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Divisor :: isCartier

isCartier -- whether 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);
i2 : D = divisor({2, -3}, {ideal(x, u), ideal(y, v)})

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

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 = 2*Div(y) + Div(x)

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

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

See also

Ways to use isCartier :