Analysis of embedded points

[1,1,12,28,42] (d=84) with x^84 + y^84 + z^7 + u^3 + t^2
d:=84; A:=[1,1,12,28,42]; B:=[];
D:=P(d,A)-PA(A)-Pb(B);
  // the main feature is a curve of 1/2 of deg 2/(12*14) = 1/168, with
  // 1/14(1,1,12), 1/4(1,1,2), 1/6(1,1,4)
D:=P(d,A)-PA(A)-Pb(B);
D1 := D-&+[t^(2*i) : i in [2..7]]/&*[1-t^i : i in [1,1,2,14]]; // deals with the 1/14
D2 := D1-&+[t^(2*i) : i in [2..3]]/&*[1-t^i : i in [1,1,2,6]]; // deals with the 1/6
D3 := D2-&+[t^(2*i) : i in [2..2]]/&*[1-t^i : i in [1,1,2,4]];

D4:=P(d,A)-PA(A)-Pb(B)-1/168*Phalfc()-Pemb2(7)-Pemb2(3)-Pemb2(2);
 // 0 -- this confirms that the emb 1/2r(1,1,2r-2) are programmed correctly

[1,2,6,16,23]  (d=48) with x^48 + y^24 + z^8 + u^3 + t^2*y
d := 48; A:=[1,2,6,16,23]; B := [[23,1,6,16]];
 // singular curve of 1/2 of deg 48/(2*6*16) = 1/4 with no emb
D:=P(d,A)-PA(A)-Pb(B) -1/4*Phalfc(); // 0

[1,1,1,4,6,13],
d:=13; A:=[1,1,1,4,6]; B:=[]; // sing curve of deg 1/(4*6) with emb 1/4, 1/6
D := P(d,A)-PA(A)-Pb(B)-1/24*Phalfc()-Pemb2(2)-Pemb2(3); // 0

[1,1,2,3,6,13],
 // curve of 1/2 of deg 1/(2*6)
 // curve of 1/3 of deg 1/(3*6)
 // emb 1/6(1,2,3)
 // calc E1 = Oh(13-1-1) = Oh(11), E2 = Oh(-2), E1-E2 = Oh(13)
 // deg 13; (13 + 2)/2*r = 15/12 = a = 4/3
d:=13; A:=[1,1,2,3,6]; B:=[];
D := P(d,A)-PA(A)-Pb(B)-1/3*Ppc(2,0,1)-1/2*Ppc(3,3/2,1);
 // (7/12*t^5 - 5/12*t^4 + 7/12*t^3)
 // inconclusive

[1,1,2,6,9,19],
d:=19; A:=[1,1,2,6,9]; B:=[]
 // curve of 1/2 of deg 1/(2*6)
 // curve of 1/3 of deg 1/(6*9) with emb 1/9(1,3,6)
 // emb 1/6(1,2,3)
 // calc E1 = Oh(19-1-1) = Oh(17), E2 = Oh(-2), E1-E2 = Oh(19)
 // deg 19; (19 + 2)/2*r = a = 3/2
D := P(d,A)-PA(A)-Pb(B)-1/3*Ppc(2,0,1)-1/6*Ppc(3,3/2,1);

[1,1,2,5,8,17],
 // curve of 1/2 with deg 1/(2*8) with emb 1/8(1,2,5)
d:=17; A:=[1,1,2,5,8]; B:=[[5,1,1,3]];
D := P(d,A)-PA(A)-Pb(B)-1/4*Ppc(2,0,1);
D := P(d,A)-PA(A)-Pb(B)-Ppc(2,0,1);
 // t^5 - t^6 + t^7 / denom(1,1,2,8)
 // i.e., we think of 1/8(1,2,5) as adding deg 3*1/2^2 to deg Ga
 // then subtracting t^5-t^6+t^7 / denom(1,1,2,8)

[1,1,2,7,10,21],
 // curve of 1/2 of deg 1/(2*10) with emb 1/10(1,2,7)
d := 21; A := [1,1,2,7,10]; B :=[];
D := P(d,A)-PA(A)-Pb(B) - 1/5*Ppc(2,0,1);
D := P(d,A)-PA(A)-Pb(B) - Ppc(2,0,1);
 // (t^11+t^10+t^9+2*t^7+t^5+t^4+t^3) / denom(1,1,2,10)

[1,1,3,4,8,17],
 // curve of 1/4 of deg 1/(4*8) with emb 1/8(1,3,4)
d:=17; A:=[1,1,3,4,8]; B:=[[3,1,1,1]];
D := P(d,A)-PA(A)-Pb(B) - 1/2*Ppc(4,5,1);
D := P(d,A)-PA(A)-Pb(B) - Ppc(4,4,1);
 // (-2*t^11-5*t^10-4*t^9-4*t^8-5*t^7-4*t^6-4*t^5-5*t^4-2*t^3)
  /denom(1,1,4,8)
 or
  D := P(d,A)-PA(A)-Pb(B) - Ppc(4,5,1);
 // (-3*t^11-7*t^10-6*t^9-6*t^8-7*t^7-6*t^6-6*t^5-7*t^4-3*t^3)
  /denom(1,1,4,8)

[1,1,4,6,6,18],
 // curve of 1/2 of deg 18/(4*6*6) = 1/8 with emb 3 x 1/6(1,1,4) and 1/4(1,1,2)
d := 18; A:=[1,1,4,6,6]; B:=[];
D := P(d,A)-PA(A)-Pb(B)-1/2*Ppc(2,0,1)-3*Pemb2(3)-Pemb2(2);

[1,1,4,6,8,20],
 // curve of 1/2 of deg 20/(4*6*8) = 5/48 with emb 2 x 1/4(1,1,2)
 // and 1/6(1,1,4) and 1/8(1,1,6)
d := 20; A:=[1,1,4,6,8]; B:=[];
D := P(d,A)-PA(A)-Pb(B)-5/12*Ppc(2,0,1)-2*Pemb2(2)-Pemb2(3)-Pemb2(4);

[1,1,4,6,12,24],
 // curve of 1/2 of deg 24/(4*6*12) = 1/12 with emb 2 x 1/4(1,1,2), 2 x 1/6(1,1,4)
d := 24; A:=[1,1,4,6,12]; B:=[];
D := P(d,A)-PA(A)-Pb(B)-1/3*Ppc(2,0,1)-2*Pemb2(2)-2*Pemb2(3);