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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 : setRandomSeed("alpha");
i2 : R=ZZ/32003[x_0..x_2];
i3 : J=(random nodalPlaneCurve)(6,3,R);

o3 : Ideal of R
i4 : D={J+ideal random(R^1,R^{1:-3}),J+ideal 1_R};
i5 : l=completeLinearSystemOnNodalPlaneCurve(J,D)

                                                        
o5 = (| x_1^2x_2^3+10679x_0x_2^4-7126x_1x_2^4-10050x_2^5
                                                        
     ------------------------------------------------------------------------
                                                                    
     x_1^3x_2^2+10679x_0x_1x_2^3-4580x_0x_2^4-1165x_1x_2^4+6414x_2^5
                                                                    
     ------------------------------------------------------------------------
                                                                
     x_0x_1^2x_2^2+10679x_0^2x_2^3-7126x_0x_1x_2^3-10050x_0x_2^4
                                                                
     ------------------------------------------------------------------------
                                                                             
     x_1^4x_2-14352x_0^2x_2^3-9160x_0x_1x_2^3+9759x_0x_2^4-6599x_1x_2^4+4848x
                                                                             
     ------------------------------------------------------------------------
                                                                             
     _2^5 x_0x_1^3x_2+10679x_0^2x_1x_2^2-4580x_0^2x_2^3-1165x_0x_1x_2^3+6414x
                                                                             
     ------------------------------------------------------------------------
                                                                            
     _0x_2^4 x_0^2x_1^2x_2+10679x_0^3x_2^2-7126x_0^2x_1x_2^2-10050x_0^2x_2^3
                                                                            
     ------------------------------------------------------------------------
                                                                             
     x_1^5-14352x_0^2x_1x_2^2-13531x_0^2x_2^3-10284x_0x_1x_2^3+14746x_0x_2^4-
                                                                             
     ------------------------------------------------------------------------
                                                                             
     7219x_1x_2^4-9734x_2^5 x_0x_1^4-14352x_0^3x_2^2-9160x_0^2x_1x_2^2+9759x_
                                                                             
     ------------------------------------------------------------------------
                                          
     0^2x_2^3-6599x_0x_1x_2^3+4848x_0x_2^4
                                          
     ------------------------------------------------------------------------
                                                                             
     x_0^2x_1^3+10679x_0^3x_1x_2-4580x_0^3x_2^2-1165x_0^2x_1x_2^2+6414x_0^2x_
                                                                             
     ------------------------------------------------------------------------
                                                                 
     2^3 x_0^3x_1^2+10679x_0^4x_2-7126x_0^3x_1x_2-10050x_0^3x_2^2
                                                                 
     ------------------------------------------------------------------------
                                                                             
     x_0^4x_1-7195x_0^4x_2-4690x_0^3x_1x_2-6614x_0^3x_2^2-11252x_0^2x_1x_2^2-
                                                                             
     ------------------------------------------------------------------------
                                                                          
     13642x_0^2x_2^3+11845x_0x_1x_2^3-15169x_0x_2^4+8041x_1x_2^4+8826x_2^5
                                                                          
     ------------------------------------------------------------------------
                                                                          
     x_0^5-15424x_0^4x_2+9295x_0^3x_1x_2-8216x_0^3x_2^2+7900x_0^2x_1x_2^2+
                                                                          
     ------------------------------------------------------------------------
                                                                           
     12540x_0^2x_2^3+11623x_0x_1x_2^3+13642x_0x_2^4+15790x_1x_2^4+1405x_2^5
                                                                           
     ------------------------------------------------------------------------
         3 2        2 3          4         5         4          3      
     |, x x  - 6082x x  + 5398x x  + 11037x  + 10679x x  + 9286x x x  -
         0 1        0 1        0 1         1         0 2        0 1 2  
     ------------------------------------------------------------------------
          2 2           3           4          3 2        2   2          2 2
     4409x x x  - 269x x x  - 14027x x  + 2769x x  - 9148x x x  - 3871x x x 
          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  
     - 5459x x  - 15497x x  + 7665x x x  - 12784x x  - 2228x x  + 245x x  -
            1 2         0 2        0 1 2         1 2        0 2       1 2  
     ------------------------------------------------------------------------
           5
     12197x )
           2

o5 : Sequence
i6 : C=imageUnderRationalMap(J,l_0);

                ZZ
o6 : Ideal of -----[x , x , x , x , x , x , x , x , x , x , x  , x  ]
              32003  0   1   2   3   4   5   6   7   8   9   10   11
i7 : (dim C, degree C, genus C)

o7 = (2, 18, 7)

o7 : Sequence

See also

Ways to use completeLinearSystemOnNodalPlaneCurve :