# singularLocus -- singular locus

## 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, y, 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, y, 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])```

## Ways to use singularLocus :

• singularLocus(AffineVariety)
• singularLocus(Ideal)
• singularLocus(ProjectiveVariety)
• singularLocus(Ring)