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isQLinearEquivalent -- whether two Q-divisors are linearly equivalent

Synopsis

Description

Given two rational divisors, this method returns true if they linearly equivalent after clearing denominators or if some further multiple up to n makes them linearly equivalent. Otherwise it returns false.

i1 : R = QQ[x, y, z] / ideal(x * y - z^2);
 
i2 : D = divisor({1/2, 3/4}, {ideal(x, z), ideal(y, z)}, CoefficientType => QQ)

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

o2 : QWeilDivisor on R
 
i3 : E = divisor({3/4, 5/2}, {ideal(y, z), ideal(x, z)}, CoefficientType => QQ)

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

o3 : QWeilDivisor on R
 
i4 : isQLinearEquivalent(10, 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. However, if we don't set n high enough, the function will return false.

i5 : R = QQ[x,y,z] / ideal(x * y - z^2);
 
i6 : D = divisor(x);

o6 : WeilDivisor on R
 
i7 : E = divisor(ideal(x,z));

o7 : WeilDivisor on R
 
i8 : isQLinearEquivalent(1, D, E)

o8 = false
 
i9 : isQLinearEquivalent(3, D, E)

o9 = true

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

i10 : R = QQ[x, y, z] / ideal(x * y - z^2);
 
i11 : D = divisor({1/2, 3/4}, {ideal(x, z), ideal(y, z)}, CoefficientType => QQ)

o11 = 1/2*Div(x, z) + 3/4*Div(y, z)

o11 : QWeilDivisor on R
 
i12 : E = divisor({3/2, -1/4}, {ideal(y, z), ideal(x, z)}, CoefficientType => QQ)

o12 = -1/4*Div(x, z) + 3/2*Div(y, z)

o12 : QWeilDivisor on R
 
i13 : isQLinearEquivalent(10, D, E, IsGraded => true)

o13 = true
 
i14 : isQLinearEquivalent(10, 3*D, E, IsGraded => true)

o14 = false

See also

Ways to use isQLinearEquivalent:

For the programmer

The object isQLinearEquivalent is a method function with options.