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ramificationDivisor -- compute the ramification divisor of a finite inclusion of normal domains or a blowup over a smooth base

Synopsis

Description

Compute the ramification (relative canonical) divisor corresponding the finite inclusion of normal domains. If you pass it a non-finite map, it will compute the divisorial part of the locus where the map is not smooth. If IsGraded is set to false (the default value), then the coefficient ring of both the source and target of $f$ must be equal.

i1 : R = QQ[x];
 
i2 : S = QQ[y];
 
i3 : f = map(S, R, {y^3});

o3 : RingMap S <-- R
 
i4 : ramificationDivisor(f)

o4 = 2*Div(y)

o4 : WeilDivisor on S

The next example is a Veronese which is etale in codimension 1.

i5 : R = QQ[x,y];
 
i6 : T = QQ[a,b,c,d];
 
i7 : h = map(R, T, {x^3, x^2*y, x*y^2, y^3});

o7 : RingMap R <-- T
 
i8 : S = T/ker h;
 
i9 : f = map(R, S, {x^3, x^2*y, x*y^2, y^3});

o9 : RingMap R <-- S
 
i10 : ramificationDivisor(f)

o10 = 0, the zero divisor

o10 : WeilDivisor on R

Here is an example with wild ramification.

i11 : R = ZZ/2[t];
 
i12 : S = ZZ/2[x];
 
i13 : f = map(S, R, {x^2*(1+x)});

o13 : RingMap S <-- R
 
i14 : ramificationDivisor(f)

o14 = 2*Div(x)

o14 : WeilDivisor on S

If the option IsGraded is set to true, then the function will assume that the source of f is affine, and the target is projective over the source. In this case, the coefficient ring of the target must be equal to the source ring. This can be useful when computing things like relative canonical divisors over regular bases (it may not give the expected answer over non-regular bases). For example, this is useful if you want to compute the relative canonical divisor of a blowup.

i15 : R = QQ[x,y];
 
i16 : S = reesAlgebra(ideal(x,y^2));
 
i17 : f = map(S, R);

o17 : RingMap S <-- R
 
i18 : ramificationDivisor(f,IsGraded=>true)

o18 = 2*Div(y, x)

o18 : WeilDivisor on S

See also

Ways to use ramificationDivisor :

For the programmer

The object ramificationDivisor is a method function with options.