specialFamiliesOfSommeseVandeVen -- produce a member of the special family

Synopsis

Usage:

Y=specialFamiliesOfSommeseVandeVen(kk,fam)
Inputs:
- kk, a ring, the ground field
- fam, an integer, number of the family
Outputs:
- Y, an ideal, the ideal of a surface

Description

By [SVdV] four families of smooth surfaces Y behave differently in the adjunction process: |H+K_Y| defines a morphism Y -> Y_1 which is generically finite to 1 instead of birational. These families are

1) P2(6;2p_1,...,2p_7)

2) P2(6;2p_1,...,2p_7,p_8)

3) P2(9;3p_1,...,3p_8)

4) Y=P(E) where E is a indecomposabe rank 2 vector bundle over an elliptic curve and H=3B, where B in Y is the section with B^2=1.

i1 : kk=ZZ/nextPrime(10^3)

o1 = kk

o1 : QuotientRing

i2 : Y=specialFamiliesOfSommeseVandeVen(kk,1);

o2 : Ideal of kk[x ..x ]
                  0   6

i3 : betti(fY= res Y)

            0 1  2  3 4
o3 = total: 1 7 14 11 3
         0: 1 .  .  . .
         1: . 7  8  3 .
         2: . .  6  8 3

o3 : BettiTally

i4 : betti (fib=trim(ideal(fY.dd_4*random(kk^3,kk^1))))

            0 1
o4 = total: 1 6
         0: 1 5
         1: . 1

o4 : BettiTally

i5 : dim fib, degree fib

o5 = (1, 2)

o5 : Sequence

i6 : ll=adjunctionProcess Y;

i7 : ll_0,ll_1,ll_2, minimalBetti ll_3

                                  0 1  2  3 4
o7 = ({(6, 8, 3)}, {}, {}, total: 1 7 14 11 3)
                               0: 1 .  .  . .
                               1: . 7  8  3 .
                               2: . .  6  8 3

o7 : Sequence

The adjunction map |H+K_Y|: Y -> P2 is 2:1 in case of family 1.

i8 : Y=specialFamiliesOfSommeseVandeVen(kk,2);

o8 : Ideal of kk[x ..x ]
                  0   5

i9 : betti(fY= res Y)

            0 1 2 3
o9 = total: 1 6 8 3
         0: 1 . . .
         1: . 3 2 .
         2: . 3 6 3

o9 : BettiTally

i10 : betti (fib=trim(ideal(fY.dd_3*random(kk^3,kk^1))))

             0 1
o10 = total: 1 5
          0: 1 4
          1: . 1

o10 : BettiTally

i11 : dim fib, degree fib

o11 = (1, 2)

o11 : Sequence

The adjunction map |H+K_Y|: Y -> P2 is 2:1 in case of family 2.

i12 : Y=specialFamiliesOfSommeseVandeVen(kk,3);

o12 : Ideal of kk[x ..x ]
                   0   6

i13 : betti(fY=res Y)

             0  1  2  3 4
o13 = total: 1 10 20 15 4
          0: 1  .  .  . .
          1: .  6  8  3 .
          2: .  4 12 12 4

o13 : BettiTally

i14 : P6=ring Y,dim P6==7

o14 = (P6, true)

o14 : Sequence

i15 : (Q,adj)=slowAdjunctionCalculation(Y,fY.dd_4,symbol u);

i16 : dim Q, degree Q

o16 = (3, 2)

o16 : Sequence

i17 : P3=ring Q; dim P3==4

o18 = true

i19 : while (L=ideal random(P3^1,P3^{2:-1});
          pts=decompose (L+Q);
          #pts<2) do ()

i20 : pt=sub(syz transpose jacobian first pts,kk)

o20 = | -233 |
      | -191 |
      | 362  |
      | 1    |

               4       1
o20 : Matrix kk  <-- kk

i21 : betti(fib= trim ideal(fY.dd_4*pt))

             0 1
o21 = total: 1 6
          0: 1 5
          1: . 1

o21 : BettiTally

i22 : dim fib, degree fib

o22 = (1, 2)

o22 : Sequence

The adjunction map |H+K_Y|: Y -> Q is 2:1 onto a quadric in P3 in case of family 3.

i23 : Y=specialFamiliesOfSommeseVandeVen(kk,4);

o23 : Ideal of kk[x ..x ]
                   0   5

i24 : betti(fY=res Y)

             0  1  2 3 4
o24 = total: 1 11 18 9 1
          0: 1  .  . . .
          1: .  .  . . .
          2: . 11 18 9 1

o24 : BettiTally

i25 : P5=ring Y, dim P5==6

o25 = (P5, true)

o25 : Sequence

i26 : betti(omegaY=Ext^2(module Y,P5^{-6}))

             0  1
o26 = total: 3 12
          1: 3  3
          2: .  9

o26 : BettiTally

i27 : betti(fib=trim ideal (random(kk^1,kk^3)*presentation omegaY))

             0 1
o27 = total: 1 6
          0: 1 3
          1: . 3

o27 : BettiTally

i28 : dim fib, degree fib

o28 = (1, 3)

o28 : Sequence

The adjunction map |H+K_Y|: Y -> P2 is 3:1 in case of family 4.

Ways to use specialFamiliesOfSommeseVandeVen :

specialFamiliesOfSommeseVandeVen(Ring,ZZ)

For the programmer

The object specialFamiliesOfSommeseVandeVen is a method function.