completeLinearSystemOnNodalPlaneCurve -- Compute the complete linear system of a divisor on a nodal plane curve

Synopsis

Usage:
(L,h)=completeLinearSystemOnNodalPlaneCurve(I,D)
Inputs:
- I, an ideal, of a nodal plane curve C,
- D, a list, {D₀,D₁} of ideals representing effective divisors on C
Outputs:
- L, a matrix, of homogeneous forms with 1 row and with number of columns equal to h⁰(D₀-D₁)
- h, a ring element, such that L_(0,i)/h represents a basis of H⁰ O(D₀-D₁)

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_2^3-38x_0x_2^4+14x_1x_2^4-x_2^5
                                              
     ------------------------------------------------------------------------
                                                          
     x_1^3x_2^2-38x_0x_1x_2^3+27x_0x_2^4+5x_1x_2^4+14x_2^5
                                                          
     ------------------------------------------------------------------------
                                                      
     x_0x_1^2x_2^2-38x_0^2x_2^3+14x_0x_1x_2^3-x_0x_2^4
                                                      
     ------------------------------------------------------------------------
                                                                     
     x_1^4x_2-30x_0^2x_2^3-47x_0x_1x_2^3-50x_0x_2^4+45x_1x_2^4+5x_2^5
                                                                     
     ------------------------------------------------------------------------
                                                                     
     x_0x_1^3x_2-38x_0^2x_1x_2^2+27x_0^2x_2^3+5x_0x_1x_2^3+14x_0x_2^4
                                                                     
     ------------------------------------------------------------------------
                                                          
     x_0^2x_1^2x_2-38x_0^3x_2^2+14x_0^2x_1x_2^2-x_0^2x_2^3
                                                          
     ------------------------------------------------------------------------
                                                                           
     x_1^5-30x_0^2x_1x_2^2+32x_0^2x_2^3+2x_0x_1x_2^3+47x_0x_2^4-19x_1x_2^4+
                                                                           
     ------------------------------------------------------------------------
                                                                             
     45x_2^5 x_0x_1^4-30x_0^3x_2^2-47x_0^2x_1x_2^2-50x_0^2x_2^3+45x_0x_1x_2^3
                                                                             
     ------------------------------------------------------------------------
                                                                             
     +5x_0x_2^4 x_0^2x_1^3-38x_0^3x_1x_2+27x_0^3x_2^2+5x_0^2x_1x_2^2+14x_0^2x
                                                                             
     ------------------------------------------------------------------------
                                                        
     _2^3 x_0^3x_1^2-38x_0^4x_2+14x_0^3x_1x_2-x_0^3x_2^2
                                                        
     ------------------------------------------------------------------------
                                                                             
     x_0^4x_1-35x_0^4x_2+25x_0^3x_1x_2+31x_0^3x_2^2+22x_0^2x_1x_2^2-22x_0^2x_
                                                                             
     ------------------------------------------------------------------------
                                                    
     2^3+33x_0x_1x_2^3-14x_0x_2^4-39x_1x_2^4-21x_2^5
                                                    
     ------------------------------------------------------------------------
                                                                             
     x_0^5+49x_0^4x_2-24x_0^3x_1x_2-49x_0^3x_2^2+25x_0^2x_1x_2^2-32x_0^2x_2^3
                                                                             
     ------------------------------------------------------------------------
                                                      3 2      2 3        4  
     +38x_0x_1x_2^3-11x_0x_2^4+22x_1x_2^4+41x_2^5 |, x x  - 15x x  - 18x x  -
                                                      0 1      0 1      0 1  
     ------------------------------------------------------------------------
        5      4        3          2 2          3        4        3 2  
     38x  - 38x x  - 22x x x  + 49x x x  + 12x x x  - 10x x  - 11x 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  
     20x x x  + 42x x x  + 49x x  - 32x x  - 14x x x  - 12x x  + 24x x  +
        0 1 2      0 1 2      1 2      0 2      0 1 2      1 2      0 2  
     ------------------------------------------------------------------------
          4      5
     46x x  + 49x )
        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

Ways to use `completeLinearSystemOnNodalPlaneCurve` :

completeLinearSystemOnNodalPlaneCurve(Ideal,List)

completeLinearSystemOnNodalPlaneCurve -- Compute the complete linear system of a divisor on a nodal plane curve

Synopsis

Description

See also

Ways to use completeLinearSystemOnNodalPlaneCurve :

Ways to use `completeLinearSystemOnNodalPlaneCurve` :