# Points -- A package for making and studying points in affine and projective spaces

## Description

The package has routines for points in affine and projective spaces. The affine code, some of which uses the Buchberger-Moeller algorithm to more quickly compute the ideals of points in affine space, was written by Stillman, Smith and Stromme. The projective code was written separately by Eisenbud and Popescu.

The purpose of the projective code was to find as many counterexamples as possible to the minimal resolution conjecture; it was of use in the research for the paper "Exterior algebra methods for the minimal resolution conjecture", by David Eisenbud, Sorin Popescu, Frank-Olaf Schreyer and Charles Walter (Duke Mathematical Journal. 112 (2002), no.2, 379-395.) The first few of these counterexamples are: (6,11), (7,12), (8,13), (10,16), where the first integer denotes the ambient dimension and the second the number of points. Examples are known in every projective space of dimension >=6 except for P^9.

In version 3.0, F. Galetto and J.W. Skelton added code to compute ideals of fat points and projective points using the Buchberger-Moeller algorithm.

## Version

This documentation describes version 3.0 of Points.

## Source code

The source code from which this documentation is derived is in the file Points.m2.

## Exports

• Functions and commands
• Methods
• "affineFatPoints(Matrix,List,Ring)" -- see affineFatPoints -- produces the ideal and initial ideal from the coordinates of a finite set of fat points
• "affineFatPointsByIntersection(Matrix,List,Ring)" -- see affineFatPointsByIntersection -- computes ideal of fat points by intersecting powers of maximal ideals
• "affineMakeRingMaps(Matrix,Ring)" -- see affineMakeRingMaps -- evaluation on points
• "affinePoints(Matrix,Ring)" -- see affinePoints -- produces the ideal and initial ideal from the coordinates of a finite set of points
• "affinePointsByIntersection(Matrix,Ring)" -- see affinePointsByIntersection -- computes ideal of point set by intersecting maximal ideals
• "affinePointsMat(Matrix,Ring)" -- see affinePointsMat -- produces the matrix of values of the standard monomials on a set of points
• "projectiveFatPoints(Matrix,List,Ring)" -- see projectiveFatPoints -- produces the ideal and initial ideal from the coordinates of a finite set of fat points
• "projectiveFatPointsByIntersection(Matrix,List,Ring)" -- see projectiveFatPointsByIntersection -- computes ideal of fat points by intersecting powers of point ideals
• "projectivePoints(Matrix,Ring)" -- see projectivePoints -- produces the ideal and initial ideal from the coordinates of a finite set of projective points
• "projectivePointsByIntersection(Matrix,Ring)" -- see projectivePointsByIntersection -- computes ideal of projective points by intersecting point ideals
• "randomPointsMat(Ring,ZZ)" -- see randomPointsMat -- matrix of homogeneous coordinates of random points in projective space
• Symbols

## For the programmer

The object Points is .