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AdjointIdeal :: adjointIdeal

adjointIdeal -- Compute the adjoint ideal of a plane curve

Synopsis

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.

See also

Ways to use adjointIdeal :

For the programmer

The object adjointIdeal is a method function.