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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_2^3-4x_0x_2^4-22x_1x_2^4+34x_2^5
                                               
     ------------------------------------------------------------------------
                                                          
     x_1^3x_2^2-4x_0x_1x_2^3+13x_0x_2^4-46x_1x_2^4+41x_2^5
                                                          
     ------------------------------------------------------------------------
                                                       
     x_0x_1^2x_2^2-4x_0^2x_2^3-22x_0x_1x_2^3+34x_0x_2^4
                                                       
     ------------------------------------------------------------------------
                                                                      
     x_1^4x_2-16x_0^2x_2^3+26x_0x_1x_2^3-48x_0x_2^4+39x_1x_2^4+49x_2^5
                                                                      
     ------------------------------------------------------------------------
                                                                     
     x_0x_1^3x_2-4x_0^2x_1x_2^2+13x_0^2x_2^3-46x_0x_1x_2^3+41x_0x_2^4
                                                                     
     ------------------------------------------------------------------------
                                                           
     x_0^2x_1^2x_2-4x_0^3x_2^2-22x_0^2x_1x_2^2+34x_0^2x_2^3
                                                           
     ------------------------------------------------------------------------
                                                                             
     x_1^5-16x_0^2x_1x_2^2+3x_0^2x_2^3+19x_0x_1x_2^3-21x_0x_2^4-2x_1x_2^4-13x
                                                                             
     ------------------------------------------------------------------------
                                                                           
     _2^5 x_0x_1^4-16x_0^3x_2^2+26x_0^2x_1x_2^2-48x_0^2x_2^3+39x_0x_1x_2^3+
                                                                           
     ------------------------------------------------------------------------
                                                                             
     49x_0x_2^4 x_0^2x_1^3-4x_0^3x_1x_2+13x_0^3x_2^2-46x_0^2x_1x_2^2+41x_0^2x
                                                                             
     ------------------------------------------------------------------------
                                                         
     _2^3 x_0^3x_1^2-4x_0^4x_2-22x_0^3x_1x_2+34x_0^3x_2^2
                                                         
     ------------------------------------------------------------------------
                                                                             
     x_0^4x_1+48x_0^4x_2-22x_0^3x_1x_2+9x_0^3x_2^2+17x_0^2x_2^3+27x_0x_1x_2^3
                                                                             
     ------------------------------------------------------------------------
                                 
     +3x_0x_2^4+21x_1x_2^4+9x_2^5
                                 
     ------------------------------------------------------------------------
                                                                             
     x_0^5-12x_0^4x_2+39x_0^3x_1x_2-21x_0^3x_2^2-21x_0^2x_1x_2^2-18x_0^2x_2^3
                                                                             
     ------------------------------------------------------------------------
                                                     3 2     2 3       4  
     +5x_0x_1x_2^3+42x_0x_2^4+28x_1x_2^4-33x_2^5 |, x x  - 8x x  - 2x x  -
                                                     0 1     0 1     0 1  
     ------------------------------------------------------------------------
        5     4        3          2 2          3       4       3 2      2   2
     13x  - 4x x  + 10x x x  + 14x x x  + 13x x x  - 5x x  + 7x x  - 38x x x 
        1     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
     + 19x x x  - 36x x  - 26x x  - 14x x x  + 28x x  + 4x x  - 38x x  + 6x )
          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 :