totalBetti -- a hash table containing the graded Betti numbers of a Veronese embedding

Synopsis

Usage:

totalBetti(d,n,b)
Inputs:
- d, an integer,
- n, an integer,
- b, an integer,
Outputs:
- a hash table,

Description

This is a hash table for the total numbers of $\mathcal{O}(b)$ under the embedding by $\mathcal{O}(d)$. The keys of the hash table $H$ are pairs $(p,q)$ where $H#(p,q)$ gives the rank of $K_{p,q}(\mathbb{P}^n, d;b)$. This equals the Betti number $\beta_{p,p+q}(d,\mathbb{P}^n,b)$.Some tables are incomplete and we mark unknown entries with infinity.

Note that totalBetti differs from totalBettiTally only in that the output is a hash table instead of a Betti tally. One can convert the output of totalBetti into a Betti tally via the makeBettiTally function.

In example below we generate a hash table showing the total graded Betti numbers of $\mathbb{P}^{2}$ embedded by $\mathcal{O}(3)$.

i1 : B = totalBetti(3,2,0)

o1 = HashTable{(0, 0) => 1  }
               (0, 1) => 0
               (0, 2) => 0
               (1, 0) => 0
               (1, 1) => 27
               (1, 2) => 0
               (2, 0) => 0
               (2, 1) => 105
               (2, 2) => 0
               (3, 0) => 0
               (3, 1) => 189
               (3, 2) => 0
               (4, 0) => 0
               (4, 1) => 189
               (4, 2) => 0
               (5, 0) => 0
               (5, 1) => 105
               (5, 2) => 0
               (6, 0) => 0
               (6, 1) => 27
               (6, 2) => 0
               (7, 0) => 0
               (7, 1) => 0
               (7, 2) => 1

o1 : HashTable

If we wish to view these graded Betti numbers in the usual fashion, we can use makeBettiTally to convert the hash table above to a Betti tally.

i2 : makeBettiTally B

            0  1   2   3   4   5  6 7
o2 = total: 1 27 105 189 189 105 27 1
         0: 1  .   .   .   .   .  . .
         1: . 27 105 189 189 105 27 .
         2: .  .   .   .   .   .  . 1

o2 : BettiTally

Ways to use totalBetti :

totalBetti(ZZ,ZZ,ZZ)

For the programmer

The object totalBetti is a method function.