///////////////////////////////////////////////
//
// Primary Burniat surface with K^2=6
//
///////////////////////////////////////////////

X:=rec<Surface|>;

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

n:=[2:i in [1..8]];
Gt:=AbelianGroup(n);
G:=sub<Gt|Gt.1,Gt.2>;
T:=sub<Gt|Gt.3,Gt.4,Gt.5,Gt.6,Gt.7,Gt.8>;

X`Name:="Burniat 6";

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

X`branch:=[
Pic![1,-1,-1,0],Pic![1,-1,0,0],Pic![1,-1,0,0], // A_i
Pic![1,0,-1,-1],Pic![1,0,-1,0],Pic![1,0,-1,0], // B_i
Pic![1,-1,0,-1],Pic![1,0,0,-1],Pic![1,0,0,-1], // C_i
Pic![0,1,0,0],Pic![0,0,1,0],Pic![0,0,0,1]]; // E_i

X`phi:=[Gt.1,Gt.1+Gt.3,Gt.1+Gt.4,
      Gt.2,Gt.2+Gt.5,Gt.2+Gt.6,
      Gt.1+Gt.2+Gt.7,Gt.1+Gt.2+Gt.8,
          Gt.1+Gt.2+Gt.3+Gt.4+Gt.5+Gt.6+Gt.7+Gt.8];

X`phi[10]:=X`phi[1]+X`phi[2]+X`phi[3]+X`phi[7];
X`phi[11]:=X`phi[4]+X`phi[5]+X`phi[6]+X`phi[1];
X`phi[12]:=X`phi[7]+X`phi[8]+X`phi[9]+X`phi[4];


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

X`KY:=Pic![-3,1,1,1];
X`K:=Pic![3,-1,-1,-1];

lines:=#X`branch;

X`Lchi:=calculateLchi(X);


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


findCoordinates(X,[0,1,0,0,0,0,0,0,0,0,0,0]);

// check Del_1^1 vs Del_1^2 etc:
pushforward(X,[0,0,1,0,0,0,-1,0,0,0,0,-1],-T.1-T.5); 

X`Ktors:=X`Tors![0,0,0,0,0,1];


// -1 curves on the degree 6 dP
CC:=[Pic![0,1,0,0],Pic![0,0,1,0],Pic![0,0,0,1],
     Pic![1,0,-1,-1],Pic![1,-1,0,-1],Pic![1,-1,-1,0]];
// effective divisors on Y
EffY:={};
for a in CartesianProduct([[0..5]:i in [1..#CC]]) do
   Include(~EffY,&+[a[i]*CC[i]:i in [1..#CC]]);
end for;
X`EffY:=EffY;


LL:=[Pic![0,-1,0,0],Pic![0,0,-1,0],Pic![0,0,0,-1],
     Pic![-1,0,0,0],Pic![-2,0,0,0]]; // "standard"


Excl:=findExceptional(X,LL); // 81332 in 57million time 1122s

// samples in article are:
//<T.1 + T.3, T.4, T.2 + T.6, T.1 + T.2 + T.3 + T.4, T.1 + T.2 + T.3 + T.4 + 
    T.6>
//<T.1 + T.3, T.4, T.2 + T.6, T.6, T.1 + T.2 + T.3 + T.4 + T.6>