This notebook contains intersection calculations on the putative Dolgachev surface Y. 
The generators of the Picard group are given in the order  A,B,C,A1,B1,C1,A2,B2,C2,S,S1,S2,F3,F2.

(* setting up the intersection matrix *)
M=Table[0,{i,1,14},{j,1,14}];
For[i=1,i≤9,i++,M[[i]][[i]]=-2];
For[i=10,i≤12,i++,M[[i]][[i]]=-3];
For[i=1,i≤9,i++,j=Mod[i,9]+1;M[[i]][[j]]=1;M[[j]][[i]]=1;
];
M[[2]][[10]]=1;M[[10]][[2]]=1;
M[[5]][[11]]=1;M[[11]][[5]]=1;
M[[8]][[12]]=1;M[[12]][[8]]=1;
M[[1]][[10]]=a;M[[10]][[1]]=a;
M[[4]][[11]]=a;M[[11]][[4]]=a;
M[[7]][[12]]=a;M[[12]][[7]]=a;
M[[1]][[11]]=b;M[[11]][[1]]=b;
M[[4]][[12]]=b;M[[12]][[4]]=b;
M[[7]][[10]]=b;M[[10]][[7]]=b;
M[[1]][[12]]=c;M[[12]][[1]]=c;
M[[4]][[10]]=c;M[[10]][[4]]=c;
M[[7]][[11]]=c;M[[11]][[7]]=c;

M[[13]][[13]]=0;
M[[13]][[14]]=0;
M[[13]][[10]]=3;M[[10]][[13]]=3;
M[[13]][[11]]=3;M[[11]][[13]]=3;
M[[13]][[12]]=3;M[[12]][[13]]=3;
M[[14]][[10]]=2;M[[10]][[14]]=2;
M[[14]][[11]]=2;M[[11]][[14]]=2;
M[[14]][[12]]=2;M[[12]][[14]]=2;

(* visualization *)
M//MatrixForm

(* finding possible intersections of S with A-s *)
Unset[{a,b,c,aa,bb,cc}];
For[aa=0,aa≤5,aa++,For[bb=0,bb≤5-aa,bb++,
cc=5-aa-bb;
rk=MatrixRank[M/.{a->aa,b->bb,c->cc}];
If[rk≤10,Print[rk," ",{aa,bb,cc}]];
]];

(* this choice is the one that happens in this case *)
M1=M/.{a->1,b->3,c->1}
nm1=NullSpace[M1];
nm1//MatrixForm

M1[[10]]
relS2=-(nm1[[3]]/3 -4  nm1[[1]]/3 )
relS1=-(nm1[[4]]/3 - 5 nm1[[1]]/3 )
(* this looks like having a (1,8) pass through S2 and (1,10) through S1.*)
(*
For the (1,8) we see that it must vanish at the conic through the smooth point, and have triple vanishing at the line through two (1/3)(1,2). *)

relg7=-(relS2 + 2 relS1 -7 nm1[[1]])
(* this looks like a divisor of a rational function whose seventh root gives the cover
It will be (Eq18/(w1 u0^4))^2 (Eq110/(w1 u0^5)) u0^6/(u0^3-u1^2)^2
 *)

(* here we look to compute the conditions on the pushforwards of the sections of the 
2K from the fake projective plane *)
KSing={0,2/7,1/7,0,2/7,1/7,0,2/7,1/7,3/7,3/7,3/7,0,1/2};
Divg=relg7/7
For[m=-6,m≤3,m++,
d=-Ceiling[-2KSing- m Divg];
d2=d.M1.d;
dk=d.M1.{0,0,0,0,0,0,0,0,0,0,0,0,0,1/2};
Print[{d,m,(d2-dk)/2+1}];
]

{0,1,0,0,-1,-2,-3,-2,-1,2,0,0,0,4} + nm2[[1]]