y - u * v); |
i2 : D = divisor({2, -3}, {ideal(x, u), ideal(y, v)})
o2 = 2*Div(x, u) + -3*Div(y, v)
o2 : WeilDivisor on R
|
i3 : isCartier( D ) o3 = false |
Neither is this divisor.
i4 : R = QQ[x, y, z] / ideal(x * y - z^2 ); |
i5 : D = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
o5 = 2*Div(y, z) + Div(x, z)
o5 : WeilDivisor on R
|
i6 : isCartier( D ) o6 = false |
Of course the next divisor is Cartier.
i7 : R = QQ[x, y, z]; |
i8 : D = divisor({1, 2}, {ideal(x), ideal(y)})
o8 = Div(x) + 2*Div(y)
o8 : WeilDivisor on R
|
i9 : isCartier( D ) o9 = true |
If the option IsGraded is set to true (it is false by default), this will check as if D is a divisor on the $Proj$ of the ambient graded ring.
i10 : R = QQ[x, y, u, v] / ideal(x * y - u * v); |
i11 : D = divisor({2, -3}, {ideal(x, u), ideal(y, v)})
o11 = 2*Div(x, u) + -3*Div(y, v)
o11 : WeilDivisor on R
|
i12 : isCartier(D, IsGraded => true) o12 = true |
i13 : R = QQ[x, y, z] / ideal(x * y - z^2); |
i14 : D = divisor({1, 2}, {ideal(x, z), ideal(y, z)})
o14 = 2*Div(y, z) + Div(x, z)
o14 : WeilDivisor on R
|
i15 : isCartier(D, IsGraded => true) o15 = true |
The output value of this function is stored in the divisor's cache with the value of the last IsGraded option. If you change the IsGraded option, the value will be recomputed.
The object isCartier is a method function with options.