# 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:
• M,
• 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