Clear[m1,m3,c1,c3,thd,cbc,cbc0,np,T];
m1[{x_,y_}]:={x,y,1};
m3[{x_,y_}]:={x^3,x^2 y,x y^2,y^3,x^2,x y,y^2,x,y,1};
c1[p_]:=Det[Map[m1[#]&,Append[p,{x,y}]]];
c3[p_]:=Det[Map[m3[#]&,Append[p,{x,y}]]];
thd[cubic_,p_]:=Complement[{x,y}/.Solve[{cubic,c1[p]}==0,{x,y}],p][[1]];
cbc[p_,i_]:=cbc[p,i]=c3[Drop[p,{i}]];
cbc0:=cbc[pp,10];
np[p_,i_,j_,j_]:=np[p,i,j,j]=Complement[{x,y}/.Solve[{cbc[p,j],cbc0}==0,{x,y}],p][[1]];
np[p_,i_,j_,10]:=thd[cbc[p,j],{thd[cbc[p,j],{p[[i]],p[[10]]}],np[p,i,j,j]}];
np[p_,i_,j_,i_]:=thd[cbc0,{thd[cbc0,{p[[i]],p[[j]]}],np[p,i,j,j]}];
np[p_,i_,j_,k_]:=p[[k]]/;(k=!=i||k=!=j||k=!=10);
T[{i_,j_},p_]:=T[{i,j},p]=Table[np[p,i,j,k],{k,1,10}];
(* 
pp={{2,1},{7/4,5/4},{1/5,1},{2/5,4/5},{0,1},{1,0},{1/3,4/3},
{3/2,5/2},{3,3/7},{1/4,7/4}}; 
T[{1,2},pp]
*)