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Macaulay2Doc :: singularLocus

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 :