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relativeResolutionTwists -- compute the twists in the relative resolution

Synopsis

Description

We compute the twists of the relative resolution in the resonance scroll of a degenerate K3 X_e(a,b) in case of k resonance after re-embedding the resonance scroll with |H+jR| for j=am-a_0=bm-b_0 where $\{a_i\}|\{b_j\}$ is the splitting type of the resonance scroll.

i1 : F = relativeResolution(5,4,3);
 
i2 : (as,bs)=resonanceScroll(5,4,3)

o2 = ({1, 1, 1}, {1, 1, 0})

o2 : Sequence
 
i3 : betti F

            0 1  2 3 4
o3 = total: 1 9 16 9 1
         0: 1 .  . . .
         1: . 1  . . .
         2: . 7  8 1 .
         3: . 1  8 7 .
         4: . .  . 1 .
         5: . .  . . 1

o3 : BettiTally
 
i4 : L = relativeResolutionTwists(as_0,bs_0,F);
 
i5 : netList apply(L,c-> tally c)

     +-------------------+
o5 = |Tally{{0, 0} => 1} |
     +-------------------+
     |Tally{{2, -1} => 7}|
     |      {2, -2} => 1 |
     |      {2, 0} => 1  |
     +-------------------+
     |Tally{{3, -1} => 8}|
     |      {3, -2} => 8 |
     +-------------------+
     |Tally{{4, -1} => 1}|
     |      {4, -2} => 7 |
     |      {4, -3} => 1 |
     +-------------------+
     |Tally{{6, -3} => 1}|
     +-------------------+
 
i6 : L = relativeResolutionTwists(as_0+2,bs_0+1,F);
 
i7 : netList apply(L,c-> tally c)

     +--------------------+
o7 = |Tally{{0, 0} => 1}  |
     +--------------------+
     |Tally{{2, -2} => 1} |
     |      {2, -3} => 2  |
     |      {2, -4} => 3  |
     |      {2, -5} => 2  |
     |      {2, -6} => 1  |
     +--------------------+
     |Tally{{3, -4} => 2} |
     |      {3, -5} => 4  |
     |      {3, -6} => 4  |
     |      {3, -7} => 4  |
     |      {3, -8} => 2  |
     +--------------------+
     |Tally{{4, -6} => 1 }|
     |      {4, -7} => 2  |
     |      {4, -8} => 3  |
     |      {4, -9} => 2  |
     |      {4, -10} => 1 |
     +--------------------+
     |Tally{{6, -12} => 1}|
     +--------------------+

See also

Ways to use relativeResolutionTwists:

For the programmer

The object relativeResolutionTwists is a method function.