computes the lifting of the syzygy matrices in the minimal free resolution fX by solving iteratively the flatness condition. In fact, let (DfX.dd1)i=fX.dd1+ Tit be a deformed equation of X by the i-th column of T, and b0,...,bt are the deformation parameteres, where t is the number of columns of T. To find a lift (DfX.dd2)i=fX.dd2+ (F.dd2)i of the syzygy matrix fX.dd2, we use the neccessary condition that (DfX.dd2)i(DfX.dd1)i=0 mod (b1,...,bt)². This leads to (fX.dd1)(F.dd2)i= -Tit(fX.dd2), and by matrix quotient we have (F.dd2)i=-Tit(fX.dd2)// fX.dd1. Iterating this process for all the remaing maps in the resolution fX, and all the basis column vector of the matrix T, this function computes the chain complex F, which lifts the resolution fX to first order deformed resolution DfX. Indeed, we have DfX.ddi=fX.ddi+ F.ddi over the polynomial ring ( kk[b1,...,bt]/(b1,...,bt)²)[x0,...,xn] where kk[x0,...,xn] is the ring of fX,
i1 : loadPackage("RandomPlaneCurves",Reload=>true)
o1 = RandomPlaneCurves
o1 : Package
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i2 : kk=ZZ/101 o2 = kk o2 : QuotientRing |
i3 : R=kk[t_0..t_2] o3 = R o3 : PolynomialRing |
i4 : delta=ideal(t_0*t_1,t_0*t_2,t_1*t_2); o4 : Ideal of R |
i5 : adjointSystem=gens truncate(3,delta);
1 7
o5 : Matrix R <--- R
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i6 : C=ideal(gens saturate(delta^2)*random(source gens saturate(delta)^2,R^{ -6}));--nodal plane sextic curve
o6 : Ideal of R
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i7 : Ican=imageUnderRationalMap(C,adjointSystem);
o7 : Ideal of kk[x , x , x , x , x , x , x ]
0 1 2 3 4 5 6
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i8 : genus Ican == 7 o8 = true |
i9 : betti(fX=res(Ican,LengthLimit=>3))
0 1 2 3
o9 = total: 1 10 25 25
0: 1 . . .
1: . 10 16 9
2: . . 9 16
o9 : BettiTally
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i10 : T=firstOrderDeformation fX;
10 18
o10 : Matrix (kk[x , x , x , x , x , x , x ]) <--- (kk[x , x , x , x , x , x , x ])
0 1 2 3 4 5 6 0 1 2 3 4 5 6
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i11 : (F,L)=liftDeformationToFreeResolution(fX,T); |
i12 : mF=(F.dd_3)^{16..24}_{0..8}
o12 = | b_17 0 0 0 0 0
| 0 0 0 b_16 0 0
| 0 34b_16-34b_17 -b_16 0 b_16 0
| 0 43b_16-43b_17 10b_17 0 0 0
| 0 0 10b_17 0 0 0
| 0 -34b_16+34b_17 0 0 -b_17 0
| 0 3b_16-3b_17 34b_16-34b_17 0 -34b_16+34b_17 -b_16+b_17
| 0 0 43b_16 0 -43b_16+43b_17 10b_16-10b_17
| 0 0 -3b_16+3b_17 0 3b_16-3b_17 34b_16-34b_17
-----------------------------------------------------------------------
0 0 0 |
0 0 0 |
0 -2b_16 0 |
0 10b_16 b_16 |
0 10b_16 -10b_16 |
0 b_16 0 |
0 0 0 |
-b_17 43b_16 -34b_16 |
0 0 0 |
9 9
o12 : Matrix (kk[b , b , b , b , b , b , b , b , b , b , b , b , b , b , b , b , b , b ][x , x , x , x , x , x , x ]) <--- (kk[b , b , b , b , b , b , b , b , b , b , b , b , b , b , b , b , b , b ][x , x , x , x , x , x , x ])
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 0 1 2 3 4 5 6
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