# bettiStrataExamples -- a hash table consisting of examples for each of the 19 Betti strata

## Synopsis

• Usage:
bettiStrataExamples S
• Inputs:
• S, a ring, a polynomial ring with 4 variables
• Outputs:
• , Whose keys are strings representing each Betti table strata, and whose values are matrices of scalars over the ring $S$

## Description

The result is a hash table whose keys are the names of the 19 Betti table strata for quaternary quartics. For each, the value is a matrix whose columns represent points. The quartic corresponding to this matrix is the sum of the 4th powers of the corresponding linear forms.

 i1 : S = ZZ/101[a..d] o1 = S o1 : PolynomialRing i2 : bettiStrataExamples S o2 = HashTable{[000] => {| 1 0 0 0 1 22 2 -37 -18 32 |, 10 general points} } | 0 1 0 0 1 -47 29 -13 39 -9 | | 0 0 1 0 1 -23 -47 -10 27 -32 | | 0 0 0 1 1 -7 15 30 -22 -20 | [100] => {| 1 0 0 0 1 39 48 -38 46 |, 9 general points} | 0 1 0 0 1 43 36 33 -28 | | 0 0 1 0 1 -17 35 40 1 | | 0 0 0 1 1 -11 11 11 -3 | [200] => {| 1 0 0 0 1 16 -48 -16 |, 8 general points} | 0 1 0 0 1 22 -47 7 | | 0 0 1 0 1 45 47 15 | | 0 0 0 1 1 -34 19 -23 | [210] => {| 1 0 0 0 1 1 0 1 |, 8 points with 6 in a plane, or five in a plane and three in a line} | 0 1 0 0 1 0 1 1 | | 0 0 1 0 0 1 1 1 | | 0 0 0 1 0 0 0 1 | [300a] => {| 1 1 1 1 1 1 1 1 |, 8 points which forms a CI} | 2 2 2 2 -2 -2 -2 -2 | | 3 3 -3 -3 3 3 -3 -3 | | 1 -1 1 -1 1 -1 1 -1 | [300b] => {| 1 0 0 0 1 19 -8 |, 7 general points} | 0 1 0 0 1 19 -22 | | 0 0 1 0 1 -10 -29 | | 0 0 0 1 1 -29 -24 | [300c] => {| 1 0 1 -38 34 -18 -28 |, 7 points, 3 on a line} | 0 1 1 -16 19 -13 -47 | | 0 0 0 39 -47 -43 38 | | 0 0 0 21 -39 -15 2 | [310] => {| 1 0 0 0 1 1 1 |, 7 points with 5 on a plane} | 0 1 0 0 1 1 0 | | 0 0 1 0 1 1 0 | | 0 0 0 1 0 1 1 | [320] => {| 1 0 0 0 1 1 1 |, 7 points on a twisted cubic curve} | 0 1 0 0 1 0 0 | | 0 0 1 0 0 1 0 | | 0 0 0 1 0 0 1 | [331] => {| 1 0 0 0 24 -15 33 |, 7 points with 6 on a plane} | 0 1 0 0 -30 39 -49 | | 0 0 1 0 -48 0 -33 | | 0 0 0 1 0 0 0 | [420] => {| 1 0 0 0 1 24 |, 6 general points} | 0 1 0 0 1 -36 | | 0 0 1 0 1 -30 | | 0 0 0 1 1 -29 | [430] => {| 1 0 0 0 1 1 |, 6 points, 3 on a line} | 0 1 0 0 1 0 | | 0 0 1 0 0 1 | | 0 0 0 1 0 1 | [441a] => {| 1 0 0 0 1 1 |, 6 points, 5 on a plane} | 0 1 0 0 1 0 | | 0 0 1 0 0 1 | | 0 0 0 1 0 0 | [441b] => {| 1 0 0 0 1 0 |, 6 points, 3 each on 2 skew lines} | 0 1 0 0 1 0 | | 0 0 1 0 0 1 | | 0 0 0 1 0 1 | [550] => {| 1 0 0 0 1 |, 5 general points} | 0 1 0 0 1 | | 0 0 1 0 1 | | 0 0 0 1 1 | [551] => {| 1 0 0 0 1 |, 5 points, 4 on a plane} | 0 1 0 0 1 | | 0 0 1 0 1 | | 0 0 0 1 0 | [562] => {| 1 0 0 0 1 |, 5 points, 3 on a line} | 0 1 0 0 1 | | 0 0 1 0 0 | | 0 0 0 1 0 | [683] => {| 1 0 0 0 |, 4 general points} | 0 1 0 0 | | 0 0 1 0 | | 0 0 0 1 | o2 : HashTable

## Ways to use bettiStrataExamples :

• "bettiStrataExamples(Ring)"

## For the programmer

The object bettiStrataExamples is .