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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_2^3-37x_0x_2^4-44x_1x_2^4+7x_2^5
                                               
     ------------------------------------------------------------------------
                                                          
     x_1^3x_2^2-37x_0x_1x_2^3-12x_0x_2^4-10x_1x_2^4+5x_2^5
                                                          
     ------------------------------------------------------------------------
                                                       
     x_0x_1^2x_2^2-37x_0^2x_2^3-44x_0x_1x_2^3+7x_0x_2^4
                                                       
     ------------------------------------------------------------------------
                                                                      
     x_1^4x_2+45x_0^2x_2^3-24x_0x_1x_2^3-10x_0x_2^4-31x_1x_2^4-31x_2^5
                                                                      
     ------------------------------------------------------------------------
                                                                     
     x_0x_1^3x_2-37x_0^2x_1x_2^2-12x_0^2x_2^3-10x_0x_1x_2^3+5x_0x_2^4
                                                                     
     ------------------------------------------------------------------------
                                                           
     x_0^2x_1^2x_2-37x_0^3x_2^2-44x_0^2x_1x_2^2+7x_0^2x_2^3
                                                           
     ------------------------------------------------------------------------
                                                                            
     x_1^5+45x_0^2x_1x_2^2+21x_0^2x_2^3+45x_0x_1x_2^3+31x_0x_2^4+19x_1x_2^4+
                                                                            
     ------------------------------------------------------------------------
                                                                             
     15x_2^5 x_0x_1^4+45x_0^3x_2^2-24x_0^2x_1x_2^2-10x_0^2x_2^3-31x_0x_1x_2^3
                                                                             
     ------------------------------------------------------------------------
                                                                            
     -31x_0x_2^4 x_0^2x_1^3-37x_0^3x_1x_2-12x_0^3x_2^2-10x_0^2x_1x_2^2+5x_0^
                                                                            
     ------------------------------------------------------------------------
                                                           
     2x_2^3 x_0^3x_1^2-37x_0^4x_2-44x_0^3x_1x_2+7x_0^3x_2^2
                                                           
     ------------------------------------------------------------------------
                                                                             
     x_0^4x_1-50x_0^4x_2-46x_0^3x_1x_2+6x_0^3x_2^2+32x_0^2x_1x_2^2+14x_0^2x_2
                                                                             
     ------------------------------------------------------------------------
                                                  
     ^3+15x_0x_1x_2^3-6x_0x_2^4+14x_1x_2^4+39x_2^5
                                                  
     ------------------------------------------------------------------------
                                                                             
     x_0^5-27x_0^4x_2+39x_0^3x_1x_2-2x_0^3x_2^2-41x_0^2x_1x_2^2-27x_0^2x_2^3+
                                                                             
     ------------------------------------------------------------------------
                                                  3 2      2 3        4     5
     x_0x_1x_2^3+2x_0x_2^4+50x_1x_2^4-40x_2^5 |, x x  + 20x x  + 11x x  - 2x 
                                                  0 1      0 1      0 1     1
     ------------------------------------------------------------------------
          4        3          2 2          3        4        3 2     2   2  
     - 37x x  + 24x x x  + 25x x x  + 10x x x  + 15x x  + 44x x  - 4x x x  +
          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      5
     35x x x  - 18x x  - 11x x  - 48x x x  + 46x x  - 27x x  - 24x x  + 40x )
        0 1 2      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 :