Let R(X) be the canonical ring of a numerical Godeaux surface which we consider as a finitely generated module over the weighted polynomial ring S = k[x0,x1,y0,...,y3]. Modulo the regular sequence x0,x1, the minimal free resolution of R(X) as an S-module splits into a direct sum of three complexes whose maps depend only on the variables y0,...,y3. In this procedure we build the matrices of the individual complexes depending on the configuration of the four base points. The first complex is the minimal free resolution of the ideal of the four base points in ℙ3. The second complex is either a direct sum of the resolutions of each single base point (case "1111") or an extension of the single resolutions. Finally, the third complex is the dual of the first one.
i1 : kk = ZZ/197; |
i2 : s = "1111"; |
i3 : (A,B,subs0) = globalVariables(kk,s); |
i4 : SR = ring(A); |
i5 : D = complexModuloRegularSequence(SR,s); |
i6 : betti D
0 1 2 3
o6 = total: 8 26 26 8
0: 1 . . .
1: . . . .
2: . . . .
3: . . . .
4: 4 . . .
5: 3 6 . .
6: . 12 . .
7: . 8 8 .
8: . . 12 .
9: . . 6 3
10: . . . 4
11: . . . .
12: . . . .
13: . . . .
14: . . . 1
o6 : BettiTally
|
i7 : basePoints = ann coker D.dd_1; o7 : Ideal of SR |
i8 : netList decompose basePoints
+------------------+
o8 = |ideal (y , y , y )|
| 2 1 0 |
+------------------+
|ideal (y , y , y )|
| 3 1 0 |
+------------------+
|ideal (y , y , y )|
| 3 2 0 |
+------------------+
|ideal (y , y , y )|
| 3 2 1 |
+------------------+
|