## Synopsis

• Usage:
• Inputs:
• I, an ideal, the homogeneous ideal I of an irreducible plane algebraic curve C over K
• ib, , containing an integral basis of the closure of L[C] in L(C) where L is the algebraic closure of K given as a row matrix of length n=degree(I).
• Outputs:
• an ideal, the adjoint ideal in the ring I

## Description

Computes the adjoint ideal of an irreducible plane curve.

If ib is specified (and only then) we assume that I has the following properties:

Denote the variables of R=ring(I) by v,u,z. All singularities of C have to lie in the chart z!=0 and the curve should not contain the point (1:0:0).

Furthermore we assume that ib has the following properties: The entries are in K(u)[v] inside frac(R) where the i-th entry is of degree $i$ in v for i=0..n-1. Note that this always can be achieved.

If ib is not specified the function takes care of these conditions by applying an appropriate projective automorphism before doing the computation and afterwards applying its inverse. The algorithm will try to choose an automorphism as simple as possible, however note that this process may destroy sparseness and harm the performance.

A rational curve with three double points:

 i1 : K=QQ o1 = QQ o1 : Ring i2 : R=K[v,u,z] o2 = R o2 : PolynomialRing i3 : I=ideal(v^4-2*u^3*z+3*u^2*z^2-2*v^2*z^2) 4 3 2 2 2 2 o3 = ideal(v - 2u z - 2v z + 3u z ) o3 : Ideal of R i4 : J=adjointIdeal(I) 2 2 o4 = ideal (u - u*z, v*u - v*z, v - u*z) o4 : Ideal of R

Same example but specifying the integral basis:

 i5 : K=QQ o5 = QQ o5 : Ring i6 : R=K[v,u,z] o6 = R o6 : PolynomialRing i7 : I=ideal(v^4-2*u^3*z+3*u^2*z^2-2*v^2*z^2) 4 3 2 2 2 2 o7 = ideal(v - 2u z - 2v z + 3u z ) o7 : Ideal of R i8 : Rvu=K[v,u]; i9 : QR=frac(Rvu); i10 : ib=matrix {{1, v, (-1+v^2)/(-1+u), 1/(-1+u)/u*v^3+(-2+u)/(-1+u)/u*v}}; 1 4 o10 : Matrix QR <--- QR i11 : J=adjointIdeal(I,ib) 2 2 o11 = ideal (u - u*z, v*u - v*z, v - u*z) o11 : Ideal of R

The Cusp:

 i12 : K=QQ o12 = QQ o12 : Ring i13 : R=K[v,u,z] o13 = R o13 : PolynomialRing i14 : I=ideal(v^3-u^2*z) 3 2 o14 = ideal(v - u z) o14 : Ideal of R i15 : J=adjointIdeal(I) o15 = ideal (u, v) o15 : Ideal of R

Same example but specifying the integral basis:

 i16 : K=QQ; i17 : R=K[v,u,z]; i18 : I=ideal(v^3-u^2*z) 3 2 o18 = ideal(v - u z) o18 : Ideal of R i19 : Rvu=K[v,u]; i20 : QR=frac(Rvu); i21 : ib=matrix({{1,v,v^2/u}}); 1 3 o21 : Matrix QR <--- QR i22 : J=adjointIdeal(I,ib) o22 = ideal (u, v) o22 : Ideal of R

A curve of genus 4:

 i23 : K=QQ o23 = QQ o23 : Ring i24 : R=K[v,u,z] o24 = R o24 : PolynomialRing i25 : I=ideal(v^6+(7/5)*v^2*u^4+(6/5)*u^6+(21/5)*v^2*u^3*z+(12/5)*u^5*z+(21/5)*v^2*u^2*z^2+(6/5)*u^4*z^2+(7/5)*v^2*u*z^3) 6 7 2 4 6 6 21 2 3 12 5 21 2 2 2 6 4 2 7 2 3 o25 = ideal(v + -v u + -u + --v u z + --u z + --v u z + -u z + -v u*z ) 5 5 5 5 5 5 5 o25 : Ideal of R i26 : J=adjointIdeal(I) 3 2 2 2 2 3 o26 = ideal (u + u z, v*u + v*u*z, v u + v z, v ) o26 : Ideal of R

Same example but specifying the integral basis:

 i27 : K=QQ; i28 : R=K[v,u,z]; i29 : I=ideal(v^6+(7/5)*v^2*u^4+(6/5)*u^6+(21/5)*v^2*u^3*z+(12/5)*u^5*z+(21/5)*v^2*u^2*z^2+(6/5)*u^4*z^2+(7/5)*v^2*u*z^3); o29 : Ideal of R i30 : Rvu=K[v,u]; i31 : QR=frac(Rvu); i32 : ib=matrix({{1,v,v^2,v^3/(u+1),1/u/(u+1)*v^4,1/u^2/(u+1)*v^5-7/5*(u-1)/u*v}}); 1 6 o32 : Matrix QR <--- QR i33 : J=adjointIdeal(I,ib) 3 2 2 2 2 3 o33 = ideal (u + u z, v*u + v*u*z, v u + v z, v ) o33 : Ideal of R

 i34 : K=QQ; i35 : R=K[v,u,z]; i36 : I=ideal(v^8-u^3*(z+u)^5); o36 : Ideal of R i37 : Ruv=K[v,u]; i38 : QR=frac(Ruv); i39 : ib=matrix({{1,v,v^2/(1+u),v^3/u/(1+u),v^4/u/(1+u)^2,v^5/u/(1+u)^3,v^6/u^2/(1+u)^3,v^7/u^2/(1+u)^4}}); 1 8 o39 : Matrix QR <--- QR i40 : J=adjointIdeal(I,ib) 6 5 4 2 3 3 2 4 5 4 3 2 o40 = ideal (u + 4u z + 6u z + 4u z + u z , v*u + 3v*u z + 3v*u z + ----------------------------------------------------------------------- 2 3 2 4 2 3 2 2 2 2 3 3 3 3 2 3 2 4 2 v*u z , v u + 3v u z + 3v u z + v u*z , v u + 2v u z + v u*z , v u ----------------------------------------------------------------------- 4 5 5 6 + v u*z, v u + v z, v ) o40 : Ideal of R i41 : apply((entries gens J)#0,factor) 2 4 2 3 3 2 2 3 o41 = {(u) (u + z) , (u) (u + z) (v), (u)(u + z) (v) , (u)(u + z) (v) , (u)(u ----------------------------------------------------------------------- 4 5 6 + z)(v) , (u + z)(v) , (v) } o41 : List

## Caveat

The function so far does not cache the integral basis computation.