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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_2^3-30x_0x_2^4-45x_1x_2^4+12x_2^5
                                                
     ------------------------------------------------------------------------
                                                          
     x_1^3x_2^2-30x_0x_1x_2^3-37x_0x_2^4+7x_1x_2^4+35x_2^5
                                                          
     ------------------------------------------------------------------------
                                                        
     x_0x_1^2x_2^2-30x_0^2x_2^3-45x_0x_1x_2^3+12x_0x_2^4
                                                        
     ------------------------------------------------------------------------
                                                                     
     x_1^4x_2+9x_0^2x_2^3+27x_0x_1x_2^3-36x_0x_2^4+47x_1x_2^4+17x_2^5
                                                                     
     ------------------------------------------------------------------------
                                                                     
     x_0x_1^3x_2-30x_0^2x_1x_2^2-37x_0^2x_2^3+7x_0x_1x_2^3+35x_0x_2^4
                                                                     
     ------------------------------------------------------------------------
                                                            
     x_0^2x_1^2x_2-30x_0^3x_2^2-45x_0^2x_1x_2^2+12x_0^2x_2^3
                                                            
     ------------------------------------------------------------------------
                                                                             
     x_1^5+9x_0^2x_1x_2^2+2x_0^2x_2^3-33x_0x_1x_2^3-25x_0x_2^4+11x_1x_2^4+42x
                                                                             
     ------------------------------------------------------------------------
                                                                             
     _2^5 x_0x_1^4+9x_0^3x_2^2+27x_0^2x_1x_2^2-36x_0^2x_2^3+47x_0x_1x_2^3+17x
                                                                             
     ------------------------------------------------------------------------
                                                                             
     _0x_2^4 x_0^2x_1^3-30x_0^3x_1x_2-37x_0^3x_2^2+7x_0^2x_1x_2^2+35x_0^2x_2^
                                                                             
     ------------------------------------------------------------------------
                                                       
     3 x_0^3x_1^2-30x_0^4x_2-45x_0^3x_1x_2+12x_0^3x_2^2
                                                       
     ------------------------------------------------------------------------
                                                                             
     x_0^4x_1-2x_0^4x_2+11x_0^3x_1x_2+30x_0^3x_2^2+13x_0^2x_1x_2^2-17x_0^2x_2
                                                                             
     ------------------------------------------------------------------------
                                                  
     ^3-14x_0x_1x_2^3+3x_0x_2^4-38x_1x_2^4+32x_2^5
                                                  
     ------------------------------------------------------------------------
                                                                             
     x_0^5-5x_0^3x_1x_2+6x_0^3x_2^2-30x_0^2x_1x_2^2+22x_0^2x_2^3+16x_0x_1x_2^
                                                                             
     ------------------------------------------------------------------------
                                        3 2      2 3        4     5      4  
     3+34x_0x_2^4+12x_1x_2^4+3x_2^5 |, x x  + 29x x  + 50x x  - 8x  - 30x x 
                                        0 1      0 1      0 1     1      0 2
     ------------------------------------------------------------------------
         3         2 2          3       4        3 2      2   2       2 2  
     - 6x x x  - 8x x x  - 29x x x  + 3x x  + 33x x  - 16x x x  - 6x 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
     48x x  - 21x x  - 40x x x  + 31x x  - 6x x  + 39x x  + 23x )
        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 :