\input{ivrgpreamble}


\begin{document}

\hypersetup{pdftitle={IVRG Example: <72,40>}}
\title{IVRG Example\\$g=4$, $G=\langle 72, 40 \rangle$}

\author{David Swinarski}
\date{October 4, 2010}
\maketitle

The group $G=\langle 72, 40 \rangle$ is the automorphism group of a genus $4$ curve \cite{MSSV}.  We use \texttt{DecomposeGAction} to find equations for two curves with this automorphism group.  One is smooth, the other is singular.  

First, we compute the center of $G$ in \texttt{Magma}.  
\begin{verbatim}
> SmallGroupDatabase();           
> H:=SmallGroup(96,64);           
> Center(H);                       
GrpPC of order 1
PC-Relations:
\end{verbatim}
It is trivial.  Hence, the curve $C$ is nonhyperelliptic, and we seek one quadric and one cubic in four variables.  


Next, we find matrix generators for the action of $\Aut(C)$ on the vector space $H^{0}(C,K)$.  These are given in \cite{KKg34}, Prop. 2.4.f.2, p. 288.

Let $w= e^{2 \pi i /3}$.  Then the generators are
\begin{displaymath}
\left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & w & 0 \\
0 & 0 & 0 & w^2 
\end{array}
\right), 
\left(
\begin{array}{cccc}
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & -1 
\end{array}
\right), 
\left(
\begin{array}{cccc}
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 
\end{array}
\right)
\end{displaymath}
which we call $A$, $B$, and $C$ in the \texttt{Magma} session below.

\begin{verbatim}
> K<w>:=CyclotomicField(3);
> GL4K:=GeneralLinearGroup(4,K);
> A:=elt<GL4K | 1,0,0,0, 0,1,0,0, 0,0,w,0, 0,0,0,w^2>;
> B:=elt<GL4K | 0,1,0,0, 1,0,0,0, 0,0,-1,0, 0,0,0,-1>;
> C:=elt<GL4K | 0,0,0,1, 0,0,1,0, 0,1,0,0, 1,0,0,0>;
> G:=sub<GL4K | A,B,C>;
> IdentifyGroup(G);
<72,40>
> load "DecomposeGAction.txt";
Loading "DecomposeGAction.txt"
> S<a,b,c,d>:=PolynomialRing(K,4);
> DecomposeGAction(G,S,2);
[
    rec<recformat<CharacterRow, Dimension, Elements> | 
        CharacterRow := 1,
        Dimension := 1,
        Elements := [
            a*b + c*d
        ]
        >,
    rec<recformat<CharacterRow, Dimension, Elements> | 
        CharacterRow := 2,
        Dimension := 1,
        Elements := [
            a*b - c*d
        ]
        >,
    rec<recformat<CharacterRow, Dimension, Elements> | 
        CharacterRow := 7,
        Dimension := 4,
        Elements := [
            a^2,
            b^2,
            c^2,
            d^2
        ]
        >,
    rec<recformat<CharacterRow, Dimension, Elements> | 
        CharacterRow := 8,
        Dimension := 4,
        Elements := [
            a*c,
            a*d,
            b*c,
            b*d
        ]
        >
]

> DecomposeGAction(G,S,3);
[
    rec<recformat<CharacterRow, Dimension, Elements> | 
        CharacterRow := 2,
        Dimension := 1,
        Elements := [
            a^3 - b^3 - c^3 + d^3
        ]
        >,
    rec<recformat<CharacterRow, Dimension, Elements> | 
        CharacterRow := 3,
        Dimension := 1,
        Elements := [
            a^3 - b^3 + c^3 - d^3
        ]
        >,
    rec<recformat<CharacterRow, Dimension, Elements> | 
        CharacterRow := 5,
        Dimension := 2,
        Elements := [
            a^3 + b^3,
            c^3 + d^3
        ]
        >,
    rec<recformat<CharacterRow, Dimension, Elements> | 
        CharacterRow := 6,
        Dimension := 4,
        Elements := [
            a^2*c - b*d^2,
            a^2*d + b*c^2,
            a*c^2 - b^2*d,
            a*d^2 + b^2*c
        ]
        >,
    rec<recformat<CharacterRow, Dimension, Elements> | 
        CharacterRow := 7,
        Dimension := 4,
        Elements := [
            a^2*c + b*d^2,
            a^2*d - b*c^2,
            a*c^2 + b^2*d,
            a*d^2 - b^2*c
        ]
        >
]
\end{verbatim}

This gives us four pairs of polynomials to study: $ab+cd$ or $ab-cd$ with $a^3-b^3+c^3-d^3$ or $a^3-b^3-c^3+d^3$.  Actually, we can see that two of these pairs are the same as the other two pairs under the map which permutes $c$ and $d$, and so we really only need to study $ab+cd$ with $a^3-b^3+c^3-d^3$ and $ab-cd$ with $a^3-b^3+c^3-d^3$.

We study the schemes given by these pairs of polynomials:
\begin{verbatim}
> P3<a,b,c,d>:=ProjectiveSpace(K,3);
> X1:=Scheme(P3,[a*b+c*d,a^3-b^3+c^3-d^3]);
> IsNonsingular(X1);
true
> X2:=Scheme(P3,[a*b-c*d,a^3-b^3+c^3-d^3]);
> IsNonsingular(X2);
false
\end{verbatim}
We see that we get one smooth curve and one singular curve.  

We verify that the smooth curve is the curve we are looking for:
\begin{verbatim}
> Dimension(X1);
1
> X1:=Curve(X1);
> Genus(X1);    
4
> AutX1:=AutomorphismGroup(X1);
> Order(AutX1);
72
> PermAutX1:=PermutationRepresentation(AutX1);
> IdentifyGroup(PermAutX1);
<72, 40>
\end{verbatim}


Next, we study the singular curve $X_2$:
\begin{verbatim}
> IrreducibleComponents(X2);
[
    Scheme over K defined by
    a + w*c,
    b + (-w - 1)*d,
    Scheme over K defined by
    a + (-w - 1)*c,
    b + w*d,
    Scheme over K defined by
    a + c,
    b + d,
    Scheme over K defined by
    a + (w + 1)*d,
    b - w*c,
    Scheme over K defined by
    a - w*d,
    b + (w + 1)*c,
    Scheme over K defined by
    a - d,
    b - c
]
> L:=IrreducibleComponents(X2);
> [ IsNonsingular(Z) : Z in L];                     
[ true, true, true, true, true, true ]
> [ ArithmeticGenus(Z) : Z in L];
[ 0, 0, 0, 0, 0, 0 ]
> IsEmpty(Intersection(L[1],L[2]));                 
true
> IsEmpty(Intersection(L[1],L[3]));
true
> IsEmpty(Intersection(L[1],L[4])); 
false
> IsEmpty(Intersection(L[1],L[5]));
false
> IsEmpty(Intersection(L[1],L[6]));
false
> IsEmpty(Intersection(L[2],L[3]));
true
> IsEmpty(Intersection(L[2],L[4]));
false
> IsEmpty(Intersection(L[2],L[5]));
false
> IsEmpty(Intersection(L[2],L[6]));
false
> IsEmpty(Intersection(L[3],L[4]));
false
> IsEmpty(Intersection(L[3],L[5]));
false
> IsEmpty(Intersection(L[3],L[6]));
false
> IsEmpty(Intersection(L[4],L[5]));
true
> IsEmpty(Intersection(L[4],L[6]));
true
> IsEmpty(Intersection(L[5],L[6]));
true
> IsReduced(Intersection(L[1],L[4]));
true
> IsReduced(Intersection(L[1],L[5]));
true
> IsReduced(Intersection(L[1],L[6]));
true
> IsReduced(Intersection(L[2],L[4]));
true
> IsReduced(Intersection(L[2],L[5]));
true
> IsReduced(Intersection(L[2],L[6]));
true
> IsReduced(Intersection(L[3],L[4]));
true
> IsReduced(Intersection(L[3],L[5]));
true
> IsReduced(Intersection(L[3],L[6]));
true
> HilbertPolynomial(Ideal(Intersection(L[1],L[4])));
1
0
> HilbertPolynomial(Ideal(Intersection(L[1],L[5])));
1
0
> HilbertPolynomial(Ideal(Intersection(L[1],L[6])));
1
0
> HilbertPolynomial(Ideal(Intersection(L[2],L[4])));
1
0
> HilbertPolynomial(Ideal(Intersection(L[2],L[5])));
1
0
> HilbertPolynomial(Ideal(Intersection(L[2],L[6])));
1
0
> HilbertPolynomial(Ideal(Intersection(L[3],L[4])));
1
0
> HilbertPolynomial(Ideal(Intersection(L[3],L[5])));
1
0
> HilbertPolynomial(Ideal(Intersection(L[3],L[6])));
1
0
\end{verbatim}
From the results above, we conclude that the singular curve $X_2$ has $6$ irreducible components, each of which is the Riemann sphere.  If we call these $Z_1$ through $Z_6$, we find that $Z_i \cap Z_j \neq \emptyset $ if and only if $i \in \{1,2,3\}$ and $j \in \{ 4,5,6\}$.  When $Z_i \cap Z_j \neq \emptyset $, we see that $Z_i \cap Z_j$ is a single point.  Thus, we can describe the arrangement of the components $Z_1$ through $Z_6$ using the graph $K_{3,3}$.  






%Leave everything from here on down alone
%
\newcounter{lastbib}


\section*{References}
\begin{biblist}
	\bibselect{autc}
\end{biblist}
\setcounter{lastbib}{\value{bib}}

\section*{Software Packages Referenced}
\begin{biblist}[\setcounter{bib}{\value{lastbib}}]
	\bib{Magma}{article}{
		author={Computational Algebra Research Group,~ School of Mathematics and Statistics,~ University of Sydney},
		title={\texttt{\upshape MAGMA} computational algebra system},
		date={2008},
		note={Version 2.15-1},
		eprint={\neturl{http://magma.maths.usyd.edu.au/magma/}}
	}

\end{biblist}


\end{document}