|
Magaard, Shaska, Shpectorov, and Völklein give tables of smooth Riemann surfaces of genus \( g \leq 10\) with automorphism groups \(G\) satisfying \( \# G > 4(g-1)\). Their list is based on a computer search by Breuer.
They list a 1-parameter family of genus 5 curves with automorphism group (32,43) in the GAP library of small groups. The quotient of any surface in this family by its automorphism group has genus zero, and the quotient morphism is branched over four points with ramification indices (2,2,2,4).
We use GAP and Magma to compute equations of one member of this family, and give a conjectural description of this family.
Magma V2.21-4 Sun Sep 13 2015 09:44:36 on ace-math01 [Seed = 730530581]
Type ? for help. Type -D to quit.
> load "autcv10.txt";
Loading "autcv10.txt"
> G:=SmallGroup(32,43);
> G;
GrpPC : G of order 32 = 2^5
PC-Relations:
G.4^2 = G.5,
G.2^G.1 = G.2 * G.4,
G.3^G.1 = G.3 * G.5,
G.4^G.1 = G.4 * G.5,
G.4^G.2 = G.4 * G.5
> MatrixGens,MatrixSKG,Q,C:=RunExample(G,5,[2,2,2,4]);
Set seed to 0.
Character Table of Group G
--------------------------
-----------------------------------------
Class | 1 2 3 4 5 6 7 8 9 10 11
Size | 1 1 2 4 4 4 2 2 4 4 4
Order | 1 2 2 2 2 2 4 4 4 8 8
-----------------------------------------
p = 2 1 1 1 1 1 1 2 2 2 7 7
-----------------------------------------
X.1 + 1 1 1 1 1 1 1 1 1 1 1
X.2 + 1 1 1 -1 1 -1 1 1 1 -1 -1
X.3 + 1 1 -1 1 -1 -1 1 -1 1 1 -1
X.4 + 1 1 -1 -1 -1 1 1 -1 1 -1 1
X.5 + 1 1 -1 1 1 -1 1 -1 -1 -1 1
X.6 + 1 1 -1 -1 1 1 1 -1 -1 1 -1
X.7 + 1 1 1 1 -1 1 1 1 -1 -1 -1
X.8 + 1 1 1 -1 -1 -1 1 1 -1 1 1
X.9 + 2 2 2 0 0 0 -2 -2 0 0 0
X.10 + 2 2 -2 0 0 0 -2 2 0 0 0
X.11 + 4 -4 0 0 0 0 0 0 0 0 0
Conjugacy Classes of group G
----------------------------
[1] Order 1 Length 1
Rep Id(G)
[2] Order 2 Length 1
Rep G.5
[3] Order 2 Length 2
Rep G.3
[4] Order 2 Length 4
Rep G.2
[5] Order 2 Length 4
Rep G.1
[6] Order 2 Length 4
Rep G.2 * G.3
[7] Order 4 Length 2
Rep G.4
[8] Order 4 Length 2
Rep G.3 * G.4
[9] Order 4 Length 4
Rep G.1 * G.3
[10] Order 8 Length 4
Rep G.1 * G.2 * G.3
[11] Order 8 Length 4
Rep G.1 * G.2
Surface kernel generators: [ G.2 * G.3 * G.4 * G.5, G.1 * G.4, G.2 * G.4, G.1
* G.3 * G.5 ]
Is hyperelliptic? false
Is cyclic trigonal? false
Multiplicities of irreducibles in relevant G-modules:
I_1 =[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
S_1 =[ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1 ]
H^0(C,K) =[ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1 ]
I_2 =[ 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0 ]
S_2 =[ 2, 0, 0, 0, 1, 1, 1, 0, 2, 1, 1 ]
H^0(C,2K)=[ 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1 ]
I_3 =[ 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 3 ]
S_3 =[ 0, 1, 1, 1, 0, 0, 0, 2, 2, 1, 6 ]
H^0(C,3K)=[ 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 3 ]
I2timesS1=[ 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 3 ]
Is clearly not generated by quadrics? false
No subgroup found The error "No subgroup found" indicates that Magma encounters an error when it tries to produce matrix generators of a representation having the character \( \chi_{11}\).
We obtain these matrix generators using GAP instead of Magma. Conveniently, GAP and Magma use the same generators for this group. (To check this, we check that the GAP generators satisfy the Magma relations.)
gap> G:=SmallGroup(32,43);
gap> (G.4^2)*(G.5)^-1;
<identity> of ...
gap> (G.2^G.1)*(G.2 * G.4)^-1;
<identity> of ...
gap> (G.3^G.1)*(G.3 * G.5)^-1;
<identity> of ...
gap> (G.4^G.1)*(G.4 * G.5)^-1;
<identity> of ...
gap> (G.4^G.2)*(G.4 * G.5)^-1;
<identity> of ...
gap> IrreducibleRepresentations(G);
[ Pcgs([ f1, f2, f3, f4, f5 ]) -> [ [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ],
[ [ 1 ] ], [ [ 1 ] ] ], Pcgs([ f1, f2, f3, f4, f5 ]) ->
[ [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ],
Pcgs([ f1, f2, f3, f4, f5 ]) -> [ [ [ 1 ] ], [ [ -1 ] ], [ [ 1 ] ],
[ [ 1 ] ], [ [ 1 ] ] ], Pcgs([ f1, f2, f3, f4, f5 ]) ->
[ [ [ -1 ] ], [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ],
Pcgs([ f1, f2, f3, f4, f5 ]) -> [ [ [ 1 ] ], [ [ 1 ] ], [ [ -1 ] ],
[ [ 1 ] ], [ [ 1 ] ] ], Pcgs([ f1, f2, f3, f4, f5 ]) ->
[ [ [ -1 ] ], [ [ 1 ] ], [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ] ],
Pcgs([ f1, f2, f3, f4, f5 ]) -> [ [ [ 1 ] ], [ [ -1 ] ], [ [ -1 ] ],
[ [ 1 ] ], [ [ 1 ] ] ], Pcgs([ f1, f2, f3, f4, f5 ]) ->
[ [ [ -1 ] ], [ [ -1 ] ], [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ] ],
Pcgs([ f1, f2, f3, f4, f5 ]) ->
[ [ [ 0, 1 ], [ 1, 0 ] ], [ [ 1, 0 ], [ 0, -1 ] ],
[ [ 1, 0 ], [ 0, 1 ] ], [ [ -1, 0 ], [ 0, -1 ] ],
[ [ 1, 0 ], [ 0, 1 ] ] ], Pcgs([ f1, f2, f3, f4, f5 ]) ->
[ [ [ 0, 1 ], [ 1, 0 ] ], [ [ 1, 0 ], [ 0, -1 ] ],
[ [ -1, 0 ], [ 0, -1 ] ], [ [ -1, 0 ], [ 0, -1 ] ],
[ [ 1, 0 ], [ 0, 1 ] ] ], Pcgs([ f1, f2, f3, f4, f5 ]) ->
[
[ [ 0, 0, 0, 1 ], [ 0, 0, E(4), 0 ], [ 0, -E(4), 0, 0 ], [ 1, 0, 0, 0 ]
], [ [ 0, 1, 0, 0 ], [ 1, 0, 0, 0 ], [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ]
],
[ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, -1, 0 ], [ 0, 0, 0, -1 ] ],
[ [ E(4), 0, 0, 0 ], [ 0, -E(4), 0, 0 ], [ 0, 0, E(4), 0 ],
[ 0, 0, 0, -E(4) ] ],
[ [ -1, 0, 0, 0 ], [ 0, -1, 0, 0 ], [ 0, 0, -1, 0 ], [ 0, 0, 0, -1 ] ]
] ]
Next, I loaded these matrices into Magma, computed the characters of the representations they give, and compared them to the order of the characters in the Magma character table.
> K<i>:=CyclotomicField(4);
> rho:=function(G,K,L)
function> n:=NumberOfRows(Matrix(L[1]));
function> GLnK:=GeneralLinearGroup(n,K);
function> L:=[GLnK!Matrix(L[i]): i in [1..#L]];
function> return hom< G -> GLnK | L>;
function> end function;
> char:=function(CCLR,f)
function> return [Trace(f(CCLR[i])) : i in [1..#CCLR]];
function> end function;
> LookupCharacter:=function(T,chi)
function> for i:=1 to #T do
function|for> if T[i] eq chi then
function|for|if> return i;
function|for|if> end if;
function|for> end for;
function> end function;
> L:=[
> [ [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ],
> [ [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ],
> [ [ [ 1 ] ], [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ],
> [ [ [ -1 ] ], [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ],
> [ [ [ 1 ] ], [ [ 1 ] ], [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ] ],
> [ [ [ -1 ] ], [ [ 1 ] ], [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ] ],
> [ [ [ 1 ] ], [ [ -1 ] ], [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ] ],
> [ [ [ -1 ] ], [ [ -1 ] ], [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ] ],
> [ [ [ 0, 1 ], [ 1, 0 ] ], [ [ 1, 0 ], [ 0, -1 ] ], [ [ 1, 0 ], [ 0, 1 ] ], [\
[ -1, 0 ], [ 0, -1 ] ], [ [ 1, 0 ], [ 0, 1 ] ] ],
> [ [ [ 0, 1 ], [ 1, 0 ] ], [ [ 1, 0 ], [ 0, -1 ] ], [ [ -1, 0 ], [ 0, -1 ] ],\
[ [ -1, 0 ], [ 0, -1 ] ], [ [ 1, 0 ], [ 0, 1 ] ] ],
> [ [ [ 0, 0, 0, 1 ], [ 0, 0, i, 0 ], [ 0, -i, 0, 0 ], [ 1, 0, 0, 0 ] ], [ [ 0\
, 1, 0, 0 ], [ 1, 0, 0, 0 ], [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ] ], [ [ 1, 0, 0, 0 \
], [ 0, 1, 0, 0 ], [ 0, 0, -1, 0 ], [ 0, 0, 0, -1 ] ], [ [ i, 0, 0, 0 ], [ 0, \
-i, 0, 0 ], [ 0, 0, i, 0 ], [ 0, 0, 0, -i ] ], [ [ -1, 0, 0, 0 ], [ 0, -1, 0, \
0 ], [ 0, 0, -1, 0 ], [ 0, 0, 0, -1 ] ] ]
> ];
> G:=SmallGroup(32,43);
> CCLR:=Classes(G);
> CCLR:=[CCLR[i][3]: i in [1..#CCLR]];
> T:=CharacterTable(G);
> D:=[LookupCharacter(T,CharacterRing(G)!char(CCLR,rho(G,K,L[i]))) : i in [1..\
#L]];
> D;
[ 1, 7, 2, 8, 5, 3, 6, 4, 9, 10, 11 ]
Thus, matrix generators for the representation \( \chi_8 + \chi_{11}\) are
> MG:=[Matrix([[-1,0,0,0,0],[ 0, 0, 0, 0, 1 ], [ 0, 0, 0, i, 0 ], [ 0, 0,-i, 0\
, 0 ], [ 0, 1, 0, 0, 0 ]]), Matrix([[-1,0,0,0,0],[ 0, 0, 1, 0, 0 ], [0, 1,0, 0\
, 0 ], [ 0, 0, 0, 0, 1 ], [ 0, 0, 0, 1, 0 ]]), Matrix([[1,0,0,0,0],[ 0, 1, 0,0\
,0 ], [ 0, 0, 1, 0, 0 ], [ 0, 0, 0, -1, 0 ], [ 0, 0, 0, 0, -1 ]]),Matrix([[1,0\
,0,0,0],[ 0, i, 0, 0, 0], [ 0, 0, -i, 0, 0 ], [ 0, 0, 0, i, 0 ],[ 0, 0, 0, 0, \
-i ]]), Matrix([[1,0,0,0,0],[ 0, -1, 0, 0, 0 ], [ 0, 0,-1, 0, 0 ], [ 0, 0, 0, \
-1, 0 ], [ 0, 0, 0, 0, -1 ]])];
> MG;
[
[-1 0 0 0 0]
[ 0 0 0 0 1]
[ 0 0 0 i 0]
[ 0 0 -i 0 0]
[ 0 1 0 0 0],
[-1 0 0 0 0]
[ 0 0 1 0 0]
[ 0 1 0 0 0]
[ 0 0 0 0 1]
[ 0 0 0 1 0],
[ 1 0 0 0 0]
[ 0 1 0 0 0]
[ 0 0 1 0 0]
[ 0 0 0 -1 0]
[ 0 0 0 0 -1],
[ 1 0 0 0 0]
[ 0 i 0 0 0]
[ 0 0 -i 0 0]
[ 0 0 0 i 0]
[ 0 0 0 0 -i],
[ 1 0 0 0 0]
[ 0 -1 0 0 0]
[ 0 0 -1 0 0]
[ 0 0 0 -1 0]
[ 0 0 0 0 -1]
]
The surface kernel generators from the first block of Magma code above were [ G.2 * G.3 * G.4 * G.5, G.1 * G.4, G.2 * G.4, G.1 * G.3 * G.5 ]. Thus we obtain the following matrix surface kernel generators:
> GL5K:=GeneralLinearGroup(5,K);
> MG:=[GL5K!MG[i] : i in [1..#MG]];
> rho:=hom< G -> GL5K | MG>;
> A:=rho(G.2 * G.3 * G.4 * G.5);
> B:=rho(G.1 * G.4);
> C:=rho(G.2 * G.4);
> D:=rho(G.1 * G.3 * G.5);
> A;
[-1 0 0 0 0]
[ 0 0 i 0 0]
[ 0 -i 0 0 0]
[ 0 0 0 0 -i]
[ 0 0 0 i 0]
> B;
[-1 0 0 0 0]
[ 0 0 0 0 -i]
[ 0 0 0 -1 0]
[ 0 0 -1 0 0]
[ 0 i 0 0 0]
> C;
[-1 0 0 0 0]
[ 0 0 -i 0 0]
[ 0 i 0 0 0]
[ 0 0 0 0 -i]
[ 0 0 0 i 0]
> D;
[-1 0 0 0 0]
[ 0 0 0 0 1]
[ 0 0 0 i 0]
[ 0 0 i 0 0]
[ 0 -1 0 0 0]
We use these matrix surface kernel generators to obtain candidate polynomials.
> MatrixGens,MatrixSKG,Q,C :=RunGivenSKMatrixGenerators(32,5,[A,B,C,D]);
Set seed to 0.
Character Table of Group G
--------------------------
-----------------------------------------
Class | 1 2 3 4 5 6 7 8 9 10 11
Size | 1 1 2 4 4 4 2 2 4 4 4
Order | 1 2 2 2 2 2 4 4 4 8 8
-----------------------------------------
p = 2 1 1 1 1 1 1 2 2 2 7 7
-----------------------------------------
X.1 + 1 1 1 1 1 1 1 1 1 1 1
X.2 + 1 1 -1 1 1 -1 1 -1 -1 1 -1
X.3 + 1 1 1 -1 1 1 1 1 -1 -1 -1
X.4 + 1 1 -1 -1 1 -1 1 -1 1 -1 1
X.5 + 1 1 1 1 -1 -1 1 1 1 -1 -1
X.6 + 1 1 -1 1 -1 1 1 -1 -1 -1 1
X.7 + 1 1 1 -1 -1 -1 1 1 -1 1 1
X.8 + 1 1 -1 -1 -1 1 1 -1 1 1 -1
X.9 + 2 2 2 0 0 0 -2 -2 0 0 0
X.10 + 2 2 -2 0 0 0 -2 2 0 0 0
X.11 + 4 -4 0 0 0 0 0 0 0 0 0
Conjugacy Classes of group G
----------------------------
[1] Order 1 Length 1
Rep [1 0 0 0 0]
[0 1 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]
[0 0 0 0 1]
[2] Order 2 Length 1
Rep [ 1 0 0 0 0]
[ 0 -1 0 0 0]
[ 0 0 -1 0 0]
[ 0 0 0 -1 0]
[ 0 0 0 0 -1]
[3] Order 2 Length 2
Rep [ 1 0 0 0 0]
[ 0 -1 0 0 0]
[ 0 0 -1 0 0]
[ 0 0 0 1 0]
[ 0 0 0 0 1]
[4] Order 2 Length 4
Rep [ -1 0 0 0 0]
[ 0 0 0 0 -z^8]
[ 0 0 0 -1 0]
[ 0 0 -1 0 0]
[ 0 z^8 0 0 0]
[5] Order 2 Length 4
Rep [ -1 0 0 0 0]
[ 0 0 z^8 0 0]
[ 0 -z^8 0 0 0]
[ 0 0 0 0 -z^8]
[ 0 0 0 z^8 0]
[6] Order 2 Length 4
Rep [ -1 0 0 0 0]
[ 0 0 -z^8 0 0]
[ 0 z^8 0 0 0]
[ 0 0 0 0 -z^8]
[ 0 0 0 z^8 0]
[7] Order 4 Length 2
Rep [ 1 0 0 0 0]
[ 0 z^8 0 0 0]
[ 0 0 -z^8 0 0]
[ 0 0 0 z^8 0]
[ 0 0 0 0 -z^8]
[8] Order 4 Length 2
Rep [ 1 0 0 0 0]
[ 0 z^8 0 0 0]
[ 0 0 -z^8 0 0]
[ 0 0 0 -z^8 0]
[ 0 0 0 0 z^8]
[9] Order 4 Length 4
Rep [ -1 0 0 0 0]
[ 0 0 0 0 z^8]
[ 0 0 0 1 0]
[ 0 0 -1 0 0]
[ 0 z^8 0 0 0]
[10] Order 8 Length 4
Rep [ 1 0 0 0 0]
[ 0 0 0 1 0]
[ 0 0 0 0 z^8]
[ 0 z^8 0 0 0]
[ 0 0 -1 0 0]
[11] Order 8 Length 4
Rep [ 1 0 0 0 0]
[ 0 0 0 -1 0]
[ 0 0 0 0 -z^8]
[ 0 z^8 0 0 0]
[ 0 0 -1 0 0]
Is hyperelliptic? false
Is cyclic trigonal? false
Multiplicities of irreducibles in relevant G-modules:
I_1 =[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
S_1 =[ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1 ]
H^0(C,K) =[ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1 ]
I_2 =[ 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0 ]
S_2 =[ 2, 1, 1, 0, 0, 1, 0, 0, 2, 1, 1 ]
H^0(C,2K)=[ 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1 ]
I_3 =[ 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 3 ]
S_3 =[ 0, 0, 0, 1, 1, 0, 2, 1, 2, 1, 6 ]
H^0(C,3K)=[ 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 3 ]
I2timesS1=[ 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 3 ]
Is clearly not generated by quadrics? false
Matrix Surface Kernel Generators:
Field K Cyclotomic Field of order 32 and degree 16
[
[ -1 0 0 0 0]
[ 0 0 z^8 0 0]
[ 0 -z^8 0 0 0]
[ 0 0 0 0 -z^8]
[ 0 0 0 z^8 0],
[ -1 0 0 0 0]
[ 0 0 0 0 -z^8]
[ 0 0 0 -1 0]
[ 0 0 -1 0 0]
[ 0 z^8 0 0 0],
[ -1 0 0 0 0]
[ 0 0 -z^8 0 0]
[ 0 z^8 0 0 0]
[ 0 0 0 0 -z^8]
[ 0 0 0 z^8 0],
[ -1 0 0 0 0]
[ 0 0 0 0 1]
[ 0 0 0 z^8 0]
[ 0 0 z^8 0 0]
[ 0 -1 0 0 0]
]
Finding quadrics:
I2 contains a 1-dimensional subspace of CharacterRow 1
Dimension 2
Multiplicity 2
[
x_0^2,
x_1*x_2 - z^8*x_3*x_4
]
I2 contains a 2-dimensional subspace of CharacterRow 9
Dimension 4
Multiplicity 2
[
x_1^2,
x_2^2,
x_3^2,
x_4^2
]
The output above shows that the ideal contains quadrics from two isotypical subspaces of \(S_2\). Note that the power of z\(=\zeta_{32}\) in our equations is always a multiple of 8. Therefore in the sequel we reduce these to i\(= \zeta_{4}\).
The first isotypical subspace, which corresponds to the character \( \chi_1\) in the character table shown above, yields a polynomial of the form \[ c_1(x_0^2) + c_2(x_1 x_2 - i x_3 x_4) \] By assuming that \(c_1\) and \(c_2\) are nonzero, after scaling \(x_0\), we may assume that this polynomial is \(x_0^2 + x_1 x_2 - i x_3 x_4\).
The second isotypical subspace corresponds to the character \( \chi_{9}\) in the character table shown above. It is not obvious how to use the FindParallelBases command in this example. Therefore, instead, we compute the action of the matrix surface kernel generators on the four basis elements shown; use Magma obtain matrix generators for an action with character \( \chi_9\); and use Magma to compute generators of the space of G-module homomorphisms between these modules.
By hand, we compute that the action of \(A\), \(B\), and \(C\) on Span \( \langle x_1^2, x_2^2, x_3^2, x_4^2 \rangle\) are given by the matrices A1,B1, and C1 below
> G:=sub<GL5K | A,B,C>;
> T:=CharacterTable(G);
> GL4K:=GeneralLinearGroup(4,K);
> A1:=GL4K!Matrix([
> [0,-1,0,0],
> [-1,0,0,0],
> [0,0,0,-1],
> [0,0,-1,0]
> ]);
> B1:=GL4K!Matrix([
> [0,0,0,-1],
> [0,0,1,0],
> [0,1,0,0],
> [-1,0,0,0]
> ]);
> C1:=GL4K!Matrix([
> [0,-1,0,0],
> [-1,0,0,0],
> [0,0,0,-1],
> [0,0,-1,0]
> ]);
> rho1:=homGL4K | A1,B1,C1>;
> M1:=GModule(rho1);
> Decomposition(T,Character(M1));
[
0,
0,
0,
0,
0,
0,
0,
0,
2,
0,
0
]
( 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 )
Next we create a G-module \(M_2\) with character \( \chi_9\) in Magma, and compute two independent G-module homomorphisms from \(M_2 \rightarrow M_1\).
> M2:=GModule(T[9]);
> P:=AHom(M2,M1);
> P.1;
[ 1 0 -i 0]
[ 0 -1 0 i]
> P.2;
[ 0 1 0 i]
[-1 0 -i 0]
This output indicates that Span \(\langle x_1^2-ix_3^2, -x_2^2+ix_4^2\rangle \) and Span \(\langle x_2^2+ix_4^2, -x_1^2-ix_3^2\rangle \) form two irreducible submodules of Span \(\langle x_1^2, x_2^2, x_3^2,x_4^2\rangle \) such that the G-action is given by the same matrix generators on the basis elements shown. (This is the task we accomplish with FindParallelBases in some other examples.) Thus, candidate polynomials for this isotypical subspace have the form \[ \begin{array}{l} c_3 (x_1^2-ix_3^2) + c_4(-x_2^2+ix_4^2) \\ c_3 (x_2^2+ix_4^2) + c_4(-x_1^2-ix_3^2) \\ \end{array} \] We may assume that \(c_3\) and \(c_4\) are nonzero. After dividing by \(c_3\) we may assume \(c_3 = 1\). Thus, the candidate polynomials for this isotypical subspace are \[ \begin{array}{l} x_1^2-ix_3^2 + c_4(-x_2^2+ix_4^2) \\ x_2^2+ix_4^2 + c_4(-x_1^2-ix_3^2) \\ \end{array} \]
We are expecting to find a pencil of curves, and indeed we have one coefficient, \(c_4\), which is allowed to vary.
The value \(c_4=1\) yields a singular curve. Therefore we study \(c_4=17+i\).
> K<i>:=CyclotomicField(4);
> P4<x_0,x_1,x_2,x_3,x_4>:=ProjectiveSpace(K,4);
> c_4:=17+i;
> X:=Scheme(P4,[x_0^2+x_1*x_2 - i*x_3*x_4,
> x_1^2+i*x_3^2+c_4*(x_2^2-i*x_4^2),
> i*x_2^2-x_4^2+c_4*(i*x_1^2+x_3^2)
> ]);
> Dimension(X);
1
> IsSingular(X);
false
> HilbertPolynomial(Ideal(X));
8*$.1 - 4
2
> A:=Matrix([
> [-1,0,0,0,0],
> [0,0,i,0,0],
> [0,-i,0,0,0],
> [0,0,0,0,-i],
> [0,0,0,i,0]
> ]);
> B:=Matrix([
> [-1,0,0,0,0],
> [0,0,0,0,-i],
> [0,0,0,-1,0],
> [0,0,-1,0,0],
> [0,i,0,0,0]
> ]);
> C:=Matrix([
> [-1,0,0,0,0],
> [0,0,-i,0,0],
> [0,i,0,0,0],
> [0,0,0,0,-i],
> [0,0,0,i,0]
> ]);
> Order(A);
2
> Order(B);
2
> Order(C);
2
> Order( (A*B*C)^(-1));
4
> GL5K:=GeneralLinearGroup(5,K);
> IdentifyGroup(sub<GL5K | A,B,C>);
<32, 43>
> Automorphism(X,A);
Mapping from: Sch: X to Sch: X
with equations :
-x_0
-i*x_2
i*x_1
i*x_4
-i*x_3
and inverse
-x_0
-i*x_2
i*x_1
i*x_4
-i*x_3
> Automorphism(X,B);
Mapping from: Sch: X to Sch: X
with equations :
-x_0
i*x_4
-x_3
-x_2
-i*x_1
and inverse
-x_0
i*x_4
-x_3
-x_2
-i*x_1
> Automorphism(X,C);
Mapping from: Sch: X to Sch: X
with equations :
-x_0
i*x_2
-i*x_1
i*x_4
-i*x_3
and inverse
-x_0
i*x_2
-i*x_1
i*x_4
-i*x_3
>
> K<z_20>:=CyclotomicField(20);
> i:=z_20^5;
> P4<x_0,x_1,x_2,x_3,x_4>:=ProjectiveSpace(K,4);
> c_4:=23+z_20^3;
> X:=Scheme(P4,[x_0^2+x_1*x_2 - i*x_3*x_4,
> x_1^2+i*x_3^2+c_4*(x_2^2-i*x_4^2),
> i*x_2^2-x_4^2+c_4*(i*x_1^2+x_3^2)
> ]);
> Dimension(X);
1
> IsSingular(X);
false
> HilbertPolynomial(Ideal(X));
8*$.1 - 4
2
> Automorphism(X,A);
Mapping from: Sch: X to Sch: X
with equations :
-x_0
-z_20^5*x_2
z_20^5*x_1
z_20^5*x_4
-z_20^5*x_3
and inverse
-x_0
-z_20^5*x_2
z_20^5*x_1
z_20^5*x_4
-z_20^5*x_3
> Automorphism(X,B);
Mapping from: Sch: X to Sch: X
with equations :
-x_0
z_20^5*x_4
-x_3
-x_2
-z_20^5*x_1
and inverse
-x_0
z_20^5*x_4
-x_3
-x_2
-z_20^5*x_1
> Automorphism(X,C);
Mapping from: Sch: X to Sch: X
with equations :
-x_0
z_20^5*x_2
-z_20^5*x_1
z_20^5*x_4
-z_20^5*x_3
and inverse
-x_0
z_20^5*x_2
-z_20^5*x_1
z_20^5*x_4
-z_20^5*x_3
>