# 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) => , default value false,
• Outputs:
• contSingLoc, , 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

For a subvariety X of \PP^{n_1}x...x\PP^{n_m} and an irreducible subscheme Y of \PP^{n_1}x...x\PP^{n_m} this command tests whether X is contained in the singular locus of the reduced scheme of Y (i.e. the singular locus of the variety defined by the radical of the ideal defining Y).

 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

## Ways to use containedInSingularLocus :

• "containedInSingularLocus(Ideal,Ideal)"

## For the programmer

The object containedInSingularLocus is .