computeVloc -- computete ideals of Vpsec and Vploc

Synopsis

Usage:
(Iploc,Ipsec,Isec)=computeVloc(n,J2,I2)
(Ipsloc,Ipsec,Isec)=computeVloc(n,s,J2,I2)
Inputs:
- n, an integer, the number of variables of the homogeneous coordinate ring of P^n-1
- s, an integer, stratification number
- J2, an ideal, flattening relation in B
- I2, an ideal, of the universal unfolding in P^n-1 x Spec B
Outputs:
- Iploc, an ideal, in B of the locus V_p^loc of schemes with support in p ∈P^n-1.
- Ipsec, an ideal, in B of the locus V_p^sec of schemes Γ with p ∈Γ
- Isec, an ideal, of the universal family in P^n-1 x V_p^sec
- Ipsloc, an ideal, in B of the locus V_p,s^loc of schemes Γ∈V_p^loc with m_Γ^n-s=0 ⊂O_Γ

Description

Get an ideal of the subscheme V_p^loc ⊂Spec B, whose closed points correspond to apolar schemes Γ with support supp(Γ)=p only at our special point. In the presence of s the loci V_p,s^loc of subscheme Γ∈V_p^loc with m_Γ^n-s=0 is computed. For example: V_p^loc=V_p,0^loc and V_p^loc\V_p,1^loc is the subset of curvi-linear schemes.

i1 : p=0,n=4

o1 = (0, 4)

o1 : Sequence

i2 : (R,A,I)=unfoldingEquations(p,n);

i3 : time (lin,J0)=equationsInThePaper(n,A);
     -- used 0.00521586 seconds

i4 : (Asym,phi)=symmetryMap(n,A);

i5 : (J2,I2)=symmetricUniversalFamily(n,phi,J0,I);

i6 : time (Iploc,Vpsec,Isec):=computeVloc(n,J2,I2);
     -- used 0.0235271 seconds

i7 : (dim Vpsec,codim Vpsec,degree Vpsec, betti res Vpsec)

                      0 1
o7 = (3, 1, 2, total: 1 1)
                   0: 1 .
                   1: . 1

o7 : Sequence

i8 : Vploc=saturate(Iploc);

o8 : Ideal of QQ[a     , a     , a     , a     ]
                  2,2,2   2,2,3   2,3,3   3,3,3

i9 : (dim Vploc, degree Vploc, betti res Vploc)

                   0 1 2 3
o9 = (1, 2, total: 1 3 3 1)
                0: 1 2 1 .
                1: . 1 2 1

o9 : Sequence

i10 : p=0,n=5

o10 = (0, 5)

o10 : Sequence

i11 : (R,A,I)=unfoldingEquations(p,n);

i12 : time (lin,J0)=equationsInThePaper(n,A);
     -- used 0.0151394 seconds

i13 : (Asym,phi)=symmetryMap(n,A);

i14 : (J2,I2)=symmetricUniversalFamily(n,phi,J0,I);

i15 : (Vploc1,Ipsec,Isec)=computeVloc(n,1,J2,I2);

i16 : dim Vploc1,degree Vploc1

o16 = (2, 18)

o16 : Sequence

i17 : betti(Vploc1red=radical Vploc1)

             0  1
o17 = total: 1 18
          0: 1  3
          1: . 15

o17 : BettiTally

i18 : dim Vploc1red, degree Vploc1red, genus Vploc1red

o18 = (2, 6, 0)

o18 : Sequence

Ways to use `computeVloc` :

computeVloc(ZZ,Ideal,Ideal) (missing documentation)
computeVloc(ZZ,ZZ,Ideal,Ideal) (missing documentation)

computeVloc -- computete ideals of Vpsec and Vploc

Synopsis

Description

Ways to use computeVloc :

Ways to use `computeVloc` :