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multiBetti -- a hash table containing the multigraded Betti numbers of a Veronese embedding

Synopsis

Description

This function returns a hash table $H$ containing the multigraded Betti numbers for $\mathcal{O}(b)$ on $\mathbb{P}^{n}$ under the embedding by $d$-fold Veronese embedding given by $\mathcal{O}(d)$. The keys of the returned hash table $H$ are pairs $(p,q)$ where $H#(p,q)$ gives the the multigraded Betti decomposition of $K_{p,q}(\mathbb{P}^n, d;b)$. We record the multigraded Betti numbers via a multigraded Hilbert series. See Section 1.1 of [BEGY] for a more precise description of the multigraded Betti numbers of a Veronese embedding.

Note that the output of this function is sometimes enormous and so can take a long time to print on the screen.

i1 : totalBettiTally(3,2,0)

            0  1   2   3   4   5  6 7
o1 = total: 1 27 105 189 189 105 27 1
         0: 1  .   .   .   .   .  . .
         1: . 27 105 189 189 105 27 .
         2: .  .   .   .   .   .  . 1

o1 : BettiTally
 
i2 : multiBetti(3,2,0)

o2 = HashTable{(0, 0) => 1                                                                                                                                                                                                                                                                                                                                                                               }
               (0, 1) => 0
               (0, 2) => 0
               (1, 0) => 0
                          4 2    4        4 2    3 3     3 2       3   2    3 3    2 4     2 3       2 2 2     2   3    2 4      4         3 2       2 3        4    4 2    3 3    2 4
               (1, 1) => T T  + T T T  + T T  + T T  + 2T T T  + 2T T T  + T T  + T T  + 2T T T  + 3T T T  + 2T T T  + T T  + T T T  + 2T T T  + 2T T T  + T T T  + T T  + T T  + T T
                          0 1    0 1 2    0 2    0 1     0 1 2     0 1 2    0 2    0 1     0 1 2     0 1 2     0 1 2    0 2    0 1 2     0 1 2     0 1 2    0 1 2    1 2    1 2    1 2
               (1, 2) => 0
               (2, 0) => 0
                          6 2      6   2    5 4     5 3       5 2 2     5   3    5 4    4 5     4 4       4 3 2     4 2 3     4   4    4 5     3 5       3 4 2     3 3 3     3 2 4     3   5    2 6       2 5 2     2 4 3     2 3 4     2 2 5    2   6      6 2       5 3       4 4       3 5      2 6    5 4    4 5
               (2, 1) => T T T  + T T T  + T T  + 3T T T  + 4T T T  + 3T T T  + T T  + T T  + 4T T T  + 7T T T  + 7T T T  + 4T T T  + T T  + 3T T T  + 7T T T  + 9T T T  + 7T T T  + 3T T T  + T T T  + 4T T T  + 7T T T  + 7T T T  + 4T T T  + T T T  + T T T  + 3T T T  + 4T T T  + 3T T T  + T T T  + T T  + T T
                          0 1 2    0 1 2    0 1     0 1 2     0 1 2     0 1 2    0 2    0 1     0 1 2     0 1 2     0 1 2     0 1 2    0 2     0 1 2     0 1 2     0 1 2     0 1 2     0 1 2    0 1 2     0 1 2     0 1 2     0 1 2     0 1 2    0 1 2    0 1 2     0 1 2     0 1 2     0 1 2    0 1 2    1 2    1 2
               (2, 2) => 0
               (3, 0) => 0
                          7 4       7 3 2     7 2 3    7   4     6 5       6 4 2     6 3 3     6 2 4     6   5     5 6       5 5 2      5 4 3      5 3 4     5 2 5     5   6    4 7       4 6 2      4 5 3      4 4 4      4 3 5     4 2 6    4   7     3 7 2     3 6 3      3 5 4      3 4 5     3 3 6     3 2 7     2 7 3     2 6 4     2 5 5     2 4 6     2 3 7      7 4       6 5       5 6      4 7
               (3, 1) => T T T  + 2T T T  + 2T T T  + T T T  + 2T T T  + 5T T T  + 7T T T  + 5T T T  + 2T T T  + 2T T T  + 7T T T  + 12T T T  + 12T T T  + 7T T T  + 2T T T  + T T T  + 5T T T  + 12T T T  + 15T T T  + 12T T T  + 5T T T  + T T T  + 2T T T  + 7T T T  + 12T T T  + 12T T T  + 7T T T  + 2T T T  + 2T T T  + 5T T T  + 7T T T  + 5T T T  + 2T T T  + T T T  + 2T T T  + 2T T T  + T T T
                          0 1 2     0 1 2     0 1 2    0 1 2     0 1 2     0 1 2     0 1 2     0 1 2     0 1 2     0 1 2     0 1 2      0 1 2      0 1 2     0 1 2     0 1 2    0 1 2     0 1 2      0 1 2      0 1 2      0 1 2     0 1 2    0 1 2     0 1 2     0 1 2      0 1 2      0 1 2     0 1 2     0 1 2     0 1 2     0 1 2     0 1 2     0 1 2     0 1 2    0 1 2     0 1 2     0 1 2    0 1 2
               (3, 2) => 0
               (4, 0) => 0
                          8 5 2     8 4 3     8 3 4    8 2 5     7 6 2     7 5 3     7 4 4     7 3 5     7 2 6     6 7 2     6 6 3      6 5 4      6 4 5     6 3 6     6 2 7    5 8 2     5 7 3      5 6 4      5 5 5      5 4 6     5 3 7    5 2 8     4 8 3     4 7 4      4 6 5      4 5 6     4 4 7     4 3 8     3 8 4     3 7 5     3 6 6     3 5 7     3 4 8    2 8 5     2 7 6     2 6 7    2 5 8
               (4, 1) => T T T  + 2T T T  + 2T T T  + T T T  + 2T T T  + 5T T T  + 7T T T  + 5T T T  + 2T T T  + 2T T T  + 7T T T  + 12T T T  + 12T T T  + 7T T T  + 2T T T  + T T T  + 5T T T  + 12T T T  + 15T T T  + 12T T T  + 5T T T  + T T T  + 2T T T  + 7T T T  + 12T T T  + 12T T T  + 7T T T  + 2T T T  + 2T T T  + 5T T T  + 7T T T  + 5T T T  + 2T T T  + T T T  + 2T T T  + 2T T T  + T T T
                          0 1 2     0 1 2     0 1 2    0 1 2     0 1 2     0 1 2     0 1 2     0 1 2     0 1 2     0 1 2     0 1 2      0 1 2      0 1 2     0 1 2     0 1 2    0 1 2     0 1 2      0 1 2      0 1 2      0 1 2     0 1 2    0 1 2     0 1 2     0 1 2      0 1 2      0 1 2     0 1 2     0 1 2     0 1 2     0 1 2     0 1 2     0 1 2     0 1 2    0 1 2     0 1 2     0 1 2    0 1 2
               (4, 2) => 0
               (5, 0) => 0
                          9 5 4    9 4 5    8 7 3     8 6 4     8 5 5     8 4 6    8 3 7    7 8 3     7 7 4     7 6 5     7 5 6     7 4 7    7 3 8     6 8 4     6 7 5     6 6 6     6 5 7     6 4 8    5 9 4     5 8 5     5 7 6     5 6 7     5 5 8    5 4 9    4 9 5     4 8 6     4 7 7     4 6 8    4 5 9    3 8 7    3 7 8
               (5, 1) => T T T  + T T T  + T T T  + 3T T T  + 4T T T  + 3T T T  + T T T  + T T T  + 4T T T  + 7T T T  + 7T T T  + 4T T T  + T T T  + 3T T T  + 7T T T  + 9T T T  + 7T T T  + 3T T T  + T T T  + 4T T T  + 7T T T  + 7T T T  + 4T T T  + T T T  + T T T  + 3T T T  + 4T T T  + 3T T T  + T T T  + T T T  + T T T
                          0 1 2    0 1 2    0 1 2     0 1 2     0 1 2     0 1 2    0 1 2    0 1 2     0 1 2     0 1 2     0 1 2     0 1 2    0 1 2     0 1 2     0 1 2     0 1 2     0 1 2     0 1 2    0 1 2     0 1 2     0 1 2     0 1 2     0 1 2    0 1 2    0 1 2     0 1 2     0 1 2     0 1 2    0 1 2    0 1 2    0 1 2
               (5, 2) => 0
               (6, 0) => 0
                          9 7 5    9 6 6    9 5 7    8 8 5     8 7 6     8 6 7    8 5 8    7 9 5     7 8 6     7 7 7     7 6 8    7 5 9    6 9 6     6 8 7     6 7 8    6 6 9    5 9 7    5 8 8    5 7 9
               (6, 1) => T T T  + T T T  + T T T  + T T T  + 2T T T  + 2T T T  + T T T  + T T T  + 2T T T  + 3T T T  + 2T T T  + T T T  + T T T  + 2T T T  + 2T T T  + T T T  + T T T  + T T T  + T T T
                          0 1 2    0 1 2    0 1 2    0 1 2     0 1 2     0 1 2    0 1 2    0 1 2     0 1 2     0 1 2     0 1 2    0 1 2    0 1 2     0 1 2     0 1 2    0 1 2    0 1 2    0 1 2    0 1 2
               (6, 2) => 0
               (7, 0) => 0
               (7, 1) => 0
                          9 9 9
               (7, 2) => T T T
                          0 1 2

o2 : HashTable

Ways to use multiBetti :

For the programmer

The object multiBetti is a method function.