degenerateK3BettiTables -- compute the Betti tables of a degenerate K3 over all prime fields

Synopsis

Usage:

h = carpetBettiTables(a,b,e)
Inputs:
- a, an integer, the larger value, i.e a>=b
- b, an integer, the desired clifford index
- e, a sequence, of two small integers e=(e_1,e_2)
Outputs:
- h, a hash table, Betti tables of the degenerate K3 for various characteristics of the ground field

Description

We compute the equation and nonminimal resolution F of the degenerate K3 of type (a,b,e) where $a \ge b$ over a large finite prime field, lift the complex to the integers, which is possible if 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 : e=(-1,5)

o2 = (-1, 5)

o2 : Sequence

i3 : h=degenerateK3BettiTables(a,b,e)
 -- .00188566s elapsed
 -- .00460215s elapsed
 -- .0201942s elapsed
 -- .00729616s elapsed
 -- .00287204s elapsed

                           0  1   2   3   4   5   6   7  8 9
o3 = 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 36 160 315 318 318 315 160 36 1
                        0: 1  .   .   .   .   .   .   .  . .
                        1: . 36 160 315 288  30   .   .  . .
                        2: .  .   .   .  30 288 315 160 36 .
                        3: .  .   .   .   .   .   .   .  . 1
                           0  1   2   3   4   5   6   7  8 9
               3 => total: 1 36 160 315 298 298 315 160 36 1
                        0: 1  .   .   .   .   .   .   .  . .
                        1: . 36 160 315 288  10   .   .  . .
                        2: .  .   .   .  10 288 315 160 36 .
                        3: .  .   .   .   .   .   .   .  . 1
                           0  1   2   3   4   5   6   7  8 9
               5 => 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

o3 : HashTable

i4 : keys h

o4 = {0, 2, 3, 5}

o4 : List

i5 : elapsedTime T= minimalBetti degenerateK3(a,b,e,Characteristic=>5)
 -- .154445s elapsed

            0  1   2   3   4   5   6   7  8 9
o5 = 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

o5 : BettiTally

i6 : T-h#5

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

o6 : BettiTally

Already for fairly small values of (e_1,e_2) the result might be incorrect, because the lift to characteristic zero fails due to high powers of e_1 and e_2 in the non-minimal resolution. It would be easy to alter the program to catch these mistakes.

i7 : e=(-1,5^2)

o7 = (-1, 25)

o7 : Sequence

i8 : h=degenerateK3BettiTables(a,b,e)
 -- .00203384s elapsed
 -- .00497769s elapsed
 -- .020585s elapsed
 -- .00957248s elapsed
 -- .0039968s elapsed

                           0  1   2   3   4   5   6   7  8 9
o8 = 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 36 160 315 314 314 315 160 36 1
                        0: 1  .   .   .   .   .   .   .  . .
                        1: . 36 160 315 288  26   .   .  . .
                        2: .  .   .   .  26 288 315 160 36 .
                        3: .  .   .   .   .   .   .   .  . 1
                           0  1   2   3   4   5   6   7  8 9
               3 => total: 1 36 160 315 300 300 315 160 36 1
                        0: 1  .   .   .   .   .   .   .  . .
                        1: . 36 160 315 288  12   .   .  . .
                        2: .  .   .   .  12 288 315 160 36 .
                        3: .  .   .   .   .   .   .   .  . 1
                           0  1   2   3   4   5   6   7  8 9
               5 => total: 1 36 166 365 466 466 365 166 36 1
                        0: 1  .   .   .   .   .   .   .  . .
                        1: . 36 160 321 332 134  44   6  . .
                        2: .  .   6  44 134 332 321 160 36 .
                        3: .  .   .   .   .   .   .   .  . 1
                             0  1   2   3   4   5   6   7  8 9
               251 => total: 1 36 160 315 289 289 315 160 36 1
                          0: 1  .   .   .   .   .   .   .  . .
                          1: . 36 160 315 288   1   .   .  . .
                          2: .  .   .   .   1 288 315 160 36 .
                          3: .  .   .   .   .   .   .   .  . 1
                             0  1   2   3   4   5   6   7  8 9
               373 => total: 1 36 160 315 289 289 315 160 36 1
                          0: 1  .   .   .   .   .   .   .  . .
                          1: . 36 160 315 288   1   .   .  . .
                          2: .  .   .   .   1 288 315 160 36 .
                          3: .  .   .   .   .   .   .   .  . 1
                              0  1   2   3   4   5   6   7  8 9
               2797 => total: 1 36 160 315 289 289 315 160 36 1
                           0: 1  .   .   .   .   .   .   .  . .
                           1: . 36 160 315 288   1   .   .  . .
                           2: .  .   .   .   1 288 315 160 36 .
                           3: .  .   .   .   .   .   .   .  . 1
                               0  1   2   3   4   5   6   7  8 9
               30497 => total: 1 36 160 315 289 289 315 160 36 1
                            0: 1  .   .   .   .   .   .   .  . .
                            1: . 36 160 315 288   1   .   .  . .
                            2: .  .   .   .   1 288 315 160 36 .
                            3: .  .   .   .   .   .   .   .  . 1
                               0  1   2   3   4   5   6   7  8 9
               31627 => total: 1 36 160 315 289 289 315 160 36 1
                            0: 1  .   .   .   .   .   .   .  . .
                            1: . 36 160 315 288   1   .   .  . .
                            2: .  .   .   .   1 288 315 160 36 .
                            3: .  .   .   .   .   .   .   .  . 1
                               0  1   2   3   4   5   6   7  8 9
               35753 => total: 1 36 160 315 289 289 315 160 36 1
                            0: 1  .   .   .   .   .   .   .  . .
                            1: . 36 160 315 288   1   .   .  . .
                            2: .  .   .   .   1 288 315 160 36 .
                            3: .  .   .   .   .   .   .   .  . 1
                                0  1   2   3   4   5   6   7  8 9
               283741 => total: 1 36 160 315 289 289 315 160 36 1
                             0: 1  .   .   .   .   .   .   .  . .
                             1: . 36 160 315 288   1   .   .  . .
                             2: .  .   .   .   1 288 315 160 36 .
                             3: .  .   .   .   .   .   .   .  . 1

o8 : HashTable

i9 : keys h

o9 = {0, 30497, 2, 3, 5, 373, 35753, 31627, 251, 283741, 2797}

o9 : List

Caveat

Already for (e_1,e_2) fairly small, the algorithm might give wrong answers since the lift to characteristic zero might be incorrect. A correction is easy to implement as soon res(.,FastNonminimal=>true) allows QQ (or ZZ) as coefficient ring. Another possibility would be to use the Chinese remainder for lifting to ZZ.

Ways to use degenerateK3BettiTables:

degenerateK3BettiTables(ZZ,ZZ,Sequence)

For the programmer