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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+19x_0x_2^4+50x_1x_2^4+40x_2^5
                                                
     ------------------------------------------------------------------------
                                                           
     x_1^3x_2^2+19x_0x_1x_2^3-41x_0x_2^4-36x_1x_2^4+20x_2^5
                                                           
     ------------------------------------------------------------------------
                                                        
     x_0x_1^2x_2^2+19x_0^2x_2^3+50x_0x_1x_2^3+40x_0x_2^4
                                                        
     ------------------------------------------------------------------------
                                                                     
     x_1^4x_2+43x_0^2x_2^3+19x_0x_1x_2^3+25x_0x_2^4+2x_1x_2^4+26x_2^5
                                                                     
     ------------------------------------------------------------------------
                                                                      
     x_0x_1^3x_2+19x_0^2x_1x_2^2-41x_0^2x_2^3-36x_0x_1x_2^3+20x_0x_2^4
                                                                      
     ------------------------------------------------------------------------
                                                            
     x_0^2x_1^2x_2+19x_0^3x_2^2+50x_0^2x_1x_2^2+40x_0^2x_2^3
                                                            
     ------------------------------------------------------------------------
                                                                            
     x_1^5+43x_0^2x_1x_2^2+43x_0^2x_2^3-16x_0x_1x_2^3+10x_0x_2^4+27x_1x_2^4+
                                                                            
     ------------------------------------------------------------------------
                                                                             
     21x_2^5 x_0x_1^4+43x_0^3x_2^2+19x_0^2x_1x_2^2+25x_0^2x_2^3+2x_0x_1x_2^3+
                                                                             
     ------------------------------------------------------------------------
                                                                            
     26x_0x_2^4 x_0^2x_1^3+19x_0^3x_1x_2-41x_0^3x_2^2-36x_0^2x_1x_2^2+20x_0^
                                                                            
     ------------------------------------------------------------------------
                                                            
     2x_2^3 x_0^3x_1^2+19x_0^4x_2+50x_0^3x_1x_2+40x_0^3x_2^2
                                                            
     ------------------------------------------------------------------------
                                                                             
     x_0^4x_1-9x_0^4x_2+6x_0^3x_1x_2+5x_0^3x_2^2-14x_0^2x_1x_2^2-20x_0^2x_2^3
                                                                             
     ------------------------------------------------------------------------
                                                 
     -39x_0x_1x_2^3-21x_0x_2^4+32x_1x_2^4-41x_2^5
                                                 
     ------------------------------------------------------------------------
                                                                             
     x_0^5-49x_0^4x_2-35x_0^3x_1x_2+15x_0^3x_2^2+13x_0^2x_1x_2^2-17x_0^2x_2^3
                                                                             
     ------------------------------------------------------------------------
                                                    3 2     2 3        4  
     -7x_0x_1x_2^3+47x_0x_2^4-13x_1x_2^4-2x_2^5 |, x x  - 9x x  - 43x x  +
                                                    0 1     0 1      0 1  
     ------------------------------------------------------------------------
        5      4        3         2 2         3        4        3 2  
     44x  + 19x x  - 20x x x  - 3x x x  - 9x x x  + 19x x  + 18x 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  
     33x x x  + 17x x x  + 36x x  + 6x x  + 15x x x  + 16x x  + 34x x  +
        0 1 2      0 1 2      1 2     0 2      0 1 2      1 2      0 2  
     ------------------------------------------------------------------------
          4      5
     23x x  - 50x )
        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 :