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RandomPlaneCurves (missing documentation) :: completeLinearSystemOnNodalPlaneCurve

completeLinearSystemOnNodalPlaneCurve -- Compute the complete linear system of a divisor on a nodal plane curve

Synopsis

Description

Compute the complete linear series of D0-D1 on the normalization of C via adjoint curves and double linkage.
i1 : R=ZZ/101[x_0..x_2];
i2 : J=(random nodalPlaneCurve)(6,3,R);

o2 : Ideal of R
i3 : D={J+ideal random(R^1,R^{1:-3}),J+ideal 1_R};
i4 : l=completeLinearSystemOnNodalPlaneCurve(J,D)

                                                
o4 = (| x_1^2x_2^3-39x_0x_2^4+24x_1x_2^4-23x_2^5
                                                
     ------------------------------------------------------------------------
                                                          
     x_1^3x_2^2-39x_0x_1x_2^3+27x_0x_2^4+7x_1x_2^4+47x_2^5
                                                          
     ------------------------------------------------------------------------
                                                        
     x_0x_1^2x_2^2-39x_0^2x_2^3+24x_0x_1x_2^3-23x_0x_2^4
                                                        
     ------------------------------------------------------------------------
                                                                     
     x_1^4x_2-6x_0^2x_2^3-47x_0x_1x_2^3-18x_0x_2^4-20x_1x_2^4-41x_2^5
                                                                     
     ------------------------------------------------------------------------
                                                                     
     x_0x_1^3x_2-39x_0^2x_1x_2^2+27x_0^2x_2^3+7x_0x_1x_2^3+47x_0x_2^4
                                                                     
     ------------------------------------------------------------------------
                                                            
     x_0^2x_1^2x_2-39x_0^3x_2^2+24x_0^2x_1x_2^2-23x_0^2x_2^3
                                                            
     ------------------------------------------------------------------------
                                                                             
     x_1^5-6x_0^2x_1x_2^2-15x_0^2x_2^3-x_0x_1x_2^3-43x_0x_2^4+35x_1x_2^4+45x_
                                                                             
     ------------------------------------------------------------------------
                                                                             
     2^5 x_0x_1^4-6x_0^3x_2^2-47x_0^2x_1x_2^2-18x_0^2x_2^3-20x_0x_1x_2^3-41x_
                                                                             
     ------------------------------------------------------------------------
                                                                             
     0x_2^4 x_0^2x_1^3-39x_0^3x_1x_2+27x_0^3x_2^2+7x_0^2x_1x_2^2+47x_0^2x_2^3
                                                                             
     ------------------------------------------------------------------------
                                                     
     x_0^3x_1^2-39x_0^4x_2+24x_0^3x_1x_2-23x_0^3x_2^2
                                                     
     ------------------------------------------------------------------------
                                                                             
     x_0^4x_1-14x_0^4x_2+48x_0^3x_1x_2-46x_0^3x_2^2+36x_0^2x_1x_2^2+20x_0^2x_
                                                                             
     ------------------------------------------------------------------------
                                                    
     2^3+47x_0x_1x_2^3-37x_0x_2^4-23x_1x_2^4+34x_2^5
                                                    
     ------------------------------------------------------------------------
                                                                             
     x_0^5+22x_0^4x_2+29x_0^3x_1x_2-9x_0^3x_2^2-28x_0^2x_1x_2^2-34x_0^2x_2^3+
                                                                             
     ------------------------------------------------------------------------
                                                     3 2      2 3        4  
     47x_0x_1x_2^3+36x_0x_2^4+47x_1x_2^4-40x_2^5 |, x x  + 15x x  - 45x x  +
                                                     0 1      0 1      0 1  
     ------------------------------------------------------------------------
        5      4        3         2 2          3       4        3 2     2   2
     46x  - 39x x  + 45x x x  + 6x x x  + 12x x x  - 4x x  + 14x x  + 4x x x 
        1      0 2      0 1 2     0 1 2      0 1 2     1 2      0 2     0 1 2
     ------------------------------------------------------------------------
            2 2      3 2      2 3          3      2 3        4       4  
     + 41x x x  + 37x x  + 42x x  - 40x x x  - 43x x  + 14x x  + 8x x  +
          0 1 2      1 2      0 2      0 1 2      1 2      0 2     1 2  
     ------------------------------------------------------------------------
        5
     13x )
        2

o4 : Sequence
i5 : C=imageUnderRationalMap(J,l_0);

               ZZ
o5 : Ideal of ---[x , x , x , x , x , x , x , x , x , x , x  , x  ]
              101  0   1   2   3   4   5   6   7   8   9   10   11
i6 : (dim C, degree C, genus C)

o6 = (2, 18, 7)

o6 : Sequence

See also

Ways to use completeLinearSystemOnNodalPlaneCurve :