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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+29x_0x_2^4+15x_1x_2^4-24x_2^5
                                                
     ------------------------------------------------------------------------
                                                           
     x_1^3x_2^2+29x_0x_1x_2^3-31x_0x_2^4-47x_1x_2^4-44x_2^5
                                                           
     ------------------------------------------------------------------------
                                                        
     x_0x_1^2x_2^2+29x_0^2x_2^3+15x_0x_1x_2^3-24x_0x_2^4
                                                        
     ------------------------------------------------------------------------
                                                                      
     x_1^4x_2-33x_0^2x_2^3+39x_0x_1x_2^3+39x_0x_2^4-46x_1x_2^4-17x_2^5
                                                                      
     ------------------------------------------------------------------------
                                                                      
     x_0x_1^3x_2+29x_0^2x_1x_2^2-31x_0^2x_2^3-47x_0x_1x_2^3-44x_0x_2^4
                                                                      
     ------------------------------------------------------------------------
                                                            
     x_0^2x_1^2x_2+29x_0^3x_2^2+15x_0^2x_1x_2^2-24x_0^2x_2^3
                                                            
     ------------------------------------------------------------------------
                                                                            
     x_1^5-33x_0^2x_1x_2^2-20x_0^2x_2^3-41x_0x_1x_2^3+48x_0x_2^4-34x_1x_2^4+
                                                                            
     ------------------------------------------------------------------------
                                                                             
     7x_2^5 x_0x_1^4-33x_0^3x_2^2+39x_0^2x_1x_2^2+39x_0^2x_2^3-46x_0x_1x_2^3-
                                                                             
     ------------------------------------------------------------------------
                                                                            
     17x_0x_2^4 x_0^2x_1^3+29x_0^3x_1x_2-31x_0^3x_2^2-47x_0^2x_1x_2^2-44x_0^
                                                                            
     ------------------------------------------------------------------------
                                                            
     2x_2^3 x_0^3x_1^2+29x_0^4x_2+15x_0^3x_1x_2-24x_0^3x_2^2
                                                            
     ------------------------------------------------------------------------
                                                                             
     x_0^4x_1-6x_0^4x_2+47x_0^3x_1x_2-40x_0^3x_2^2+23x_0^2x_1x_2^2+11x_0^2x_2
                                                                             
     ------------------------------------------------------------------------
                                                 
     ^3-4x_0x_1x_2^3-16x_0x_2^4-30x_1x_2^4+9x_2^5
                                                 
     ------------------------------------------------------------------------
                                                                             
     x_0^5-47x_0^4x_2+23x_0^3x_1x_2+27x_0^3x_2^2-33x_0^2x_1x_2^2-31x_0^2x_2^3
                                                                             
     ------------------------------------------------------------------------
                                                      3 2      2 3        4  
     -23x_0x_1x_2^3-36x_0x_2^4-15x_1x_2^4-20x_2^5 |, x x  + 36x x  + 42x x  +
                                                      0 1      0 1      0 1  
     ------------------------------------------------------------------------
      5      4        3          2 2          3        4        3 2  
     x  + 29x x  + 49x x x  + 17x x x  + 11x x x  + 36x x  - 13x x  -
      1      0 2      0 1 2      0 1 2      0 1 2      1 2      0 2  
     ------------------------------------------------------------------------
        2   2        2 2     3 2     2 3         3      2 3        4        4
     18x x x  - 32x x x  + 2x x  - 3x x  + 7x x x  - 30x x  - 11x x  - 12x x 
        0 1 2      0 1 2     1 2     0 2     0 1 2      1 2      0 2      1 2
     ------------------------------------------------------------------------
          5
     + 27x )
          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 :