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Divisor :: isCartier

isCartier -- Check if a Weil divisor is Cartier

Synopsis

Description

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)

o1 = R

o1 : QuotientRing
i2 : D = divisor({2, -3}, {ideal(x, u), ideal(y, v)})

o2 = 2*Div(x, u) + -3*Div(y, v) of R

o2 : WDiv
i3 : isCartier( D )

o3 = false

Neither is this divisor.

i4 : R = QQ[x, y, z] / ideal(x * y - z^2 )

o4 = R

o4 : QuotientRing
i5 : D = divisor({1, 2}, {ideal(x, z), ideal(y, z)})

o5 = 1*Div(x, z) + 2*Div(y, z) of R

o5 : WDiv
i6 : isCartier( D )

o6 = false

Of course the next divisor is Cartier.

i7 : R = QQ[x, y, z]

o7 = R

o7 : PolynomialRing
i8 : D = divisor({1, 2}, {ideal(x), ideal(y)})

o8 = 1*Div(x) + 2*Div(y) of R

o8 : WDiv
i9 : isCartier( D )

o9 = true

If IsGraded == 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)

o10 = R

o10 : QuotientRing
i11 : D = divisor({2, -3}, {ideal(x, u), ideal(y, v)})

o11 = 2*Div(x, u) + -3*Div(y, v) of R

o11 : WDiv
i12 : isCartier(D, IsGraded => true)

o12 = true
i13 : R = QQ[x, y, z] / ideal(x * y - z^2)

o13 = R

o13 : QuotientRing
i14 : D = divisor({1, 2}, {ideal(x, z), ideal(y, z)})

o14 = 1*Div(x, z) + 2*Div(y, z) of R

o14 : WDiv
i15 : isCartier(D, IsGraded => true)

o15 = true

See also

Ways to use isCartier :