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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+50x_0x_2^4-4x_1x_2^4+22x_2^5
                                               
     ------------------------------------------------------------------------
                                                         
     x_1^3x_2^2+50x_0x_1x_2^3-2x_0x_2^4+6x_1x_2^4-13x_2^5
                                                         
     ------------------------------------------------------------------------
                                                       
     x_0x_1^2x_2^2+50x_0^2x_2^3-4x_0x_1x_2^3+22x_0x_2^4
                                                       
     ------------------------------------------------------------------------
                                                                     
     x_1^4x_2+25x_0^2x_2^3-4x_0x_1x_2^3+14x_0x_2^4+11x_1x_2^4-31x_2^5
                                                                     
     ------------------------------------------------------------------------
                                                                    
     x_0x_1^3x_2+50x_0^2x_1x_2^2-2x_0^2x_2^3+6x_0x_1x_2^3-13x_0x_2^4
                                                                    
     ------------------------------------------------------------------------
                                                           
     x_0^2x_1^2x_2+50x_0^3x_2^2-4x_0^2x_1x_2^2+22x_0^2x_2^3
                                                           
     ------------------------------------------------------------------------
                                                                             
     x_1^5+25x_0^2x_1x_2^2-2x_0^2x_2^3-2x_0x_1x_2^3+43x_0x_2^4+13x_1x_2^4-40x
                                                                             
     ------------------------------------------------------------------------
                                                                             
     _2^5 x_0x_1^4+25x_0^3x_2^2-4x_0^2x_1x_2^2+14x_0^2x_2^3+11x_0x_1x_2^3-31x
                                                                             
     ------------------------------------------------------------------------
                                                                             
     _0x_2^4 x_0^2x_1^3+50x_0^3x_1x_2-2x_0^3x_2^2+6x_0^2x_1x_2^2-13x_0^2x_2^3
                                                                             
     ------------------------------------------------------------------------
                                                    
     x_0^3x_1^2+50x_0^4x_2-4x_0^3x_1x_2+22x_0^3x_2^2
                                                    
     ------------------------------------------------------------------------
                                                                             
     x_0^4x_1-41x_0^4x_2+26x_0^3x_1x_2-32x_0^3x_2^2-31x_0^2x_1x_2^2-14x_0^2x_
                                                                             
     ------------------------------------------------------------------------
                                         
     2^3+37x_0x_1x_2^3-18x_0x_2^4-39x_2^5
                                         
     ------------------------------------------------------------------------
                                                                             
     x_0^5-22x_0^4x_2+48x_0^3x_1x_2-8x_0^3x_2^2-45x_0^2x_1x_2^2-39x_0^2x_2^3-
                                                                             
     ------------------------------------------------------------------------
                                                     2 3        4      5  
     32x_0x_1x_2^3-31x_0x_2^4+23x_1x_2^4+43x_2^5 |, x x  + 50x x  - 33x  +
                                                     0 1      0 1      1  
     ------------------------------------------------------------------------
        3          2 2          3       4        3 2     2   2        2 2  
     50x x x  + 17x x x  + 44x x x  + 2x x  - 23x x  + 2x x x  - 37x x x  -
        0 1 2      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      5
     30x x  + 47x x  + 25x x x  - 46x x  - 2x x  + x x  + 30x )
        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 :