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Divisor :: canonicalDivisor

canonicalDivisor -- Compute the canonical divisor of a ring

Synopsis

Description

Compute the canonical divisor of a ring (warning, the canonical divisor is not unique, but only unique up to linear equivalence). If the IsGraded option is set to true (default false), then it will return a canonical divisor for the Proj of R, otherwise it will return one for only the Spec. The graded version only works reliably for graded rings over a field (for instance, if you have a Rees algebra you will need to flatten the variables).

i1 : R = QQ[x,y,z]

o1 = R

o1 : PolynomialRing
i2 : canonicalDivisor(R)

o2 = 0, the zero divisor of R

o2 : WDiv
i3 : canonicalDivisor(R, IsGraded=>true)

o3 = -3*Div(x) of R

o3 : WDiv

Note the IsGraded option makes a difference. Consider now a non-Gorenstein singularity.

i4 : R = QQ[a,b,c,d]/ideal(c^2-b*d, b*c-a*d, b^2-a*c)

o4 = R

o4 : QuotientRing
i5 : canonicalDivisor(R)

o5 = -2*Div(d, c, b) of R

o5 : WDiv

Ways to use canonicalDivisor :