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carpetBettiTables -- compute the Betti tables of a carpet of given genus and Clifford index over all prime fields

Description

We compute the equation and nonminimal resolution F of the carpet of type (a,b) where $a \ge b$ over a larger finite prime field, lift the complex to the integers, which is possible since the coefficients are small. Finally we study the nonminimal strands over ZZ by computing the Smith normal form. The resulting data allow us to compute the Betti tables for arbitrary primes.

i1 : a=5,b=5

o1 = (5, 5)

o1 : Sequence
i2 : h=carpetBettiTables(a,b)
 -- .00229656s elapsed
 -- .0060891s elapsed
 -- .0228791s elapsed
 -- .0102217s elapsed
 -- .00299258s elapsed

                           0  1   2   3   4   5   6   7  8 9
o2 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1}
                        0: 1  .   .   .   .   .   .   .  . .
                        1: . 36 160 315 288   .   .   .  . .
                        2: .  .   .   .   . 288 315 160 36 .
                        3: .  .   .   .   .   .   .   .  . 1
                           0  1   2   3   4   5   6   7  8 9
               2 => total: 1 36 167 370 476 476 370 167 36 1
                        0: 1  .   .   .   .   .   .   .  . .
                        1: . 36 160 322 336 140  48   7  . .
                        2: .  .   7  48 140 336 322 160 36 .
                        3: .  .   .   .   .   .   .   .  . 1
                           0  1   2   3   4   5   6   7  8 9
               3 => total: 1 36 160 315 302 302 315 160 36 1
                        0: 1  .   .   .   .   .   .   .  . .
                        1: . 36 160 315 288  14   .   .  . .
                        2: .  .   .   .  14 288 315 160 36 .
                        3: .  .   .   .   .   .   .   .  . 1

o2 : HashTable
i3 : T= carpetBettiTable(h,3)

            0  1   2   3   4   5   6   7  8 9
o3 = total: 1 36 160 315 302 302 315 160 36 1
         0: 1  .   .   .   .   .   .   .  . .
         1: . 36 160 315 288  14   .   .  . .
         2: .  .   .   .  14 288 315 160 36 .
         3: .  .   .   .   .   .   .   .  . 1

o3 : BettiTally
i4 : J=canonicalCarpet(a+b+1,b,Characteristic=>3);

              ZZ
o4 : Ideal of --[x ..x , y ..y ]
               3  0   5   0   5
i5 : elapsedTime T'=minimalBetti J
 -- .12596s elapsed

            0  1   2   3   4   5   6   7  8 9
o5 = total: 1 36 160 315 302 302 315 160 36 1
         0: 1  .   .   .   .   .   .   .  . .
         1: . 36 160 315 288  14   .   .  . .
         2: .  .   .   .  14 288 315 160 36 .
         3: .  .   .   .   .   .   .   .  . 1

o5 : BettiTally
i6 : T-T'

            0 1 2 3 4 5 6 7 8 9
o6 = total: . . . . . . . . . .
         1: . . . . . . . . . .
         2: . . . . . . . . . .
         3: . . . . . . . . . .

o6 : BettiTally
i7 : elapsedTime h=carpetBettiTables(6,6);
 -- .00382701s elapsed
 -- .0164827s elapsed
 -- .144312s elapsed
 -- 1.05081s elapsed
 -- .393445s elapsed
 -- .0356114s elapsed
 -- .00585348s elapsed
 -- 4.8016s elapsed
i8 : keys h

o8 = {0, 2, 3, 5}

o8 : List
i9 : carpetBettiTable(h,7)

            0  1   2   3    4    5    6    7   8   9 10 11
o9 = total: 1 55 320 891 1408 1155 1155 1408 891 320 55  1
         0: 1  .   .   .    .    .    .    .   .   .  .  .
         1: . 55 320 891 1408 1155    .    .   .   .  .  .
         2: .  .   .   .    .    . 1155 1408 891 320 55  .
         3: .  .   .   .    .    .    .    .   .   .  .  1

o9 : BettiTally
i10 : carpetBettiTable(h,5)

             0  1   2   3    4    5    6    7   8   9 10 11
o10 = total: 1 55 320 891 1408 1275 1275 1408 891 320 55  1
          0: 1  .   .   .    .    .    .    .   .   .  .  .
          1: . 55 320 891 1408 1155  120    .   .   .  .  .
          2: .  .   .   .    .  120 1155 1408 891 320 55  .
          3: .  .   .   .    .    .    .    .   .   .  .  1

o10 : BettiTally

See also

Ways to use carpetBettiTables:

  • carpetBettiTables(ZZ,ZZ)

For the programmer

The object carpetBettiTables is a method function.


The source of this document is in K3Carpets.m2:1404:0.