isSmooth -- whether R mod the ideal is smooth

Synopsis

Usage:

isSmooth( I )
Inputs:
- I, an ideal,
Optional inputs:
- IsGraded => a Boolean value, default value false, specify that we should do this computation on a projective algebraic variety
Outputs:
- flag, a Boolean value,

Description

This function returns true if $R/I$ is regular where $R$ is the ambient ring of $I$, otherwise it sets to false.

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

i2 : I = ideal(x * y - z^2 )

                  2
o2 = ideal(x*y - z )

o2 : Ideal of R

i3 : isSmooth( I )

o3 = false

i4 : R = QQ[x, y, u, v];

i5 : I = ideal(x * y - u * v)

o5 = ideal(x*y - u*v)

o5 : Ideal of R

i6 : isSmooth( I )

o6 = false

i7 : R = QQ[x, y, z];

i8 : J = ideal( x )

o8 = ideal x

o8 : Ideal of R

i9 : isSmooth( J )

o9 = true

If IsGraded is set to true (default false) then it treats $I$ as an ideal on $Proj R$ (and it assumes $R$ is standard graded over a field). In particular, singularities at the origin (corresponding to the irrelevant ideal) are ignored.

i10 : R = QQ[x, y, z];

i11 : I = ideal(x * y - z^2 )

                   2
o11 = ideal(x*y - z )

o11 : Ideal of R

i12 : isSmooth(I)

o12 = false

i13 : isSmooth(I, IsGraded => true)

o13 = true

i14 : R = QQ[x, y, u, v];

i15 : I = ideal(x * y - u * v)

o15 = ideal(x*y - u*v)

o15 : Ideal of R

i16 : isSmooth(I)

o16 = false

i17 : isSmooth(I, IsGraded => true)

o17 = true

Ways to use isSmooth :

isSmooth(Ideal)

For the programmer

The object isSmooth is a method function with options.