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NumericalGodeaux :: complexModuloRegularSequence

complexModuloRegularSequence -- set-up for minimal free resolution modulo x0,x1

Synopsis

Description

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 $S = k[x_0,x_1,y_0,\ldots,y_3]$. Modulo the regular sequence $x_0,x_1$, 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 $y_0,\ldots,y_3$. 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 $\mathbb{P}^3$. 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 |
     +------------------+

Ways to use complexModuloRegularSequence :

For the programmer

The object complexModuloRegularSequence is a method function.