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

## 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 .