M06bar_Pic.magma

//this file compute cohomology of the Picard module of \bar M_{0,6} and its dual 
//the matrix m1 is the 5-cycle in S6 
//the matrix m2 is the involution of the five points in P3
//m1 and m2 generates the permuation on five points in P3
//the matrix m3 is the cremona involution
//m1,m2,m3 generate the S6

function choose(n,k)//choose k elements without permutation in a list ls
	
	if k eq 1 then 	
		return [[x]:x in [1..n]];
	end if;
	t:=choose(n,k-1);
	res:=[];
	for x in t do 
		for y in [x[k-1]+1..n] do 
			Append(~res,x cat [y]);
		end for;
	end for;
	return res;
end function;

ind:=AssociativeArray();
E:=[];
n:=0;

poss:=choose(5,2);

for i in [1..5] do 
	n+:=1;
	E[n]:=[0,i];
	ind[[0,i]]:=n;
end for;
for x in poss do 
	n+:=1;
	E[n]:=[x[1],x[2]];
	ind[[x[1],x[2]]]:=n;
end for;

E[16]:=[0,0];
ind[[0,0]]:=16;
n+:=1;
m1:=SparseMatrix(Integers(),n,n);

k:=[0,1,2,3,4,5];
c5:=map<k->k|[<0,0>,<1,2>,<2,3>,<3,4>,<4,5>,<5,1>]>;


for i in [1..5] do 
	m1[c5(i),i]:=1;
end for;

for i in [6..15] do 	
	x1:=E[i][1];
	x2:=E[i][2];
	tar:=ind[Sort([c5(x1),c5(x2)])];
	m1[tar,i]:=1;
end for;

m1[16,16]:=1;

m2:=SparseMatrix(Integers(),n,n);

c2:=map<k->k|[<0,0>,<1,2>,<2,1>,<3,3>,<4,4>,<5,5>]>;


for i in [1..5] do 
	m2[c2(i),i]:=1;
end for;

for i in [6..15] do 	
	x1:=E[i][1];
	x2:=E[i][2];
	tar:=ind[Sort([c2(x1),c2(x2)])];
	m2[tar,i]:=1;
end for;

m2[16,16]:=1;
m1:=Matrix(m1);
m2:=Matrix(m2);


m3:=SparseMatrix(Integers(),n,n);

e:=AssociativeArray();
e[1]:=[[0,1],[0,3],[0,4],[1,3],[1,4],[3,4]];
m3[16,1]:=1;
for x in e[1] do 
	tar:=ind[x];
	m3[tar,1]:=-1;
end for;

e[2]:=[[0,2],[0,3],[0,4],[2,3],[2,4],[3,4]];
m3[16,2]:=1;
for x in e[2] do 
	tar:=ind[x];
	m3[tar,2]:=-1;
end for;

e[3]:=[[0,1],[0,2],[0,3],[1,2],[1,3],[2,3]];
m3[16,3]:=1;
for x in e[3] do 
	tar:=ind[x];
	m3[tar,3]:=-1;
end for;

e[4]:=[[0,1],[0,2],[0,4],[1,2],[1,4],[2,4]];
m3[16,4]:=1;
for x in e[4] do 
	tar:=ind[x];
	m3[tar,4]:=-1;
end for;

m3[5,5]:=1;

m3[ind[[3,4]],ind[[1,2]]]:=1;
m3[ind[[1,3]],ind[[1,3]]]:=1;
m3[ind[[1,4]],ind[[1,4]]]:=1;
m3[ind[[2,5]],ind[[1,5]]]:=1;
m3[ind[[2,3]],ind[[2,3]]]:=1;
m3[ind[[2,4]],ind[[2,4]]]:=1;
m3[ind[[1,5]],ind[[2,5]]]:=1;
m3[ind[[1,2]],ind[[3,4]]]:=1;
m3[ind[[4,5]],ind[[3,5]]]:=1;
m3[ind[[3,5]],ind[[4,5]]]:=1;

m3[16,16]:=3;
m3[ind[[0,1]],16]:=-2;
m3[ind[[0,2]],16]:=-2;
m3[ind[[0,3]],16]:=-2;
m3[ind[[0,4]],16]:=-2;
m3[ind[[1,2]],16]:=-1;
m3[ind[[1,3]],16]:=-1;
m3[ind[[1,4]],16]:=-1;
m3[ind[[2,3]],16]:=-1;
m3[ind[[2,4]],16]:=-1;
m3[ind[[3,4]],16]:=-1;

m3:=Matrix(m3);

S6:=Sym(6);
S6:=sub<S6|S6!(1,2,3,4,5),S6!(1,2),S6!(1,2)(3,4)(5,6)>;

M:=GL(16,Integers());
MS6:=sub<M|m1,m2,m3>;
S6:=Sym(6);
f:=hom<S6->MS6|[S6!(1,2,3,4,5)->m1,S6!(1,2)->m2,S6!(1,2)(3,4)(5,6)->m3]>;

S5:=Sym(5);
MS5:=sub<M|m1,m2>;
S5map:=hom<S5->MS5|[S5!(1,2,3,4,5)->m1,S5!(1,2)->m2]>;
TT:=S5map(S5!(2,3)(4,5));
sixcycle:=m3*m1*TT;//Find the matrix of the 6-cycle (1,2,3,4,5,6) in S6
g:=hom<S6->MS6|[S6!(1,2,3,4,5,6)->sixcycle,S6!(1,2)->m2]>;

GM:=GModule(S6,[sixcycle,m2]);
GMd:=GModule(S6,[Transpose(sixcycle^(-1)),Transpose(m2^(-1))]);

LS:=SubgroupLattice(S6);
Pic:=CohomologyModule(S6,GM);//This is the Picard moudle
Picd:=CohomologyModule(S6,GMd);//This is the dual of the Picard module
zer:=AssociativeArray();
dlt:={@@};


for x in [2..#LS] do 	
	H:=LS[x];
	Picco:=Restriction(Pic,H);//This is Picard module restricted to the subgroup H
	Picdco:=Restriction(Picd,H);//This is the dual of Picard modulerestricted to the subgroup H
	H11:=CohomologyGroup(Picco,1);//this is H1 of Picard;
	Hd1:=CohomologyGroup(Picdco,1);//this is H1 of Picard Dual;
	H01:=CohomologyGroup(Picdco,0);//this is H0 of Picard;
	IntegerToString(x) cat ". " cat "Rank of Pic(X3)^G where G= " cat GroupName(H) cat " is " cat IntegerToString(Degree(H01));

end for;