Description
singularLocus R -- produce the singular locus of a ring, which is assumed to be integral.
This function can also be applied to an ideal, in which case the singular locus of the quotient ring is returned, or to a variety.
i1 : singularLocus(QQ[x,y] / (x^2 - y^3))
QQ[x..y]
o1 = ---------------------
3 2 2
(- y + x , 2x, -3y )
o1 : QuotientRing
|
i2 : singularLocus Spec( QQ[x,y,z] / (x^2 - y^3) )
/ QQ[x..z] \
o2 = Spec|---------------------|
| 3 2 2 |
\(- y + x , 2x, -3y )/
o2 : AffineVariety
|
i3 : singularLocus Proj( QQ[x,y,z] / (x^2*z - y^3) )
/QQ[x..z]\
o3 = Proj|--------|
| 2 |
\ (x, y )/
o3 : ProjectiveVariety
|
For rings over ZZ the locus where the ring is not smooth over ZZ is computed.
i4 : singularLocus(ZZ[x,y]/(x^2-x-y^3+y^2))
ZZ[x..y]
o4 = ----------------------------------------
3 2 2 2
(- y + x + y - x, 2x - 1, - 3y + 2y)
o4 : QuotientRing
|
i5 : gens gb ideal oo
o5 = | 11 y+3 x+5 |
1 3
o5 : Matrix (ZZ[x..y]) <-- (ZZ[x..y])
|