///////////////////////////////////////////////
//
// Secondary Burniat surface with K^2=5
//
///////////////////////////////////////////////


X:=rec<Surface|>;

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

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

X`Name:="Burniat 5";

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

X`branch:=[
Pic![1,-1,-1,0,0],Pic![1,-1,0,0,-1],Pic![1,-1,0,0,0], // A_i
Pic![1,0,-1,-1,0],Pic![1,0,-1,0,0],Pic![1,0,-1,0,-1], // B_i
Pic![1,-1,0,-1,0],Pic![1,0,0,-1,0],Pic![1,0,0,-1,-1], // C_i
Pic![0,1,0,0,0],Pic![0,0,1,0,0],Pic![0,0,0,1,0]]; // 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.4+Gt.5+Gt.7,Gt.1+Gt.2+Gt.3+Gt.6];
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],T![1,0,0,0,0],T![0,1,0,0,1],
               T![0,0,1,1,0],T![0,0,1,0,0],T![0,0,0,1,1],
               T!0,T![0,0,0,0,1],T!0,
               T!0,T!0,T!0];

X`KY:=Pic![-3,1,1,1,1];
X`K:=Pic![3,-1,-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],
        [0,0,0,0,0,0,1,0,-1,1,0,0]];

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

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

// -1 curves on the degree 5 dP
CC1:=[Pic.i:i in [2..5]];
CC2:=&cat[[Pic.1-Pic.i-Pic.j:i in [j+1..5]]:j in [2..5]];
CC:=[Eltseq(c):c in (CC1 cat CC2)];
// effective divisors on Y  -- 2 secs with max = 2, 13 with max = 3
EffY:={};
for a in CartesianProduct([[0..2]:i in [1..#CC1]]) do
   for b in CartesianProduct([[0..2]:i in [1..#CC2]]) do
      Include(~EffY,&+[a[i]*CC1[i]:i in [1..#CC1]]+&+[b[i]*CC2[i]:i in [1..#CC2]
]);
   end for;
end for;
X`EffY:=EffY;


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


time Excl:=findExceptional(X,LL); // get 2597 from 41016820 in 892s

// sample from article
ee:=<T.3 + T.4 + T.5, T.2 + T.5, T.1 + T.2 + T.3 + T.5, T.2 + T.3, T.2 + T.3 + T.5, T.1 + T.4 + T.5>;

ee in Excl; // true

H,E:=findHoms(X,LL,ee);
//[H[[i,j]]:i in [j+1..#LL],j in [0..#LL]];