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# 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..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])

## Ways to use singularLocus:

- `singularLocus(Ideal)`
- `singularLocus(Ring)`

## For the programmer

The object singularLocus is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Macaulay2Doc/doc7.m2:181:0.