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Key
"Finding the possible betti tables for points in P^3 with given geometry"
Headline
Material from Section 3 of [QQ]
Description
Text
The following code finds the ideal and betti table for a point configuration.
The point configuration is given by a matrix whose column vectors are the
coordinates of the points. The command pointideal does this for a single point,
and pointsideal does it for several points
Example
K = ZZ/101;
R = K[x_0..x_3];
Text
We check this for some special configurations in P^3, first for a set of six points consisting
of two sets of three collinear points, and second for seven points on a twisted cubic
Example
TwoSets3Points=transpose matrix{{1,0,0,0},{0,1,0,0},{1,1,0,0},{0,0,1,1},{0,0,1,0},{0,0,0,1}}**R
I = pointsIdeal TwoSets3Points
minimalBetti I
SevenPointsOnTC=transpose matrix{{1,1,1,1},{1,2,4,8},{1,3,9,27},{1,4,16,64},{1,5,25,125},{1,6,36,216},{1,7,49,343}}**R
J = pointsIdeal SevenPointsOnTC
minimalBetti J
Text
Finally we check configurations of 3 to 10 generic points in P^3, note 3 points will have a linear form
Example
netList(pack(2,apply({3,4,5,6,7,8,9,10},i->(minimalBetti pointsIdeal random(R^4,R^i)))))
SeeAlso
"[QQ]"
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