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minimalCurve -- generates a minimal curve in the biliaison class

Description

A finite length module M determines a unique biliaison class. Curves of minimal degrees in this class are called minimal curves. Given the ideal of a curve J, this function generates a random minimal curve in the biliaison class of J. Given a finite length module M, this function generates a random minimal curve in the biliaison class specified by M.

Synopsis

  • Usage:
    I = minimalCurve(M)
  • Inputs:
  • Outputs: 
i1 : R = ZZ/101[x,y,z,w];
 
i2 : M = coker vars R;
 
i3 : I = minimalCurve M

              2             2                     2                          
o3 = ideal (4x  + 3x*y + 23y  + 7x*z - 38y*z + 11z  - 41x*w - 43y*w - 19z*w +
     ------------------------------------------------------------------------
        2     2              2                      2                       
     44w , 47x  - 12x*y - 49y  + 35x*z - 25y*z + 18z  - 9x*w + 39y*w + 30z*w
     ------------------------------------------------------------------------
          2       2                              2                        
     + 46w , - 33x  - 47x*y - 21x*z - 48y*z + 18z  - 4x*w + 4y*w + 35z*w -
     ------------------------------------------------------------------------
       2   2             2                     2                    2
     2w , x  + 18x*y - 9y  + 12x*z + 45y*z + 5z  - 26x*w + z*w - 43w )

o3 : Ideal of R

Synopsis

  • Usage:
    I = minimalCurve(J)
  • Inputs:
    • J, an ideal, of a pure dimension one subscheme
  • Outputs:
    • I, an ideal, of a minimal curve in the biliaison class
i4 : R = ZZ/101[x,y,z,w];
 
i5 : J = monomialCurveIdeal(R,{1,3,4});

o5 : Ideal of R
 
i6 : I = minimalCurve J

               2              2              2                          
o6 = ideal (34x  - 25x*y - 11y  + 43y*z - 36z  + 18x*w - 45y*w + 40z*w -
     ------------------------------------------------------------------------
        2     2              2                      2                       
     42w , 44x  + 47x*y - 25y  - 40x*z + 21y*z + 45z  - 45x*w - 13y*w - 4z*w
     ------------------------------------------------------------------------
         2       2              2                      2                  
     + 2w , - 50x  - 14x*y - 30y  + 39x*z - 41y*z - 27z  + 29x*w - 50y*w +
     ------------------------------------------------------------------------
               2   2              2                     2
     34z*w - 7w , x  + 15x*y - 24y  + 3x*z + 29y*z + 31z  + 6x*w - 38y*w -
     ------------------------------------------------------------------------
     41z*w)

o6 : Ideal of R

See also

Ways to use minimalCurve:

  • minimalCurve(Curve)
  • minimalCurve(Ideal)
  • minimalCurve(Module)

For the programmer

The object minimalCurve is a method function.

The source of this document is in /__w/M2/M2/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SpaceCurves.m2:1374:0.