containedInSingularLocus -- This method tests is an irreducible variety is contained in the singular locus of the reduced scheme of an irreducible scheme

Synopsis

• Usage:

  containedInSingularLocus(IX,IY)

• Inputs:
  • IX, an ideal, a multi-homogeneous prime ideal defining a closed subscheme of \PP^{n_1}x...x\PP^{n_m}; makeProductRing builds the graded coordinate ring of \PP^{n_1}x...x\PP^{n_m}.
  • IY, an ideal, a multi-homogeneous primary ideal defining a closed subscheme of \PP^{n_1}x...x\PP^{n_m}; makeProductRing builds the graded coordinate ring of \PP^{n_1}x...x\PP^{n_m}.
• Optional inputs:
  • Verbose (missing documentation) => a Boolean value, default value false,
• Outputs:
  • contSingLoc, a Boolean value, whether or not the variety X associated to IX is contained in the singular locus of the variety associated to the radical of IY

Description

i1 : n=6

o1 = 6

i2 : R = makeProductRing({n})

o2 = R

o2 : PolynomialRing

i3 : x=gens(R)

o3 = {a, b, c, d, e, f, g}

o3 : List

i4 : m=matrix{for i from 0 to n-3 list x_i,for i from 0 to n-3 list (i+3)*x_(i+3),for i from 0 to n-3 list x_(i+2),for i from 0 to n-3 list x_(i)+(5+i)*x_(i+1)}

o4 = | a    b    c    d    |
     | 3d   4e   5f   6g   |
     | c    d    e    f    |
     | a+5b b+6c c+7d d+8e |

             4      4
o4 : Matrix R  <-- R

i5 : C=ideal mingens(minors(3,m));

o5 : Ideal of R

i6 : P=ideal(x_0,x_4,x_3,x_2,x_1)

o6 = ideal (a, e, d, c, b)

o6 : Ideal of R

i7 : containedInSingularLocus(P,C)

o7 = true