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geometricGenus -- Geometric genus of a plane curve

Synopsis

Description

Computes the geometric genus of a 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 : geometricGenus(I)

o4 = 0

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 : geometricGenus(I,ib)

o11 = 0

The Cusp:

i12 : K=QQ;
 
i13 : R=K[v,u,z];
 
i14 : I=ideal(v^3-u^2*z)

             3    2
o14 = ideal(v  - u z)

o14 : Ideal of R
 
i15 : geometricGenus(I)

o15 = 0

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 : geometricGenus(I,ib)

o22 = 0

A curve of genus 4:

i23 : K=QQ;
 
i24 : R=K[v,u,z];
 
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);

o25 : Ideal of R
 
i26 : geometricGenus(I)

o26 = 4

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 : geometricGenus(I,ib)

o33 = 4

Caveat

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

See also

Ways to use geometricGenus :

For the programmer

The object geometricGenus is a method function.