projectiveFatPoints -- produces the ideal and initial ideal from the coordinates of a finite set of fat points

Synopsis

Usage:

(inG,G) = projectiveFatPoints(M,mults,R)
Inputs:
- M, a matrix, in which each column consists of the projective coordinates of a point
- mults, a list, in which each element determines the multiplicity of the corresponding point
- R, a ring, homogeneous coordinate ring of the projective space containing the points
Optional inputs:
- VerifyPoints => ..., default value true, Option to projectiveFatPoints.
Outputs:
- inG, an ideal, initial ideal of the set of fat points
- G, a list, list of generators for Grobner basis for ideal of fat points

Description

This function uses a modified Buchberger-Moeller algorithm to compute a grobner basis for the ideal of a finite number of fat points in projective space.

i1 : R = QQ[x,y,z]

o1 = R

o1 : PolynomialRing

i2 : M = transpose matrix{{1,0,0},{0,1,1}}

o2 = | 1 0 |
     | 0 1 |
     | 0 1 |

              3        2
o2 : Matrix ZZ  <--- ZZ

i3 : mults = {3,2}

o3 = {3, 2}

o3 : List

i4 : (inG,G) = projectiveFatPoints(M,mults,R)

              2    3       2   2 3     2        2    3   3       2     3       2      3   2 3
o4 = (ideal (y z, y , x*y*z , x z ), {y z - 2y*z  + z , y  - 3y*z  + 2z , x*y*z  - x*z , x z })

o4 : Sequence

i5 : monomialIdeal G == inG

o5 = true

Caveat

For small sets of points and/or multiplicities, this method might be slower than projectiveFatPointsByIntersection.

Ways to use `projectiveFatPoints` :

"projectiveFatPoints(Matrix,List,Ring)"

For the programmer

The object projectiveFatPoints is a method function with options.