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Divisor :: isQLinearEquivalent

isQLinearEquivalent -- Check if two rational divisors are linearly equivalent

Synopsis

Description

Given two rational divisors, this method returns if they are Q-linearly equivalent. Otherwise it returns false.

i1 : R = QQ[x, y, z] / ideal(x * y - z^2)

o1 = R

o1 : QuotientRing
i2 : D = divisor({1/2, 3/4}, {ideal(x, z), ideal(y, z)}, CoeffType => QQ)

o2 = 1/2*Div(x, z) + 3/4*Div(y, z) of R

o2 : QDiv
i3 : E = divisor({3/4, 5/2}, {ideal(y, z), ideal(x, z)}, CoeffType => QQ)

o3 = 3/4*Div(y, z) + 5/2*Div(x, z) of R

o3 : QDiv
i4 : isQLinearEquivalent(D, E)

o4 = true

In the above ring, every pair of divisors is Q-linearly equivalent because the Weil divisor class group is isomorphic to Z/2.

If IsGraded=>true (the default is false), then it treats the divisors as if they are divisors on the Proj of their ambient ring.

i5 : R = QQ[x, y, z] / ideal(x * y - z^2)

o5 = R

o5 : QuotientRing
i6 : D = divisor({1/2, 3/4}, {ideal(x, z), ideal(y, z)}, CoeffType => QQ)

o6 = 1/2*Div(x, z) + 3/4*Div(y, z) of R

o6 : QDiv
i7 : E = divisor({3/4, 5/2}, {ideal(y, z), ideal(x, z)}, CoeffType => QQ)

o7 = 3/4*Div(y, z) + 5/2*Div(x, z) of R

o7 : QDiv
i8 : isQLinearEquivalent(D, E, IsGraded => true)

o8 = false

This is a more restrictive condition, and now the two divisors are not Q-linearly equivalent as they have different degrees on the corresponding projective line.

See also

Ways to use isQLinearEquivalent :