# 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, \{D_0,D_1\}\ of ideals representing effective divisors on C
• Outputs:
• L, , of homogeneous forms with 1 row and with number of columns equal to $h^0(D_0-D_1)$
• h, , such that L_{(0,i)}/h represents a basis of $H^0 O(D_0-D_1)$

## Description

Compute the complete linear series of D_0-D_1 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_2^3-15905x_0x_2^4-3127x_1x_2^4-14505x_2^5 ------------------------------------------------------------------------ x_1^3x_2^2-15905x_0x_1x_2^3-2273x_0x_2^4+284x_1x_2^4-8884x_2^5 ------------------------------------------------------------------------ x_0x_1^2x_2^2-15905x_0^2x_2^3-3127x_0x_1x_2^3-14505x_0x_2^4 ------------------------------------------------------------------------ x_1^4x_2+14690x_0^2x_2^3-4546x_0x_1x_2^3+12199x_0x_2^4+15103x_1x_2^4- ------------------------------------------------------------------------ 8967x_2^5 x_0x_1^3x_2-15905x_0^2x_1x_2^2-2273x_0^2x_2^3+284x_0x_1x_2^3- ------------------------------------------------------------------------ 8884x_0x_2^4 x_0^2x_1^2x_2-15905x_0^3x_2^2-3127x_0^2x_1x_2^2-14505x_0^2x ------------------------------------------------------------------------ _2^3 x_1^5+14690x_0^2x_1x_2^2-9353x_0^2x_2^3+6189x_0x_1x_2^3-14853x_0x_2 ------------------------------------------------------------------------ ^4+13689x_1x_2^4+8480x_2^5 ------------------------------------------------------------------------ x_0x_1^4+14690x_0^3x_2^2-4546x_0^2x_1x_2^2+12199x_0^2x_2^3+15103x_0x_1x_ ------------------------------------------------------------------------ 2^3-8967x_0x_2^4 x_0^2x_1^3-15905x_0^3x_1x_2-2273x_0^3x_2^2+284x_0^2x_1x ------------------------------------------------------------------------ _2^2-8884x_0^2x_2^3 x_0^3x_1^2-15905x_0^4x_2-3127x_0^3x_1x_2-14505x_0^3x ------------------------------------------------------------------------ _2^2 x_0^4x_1-11073x_0^4x_2-14059x_0^3x_1x_2-14725x_0^3x_2^2+15715x_0^2x ------------------------------------------------------------------------ _1x_2^2+5589x_0^2x_2^3-5681x_0x_1x_2^3+9449x_0x_2^4-752x_1x_2^4-4257x_2^ ------------------------------------------------------------------------ 5 x_0^5-11952x_0^4x_2+4264x_0^3x_1x_2-13256x_0^3x_2^2-8472x_0^2x_1x_2^2- ------------------------------------------------------------------------ 3 2 1481x_0^2x_2^3+1434x_0x_1x_2^3-x_0x_2^4-13158x_1x_2^4-15984x_2^5 |, x x 0 1 ------------------------------------------------------------------------ 2 3 4 5 4 3 2 2 + 7632x x - 14167x x + 15007x - 15905x x - 2708x x x + 2874x x x + 0 1 0 1 1 0 2 0 1 2 0 1 2 ------------------------------------------------------------------------ 3 4 3 2 2 2 2 2 3 2 10670x x x + 8250x x + 7181x x - 3x x x + 15061x x x - 15223x x - 0 1 2 1 2 0 2 0 1 2 0 1 2 1 2 ------------------------------------------------------------------------ 2 3 3 2 3 4 4 5 10388x x + 6271x x x - 2509x x + 7640x x - 14385x x - 1297x ) 0 2 0 1 2 1 2 0 2 1 2 2 o5 : Sequence i6 : C=imageUnderRationalMap(J,l_0); ZZ o6 : Ideal of -----[x ..x ] 32003 0 11 i7 : (dim C, degree C, genus C) o7 = (2, 18, 7) o7 : Sequence