totalBettiTally -- a Betti tally containing the graded Betti numbers of a Veronese embedding

Synopsis

Usage:

totalBettiTally(d,n,b)
Inputs:
- d, an integer,
- n, an integer,
- b, an integer,
Outputs:
- a Betti tally,

Description

This function returns a Betti tally for the total graded Betti numbers of $\mathcal{O}(b)$ under the embedding by $\mathcal{O}(d)$. Some tables are incomplete and we mark unknown entries with infinity.

Note that totalBettiTally differs from totalBetti only in that the output is a Betti tally instead of a hash table.

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

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

We can also produce the Betti tables of the pushforwards of line bundles. For instance, the following example computes the Betti table of the pushforward of $\mathcal{O}(1)$ under the 3-uple embedding.

i2 : totalBettiTally(3,2,1)

            0  1  2   3   4   5  6 7
o2 = total: 3 15 42 105 147 105 39 6
         0: 3 15 21   .   .   .  . .
         1: .  . 21 105 147 105 39 6
         2: .  .  .   .   .   .  . .

o2 : BettiTally

Ways to use totalBettiTally :

totalBettiTally(ZZ,ZZ,ZZ)

For the programmer

The object totalBettiTally is a method function.