next | previous | forward | backward | up | index | toc

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 .