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

## Synopsis

• Usage:
totalBettiTally(d,n,b)
• Inputs:
• Outputs:
• ,

## 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 .