Given an principal ideal I or a polynomial in the homogeneous coordinate ring of P2 the function checks whether the corresponding curve has only ordinary singularities by checking that the jacobian ideal of I + I defines a possibly empty collection of distinct points
i1 : kk=ZZ/10007; |
i2 : S=kk[x,y,z] o2 = S o2 : PolynomialRing |
i3 : ordinaryDoublePoints(z*y^2-x^3) o3 = false |
i4 : ordinaryDoublePoints(f=z*y^2-z*x^2-x^3) o4 = true |
i5 : ordinaryDoublePoints(x^2+y^2+z^2) o5 = true |
i6 : Ipts=randomPlanePoints(3,S); o6 : Ideal of S |
i7 : Ipts2=saturate(Ipts^2); o7 : Ideal of S |
i8 : betti Ipts2
0 1
o8 = total: 1 4
0: 1 .
1: . .
2: . 1
3: . 3
o8 : BettiTally
|
i9 : I=ideal( gens Ipts2*random(source gens Ipts2,S^{1:-4}))
4 3 2 2 3 4 3
o9 = ideal(- 4981x + 1818x y - 2714x y + 1617x*y - 1190y - 3171x z +
------------------------------------------------------------------------
2 2 3 2 2 2 2 2
2206x y*z - 652x*y z - 4448y z + 3342x z - 2222x*y*z + 4172y z -
------------------------------------------------------------------------
3 3 4
4538x*z + 889y*z + 3935z )
o9 : Ideal of S
|
i10 : ordinaryDoublePoints I o10 = true |