///////////////////////////////////////////////
//
// Beauville surface with G=(Z/3)^2
//
///////////////////////////////////////////////


X:=rec<Surface|>;

Pic:=LatticeWithGram(Matrix(2,[0,1,1,0]):
                        CheckPositive := false); // allows neg-def

n:=[3:i in [1..6]];
Gt:=AbelianGroup(n);
G:=sub<Gt|Gt.1,Gt.2>;
T:=sub<Gt|Gt.3,Gt.4,Gt.5,Gt.6>;

X`Name:="Beauville Z3";

X`Big:=Gt;
X`Small:=G;
X`Tors:=T;

X`branch:=[Pic![1,0],Pic![1,0],Pic![1,0],Pic![1,0],
           Pic![0,1],Pic![0,1],Pic![0,1],Pic![0,1]];

X`phi:=[Gt.1,Gt.2,2*Gt.1+Gt.3,2*Gt.2+2*Gt.3,
        Gt.1+Gt.2+Gt.4,Gt.1+2*Gt.2+Gt.5,2*Gt.1+2*Gt.2+Gt.6,
                               2*Gt.1+Gt.2+2*Gt.4+2*Gt.5+2*Gt.6];


X`branchtors:=[T!0,T![1,0,1,0],T![0,2,2,1],T![1,2,2,1],
               T!0,T![0,1,2,0],T![0,1,0,2],T![0,1,0,0]];

X`KY:=Pic![-2,-2];
X`K:=Pic![2,2];

lines:=#X`branch;

X`Lchi:=calculateLchi(X);

X`Basis:=[[1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0]];

X`Ktors:=T![1,2,2,2];

// effective divisors on Y
//CC:=[[1,0],[0,1]];
EffY:=[];
for i in [0..5] do
   for j in [0..5] do
      Append(~EffY,Pic![i,j]);
   end for;
end for;
X`EffY:=EffY;


for i in [-5..5] do
   LL:=[Pic![-1,0],Pic![i-1,-1],Pic![i-2,-1]];
   Excl:=findExceptional(X,LL);
   i,#Excl;
end for;

for i in [-5..5] do
   LL:=[Pic![0,-1],Pic![-1,i-1],Pic![-1,i-2]];
   Excl:=findExceptional(X,LL);
   i,#Excl;
end for;


////////
// interesting samples of type I_0
//////
LL:=[Pic![-1,0],Pic![0,-1],Pic![-1,-1]];
ee:=<X`Tors![ 0, 1, 0, 0 ], X`Tors![ 2, 2, 0, 0 ], X`Tors![ 1, 0, 1, 0 ]>;

extendedquiver(X,LL,ee);