/////////////////////////////////////////////// // // Beauville surface with G=(Z/5)^2 // /////////////////////////////////////////////// X:=rec<Surface|>; Pic:=LatticeWithGram(Matrix(2,[0,1,1,0]): CheckPositive := false); // allows neg-def n:=[5:i in [1..4]]; Gt:=AbelianGroup(n); G:=sub<Gt|Gt.1,Gt.2>; T:=sub<Gt|Gt.3,Gt.4>; X`Name:="Beauville Z5"; X`Big:=Gt; X`Small:=G; X`Tors:=T; X`branch:=[Pic![1,0],Pic![1,0],Pic![1,0], Pic![0,1],Pic![0,1],Pic![0,1]]; X`phi:=[Gt.1,Gt.2,4*Gt.1+4*Gt.2, Gt.1+2*Gt.2+Gt.3,3*Gt.1+4*Gt.2+Gt.4,Gt.1+4*Gt.2+4*Gt.3+4*Gt.4]; X`branchtors:=[T!0,T![1,1],T![4,2],T!0,T![1,4],T![1,0]]; X`KY:=Pic![-2,-2]; X`K:=Pic![2,2]; lines:=#X`branch; X`Lchi:=calculateLchi(X); // find torsion twists... X`Basis:=[[1,0,0,0,0,0],[0,0,0,1,0,0]]; findCoordinates(X,[0,0,0,1,0,0]); X`Ktors:=T![3,3]; // 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 [1..10] do LL:=[Pic![-1,0],Pic![i-1,-1],Pic![i-2,-1]]; Excl:=findExceptional(X,LL); end for; LL:=[Pic![0,-1],Pic![-1,-1],Pic![-1,-2]]; Excl:=findExceptional(X,LL); extendedquiver(X,LL,Excl[1]); LL:=[Pic![-1,0], Pic![0,-1], Pic![-1,-1]]; // I1 LL:=[Pic![1,-1], Pic![0,-1], Pic![-1,-2]]; // IV1 LL:=[Pic![-1,0], Pic![-2,-1], Pic![-3,-1]]; // I-1 LL:=[Pic![-1,-1], Pic![-2,-1], Pic![-1,-2]]; // IV-1 LL:=[Pic![0,-1], Pic![-1,-1], Pic![-1,-2]]; // II0 LL:=[Pic![-1,0], Pic![-1,-1], Pic![-2,-1]]; // I0