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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+17x_0x_2^4+29x_1x_2^4-39x_2^5
                                                
     ------------------------------------------------------------------------
                                                           
     x_1^3x_2^2+17x_0x_1x_2^3+12x_0x_2^4+29x_1x_2^4+20x_2^5
                                                           
     ------------------------------------------------------------------------
                                                        
     x_0x_1^2x_2^2+17x_0^2x_2^3+29x_0x_1x_2^3-39x_0x_2^4
                                                        
     ------------------------------------------------------------------------
                                                                      
     x_1^4x_2+14x_0^2x_2^3+24x_0x_1x_2^3-32x_0x_2^4-13x_1x_2^4+20x_2^5
                                                                      
     ------------------------------------------------------------------------
                                                                      
     x_0x_1^3x_2+17x_0^2x_1x_2^2+12x_0^2x_2^3+29x_0x_1x_2^3+20x_0x_2^4
                                                                      
     ------------------------------------------------------------------------
                                                            
     x_0^2x_1^2x_2+17x_0^3x_2^2+29x_0^2x_1x_2^2-39x_0^2x_2^3
                                                            
     ------------------------------------------------------------------------
                                                                             
     x_1^5+14x_0^2x_1x_2^2-4x_0^2x_2^3-21x_0x_1x_2^3+46x_0x_2^4-7x_1x_2^4-2x_
                                                                             
     ------------------------------------------------------------------------
                                                                             
     2^5 x_0x_1^4+14x_0^3x_2^2+24x_0^2x_1x_2^2-32x_0^2x_2^3-13x_0x_1x_2^3+20x
                                                                             
     ------------------------------------------------------------------------
                                                                             
     _0x_2^4 x_0^2x_1^3+17x_0^3x_1x_2+12x_0^3x_2^2+29x_0^2x_1x_2^2+20x_0^2x_2
                                                                             
     ------------------------------------------------------------------------
                                                        
     ^3 x_0^3x_1^2+17x_0^4x_2+29x_0^3x_1x_2-39x_0^3x_2^2
                                                        
     ------------------------------------------------------------------------
                                                                             
     x_0^4x_1-35x_0^4x_2+30x_0^3x_1x_2-40x_0^3x_2^2-10x_0^2x_1x_2^2+7x_0^2x_2
                                                                             
     ------------------------------------------------------------------------
                                                  
     ^3-9x_0x_1x_2^3+14x_0x_2^4+48x_1x_2^4+34x_2^5
                                                  
     ------------------------------------------------------------------------
                                                                             
     x_0^5+5x_0^4x_2+48x_0^3x_2^2+38x_0^2x_1x_2^2+47x_0^2x_2^3-12x_0x_1x_2^3+
                                                                             
     ------------------------------------------------------------------------
                                       3 2     2 3        4      5      4    
     10x_0x_2^4+18x_1x_2^4-48x_2^5 |, x x  - 2x x  - 38x x  - 43x  + 17x x  -
                                       0 1     0 1      0 1      1      0 2  
     ------------------------------------------------------------------------
       3          2 2          3        4       3 2      2   2       2 2  
     5x x x  - 42x x x  - 36x x x  + 18x x  + 4x x  + 32x x x  + 7x x x  -
       0 1 2      0 1 2      0 1 2      1 2     0 2      0 1 2     0 1 2  
     ------------------------------------------------------------------------
        3 2     2 3          3      2 3        4        4      5
     33x x  - 4x x  + 23x x x  - 48x x  + 36x x  + 41x x  + 42x )
        1 2     0 2      0 1 2      1 2      0 2      1 2      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 :