curves of transverse type 1/3(1,2)

Y12 in PP(1,2,3,3,3)

has a surface section

S12 in PP(2,3,3,3)

This is iso to dP2 S4 in PP(2,1,1,1) (write B = 3A for the new Oh(1), with the
branch curve Ga as a 1/3 locus). Then
 A = K(S,orb) = -B + 2/3 Ga
so that
 2A = 1/3 Ga,
 3A = B.

I still don't have a systematic way of thinking about the orbifold curve.
Instead, use RR on S with [nA] and K(S) = [K(S,orb)].

[nA] = nA - 2/3 Ga if n == 1 mod 3
          nA - 1/3 Ga if n == 2
          nA              if n == 0.

So setting B^2 = b gives for n = 3k, 3k+1, 3k+2

P(n) = h^0(S, kB),  h^0(S, (k-1)B)  or  h^0(S, kB).

1 if n = 0
0 if n = 1
1 + k(k+1)b/2 for n = 3k+2, k >= 0
1 + k(k+1)b/2 for n = 3k, k >= 1
1 + (k-1)k b/2 for n = 3k+1, k >= 1

S<s> := PowerSeriesRing(Q,50);
b:=1;
P := 1 + &+[(1 + k*(k+1)*b/2)*s^(3*k+2) : k in [0..20]] + &+[(1 + k*(k+1)*b/2)*s^(3*k) : k in [1..20]] + &+[(1 + (k-1)*k*b/2)*s^(3*k+1) : k in [1..20]];
P*&*[(1-s^i) : i in [2,3,3]];

Suppose I compare with the formula in terms of P1, P2.

P1 = pg, P2 = K^2 + pg + 1;
P*(1-t)^3 = 1 + (P1-3)*s+ (P2-3P1+3)*s^2 + (P1-3)*s^3 + s^4;

P1:=0; P2:=1;
Pa := (1 + (P1-3)*s+ (P2-3*P1+3)*s^2 + (P1-3)*s^3 + s^4)/(1-s)^3;

expt shows that P - Pa = b*(s^3 + s^5 + 3*s^6 + s^7 + 3*s^8 + 6*s^9 + 3*s^10 + 6*s^11 + 10*s^12 + 6*s^13
  + 10*s^14 + ..)
 = s^3*(1-s+s^2)/((1-s^3)^2*(1-s))

Pb := b*s^3*(1-s+s^2)/((1-s^3)^2*(1-s));

"size of function = order of leading term, at pole of order 3, at s=1"
It is obviously 1/9.
Magma can't do PowerSeries -> MeromorphicFunction, OrderOfPole, 
LeadingTerm, etc. Maybe they will want to take this on board?

P1:=0; P2:=1;
Pa := (1 + (P1-3)*s+ (P2-3*P1+3)*s^2 + (P1-3)*s^3 + s^4)/(1-s)^3;
  --> degree P2 - P1 - 1
b := 1;
Pb := b*s^3*(1-s+s^2)/((1-s^3)^2*(1-s));
  --> degree b*1/9
P := Pa + Pb;

> Pa := (1 + (P1-3)*s+ (P2-3*P1+3)*s^2 + (P1-3)*s^3 + s^4)/(1-s)^3;
> Pb := b*s^3*(1-s+s^2)/((1-s^3)^2*(1-s));
> P:=Pa+Pb;
> P;
1 + s^2 + 2*s^3 + s^4 + 2*s^5 + 4*s^6 + 2*s^7 + 4*s^8 + 7*s^9 + 4*s^10 + 7*s^11 
    + 11*s^12 + 7*s^13 + 11*s^14 + 16*s^15 + 11*s^16 + 16*s^17 + 22*s^18 + 
    16*s^19 + 22*s^20 + 29*s^21 + 22*s^22 + 29*s^23 + 37*s^24 + 29*s^25 + 
    37*s^26 + 46*s^27 + 37*s^28 + 46*s^29 + 56*s^30 + 46*s^31 + 56*s^32 + 
    67*s^33 + 56*s^34 + 67*s^35 + 79*s^36 + 67*s^37 + 79*s^38 + 92*s^39 + 
    79*s^40 + 92*s^41 + 106*s^42 + 92*s^43 + 106*s^44 + 121*s^45 + 106*s^46 + 
    121*s^47 + 137*s^48 + 121*s^49 + O(s^50)
> P*&*[1-s^i: i in [2,3,3]];   
1 + s^9 + O(s^50)

I try to add 1/5(2,2)

K5<ep5> := CyclotomicField(5);
 ep5^-1/((1-ep5^2)*(1-ep5^2));  // Answer 1/5*(ep5^3 - 2*ep5^2 + ep5)

So the contribution contr5 is 1/(1-t^5)*1/5*(t^2-2t^3+t^4)
 
this should add -1/5 to A^2 (as calculated from P1, P2). So

Pc := -1/5*1/(1-s)^2;
Pd := 1/(1-s^5)*1/5*(s^2-2*s^3+s^4);

> P1:=1; P2:=3;
> Pa := (1 + (P1-3)*s+ (P2-3*P1+3)*s^2 + (P1-3)*s^3 + s^4)/(1-s)^3;
> Pb := b*s^3*(1-s+s^2)/((1-s^3)^2*(1-s));
> Pc := -1/5*s^2*1/(1-s)^3;
> Pd := 1/(1-s^5)*1/5*(s^2-2*s^3+s^4);
> P:=Pa+Pb+Pc+Pd;
 // gives 1 + s + 3*s^2 + 5*s^3 + 7*s^4 + 11*s^5 + ..
> P*&*[1-s^i: i in [1,2,2,3,3,5]];
1 - s^4 - s^5 + s^6 - s^7 - 3*s^8 + 3*s^9 + s^10 - s^11 + s^12 + s^13 - s^17 + 
    O(s^50)
> P*&*[1-s^i: i in [1,2,2,3,3,5,6]];
1 - s^4 - s^5 - s^7 - 3*s^8 + 3*s^9 + 2*s^10 + 2*s^13 + 3*s^14 - 3*s^15 - s^16 -
    s^18 - s^19 + s^23 + O(s^50)

=====================

curve of type 1/4(3)

Y(16) in PP(1,3,4,4,4)

curve of 1/4(1,3)
   one 1/3(1,1,1) singularity

S(16) in PP(3,4,4,4) iso S(4) in PP(3,1,1,1) (write B for Oh(1) here).

S(4) has KS = -2B and 1/3(1,1,1) as its only singularity. B^2 = 4/3
H^0(S,kB) = 1 + 1/2k(k+2)B^2 - 1/3 (if k ==2 (mod 3) or 0.

Check:
if k = -2 then 1 + 0
if k = -1 then 1 + 1/2(-1)(+1)(4/3) - 1/3
if k = 0 then 1 + 0
if k = 1 then 1 + 1/2(1)(3)(4/3) - 0
if k = 2 then 1 + 1/2(2)(4)(4/3) - 1/3

S(16) has a 1/4 orbifold curve along Ga : (x=0) in 3B, and
the polarisation is A = KS + (3/4)Ga = -2B + (3/4)Ga == (1/4)B.
It satisfied
 3A == 1/4 Ga, because = -6B + (9/4)Ga
 4A = B
 A^2 = 1/12.

PolyRR(S,A) written in terms of A, K(S,orb) gives
 chi(nA) = 1 + n(n-1)A^2/2 = 1 + n(n-1)/24.

The actual values come from [nA] = nA -(3/4)Ga, -(1/2)Ga, -(1/4)Ga, 0
and KS = [K(S,orb)] = K(S,orb) - (3/4)Ga. Thus
H^0(S(16),nA) =
1 if n = 0
0 if n = 1, because = H^0(S,-2B)
0 if n = 2, because = H^0(S,-4B+(6/4)Ga-(1/2)Ga) = H^0(S,-B)
1 if n = 3, because = H^0(S,-6B+(9/4)Ga-(1/4)Ga) = H^0(S,Oh)

H^0(S,-2nB+(3n/4)Ga -(3/4)Ga, -(1/2)Ga, -(1/4)Ga, 0) for n >= 2

which means

H^0(S,(-8k-2+(12k+3)/4(3B)-(3/4)3B)) = H^0(S,(k-2)B) if n = 4k+1

H^0(S,(-8k-4+(12k+6)/4(3B)-(2/4)3B)) = H^0(S,(k-1)B) if n = 4k+2

H^0(S,(-8k-6+(12k+9)/4(3B)-(1/4)3B)) = H^0(S,kB) if n = 4k+3

H^0(S,kB) if n = 4k.

Write this as ActRR([nA])

  = 1 + (k-2)kB^2/2        if n = 4k+1
    1 + (k-1)(k+1)B^2/2    if n = 4k+2
    1 + k(k+2)B^2/2        if n = 4k+3
    1 + k(k+2)B^2/2        if n = 4k

or same -1/3 if the reduced value is 2 mod 3. (Just means round down.)

S<s> : PowerSeriesRing(Q,50);

BB := 4/3; // this is the degree of 4A or the degree of S(4) in PP(3,1,1,1)

P := 1
 + &+ [(1 + Floor(k*(k+2)*BB/2))*s^(4*k) : k in [1..20]]
 + &+ [(1 + Floor((k-2)*k*BB/2))*s^(4*k+1) : k in [1..20]]
 + &+ [(1 + Floor((k-1)*(k+1)*BB/2))*s^(4*k+2) : k in [0..20]]
 + &+ [(1 + Floor(k*(k+2)*BB/2))*s^(4*k+3) : k in [0..20]]
  ;

/ / 1 + s^3 + 3*s^4 + s^6 + 3*s^7 + ..; Looks nice enough
P* &*[1-s^i : i in [3,4,4,4]]; // 1 - s^16 + O(s^50)

The formula in terms of P1, P2 is

P1 := 0; P2 := 0;
Pa := (1+(P1-3)*s+(P2-3*P1+3)*s^2+(P1-3)*s^3+s^4)/(1-s)^3;

The difference is the sum of orbifold corrections:

Pcor := P-Pa;
 // Pcor is 3*s^3 + 8*s^4 + 9*s^5 + 15*s^6 + 

Pcor *&*[1-s^i: i in [3,4,4]];
3*s^3 + 8*s^4 + 9*s^5 + 12*s^6 + 9*s^7 + 8*s^8 + 3*s^9
  // it needs the whole denom [3,4,4]

K3<ep3> := CyclotomicField(3);
 ep3^-1*1/(1-ep3)^2;
  // effect of having 1/3(1,1) sing on plurigenera is
  // 1/3*s^2/(1-s)^3-1/3*s^2*/(1-s^3), which equals s^3/((1-s)^2*(1-s^3))

P3cor := s^3/((1-s)^2*(1-s^3));
P4cont := P-Pa-P3cor;
P4cont*(1-s)*(1-s^4)^2;
 // 2*s^3 + 4*s^4 + 4*s^6 + 2*s^7
 
Conclusion: the effect of having 1/3(1,1) is to add
  s^3/((1-s)^2*(1-s^3))
 The effect of having a curve Ga of 1/4(3) of that degree is to add
  (2*s^3 + 4*s^4 + 4*s^6 + 2*s^7)/((1-s)*(1-s^4)^2);

Try it out from scratch

P1 := 0; P2 := 0;
Pa := (1+(P1-3)*s+(P2-3*P1+3)*s^2+(P1-3)*s^3+s^4)/(1-s)^3;
n := 2; // number of P(1/3(1,1))
Pb := n*s^3/((1-s)^2*(1-s^3));
b := 1; // degree Ga(1/4(3)) eq b*1/8
Pc := b*(s^3 + 2*s^4 + 2*s^6 + s^7)/((1-s)*(1-s^4)^2);

P:=Pa+Pb+Pc;
P;
P* &* [(1-s^i) : i in [3,4,4]];
================

Now let's do some curves of 1/5(4). It's even easier that the 1/4(3) case.

[ 1, 4, 5, 5, 5, 20 ],
[ 1, 4, 5, 5, 10, 25 ],
[ 1, 4, 5, 10, 10, 30 ],
[ 1, 4, 5, 10, 20, 40 ],
[ 1, 4, 5, 15, 20, 45 ],
[ 1, 4, 5, 20, 30, 60 ],

Y(20) in PP(1,4,5,5,5)

pass to the surface S(20) in PP(4,5,5,5) iso S(4) in PP(4,1,1,1), that is iso
PP^2 with a marked quartic Ga as 1/5(4) curve.

KS = -3B, whereas

A = K(S,orb) = -3+(4/5)Ga == (-3 + 16/5)B == (1/5)B

A has 4A = (1/4)Ga, and 5A = B. Also A^2 = 1/25.

So PolyRR(nA) = 1 + 1/2*n(n-1)*1/25.
[nA] = nA - (4/5)Ga or -(3/5)Ga or -(2/5)Ga or -(1/5)Ga or 0.

We know this gives (1-t^20)/(1-t^4)(1-t^5)^3.

P := (1-s^20)/((1-s^4)*(1-s^5)^3));
P1 := 0; P2 := 0;
Pa :=  (1+(P1-3)*s+(P2-3*P1+3)*s^2+(P1-3)*s^3+s^4)/(1-s)^3;
P-Pa;

gives

(2*s^3 + 6*s^4 + 12*s^5 + 14*s^6 + 18*s^7
   + 18*s^8 + 14*s^9 + 12*s^10 + 6*s^11 + 2*s^12)/(1-s^4)*(1-s^5)^2;

================Scrap:

I suspect that the coeffs 8 should be 9, and the factor
(1-t^3) in the denom is not needed:

Factorization(t^6+3*t^5+3*t^4+4*t^3+3*t^2+3*t+1);

[
    <t^2 + t + 1, 1>,
    <t^4 + 2*t^3 + 2*t + 1, 1>
]

PP := Ptemp - 3*s^3*(1+2*s+2*s^2+s^3)/((1-s)*(1-s^4)^2);

-t^4 - 6*t^5 - 3*t^6 + 5*t^8 + 3*t^9

PP := Ptemp + 3*s^3*(1+2*s+2*s^2+s^3)/((1-s)*(1-s^4)^2);

(3*t^3 + 8*t^4 + 9*t^5 + 12*t^6 + 9*t^7 + 8*t^8 + 3*t^9);

(t^3 + t^5 + 2*t^6 + 6*t^7 + 4*t^8 + 4*t^9 + 2*t^10 + 6*t^11 + 4*t^12 + 4*t^13 + 
    2*t^14 + 5*t^15 + 4*t^16 + 3*t^17)
    
    
    P4cont*(1-s)*(1-s^4)^2;
Factorization(t^3 + 2*t^4 + 2*t^6 + t^7);x

=============================
start again with S(12) in PP(2,3,3,3)
P1 := 0, P2 := 1;
Pa := (1 + (P1-3)*s+ (P2-3*P1+3)*s^2 + (P1-3)*s^3 + s^4)/(1-s)^3;
P := (1-s^12)/&*[1-s^i : i in [2,3,3,3]];
P-Pa;
1/2*(P-Pa);
(1/2*(P-Pa)) * &* [1-s^i : i in [1,3,3]];  // s^3 -s^4 + s^5

  i.e. the effect of adding this 1/3(2) curve is an integer multiple
  of  s^3(1 -s + s^2)/(1-s)*(1-s^3)^2.  This has deg 1/9.

========
S(16) in PP(3,4,4,4)
P1:=0; P2:=0;
Pa := (1 + (P1-3)*s+ (P2-3*P1+3)*s^2 + (P1-3)*s^3 + s^4)/(1-s)^3;
P := (1-s^16)/&*[1-s^i : i in [3,4,4,4]];

P3corr := s^3/((1-s)^2*(1-s^3)); // corrects for 1/3(1,1)
1/2*(P-Pa-P3cor);
1/2*(P-Pa-P3cor)* &*[1-s^i : i in [1,4,4]]; // s^3 + 2*s^4 + 2*s^6 + s^7

  i.e. the effect of adding this 1/4(3) curve is an integer multiple
  of  s^3(1 +2s + 2s^3 + s^4)/(1-s)*(1-s^4)^2.  This has deg 6/16.

========
S(20) in PP(4,5,5,5)
P1:=0; P2:=0;
Pa := (1 + (P1-3)*s+ (P2-3*P1+3)*s^2 + (P1-3)*s^3 + s^4)/(1-s)^3;
P := (1-s^20)/&*[1-s^i : i in [4,5,5,5]];

P-Pa;
1/2*(P-Pa)* &*[1-s^i : i in [1,5,5]];
  // s^3 + 2*s^4 + 3*s^5 + s^6 + 3*s^7 + 2*s^8 + s^9

  i.e. the effect of adding this 1/5(4) curve is an integer multiple
  of  s^3(1+2*s+3*s^2+s^3+3*s^4+2*s^5+s^6)/(1-s)*(1-s^5)^2.
  = s^3 (1 - s + s^2) (1 + 3t + 5t^2 + 3t^3 + t^4)

Factorization(1+2*t+3*t^2+t^3+3*t^4+2*t^5+t^6);
[
    <t^2 - t + 1, 1>,
    <t^4 + 3*t^3 + 5*t^2 + 3*t + 1, 1>
]
========
S(36) in PP(5,6,6,18)
P1:=0; P2:=0;
Pa := (1 + (P1-3)*s+ (P2-3*P1+3)*s^2 + (P1-3)*s^3 + s^4)/(1-s)^3;
P := (1-s^36)/&*[1-s^i : i in [5,6,6,18]];

Need to correct for 1/5(1,3):
K5<ep5> := CyclotomicField(5);
ep5^-1/((1-ep5)*(1-ep5^3)); // 1/5*(-2*ep5^3 - ep5^2 - 2*ep5)

This means I need to take off something like
P5corr := s^3/((1-s)*(1-s^3)*(1-s^5));
 //  - 1/5*(1/(1-s^5))*(-s^2+2*s^3+-s^4)

    // S11 in PP(1,1,3,5) has 1/3(1,1) and 1/5(1,3)
    P1:=2; P2:=3;
    Pa := (1 + (P1-3)*s+ (P2-3*P1+3)*s^2 + (P1-3)*s^3 + s^4)/(1-s)^3;
    P := (1-s^11)/&*[1-s^i: i in [1,1,3,5]];
    P-Pa;
    P-Pa-s^3/((1-s)^2*(1-s^3));
    (P-Pa-s^3/((1-s)^2*(1-s^3)))*[1-s^i : i in [1,3,5]];
    (P-Pa-s^3/((1-s)^2*(1-s^3)))* &*[1-s^i : i in [1,3,5]];
    Factorization(t^3+t^4+2*t^5+t^6+t^7); // t^3*(1+t^2)*(1+t+t^2)

P5corr := s^3*(1+s^2)*(1+s+s^2)/&*[1-s^i : i in [1,3,5]];

(P-Pa-P5corr)*&*[1-s^i : i in [1,6,6]];
 // s^3 + 2*s^4 + 3*s^5 + 4*s^6 + 2*s^7 + 4*s^8 + 3*s^9 + 2*s^10 + s^11
Factorization(t^3+2*t^4+3*t^5+4*t^6+2*t^7+4*t^8+3*t^9+2*t^10+t^11);

   i.e. the effect of adding this 1/6(5) curve is an integer multiple
   of s^3(1+2*s+3*s^2+4*s^3+2*s^4+4*s^5+3*s^6+2*s^7+s^8)/(1-s)(1-s^6)^2;

========
S(42) in PP(6,7,7,21)

    // This has 2 x 1/3(1,1) away from a curve Ga of 1/7(6).
    // S7 in PP(1,1,1,3) has 1/3(1,1)
    P1:=3; P2:=6;
    Pa := (1 + (P1-3)*s+ (P2-3*P1+3)*s^2 + (P1-3)*s^3 + s^4)/(1-s)^3;
    P := (1-s^7)/&*[1-s^i: i in [1,1,1,3]];
    P-Pa;
    P-Pa-s^3/((1-s)^2*(1-s^3));

P3corr := s^3/((1-s)^2*(1-s^3));
P1:=0; P2:=0;
Pa := (1 + (P1-3)*s+ (P2-3*P1+3)*s^2 + (P1-3)*s^3 + s^4)/(1-s)^3;
P := (1-s^42)/&*[1-s^i : i in [6,7,7,21]];

(P-Pa-P3corr)*&*[1-s^i: i in [1,7,7]];

Factorization(t^3+2*t^4+3*t^5+4*t^6+5*t^7+3*t^8+5*t^9+4*t^10+3*t^11+2*t^12+
t^13);

// s^3*(1+2*s+3*s^2+4*s^3+5*s^4+3*s^5+5*s^6+4*s^7+3*s^8+2*s^9+s^10)

   i.e. the effect of adding this 1/7(6) curve is an integer multiple
   of s^3(1+2*s+3*s^2+4*s^3+5*s^4+3*s^5+5*s^6+4*s^7+3*s^8+2*s^9+s^10)/(1-s)(1-s^7)^2;

========
  I begin to detect a pattern.

s^3/(1-s)(1-s^r)^2

times

r=3: 1-s+s^2                                         deg 1
r=4: 1+2s+2s^3+s^4                                   deg 6
r=5: 1+2*s+3*s^2+s^3+3*s^4+2*s^5+s^6                 deg 13
r=6: 1+2*s+3*s^2+4*s^3+2*s^4+4*s^5+3*s^6+2*s^7+s^8   deg 22
r=7: 1+2*s+3*s^2+4*s^3+5*s^4+3*s^5+5*s^6+4*s^7+3*s^8+2*s^9+s^10  deg 33

I guess the general form for general r:

 1 + 2s + 3s^2 + .. (r-2)s^(r-3) + (r-4)s^(r-2) 
               + (r-2)s^(r-1) + (r-1)s^r + .. down to 2s^(2r-5) + s^(2r-4)

r:= 8; temp:=&+[(i+1)*s^i:i in [0..r-3]]+(r-4)*s^(r-2)
                             +&+[(2*r-3-i)*s^i:i in [r-1..2*r-3]];
Pccorr8 := s^3*temp*1/((1-s)*(1-s^8)^2);

Can't find any suitable surfaces, but try the CY 3-fold

[ 3, 5, 8, 8, 24, 48 ],

This has 2 x 1/3 (2,2,2), 1/5(3,3,4) and Ga of type 1/8(3.5).

========
S(80) in PP(7,8,24,40)
    
This has 1/7(8,40) = 1/7(1,5) and 1/24(7,40) = 1/24(7,16) in addition to the
desired curve Ga of 1/8(7).
 // Digression: Treat S15 in P(1,1,5,7) to find P7corr
 P1 := 2; P2 := 3;
 Pa := (1 + (P1-3)*s+ (P2-3*P1+3)*s^2 + (P1-3)*s^3 + s^4)/(1-s)^3;
 P := (1-s^15)/&*[1-s^i : i in [1,1,5,7]];
 (P-Pa)*&*[1-s^i : i in [1,5,7]];
 P eq Pa + s^3*S!(1+t+2*t^2+2*t^3+3*t^4+2*t^5+2*t^6+t^7+t^8)/&*[1-s^i : i in [1,5,7]];
 Pcorr7 := s^3*(1+s+2*s^2+2*s^3+3*s^4+2*s^5+2*s^6+s^7+s^8)/&*[1-s^i : i in [1,5,7]];

r := 8; curvecorr := 

============
Start again:
[ 1, 1, 1, 1, 3, 7 ],

P1 := 3; P2 := 6; // these are the P1, P2 of the surface section
Pa := (1 + (P1-3)*s+ (P2-3*P1+3)*s^2 + (P1-3)*s^3 + s^4)/(1-s)^4;
P := (1-s^7)/&*[1-s^i : i in [1,1,1,1,3]];
P-Pa;
s^3 + 3*s^4 + 6*s^5 + 11*s^6 + ..
(P-Pa)*&*[1-s^i: i in [1,1,1,3]];  // s^3

i.e. the sing 1/3(1,1,1) contributes s^3/(1-s)^3(1-s^3)

[ 1, 1, 1, 2, 2, 7 ]

P1 := 2; P2 := 5; // these are the P1, P2 of the surface section
Pa := (1 + (P1-3)*s+ (P2-3*P1+3)*s^2 + (P1-3)*s^3 + s^4)/(1-s)^4;
P := (1-s^7)/&*[1-s^i : i in [1,1,1,2,2]];
P-Pa;
(P-Pa)*&*[1-s^i : i in [1,1,2,2]]; // -s^3

i.e. the line Ga of 1/2(1,1) with degree 1/4 contributes
 -s^3/(1-s)^2(1-s^2)^2 // deg -3/4

[ 1, 1, 1, 2, 4, 9 ],
This has a line Ga of 1/2, with deg Ga = 1/8 and 1/4(1,1,2) on it.

P1 := 2; P2 := 4; // these are the P1, P2 of the surface section
Pa := (1 + (P1-3)*s+ (P2-3*P1+3)*s^2 + (P1-3)*s^3 + s^4)/(1-s)^4;
P := (1-s^9)/&*[1-s^i : i in [1,1,1,2,4]];
P-Pa;
(P-Pa)*&*[1-s^i : i in [1,1,2,4]]; // s^4
 // s^4/(1-t)^2(1-t^2)(1-t^4);
 // N.B. It is not clear from this how it breaks down into a contr from the
 line and a separate contr from the nonisolated 1/4.
 Preferred answer because no t^3 in it. It means that we calc the contr of
 1/4(1,1,2) including curve of 2 of deg 1/8. //

[ 1, 1, 1, 3, 5, 11 ],
This has 1/3(1,1,1) and 1/5(1,1,3); we already know the first contr.

P1 := 2; P2 := 3; // these are the P1, P2 of the surface section
Pa := (1 + (P1-3)*s+ (P2-3*P1+3)*s^2 + (P1-3)*s^3 + s^4)/(1-s)^4;
P := (1-s^11)/&*[1-s^i : i in [1,1,1,3,5]];
P3corr := -s^3/((1-s)^3*(1-s^3));
P-Pa+P3corr;
(P-Pa+P3corr)*&*[1-s^i : i in [1,1,1,5]]; // s^3 + s^5
  // this is the effect of having a divisor 2/5 P on nonsingular curve C.

NB. Can do the general 1/r(1,1,r-2) case from the orbifold curve case:
For r odd: Contribution of 1/r(1,1,r-2) is
  // (s^3 + s^5 + .. s^r)/(1-s)^3(1-s^r)

h:=1; g :=1;
Pa := S!((1+h*t+g*t^2+h*t^3+t^4)/(1-t)^2);
r := 7; a := 3;
Pb := &+[Floor(i*a/r)*s^i : i in [3..60]];
  // (s^3 + s^5 + s^7)/(1-s)(1-s^7)

[ 1, 1, 1, 5, 7, 15 ],
This is a check; I predict that the only contribution
is (s^3 + s^5 + .. s^r)/(1-s)^3(1-s^r)
P1 := 2; P2 := 3; // these are the P1, P2 of the surface section
Pa := (1 + (P1-3)*s+ (P2-3*P1+3)*s^2 + (P1-3)*s^3 + s^4)/(1-s)^4;
P7corr := (s^3 + s^5 + s^7)/((1-s)^3*(1-s^r));
(1-s^15)/&*[1-s^i:i in [1, 1, 1, 5, 7]] eq Pa + P7corr; // true

[ 1, 1, 1, 4, 6, 13 ]
similar to above [ 1, 1, 1, 2, 4, 9 ], has a line Ga of 1/2, with
deg Ga = 1/(2*3*13) plus nonisolated 1/6(1,1,4) and 1/4(1,1,2) on it.
P1 := 2; P2 := 3; // these are the P1, P2 of the surface section
Pa := (1 + (P1-3)*s+ (P2-3*P1+3)*s^2 + (P1-3)*s^3 + s^4)/(1-s)^4;
P := (1-s^13)/&*[1-s^i : i in [1,1,1,4,6]];
P-Pa;
(P-Pa)*&*[1-s^i : i in [1,1,4,6]]; // s^3 + 2*s^4 + 2*s^5 + 3*s^6 + 2*s^7 + 2*s^8 + s^9

 i.e. the contribution from this curve and it nonisolated qt points is
s^3(1+2*s+2*s^2+3*s^3+2*s^4+2*s^5+s^6)/(1-s)^2(1-s^4)(1-s^6)

P eq Pa + s^3*(1 + 2*s + 2*s^2 + 3*s^3 + 2*s^4 + 2*s^5 + s^6)/((1-s)^2*(1-s^4)*(1-s^6)); // true

[ 1, 1, 2, 2, 2, 8 ],
This is the simplest example of a nonisolated locus -- a curve of 1/2 having
degree 8/2^3 = 4/4.

P1:=1; P2:=4;
Pa := (1 + (P1-3)*s+ (P2-3*P1+3)*s^2 + (P1-3)*s^3 + s^4)/(1-s)^4;
P := (1-s^8)/&*[1-s^i : i in [1,1,2,2,2]];
P-Pa;
1/4*(P-Pa)*&*[1-s^i : i in [1,1,2,2]]; // -s^3

This curve contributes -4*s^3/(1-s)^2(1-s^2)^2.

[ 1, 1, 2, 2, 3, 9 ],
 The singularity is a line Ga of 1/2(1,1) with degree 1/4, and as with the 7 case
 above, it should contributes
 -s^3(1+s+s^2)^2/(1-s)^2(1-s^2)2 // deg 1/4
P1:=1; P2:=3;
Pa := (1 + (P1-3)*s+ (P2-3*P1+3)*s^2 + (P1-3)*s^3 + s^4)/(1-s)^4;
P := (1-s^9)/&*[1-s^i : i in [1,1,2,2,3]];
P eq Pa + -s^3/((1-s)^2*(1-s^2)^2); // true

[ 1, 1, 2, 2, 4, 10 ],
This has a line of 1/2 of degree 10/(2.2.4) = 5/8 with a nonisolated
1/4(1,1,2) on it. Similar to [ 1, 1, 1, 2, 4, 9 ] above, but not exactly.
P1:=1; P2:=3;
Pa := (1 + (P1-3)*s+ (P2-3*P1+3)*s^2 + (P1-3)*s^3 + s^4)/(1-s)^4;
P := (1-s^10)/&*[1-s^i : i in [1,1,2,2,4]];
P-Pa;
(P-Pa)*&*[1-s^i : i in [1,1,2,4]]; //  -2*s^3 + s^4 - 2*s^5

[ 1, 1, 2, 2, 5, 11 ]
 This has 1/5(1,2,2) and a line of 1/2 of degree 1/4.
P1:=1; P2:=3;
Pa := (1 + (P1-3)*s+ (P2-3*P1+3)*s^2 + (P1-3)*s^3 + s^4)/(1-s)^4;
P := (1-s^11)/&*[1-s^i : i in [1,1,2,2,5]];
P-Pa -s^3/((1-s)^2*(1-s^2)^2);
(P-Pa -s^3/((1-s)^2*(1-s^2)^2))*&*[1-s^i : i in [1,2,2,5]]; // -3*s^3 - 3*s^4 - 2*s^5 - 3*s^6 - 3*s^7
Factorization(-3*t^3 - 3*t^4 - 2*t^5 - 3*t^6 - 3*t^7);
  This seems to say that 1/5(1,2,2) makes the contr
   -s^3(3 + 3*s + 2*s^2 + 3*s^3 + 3*s^4)/(1-s)(1-s^2)^2(1-s^5)
  
[ 1, 1, 2, 3, 3, 10 ]
  A curve Ga of 1/3(1,2), of degree 1/9
P1:=1; P2:=2;
Pa := (1 + (P1-3)*s+ (P2-3*P1+3)*s^2 + (P1-3)*s^3 + s^4)/(1-s)^4;
P := (1-s^10)/&*[1-s^i : i in [1,1,2,3,3]];
P-Pa;
(P-Pa)*&*[1-s^i : i in [1,1,3,3]];  // 2*s^3 + s^4 + 2*s^5
  The curve Ga of 1/3(1,2), of degree 1/9 contributes 
  s^3(2 + s + 2*s^2)/(1,1,3,3)

[ 1, 1, 2, 4, 4, 12 ],
   A curve Ga of 1/2, of degree 12/2.4.4 = 3/8 and emb 3 x 1/4(1,1,2).
P1:=1; P2:=2;
Pa := (1 + (P1-3)*s+ (P2-3*P1+3)*s^2 + (P1-3)*s^3 + s^4)/(1-s)^4;
P := (1-s^12)/&*[1-s^i : i in [1,1,2,4,4]];
P-Pa;
1/3*(P-Pa)*&*[1-s^i : i in [1,1,2,4]];  // s^4

  i.e. curious, the curve contr 3*s^4/(1,1,2,4), exact 3 times the contr of the
  line Ga of 1/2, with deg Ga = 1/8 and 1/4(1,1,2) on [ 1, 1, 1, 2, 4, 9 ].

[ 1, 1, 2, 4, 7, 15 ],
   The same line Ga of 1/2 with deg Ga = 1/8 and a single 1/4(1,1,2).
   Try to use it to determine contr of 1/7(1,2,4).
P1:=1; P2:=2;
Pa := (1 + (P1-3)*s+ (P2-3*P1+3)*s^2 + (P1-3)*s^3 + s^4)/(1-s)^4;
P := (1-s^15)/&*[1-s^i : i in [1,1,2,4,7]];
P-Pa-s^4/&*[1-s^i : i in [1,1,2,4]];
 (P-Pa-s^4/&*[1-s^i : i in [1,1,2,4]])*&*[1-s^i: i in [1,1,2,7]];
 // s^4 + s^5 + s^6 + 2*s^7 + s^8 + s^9 + s^10
 So guess that 1/7(1,2,4) contributes
 s^4*(1 + s^3)/[1,1,2,7]

with hindsight:
d := 15; A := [1,1,2,4,7]; B := [[7,1,2,4]];
(P(d,A) - PA(A) - Pb(B) )*&*[1-t^i : i in [1,1,2,4]]; // t^4
 
[ 1, 1, 2, 5, 8, 17 ],
   d:=17; A:=[1, 1, 2, 5, 8]; B := [[5,1,1,3]];
   P(d,A) - PA(A) - Pb(B) - 1/16*c2;
   As well as isolated 1/5(1,1,3), this has a line Ga of 1/2, with deg Ga =
   1/16, and a single 1/8(1,2,5)

with hindsight:
d := 17; A := [1,1,2,5,8]; B := [[5,1,1,3]];
D := P(d,A) - PA(A) - Pb(B); D*&*[1-t^i : i in [1,1,2,8]]; // -t^9 - t^6 - t^3

[ 1, 1, 2, 6, 8, 18 ]
   a line Ga of 1/2, with deg Ga = 3/16, and a single 1/8(1,1,6)

[ 1, 1, 2, 7, 11, 22 ]
   only isolated 1/7(1,2,11) = 1/7(1,2,4)

K7<ep7> := CyclotomicField(7);
ep7^-1/&*[1-ep7^i : i in [1,2,4]]; //1/7*(-ep7^5-ep7^4+ep7^3-ep7^2+ep7+1)
Pcor7 := 1/7*1/(1-s^7)*(s+s^2-s^3+s^4-s^5-s^6);

1/7*(s-3*s^2+s^3)/(1-s)^4-Pcor7; // -s^4-2*s^5-4*s^6-7*s^7-11*s^8-16*s^9

P1:=1; P2:=2;
Pa := (1 + (P1-3)*s+ (P2-3*P1+3)*s^2 + (P1-3)*s^3 + s^4)/(1-s)^4;
P := (1-s^22)/&*[1-s^i : i in [1,1,2,7,11]];
P eq Pa - 1/7*(s-3*s^2+s^3)/(1-s)^4 + Pcor7; // true

[ 1, 1, 2, 8, 12, 24 ]
   This has curve Ga of 1/2 of deg 1/8 with nonisolated 1/4(1,1,2)

P1:=1; P2:=2;
Pa := (1 + (P1-3)*s+ (P2-3*P1+3)*s^2 + (P1-3)*s^3 + s^4)/(1-s)^4;
P := (1-s^24)/&*[1-s^i : i in [1,1,2,8,12]];
P eq Pa + s^4/((1-s)^2*(1-s^2)*(1-s^4)); // true

[ 1, 1, 3, 3, 7, 15 ],

From above: the sing 1/3(1,1,1) contributes s^3/(1-s)^3(1-s^3)

K7<ep7> := CyclotomicField(7);
ep7^-1/&*[1-ep7^i : i in [1,3,3]]; // 1/7*(ep7^5-ep7^4-2*ep7^3+2*ep7^2+ep7-1)
Pcor7 := 1/7*1/(1-s^7)*(-s+s^2+2*s^3-2*s^4-s^5+s^6);
1/7*(s-5*s^2+s^3)/(1-s)^4 + Pcor7; // 
Pcor3 := s^3/((1-s)^3*(1-s^3));

P1:=1; P2:=1;
Pa := (1 + (P1-3)*s+ (P2-3*P1+3)*s^2 + (P1-3)*s^3 + s^4)/(1-s)^4;
P := (1-s^15)/&*[1-s^i : i in [1,1,3,3,7]];
P eq Pa + 1/7*(s-5*s^2+s^3)/(1-s)^4 + Pcor7 + 5*Pcor3;

===============
General rule: the periodic fractional contribution of 1/r(a,b,c) is the
coefficients of 1/(1-ep^a)(1-ep^b)(1-ep^c). The degree contribution is
1/r(a1*s+a2*s^2+a1*s^3)/(1-s)^4 where a1 and a2 cancel the frac contribution
to P1 and P2 and the numerator has pleasing symmetric shape.

e.g. the sing 1/3(1,1,1) contributes s^3/((1-s)^3*(1-s^3))

K3<ep3> := CyclotomicField(3);
ep3^-1/(1-ep3)^3;
 ep3^-1/(1-ep3)^3; // 1/9*(-ep3 + 1)
 // therefore frac contrib = 1/9*1/(1-s^3)*(s-s^2)
Pcor3 := 1/9*1/(1-s^3)*(s-s^2); // 1/9*s - 1/9*s^2 + 1/9*s^4 - 1/9*s^5
From the first two coefficients only,
Pd3 := 1/9*(-s+5*s^2-s^3)/(1-s)^4;
 Pcor3 + Pd3 eq s^3/((1-s)^3*(1-s^3)); // true

1/11(1,1,9)
We know the contr will be
(s^3 + s^5 + s^7 + s^9 + s^11)/(1-s)^3(1-s^11);

K11<ep11> := CyclotomicField(11);
ep11^-1/((1-ep11)^2*(1-ep11^9));
 Pc11 := 1/11*1/(1-s^11)*(5*s+5*s^2+6*s^3+3*s^4+2*s^5-2*s^6-3*s^7-6*s^8-5*s^9-5*s^10);
-5*ep11^9 - 5*ep11^8 - 6*ep11^7 - 3*ep11^6 - 2*ep11^5 + 2*ep11^4 + 
   3*ep11^3 + 6*ep11^2 + 5*ep11 + 5

Pc11 + 1/11*(-5*s+15*s^2-5*s^3)/(1-s)^4
eq
(s^3 + s^5 + s^7 + s^9 + s^11)/((1-s)^3*(1-s^11));  // true

===============

[ 1, 1, 3, 9, 13, 27 ],
   This has 3 x 1/3(1,1,1) and 1/13(1,3,9)

P1:=1; P2:=1;
Pa := (1 + (P1-3)*s+ (P2-3*P1+3)*s^2 + (P1-3)*s^3 + s^4)/(1-s)^4;
P := (1-s^27)/&*[1-s^i : i in [1,1,3,9,13]];

K13<ep13> := CyclotomicField(13);
ep13^-1/&*[1-ep13^i : i in [1,3,9]];
 /*
1/13*(-2*ep13^11 - 3*ep13^10 - 2*ep13^9 + 2*ep13^8 - 3*ep13^7 - 3*ep13^6 + 
   3*ep13^5 + 3*ep13^4 - 2*ep13^3 + 2*ep13^2 + 3*ep13 + 2)
 */

Pc13 := 1/13*1/(1-s^13)*(2*s^1 + 3*s^2 + 2*s^3 - 2*s^4 + 3*s^5 + 3*s^6
                      - 3*s^7 - 3*s^8 + 2*s^9 - 2*s^10 - 3*s^11 - 2*s^12);
Pd13 := 1/13*(-2*s+5*s^2-2*s^3)/(1-s)^4;

Pc13+Pd13; // s^5 + 2*s^6 + 3*s^7 + 5*s^8 + 8*s^9 + 11*s^10 + ..

P eq Pa + 3*s^3/((1-s)^3*(1-s^3)) + Pc13+Pd13; // true

=============================================

/* This function computes the PERIODIC contribution to Hilbert series of an
isolated orbifold point 1/r(a,b..). It assumes that the univariate polynomial
ring Q[t] is defined and that the input list L = [a1,a2,.. ] consists of
coprime elements. */

function Pc(r,L)
  if &or[GreatestCommonDivisor([r,a]) ne 1 : a in L] // other mugtrap
     then error "Error: Not Coprime";
  end if;
A := (t^r-1) div (t-1);
B := t*&*[1-t^i : i in L];
h_throwaway, al_throwaway, be := XGCD(A,B);
 return r*t*be;
end function;

/* This function computes the GROWING contribution to Hilbert series of an
isolated orbifold point 1/r(a,b..). It assumes that the univariate polynomial
ring Q[t] is defined and that the input list L = [a1,a2,.. ] consists of
coprime elements. */

function Pd(r,L)
n := #L; f := (1-t)^(n+1)*Pc(r,L);
return
&+ [-Coefficient(f,k)*t^i where k is Min(i,n+1-i) : i in [1..n] ]
 / (1-t)^(n+1);
end function;

=============================================

// Want a CY 4-fold with 1/4(1,1,3,3) and 1/5(1,2,3,4)?

a1 := -3; a2 := 3; B := [[4,1,1,3,3],[5,1,2,3,4]];

Pa := (1 + a1*t + a2*t^2 + a2*t^3 + a1*t^4 + t^5)/(1-t)^5;

P := Pa + &+[1/r*(1/(1-t^r)*Pc(r,L) + Pd(r,L))
                       where r is b[1] where L is b[2..#b] : b in B];

=============================================

// Want an orbicurve with 1/8(1) and 1/15(2)?

a1 := -1; a2 := 0; a3 := 0; a4 := 0;
Pa := (1 + a1*t + a2*t^2 + a3*t^3 + a4*t^4
                          + a4*t^5 + a3*t^6 + a2*t^7 +a1*t^8 + t^9)/(1-t)^2;

B := [[8,1],[15,8]];

P := Pa + &+[1/r*(1/(1-t^r)*Pc(r,L) + Pd(r,L))
                       where r is b[1] where L is b[2..#b] : b in B];
 S!P;

That is, C(31) in PP(1,8,15). It is a nonsingular elliptic curves with
K{C,orb} = 7A, where A = 1/8 P + 2/15 Q

========
// CY 3-fold with 3 x 1/3(1,1,1) and 1/13(1,3,9)

P1:=1; P2:=1;
Pa := (1 + (P1-3)*t+ (P2-3*P1+3)*t^2 + (P1-3)*t^3 + t^4)/(1-t)^4;

B := [[3,1,1,1],[3,1,1,1],[3,1,1,1],[13,1,3,9]];

P := Pa + &+[1/r*(1/(1-t^r)*Pc(r,L) + Pd(r,L))
                       where r is b[1] where L is b[2..#b] : b in B];
 S!P;
P*&*[1-t^i : i in [1,1,3,9,13]];  // -t^27 + 1

=====
Y(5) in PP(1,1,1,1,1)

P1 := 5; P2 := 15; // so a1, a2 come from
(1+5*t+15*t^2)*(1-t)^4; // 1 + t + t^2 - 34*t^3 ..
a1 := 1; a2 := 1; // to match this up to term 2
Pa := (1 + a1*t+ a2*t^2 + a1*t^3 + t^4)/(1-t)^4;

S!Pa;
Pa*&*[1-t^i : i in [1,1,1,1,1]]; // gives 1 - t^5;

=====
Y(6) in PP(1, 1, 1, 1, 2)

(1+4*t+11*t^2)*(1-t)^4; // 1 + t^2 - 24*t^3 ..
a1 := 0; a2 := 1;
Pa := (1 + a1*t+ a2*t^2 + a1*t^3 + t^4)/(1-t)^4;

S!Pa;
Pa*&*[1-t^i : i in [1,1,1,1,2]]; // gives 1 - t^6;

=====
Y(7) in PP(1, 1, 1, 1, 3)
(1+4*t+10*t^2)*(1-t)^4; // 1 - 20*t^3 ..
a1 := 0; a2 := 0;
Pa := (1 + a1*t+ a2*t^2 + a1*t^3 + t^4)/(1-t)^4;

B := [[3,1,1,1]]; // basket of 1/3(1,1,1)
P := Pa + &+[1/r*(1/(1-t^r)*Pc(r,L) + Pd(r,L)) where r is b[1] where L is b[2..#b] : b in B];
S!P;
P*&*[1-t^i : i in [1,1,1,1,3]];  // -t^7 + 1

=====
Y(8) in PP(1,1,1,1,4);
a1 := 0; a2 := 0; B := [];
Pa := (1 + a1*t+ a2*t^2 + a1*t^3 + t^4)/(1-t)^4;
P := Pa + &+S where S := [ FieldOfFractions(R) | 1/r*(1/(1-t^r)*Pc(r,L) + Pd(r,L)) where r is b[1] where L is b[2..#b] : b in B];

P*&*[1-t^i : i in [1,1,1,1]]; // t^4 + 1
P*&*[1-t^i : i in [1,1,1,1,4]]; // -t^8 + 1

=====
Y(7) in PP(1,1,1,2,2)
(1+3*t+8*t^2)*(1-t)^4; // 1 - t + 2*t^2 - 18*t^3 ..
a1 := -1; a2 := 2;
Pa := (1 + a1*t+ a2*t^2 + a1*t^3 + t^4)/(1-t)^4;

In this case we don't know the formula, but we know the answer

a1 := -1; a2 := 2;
Pa := (1 + a1*t+ a2*t^2 + a1*t^3 + t^4)/(1-t)^4;
P := (1-t^7)/&*[1-t^i : i in [1,1,1,2,2]];
P-Pa; // this is the contr from the curve
S!(P-Pa); // -s^3 - 2*s^4 - 5*s^5 - 8*s^6 - 14*s^7 - 20*s^8
(P-Pa)*&*[1-t^i : i in [1,1,2,2]]; // -t^3

Conclude: a curve Ga of 1/2 with deg 1/4 and genus 0 contributes
 -t^3/PP(1,1,2,2)

====
// with hindsight
// Y(7) in PP(1,1,1,2,2)
d:=7; A:=[1,1,1,2,2]; B := [];
D := P(d,A) - PA(A) - Pb(B);
 -t^3/&*[1-t^i : i in [1,1,2,2]]; // contr of curve Ga of deg 1/2^2
====
Y(8) in PP(1, 1, 1, 2, 3)
a1 := -1; a2 := 1; B := [[3,1,1,1]];

Pa := (1 + a1*t+ a2*t^2 + a1*t^3 + t^4)/(1-t)^4;
P := Pa + &+S where S := [ FieldOfFractions(R) | 1/r*(1/(1-t^r)*Pc(r,L) + Pd(r,L)) where r is b[1] where L is b[2..#b] : b in B];
P*&*[1-t^i : i in [1,1,1,2,3]]; // -t^8 + 1

====
Y(9) in PP(1, 1, 1, 2, 4)

a1 := -1; a2 := 1;
Pa := (1 + a1*t+ a2*t^2 + a1*t^3 + t^4)/(1-t)^4;

Curve Ga of 1/2 with emb 1/4(1,1,2). Need to calculate the contrib
P := (1-t^9)/&*[1-t^i : i in [1,1,1,2,4]];

(P-Pa)*&*[1-t^i : i in [1,1,2,4]]; // t^4

The conclusion is that a curve Ga of 1/2 with deg 1/8 and genus 0
and emb 1/4(1,1,2) makes contr +t^4/PP(1,1,2,4)

====
// with hindsight
// Y(9) in PP(1,1,1,2,4)
d:=9; A:=[1,1,1,2,4]; B := [];
D := P(d,A) - PA(A) - Pb(B); // t^4/&*[1-t^i : i in [1,1,2,4]];
 // This is the contr of curve of 2 of deg 1/8 with emb 1/4. I think of it
 // as including 1/8 of deg of curve, so curve now counts as 0.
====
Y(10) in PP(1, 1, 1, 2, 5)

a1 := -1; a2 := 1; B := [];
Pa := (1 + a1*t+ a2*t^2 + a1*t^3 + t^4)/(1-t)^4;
P := Pa + &+S where S := [ FieldOfFractions(R) | 1/r*(1/(1-t^r)*Pc(r,L) + Pd(r,L)) where r is b[1] where L is b[2..#b] : b in B];

P*&*[1-t^i : i in [1,1,1,2,5]]; // -t^10 + 1

====
Y(9) in PP(1, 1, 1, 3, 3)
a1 := -1; a2 := 0; B := [[3,1,1,1],[3,1,1,1],[3,1,1,1]];
Pa := (1 + a1*t+ a2*t^2 + a1*t^3 + t^4)/(1-t)^4;
P := Pa + &+S where S := [ FieldOfFractions(R) | 1/r*(1/(1-t^r)*Pc(r,L) + Pd(r,L)) where r is b[1] where L is b[2..#b] : b in B];
P*&*[1-t^i : i in [1,1,1,3,3]]; // -t^9 + 1

====
Y(11) in PP(1, 1, 1, 3, 5)
a1 := -1; a2 := 0; B := [[3,1,1,1],[5,3,1,1]];
Pa := (1 + a1*t+ a2*t^2 + a1*t^3 + t^4)/(1-t)^4;
P := Pa + &+S where S := [ FieldOfFractions(R) | 1/r*(1/(1-t^r)*Pc(r,L) + Pd(r,L)) where r is b[1] where L is b[2..#b] : b in B];
P*&*[1-t^i : i in [1,1,1,3,3]]; // -t^11 + 1

====
Y(12) in PP(1, 1, 1, 3, 6)
a1 := -1; a2 := 0; B := [[3,1,1,1],[3,1,1,1]];
Pa := (1 + a1*t+ a2*t^2 + a1*t^3 + t^4)/(1-t)^4;
P := Pa + &+S where S := [ FieldOfFractions(R) | 1/r*(1/(1-t^r)*Pc(r,L) + Pd(r,L)) where r is b[1] where L is b[2..#b] : b in B];
P*&*[1-t^i : i in [1,1,1,3,6]]; // -t^12 + 1

====
Y(13) in PP(1, 1, 1, 4, 6)
 A curve Ga of 1/2 of degree 1/24 with emb 1/4(1,1,2) and 1/6(1,1,4)
a1 := -1; a2 := 0;
Pa := (1 + a1*t+ a2*t^2 + a1*t^3 + t^4)/(1-t)^4;
P := (1-t^13)/&*[1-t^i : i in [1,1,1,4,6]];
(P-Pa)*&*[1-t^i : i in [1,1,4,6]]; // t^9 + 2*t^8 + 2*t^7 + 3*t^6 + 2*t^5 + 2*t^4 + t^3
 Thus the curve contributes (t^9 + 2*t^8 + 2*t^7 + 3*t^6 + 2*t^5 + 2*t^4 + t^3)/PP(1,1,4,6)
 ?? This has degree 13/24 whereas Ga only has deg 1/24. ??
 ?? Note that Pa itself only gives deg 0.
====
Y(13) in PP(1, 1, 1, 4, 6)
with hindsight
d := 13; A := [1,1,1,4,6]; B := [];
D := P(d,A)-PA(A)-Pb(B);
 // (t^9 + 2*t^8 + 2*t^7 + 3*t^6 + 2*t^5 + 2*t^4 + t^3)/&*[1-t^i : i in [1,1,4,6]];
D - t^4/&*[1-t^i : i in [1,1,2,4]];
 // (t^7 + t^6 + t^5 + t^4 + t^3)/&*[1-t^i : i in [1,1,2,6]];
====
Y(15) in PP(1, 1, 1, 5, 7)
a1 := -1; a2 := 0; B := [[7,1,1,5]];
Pa := (1 + a1*t+ a2*t^2 + a1*t^3 + t^4)/(1-t)^4;
P := Pa + &+S where S := [ FieldOfFractions(R) | 1/r*(1/(1-t^r)*Pc(r,L) + Pd(r,L)) where r is b[1] where L is b[2..#b] : b in B];
P*&*[1-t^i : i in [1,1,1,5,7]]; // -t^15 + 1

====
Y(16) in PP(1, 1, 1, 5, 8)
a1 := -1; a2 := 0; B := [[5,1,1,3]];
Pa := (1 + a1*t+ a2*t^2 + a1*t^3 + t^4)/(1-t)^4;
P := Pa + &+S where S := [ FieldOfFractions(R) | 1/r*(1/(1-t^r)*Pc(r,L) + Pd(r,L)) where r is b[1] where L is b[2..#b] : b in B];
P*&*[1-t^i : i in [1,1,1,5,8]]; // -t^16 + 1

====
Y(18) in PP(1, 1, 1, 6, 9)
a1 := -1; a2 := 0; B := [[3,1,1,1]];
Pa := (1 + a1*t+ a2*t^2 + a1*t^3 + t^4)/(1-t)^4;
P := Pa + &+S where S := [ FieldOfFractions(R) | 1/r*(1/(1-t^r)*Pc(r,L) + Pd(r,L)) where r is b[1] where L is b[2..#b] : b in B];
P*&*[1-t^i : i in [1,1,1,6,9]]; // -t^18 + 1

====
Y(8) in PP(1, 1, 2, 2, 2)
(1+2*t+6*t^2)*(1-t)^4;
 A curve Ga of 1/2 of degree 8/2^3 = 1 and genus 3. We don't know what
contr it makes

a1 := -2; a2 := 4;
Pa := (1 + a1*t+ a2*t^2 + a1*t^3 + t^4)/(1-t)^4;
P := (1-t^8)/&*[1-t^i : i in [1,1,2,2,2]];
P-Pa;
S!(P-Pa);
(P-Pa)*&*[1-t^i : i in [1,1,2,2]]; // -4*t^3

The conclusion is that the curve Ga of 1/2 of degree 8/2^3 = 1 and genus 3
contributes (-4*t^3)/PP(1,1,2,2)

====
Y(9) in PP(1, 1, 2, 2, 3)
(1+2*t+2*t^2)*(1-t)^4; // 1 - 2*t + 5*t^4
a1 := -2; a2 := 3;
Pa := (1 + a1*t+ a2*t^2 + a1*t^3 + t^4)/(1-t)^4;
P := (1-t^9)/&*[1-t^i : i in [1,1,2,2,3]];
(P-Pa)*&*[1-t^i : i in [1,1,2,2]]; // -t^3

Here the curve Ga of 1/2 of deg 1/4 and genus 0 contributes -t^3/PP(1,1,2,2)

====
Y(10) in PP(1, 1, 2, 2, 4)
a1 := -2; a2 := 3;
Pa := (1 + a1*t+ a2*t^2 + a1*t^3 + t^4)/(1-t)^4;
P := (1-t^10)/&*[1-t^i : i in [1,1,2,2,4]];
(P-Pa)*&*[1-t^i : i in [1,1,2,4]]; // -2*t^3 + t^4 - 2*t^4

The curve Ga of 1/2 of deg 5/8 and genus 2 (probably?) with emb
1/4(1,1,2) contributes (-2*t^3 + t^4 - 2*t^4)/PP(1,1,2,4)

====
Y(11) in PP(1, 1, 2, 2, 5)
a1 := -2; a2 := 3; B := [[5,1,2,2]];
Pa := (1 + a1*t+ a2*t^2 + a1*t^3 + t^4)/(1-t)^4;
P := (1-t^11)/&*[1-t^i : i in [1,1,2,2,5]];
(P-Pa-&+S where S := [ FieldOfFractions(R) | 1/r*(1/(1-t^r)*Pc(r,L) + Pd(r,L)) where r is b[1] where L is b[2..#b] : b in B])*&*[1-t^i : i in [1,1,2,2]];
// -t^3

The curve Ga of 1/2 of deg 1/4 contributes -t^3/PP(1,1,2,2).

====
Y(12) in PP(1, 1, 2, 2, 6)
a1 := -2; a2 := 3;
Pa := (1 + a1*t+ a2*t^2 + a1*t^3 + t^4)/(1-t)^4;
P := (1-t^12)/&*[1-t^i : i in [1,1,2,2,6]];
(P-Pa)*&*[1-t^i : i in [1,1,2,2]]; // -2*t^3

The curve Ga of 1/2 of deg 1/2 contributes -2*t^3/PP(1,1,2,2).

====
Y(10) in PP(1, 1, 2, 3, 3)
a1 := -2; a2 := 2;
Pa := (1 + a1*t+ a2*t^2 + a1*t^3 + t^4)/(1-t)^4;
P := (1-t^10)/&*[1-t^i : i in [1,1,2,3,3]];
(P-Pa)*&*[1-t^i : i in [1,1,3,3]]; // 2*t^3 + t^4 + 2*t^5

The curve Ga of 1/3(1,2) of deg 1/9 contributes
 (2*t^3 + t^4 + 2*t^5)/PP(1,1,3,3).

====
Y(11) in PP(1, 1, 2, 3, 4)

a1 := -2; a2 := 2; B := [[3,1,1,1]];
Pa := (1 + a1*t+ a2*t^2 + a1*t^3 + t^4)/(1-t)^4;
P := (1-t^11)/&*[1-t^i : i in [1,1,2,3,4]];
(P-Pa-&+S where S := [ FieldOfFractions(R) | 1/r*(1/(1-t^r)*Pc(r,L) + Pd(r,L)) where r is b[1] where L is b[2..#b] : b in B])*&*[1-t^i : i in [1,1,2,4]];
 // t^4
Curve Ga of 1/2 of deg 1/8 with one emb 1/4(1,1,2) contributes t^4/PP(1,1,2,4)

====
Y(12) in PP(1, 1, 2, 3, 5)

a1 := -2; a2 := 2; B := [[5,1,1,3]];
Pa := (1 + a1*t+ a2*t^2 + a1*t^3 + t^4)/(1-t)^4;
P := (1-t^12)/&*[1-t^i : i in [1,1,2,3,5]];
P eq Pa + &+S where S := [ FieldOfFractions(R) | 1/r*(1/(1-t^r)*Pc(r,L) + Pd(r,L)) where r is b[1] where L is b[2..#b] : b in B];
 // true

====
Y(13) in PP(1, 1, 3, 3, 5)

a1 := -2; a2 := 1; B := [];
Pa := (1 + a1*t+ a2*t^2 + a1*t^3 + t^4)/(1-t)^4;
Pb := &+S where S := [ FieldOfFractions(R) | 1/r*(1/(1-t^r)*Pc(r,L) + Pd(r,L)) where r is b[1] where L is b[2..#b] : b in B];
P := (1-t^13)/&*[1-t^i : i in [1,1,3,3,5]];
(P-Pa-Pb)*&*[1-t^i : i in [1,1,3,3]]; // 3*t^5 + 2*t^4 + 3*t^3

The curve Ga of 1/3(1,2) of deg 1/9 contributes
 (3*t^3 + 2*t^4 + 3*t^5)/PP(1,1,3,3).

====
Y(16) in PP(1, 1, 3, 3, 8)

a1 := -2; a2 := 1; B := [];
Pa := (1 + a1*t+ a2*t^2 + a1*t^3 + t^4)/(1-t)^4;
Pb := &+S where S := [ FieldOfFractions(R) | 1/r*(1/(1-t^r)*Pc(r,L) + Pd(r,L)) where r is b[1] where L is b[2..#b] : b in B];
P := (1-t^16)/&*[1-t^i : i in [1,1,3,3,8]];
(P-Pa-Pb)*&*[1-t^i : i in [1,1,3,3]];

====
Y(13) in PP(1, 1, 3, 4,4)
a1 := -2; a2 := 1; B := [[3,1,1,1]];
Pa := (1 + a1*t+ a2*t^2 + a1*t^3 + t^4)/(1-t)^4;
Pb := &+S where S := [ FieldOfFractions(R) | 1/r*(1/(1-t^r)*Pc(r,L) + Pd(r,L)) where r is b[1] where L is b[2..#b] : b in B];
P := (1-t^13)/&*[1-t^i : i in [1,1,3,4,4]];
(P-Pa-Pb)*&*[1-t^i : i in [1,1,4,4]];
 // 2*t^7 + 4*t^6 + 3*t^5 + 4*t^4 + 2*t^3
Curve Ga of 1/4(1,3) degree 1/4^2 contributes
 2*t^7 + 4*t^6 + 3*t^5 + 4*t^4 + 2*t^3

(P(13,[1, 1, 3, 4,4]) - Pa(2,0) - Pb([[3,1,1,1]]) )*&*[1-t^i : i in [1,1,4,4]];

2*t^7 + 4*t^6 + 3*t^5 + 4*t^4 + 2*t^3

====
Y(17) in PP(1, 1, 3, 4, 8)
d := 17; A := [1, 1, 3, 4, 8]; B := [[3,1,1,1]];
(P(d,A) - Pa(2,0) - Pb(B) )*&*[1-t^i : i in [1,1,4,8]];

Curve of 1/4(1,3) of degree 1/(4*8) with emb 1/8(1,3,4)
contr
(2*t^11+3*t^10+3*t^9+4*t^8+3*t^7
  +4*t^6+3*t^5+3*t^4+2*t^3)/PP(1,1,4,8)

or

(2*t^10 - t^9 + 3*t^8 - t^7 + 3*t^6 - t^5 + 2*t^4)/PP(1,1,4,8)

====
Y(15) in PP(1, 1, 3, 5, 5)
d := 15; A := [1, 1, 3, 5, 5];
 B := [[5,1,1,3] : i in [1..3]];
P(d,A) - Pa(2,0) - Pb(B); // 0

====
Y(17) in PP(1, 1, 3, 5, 7)
d := 17; A := [1, 1, 3, 5, 7];
 B := [[3,1,1,1], [5,1,1,3], [7,1,1,5]];
P(d,A) - Pa(2,0) - Pb(B); // 0

====
Y(19) in PP(1, 1, 3, 5, 9)
d := 19; A := [1, 1, 3, 5, 9];
 B := [[5,1,1,3]];
(P(d,A) - Pa(2,0) - Pb(B) )*&*[1-t^i : i in [1,1,3,9]];

Curve of 1/3(1,2) of degree 1/(3*9) with emb 1/9(1,3,5)
contributes
(2*t^11 + 2*t^10 + 3*t^9 + 2*t^8 + 2*t^7
   + 2*t^6 + 3*t^5 + 2*t^4 + 2*t^3)/PP(1,1,3,9)

====
Y(19) in PP(1, 1, 3, 6, 8)
d := 19; A := [1, 1, 3, 6, 8];
 B := [];
(P(d,A) - Pa(2,0) - Pb(B) )*&*[1-t^i : i in [1,3,6,8]];

Curve of 1/3(1,2) of deg 1/(3*6) with emb 1/6(1,2,3)
plus curve of 1/2 of deg 1/(6*8) with emb 1/6(1,2,3)
together contribute
(3*t^15 + 6*t^14 + 10*t^13 + 14*t^12 + 17*t^11
 + 21*t^10 + 21*t^9 + 21*t^8
 + 17*t^7 + 14*t^6 + 10*t^5 + 6*t^4 + 3*t^3)/PP(1,3,6,8)

====
Y(7) in PP(1,1, 1, 2, 2)
d := 7; A := [1,1,1,2,2]; B := [];
(P(d,A) - Pa(3,2) - Pb(B) )*&*[1-t^i : i in [1,1,2,2]];

Curve of 1/2 of degree 1/(2*2) contributes -t^3/PP(1,1,2,2)

====
Y(9) in PP(1,1, 1, 2, 4)
d := 9; A := [ 1, 1, 1, 2, 4 ]; B := [];
(P(d,A) - Pa(3,1) - Pb(B) )*&*[1-t^i : i in [1,1,2,4]];

Curve of 1/2 of degree 1/(2*4) with emb 1/4(1,1,2)
contributes t^4/PP(1,1,2,4)

 P(d,A) eq PA(A) + Pb(B) + 1/(2*4)*Phalfc() + Pemb2(2); // true

====
Y(13) in PP(1,1, 1, 4, 6)
d := 13; A := [ 1, 1, 1, 4, 6 ]; B := [];
(P(d,A) - PA(A) - Pb(B) )*&*[1-t^i : i in [1,1,4,6]];

Curve of 1/2 of degree 1/(4*6) with emb 1/4(1,1,2)
and 1/6(1,1,4) contributes
(t^9+2*t^8+2*t^7+3*t^6+2*t^5+2*t^4+t^3)/PP(1,1,4,6)

 P(d,A) eq PA(A) + Pb(B) + 1/(4*6)*Phalfc() + Pemb2(2) + Pemb2(3);
 // true

====
Y(14) in PP(1, 1, 2, 3, 7)
d := 14; A := [1, 1, 2, 3, 7]; B := [[3,1,1,1]];
P(d,A) - PA(A) - Pb(B); // 0

====
Y(13) in PP(1, 1, 2, 3, 6)
d := 13; A := [1, 1, 2, 3, 6]; B := [];
(P(d,A) - PA(A) - Pb(B) )*&*[1-t^i : i in [1,2,3,6]];

Curve of 1/3(1,2) of deg 1/(3*6) with emb 1/6(1,2,3)
plus curve of 1/2 of deg 1/(6*8) with emb 1/6(1,2,3)
together contribute
(t^9 + 2*t^8 + 2*t^7 + 3*t^6 + 2*t^5 + 2*t^4 + t^3)/PP(1,2,3,6)

====
Y(12) in PP(1, 1, 2, 4, 4)
d := 12; A := [1, 1, 2, 4, 4]; B := [];
(P(d,A) - PA(A) - Pb(B) )*&*[1-t^i : i in [1,1,2,4]];

Curve of 1/2 of deg 12/(2*4*4) with 3 x emb 1/4(1,1,2)
contrib 3*t^4/PP(1,1,2,4)

 P(d,A) eq PA(A) + Pb(B) + 12/(2*4*4)*Phalfc() + 3*Pemb2(2); // true

====
Y(14) in PP(1, 1, 2, 4, 6)
d := 14; A := [1, 1, 2, 4, 6]; B := [];
(P(d,A) - PA(A) - Pb(B) )*&*[1-t^i : i in [1,1,4,6]];

Curve of 1/2 of deg 14/(2*4*6) with emb 1/4(1,1,2)
and 1/6(1,1,4) contrib (2*t^8 + 3*t^6 + 2*t^4)/PP(1,1,4,6)

 P(d,A) eq PA(A) + Pb(B) + 14/(2*4*6)*Phalfc() + Pemb2(2) + Pemb2(3); // true
====
Y(15) in PP(1, 1, 2, 4, 7)
d := 15; A := [1, 1, 2, 4, 7]; B := [[7,1,2,4]];
(P(d,A) - PA(A) - Pb(B) )*&*[1-t^i : i in [1,1,2,4]];

Curve of 1/2 of deg 1/(2*4) with emb 1/4(1,1,2)
contrib (t^4)/PP(1,1,2,4)

====
Y(16) in PP(1, 1, 2, 4, 8)
d := 16; A := [1, 1, 2, 4, 8]; B := [];
(P(d,A) - PA(A) - Pb(B) )*&*[1-t^i : i in [1,1,2,4]];

Curve of 1/2 of deg 16/(2*4*8) = 1/4 with 2 x emb 1/4(1,1,2)
contrib (2*t^4)/PP(1,1,2,4)

====
Y(16) in PP(1, 1, 2, 5, 7)
d := 16; A := [1, 1, 2, 5, 7]; B := [[5,1,2,2],[7,1,1,5]];
P(d,A) - PA(A) - Pb(B); // 0

====
Y(17) in PP(1, 1, 2, 5, 8)
d := 17; A := [1, 1, 2, 5, 8]; B := [[5,1,1,3]];
(P(d,A) - PA(A) - Pb(B) )*&*[1-t^i : i in [1,1,2,8]];

Curve of 1/2 of deg 1/(2*8) with emb 1/8(1,2,5)
contrib (-t^9 - t^6 - t^3)/PP(1,1,2,8)

====
Y(18) in PP(1, 1, 2, 6, 8)
d := 18; A := [1, 1, 2, 6, 8]; B := [];
(P(d,A) - PA(A) - Pb(B) )*&*[1-t^i : i in [1,1,2,8]];

Curve of 1/2 of deg 18/(2*6*8) = 3/16 with emb 1/8(1,1,6)
contrib (t^8 + t^6 + t^4)/PP(1,1,2,8)

====
Y(20) in PP(1, 1, 3, 5, 10)
d := 20; A := [1, 1, 3, 5, 10]; B := [[3,1,1,1], [5,1,1,3], [5,1,1,3]];
P(d,A) eq PA(A) + Pb(B);

====
Y(24) in PP(1, 1, 2, 8, 12)
d := 24; A := [1, 1, 2, 8, 12]; B := [];
(P(d,A) - PA(A) - Pb(B) )*&*[1-t^i : i in [1,1,2,4]]; // t^4

Curve of 1/2 of deg 24/(2*8*24) = 1/8 with emb 1/4(1,1,2);
Same contrib as Y(9) in PP(1, 1, 1, 2, 4)

====
Y(21) in PP(1, 1, 2, 7, 10)
d := 21; A := [1, 1, 2, 7, 10]; B := [];
(P(d,A) - PA(A) - Pb(B) )*&*[1-t^i : i in [1,1,2,10]]; // t^10 + t^7 + t^4

Curve of 1/2 of deg 1/(2*10) with emb 1/10(1,2,7) contrib t^10 + t^7 + t^4

====
Y(20) in PP(1, 1, 2, 6, 10)
d := 20; A := [1, 1, 2, 6, 10]; B := [];
(P(d,A) - PA(A) - Pb(B) )*&*[1-t^i : i in [1,1,2,6]]; // t^6 + t^4

Curve of 1/2 of deg 20/(2*6*10) with emb 1/6(1,1,4) contrib // t^6 + t^4

====
Y(32) in PP(1, 1, 6, 8, 16)
d := 32; A := [1, 1, 6, 8, 16]; B := [];
(P(d,A) - PA(A) - Pb(B) )*&*[1-t^i : i in [1,1,6,8]];
 /* 2*t^13+3*t^12+4*t^11+6*t^10+6*t^9+8*t^8+6*t^7+6*t^6+4*t^5+
3*t^4+2*t^3  */

Curve of 1/2 of deg 32/(6*8*16) = 1/24 with emb 1/6(1,1,4) and
2 x 1/8(1,1,6)

====
Y(24) in PP(1, 1, 6, 8, 8)
d := 24; A := [1, 1, 6, 8, 8]; B := [];
(P(d,A) - PA(A) - Pb(B) )*&*[1-t^i : i in [1,1,2,8]];
 // 2*t^9 + 3*t^8 + 2*t^7 + 3*t^6 + 2*t^5 + 3*t^4 + 2*t^3

Curve of deg 24/(6*8*8) = 1/16 with emb 3 x 1/8(1,1,6) contrib
(2*t^9 + 3*t^8 + 2*t^7 + 3*t^6 + 2*t^5 + 3*t^4 + 2*t^3)/PP(1,1,2,8)

====
[ 1, 1, 5, 8, 10, 25 ],

d := 25; A := [ 1, 1, 5, 8, 10]; B := [[5,1,1,3], [5,1,1,3]];
(P(d,A) - PA(A) - Pb(B) )*&*[1-t^i : i in [1,1,8,10]];
 /*
 t^16 + t^15 + t^14 + 2*t^13 + 2*t^12 + 2*t^11 + 3*t^10 + 2*t^9 + 2*t^8 + 2*t^7 +
   t^6 + t^5 + t^4
   eq t^4*(t^9-1)/(t^3-1) * (t^7-1)/(t-1) /PP(1,1,8,10);
  */

Curve of 1/2 of deg 1/(8*10) with emb 1/8(1,2,5) and 1/10(1,1,8)

=======
d := 28; A := [ 1, 1, 4, 8, 14]; B := [];
(P(d,A) - PA(A) - Pb(B) )*&*[1-t^i : i in [1,1,2,8]];

Curve of 1/2 of deg 28/(4*8*14) = 1/16 with emb 3 x 1/4(1,1,2) and
1/8(1,1,6)

// 2*t^9 + 4*t^8 + 2*t^7 + t^6 + 2*t^5 + 4*t^4 + 2*t^3 //

[ 1, 1, 4, 8, 14, 28 ],

=======
[ 1, 1, 4, 4, 6, 16 ],

d := 16; A := [1, 1, 4, 4, 6]; B := [];
(P(d,A) - PA(A) - Pb(B) )*&*[1-t^i : i in [1,1,4,6]];
 // (2*t^9 + 5*t^8 + 4*t^7 + 6*t^6 + 4*t^5 + 5*t^4 + 2*t^3)

Curve of 1/2 of deg 16/(4*4*6) = 1/6 with emb 1/6(1,1,4)
and 4 x 1/4(1,1,2)

=======
[ 1, 1, 1, 4, 6, 13 ],
d := 13; A := [ 1, 1, 1, 4, 6];
(P(d,A) - PA(A) - Pb(B) )*&*[1-t^i : i in [1,1,4,6]];

// (t^9 + 2*t^8 + 2*t^7 + 3*t^6 + 2*t^5 + 2*t^4 + t^3)

A curve Ga of 1/2 of degree 1/24 with emb 1/4(1,1,2) and 1/6(1,1,4)

=======
[ 1, 1, 4, 6, 6, 18 ],
d := 18; A := [ 1, 1, 4, 6, 6]; B := [];
(P(d,A) - PA(A) - Pb(B) )*&*[1-t^i : i in [1,1,4,6]];

A curve of 1/2 of deg 18/(4*6*6) = 1/8 and emb 1/4(1,1,2)
and 3 x 1/6(1,1,4)

// 2*t^9 + 4*t^8 + 4*t^7 + 7*t^6 + 4*t^5 + 4*t^4 + 2*t^3/ [1,1,4,6]
=======
[ 1, 1, 4, 12, 18, 36 ],

d := 36; A := [ 1, 1, 4, 12, 18]; B := [];
(P(d,A) - PA(A) - Pb(B) )*&*[1-t^i : i in [1,1,4,6]];

A curve of 1/2 of deg 36/(4*12*18) = 1/24 with emb 3 x 1/4(1,1,2) 
and 1/6(1,1,4) contrib

(2*t^9 + 4*t^8 + 4*t^7 + 5*t^6 + 4*t^5 + 4*t^4 + 2*t^3)/[1,1,4,6]

=======
 [ 1, 1, 1, 2, 4, 9 ],
This has a line Ga of 1/2, with deg Ga = 1/8 and emb 1/4(1,1,2) on it.
d := 9; A := [ 1, 1, 1, 2, 4]; B := [];
(P(d,A) - PA(A) - Pb(B) )*&*[1-t^i : i in [1,1,2,4]];
 // t^4/[1,1,2,4]

=================
comparing
[ 1, 1, 4, 6, 6, 18 ],
[ 1, 1, 1, 2, 4, 9 ],
it is possible that 3 x 1/6(1,1,4) contr
2*t^9 + 3*t^8 + 4*t^7 + 6*t^6 + 4*t^5 + 3*t^4 + 2*t^3/ [1,1,4,6]

=======
[ 1, 2, 2, 3, 7, 15 ],
Line of 1/2 with deg 1/4 and no emb
d := 15; A := [ 1, 2, 2, 3, 7]; B := [[7,2,2,3]];
(P(d,A) - PA(A) - Pb(B) )*&*[1-t^i : i in [1,1,2,2]];
 // Line of 1/2 with deg 1/4 and no emb -t^3/[1,1,2,2]

=======
[ 1, 1, 6, 10, 12, 30 ],
d := 30; A := [ 1, 1, 6, 10, 12]; B := [];
(P(d,A) - PA(A) - Pb(B) )*&*[1-t^i : i in [1,1,2,12]];

deg Ga = 30/(6*10*12) = 1/24, emb 2 x 1/6(1,1,4) and 1/12(1,1,10)
contr (2*t^13 + 3*t^12 + 2*t^11 + 3*t^10 + 2*t^9 + t^8
                     + 2*t^7 + 3*t^6 + 2*t^5 + 3*t^4 + 2*t^3) / [1,1,2,12]

=======
[ 1, 2, 2, 8, 11, 24 ],
d := 24; A := [1, 2, 2, 8, 11]; B := [[11,1,2,8]];
(P(d,A) - PA(A) - Pb(B) )*&*[1-t^i : i in [1,1,2,2]];
 // -3*t^3 Ga has deg 3/4.
 
=======
[ 1, 2, 3, 4, 7, 17 ],
d := 17; A := [ 1, 2, 3, 4, 7]; B := [[3,1,1,1],[7,1,2,4]];
(P(d,A) - PA(A) - Pb(B) )*&*[1-t^i : i in [1,1,2,4]];
 // -t^5 - t^4 - t^3
 Ga has deg 1/(2*4) and 1/4(2,3,3)

=======
[ 1, 2, 3, 4, 10, 20 ]
d := 20; A := [ 1, 2, 3, 4, 10]; B := [[3,1,1,1]];
(P(d,A) - PA(A) - Pb(B) )*&*[1-t^i : i in [1,1,2,2]]; // -t^3

 // Ga has deg 20/(2*4*10) = 1/4 and no emb, contrib -t^3/[1,1,2,2]

=======
[ 1, 2, 3, 8, 13, 27 ],
d := 27; A := [ 1, 2, 3, 8, 13 ]; B := [[13,2,3,8]];
(P(d,A) - PA(A) - Pb(B) )*&*[1-t^i : i in [1,1,2,8]];
// (-t^9 - t^6 - t^3)/PP(1,1,2,8)
Curve of 1/2 deg 1/(2*8) = 1/16 with emb 1/8(1,2,5)
Same as [ 1, 1, 2, 5, 8, 17 ].

=======
[ 1, 2, 3, 11, 16, 33 ],
d := 33; A := [ 1, 2, 3, 11, 16]; B := [];
(P(d,A) - PA(A) - Pb(B) )*&*[1-t^i : i in [1,1,2,16]];
 // t^14 - t^13 + t^11 - t^10 + t^9 - t^7 + t^6

=======
[ 1, 2, 4, 4, 5, 16 ],
d := 16; A := [ 1, 2, 4, 4, 5]; B := [[5,2,4,4]];
(P(d,A) - PA(A) - Pb(B) )*&*[1-t^i : i in [1,1,2,4]];
 // 4*t^4 curve has deg 1/2, 4 x 1/4(1,1,2)

=======
[ 1, 2, 2, 8, 13, 26 ],
d := 26; A := [1, 2, 2, 8, 13]; B := [];
(P(d,A) - PA(A) - Pb(B) )*&*[1-t^i : i in [1,1,2,8]];
 // Curve of deg 26/(2*2*8) = 13/16 with 1/8(1,2,13)
 contr (-4*t^9 - 3*t^7 - t^6 - 3*t^5 - 4*t^3)/[1,1,2,8]

=======
[ 1, 1, 3, 6, 10, 21 ],
d := 21; A := [ 1, 1, 3, 6, 10]; B := [[3,1,1,1] : i in [1..3]];
-10*(P(d,A) - (PA(A) + Pb(B) + 1/(6*10)*c2 + c6))*&*[1-t^i: i in [1,1,1,10]];
P(d,A) eq PA(A) + Pb(B) + 1/(6*10)*c2 + c6 + c10136;

=======
[ 1, 1, 4, 4, 6, 16 ],
d:=16; A:=[ 1, 1, 4, 4, 6]; B := [];
P(d,A) eq PA(A) + Pb(B) + 16/(4*4*6)*c2 + 4*c4 + c6;

=======
[ 1, 1, 4, 4, 10, 20 ],
d := 20; A := [ 1, 1, 4, 4, 10]; B := [];
P(d,A) eq PA(A) + Pb(B) + 20/(4*4*10)*c2 + 5*c4;

=======
[ 1, 1, 4, 6, 6, 18 ],
d := 18; A := [ 1, 1, 4, 6, 6]; B := [];
P(d,A) eq PA(A) + Pb(B) + 18/(4*6*6)*c2 + c4 + 3*c6;

=======
[ 1, 1, 4, 6, 8, 20 ],
d := 20; A := [ 1, 1, 4, 6, 8]; B := [];
P(d,A) eq PA(A) + Pb(B) + 20/(4*6*8)*c2 + 2*c4 + c6 + c8;

=======
[ 1, 1, 4, 6, 12, 24 ],
d:=24; A:=[ 1, 1, 4, 6, 12]; B:=[];
P(d,A) eq PA(A) + Pb(B) + 24/(4*6*12)*c2 + 2*c4 + 2*c6;

=======
[ 1, 1, 4, 8, 10, 24 ],
d:=24; A:=[ 1, 1, 4, 8, 10]; B:=[];
10*(P(d,A) - (PA(A) + Pb(B) + 24/(4*8*10)*c2 + 3*c4))*&*[1-t^i: i in [1,1,1,10]];
P(d,A) eq PA(A) + Pb(B) + 24/(4*8*10)*c2 + 3*c4 + c10;

=======
[ 1, 1, 12, 16, 18, 48 ],
Curve of 1/2 of degree 48/(12*16*18) = 1/72 with emb 1/4(1,1,2) at (y^4+z^3) and
1/6(1,1,4) at y(y^3+z^2) and 1/18(1,1,16)
d:=48; A:=[ 1, 1, 12, 16, 18]; B := [];
18*(P(d,A) - (PA(A) + Pb(B) + 48/(12*16*18)*c2 + c4 + c6))*&*[1-t^i: i in [1,1,2,18]];
P(d,A) eq PA(A) + Pb(B) + 48/(12*16*18)*c2 + c4 + c18;

=======
[ 1, 1, 9, 12, 22, 45 ],
d:=45; A:=[ 1, 1, 9, 12, 22]; B:=[[3,1,1,1] : i in [1..3]];
c221912 :=
(P(d,A) - (PA(A) + Pb(B) + 1/(12*22)*c2 + Pemb2(6)));
(-5/11*c2 + (P(d,A) - (PA(A) + Pb(B) + 1/(12*22)*c2 + Pemb2(6))))
*&*[1-t^i: i in [1,2,3,22]];
P(d,A) eq PA(A) + Pb(B) + 1/(12*22)*Phalfc() + Pemb2(6) + c221912;

=======
[ 1, 1, 8, 11, 20, 41 ],
d := 41; A :=[ 1, 1, 8, 11, 20]; B := [[11,1,1,9]];
(P(d,A) - (PA(A) + Pb(B) + 1/(8*20)*Phalfc() + ??);
this has curve of 1/4(1,3), emb 1/8(1,3,4) and 1/20(1,8,11)

=======
[ 1, 1, 2, 7, 10, 21 ],
Curve of 1/2 of deg 1/(2*10) with emb 1/10(1,2,7) contrib t^10 + t^7 + t^4
d := 21; A :=[ 1, 1, 2, 7, 10]; B := [];
P(d,A) - (PA(A) + Pb(B) + 1/(2*10)*Phalfc() + c10127); // 0

=======
[ 1, 1, 12, 16, 30, 60 ],
d := 60; A := [ 1, 1, 12, 16, 30]; B := [];
P(d,A) eq PA(A) + Pb(B) + 60/(12*16*30)*Phalfc() + Pemb2(2) + Pemb2(3) + Pemb2(8);

=======
[ 1, 1, 5, 8, 14, 29 ],
d := 29; A := [ 1, 1, 5, 8, 14]; B := [[5,1,1,3]];
P(d,A) - (PA(A) + Pb(B) + 1/(8*14)*Phalfc() + Pemb2(4) - 1/14*Phalfc()
 +(t^14+t^13+t^10+t^8+t^5+t^4)/&*[1-t^i : i in [1,1,2,14]]);

=======
[ 1, 2, 17, 40, 60, 120 ],
d := 120; A := [ 1, 2, 17, 40, 60]; B := [[17,2,6,9]];
P(d,A) - (PA(A) + Pb(B) + 120/(2*40*60)*Phalfc() + c201217);

=======
[ 1, 2, 10, 13, 26, 52 ],
d := 52; A := [ 1, 2, 10, 13, 26]; B := [[13,1,2,10], [13,1,2,10]];
P(d,A) - (PA(A) + Pb(B) + 52/(2*13*26)*Phalfc() +9/52*Phalfc());

=======
[ 1, 2, 10, 14, 17, 44 ],
d := 44; A :=[ 1, 2, 10, 14, 17]; B := [[17,1,2,14]];
P(d,A) - (PA(A) + Pb(B) + 44/(2*10*14)*Phalfc() + c10127 - 11/70*Phalfc());

=======
[ 1, 2, 10, 23, 34, 70 ],
d := 70; A := [ 1, 2, 10, 23, 34]; B := [[23,2,10,11]];
P(d,A) - (PA(A) + Pb(B) + 70/(2*10*34)*Phalfc() - 7/68*Phalfc());

c3411023 :=
(t^34 + t^33 + t^32 + t^30 + t^29 + t^28 + t^26 + t^23 + t^22 + t^20 + t^19 + 
   t^18 + t^16 + t^15 + t^12 + t^10 + t^9 + t^8 + t^6 + t^5 + t^4)&*[1-t^i : i in [1,1,2,34]];

=======
[ 1, 2, 11, 14, 28, 56 ],
d := 56; A := [ 1, 2, 11, 14, 28]; B := [[11,2,3,6]];
P(d,A) - (PA(A) + Pb(B) + 56/(2*14*28)*Phalfc() + 3/7*Phalfc());

=======
[ 1, 2, 14, 18, 21, 56 ],
d := 56; A := [ 1, 2, 14, 18, 21];
 also curve of 1/3

=======
[ 1, 2, 14, 33, 50, 100 ],
d := 100; A := [ 1, 2, 14, 33, 50]; B := [[33,2,14,17]];
P(d,A) - (PA(A) + Pb(B) + 100/(2*14*50)*Phalfc() - 1/14*Phalfc());

=======
[ 1, 1, 8, 12, 14, 36 ],
d := 36; A := [ 1, 1, 8, 12, 14];
P(d,A) - (PA(A) + Pb(B) + 36/(8*12*14)*Phalfc() - 3/112*Phalfc());
 emb 1/4 and 1/8 seem to cancel!

=======
[ 1, 2, 3, 10, 14, 30 ],
d := 30; A := [ 1, 2, 3, 10, 14]; B := [];
P(d,A) - (PA(A) + Pb(B) + 30/(2*10*14)*Phalfc() - 3/28*Phalfc());

=======
[ 1, 4, 4, 9, 14, 32 ],
d := 32; A := [ 1, 4, 4, 9, 14]; B := [[9,1,4,4]];
P(d,A) - (PA(A) + Pb(B) + 8*Pemb2(2) + 32/(4*4*14)*Phalfc() + 3/28*Phalfc());

=======
[ 1, 4, 8, 9, 14, 36 ],
d := 36; A := [ 1, 4, 8, 9, 14]; B := [];
P(d,A) - (PA(A) + Pb(B) + Pemb2(4) + 4*Pemb2(2) + 36/(4*8*14)*Phalfc() + 3/28*Phalfc());

=======
[ 1, 4, 14, 33, 51, 103 ],
d := 103; A := [ 1, 4, 14, 33, 51]; B := [[33,1,14,18],[51,4,14,33]];
 also has a line of 1/3

=======
[ 1, 5, 8, 14, 28, 56 ],
d := 56; A := [ 1, 5, 8, 14, 28]; B := [[5,3,4,3]];
P(d,A) - (PA(A) + Pb(B) + Pemb2(2)+ 56/(8*14*28)*Phalfc() - 1/7*Phalfc());
 gives right contr at 2 x emb 1/14(1,5,8)

=======
[ 1, 7, 11, 14, 16, 49 ]
d := 49; A := [ 1, 7, 11, 14, 16]; B := [[11,1,3,7],[7,1,4,2],[7,1,4,2],[7,1,4,2]];
P(d,A) - (PA(A) + Pb(B) + Pemb2(2)+ 56/(8*14*28)*Phalfc() - 1/7*Phalfc());
 1/16(7,11,14) and 1/14(1,11,2)

=======
[ 1, 8, 9, 14, 32, 64 ],
d := 64; A := [ 1, 8, 9, 14, 32]; B := [[9,8,5,5]];
P(d,A) - (PA(A) + Pb(B) + 2*Pemb2(4)+ 64/(8*14*32)*Phalfc() + 3/28*Phalfc());
 gives right contr at 1/14(1,4,9)

=======
[ 1, 8, 14, 16, 25, 64 ],
d := 64; A := [ 1, 8, 14, 16, 25]; B := [[25,1,8,16]];
P(d,A) - (PA(A) + Pb(B) + 4*Pemb2(4)+ 64/(8*14*16)*Phalfc() + 6/28*Phalfc());
 gives right contr at 1/14(1,2,11)

=======
[ 1, 8, 14, 19, 23, 65 ],
d := 65; A := [ 1, 8, 14, 19, 23]; B := [[19,1,14,4],[23,1,8,14]];
P(d,A) - (PA(A) + Pb(B) + Pemb2(4)+ 1/(8*14)*Phalfc() + 0/28*Phalfc());
 has emb 1/8(6,3,7) and 1/14(1,5,8)

=======
[ 1, 1, 12, 16, 18, 48 ],
d := 48; A := [ 1, 1, 12, 16, 18]; B := [];
P(d,A) - (PA(A) + Pb(B) + Pemb2(2) + Pemb2(3) + Pemb2(9) + 48/(12*16*18)*Phalfc());

=======
[ 1, 2, 7, 10, 18, 38 ],
d := 38; A := [ 1, 2, 7, 10, 18]; B := [[7,1,2,4]];
P(d,A) - (PA(A) + Pb(B) + c10127 + 38/(2*10*18)*Phalfc() + 13/90*Phalfc());
 gives
 c181710:= (-t^4+t^5-t^8+t^9-t^11+t^13-t^14+t^17-t^18)/&*[1-t^i : i in [1,1,2,18]];

=======
[ 1, 2, 7, 18, 28, 56 ]
d := 56; A := [ 1, 2, 7, 18, 28]; B := [[7,1,2,4] : i in [1,2]];
P(d,A) - (PA(A) + Pb(B) + 56/(2*18*28)*Phalfc() + 7/36*Phalfc());
 gives same for c181710

=======
[ 1, 3, 18, 25, 46, 93 ]
d := 93; A := [ 1, 3, 18, 25, 46]; B := [[3,1,1,1],[3,1,1,1],[3,1,1,1],[3,1,1,1],[3,1,1,1],[25,1,3,21]];
P(d,A) - (PA(A) + Pb(B) + c181710 + 1/(18*46)*Phalfc() + 413/828*Phalfc());

> marker
==============
Jobs to be continued:
==============

[ 1, 2, 10, 23, 34, 70 ],
[ 1, 2, 10, 24, 37, 74 ],
[ 1, 2, 10, 25, 37, 75 ],
[ 1, 2, 11, 14, 27, 55 ],
[ 1, 2, 11, 14, 28, 56 ],
[ 1, 2, 11, 15, 27, 56 ],
[ 1, 2, 11, 16, 19, 49 ],
[ 1, 2, 11, 25, 38, 77 ],
[ 1, 2, 11, 26, 38, 78 ],
[ 1, 2, 11, 27, 41, 82 ],
[ 1, 2, 12, 15, 18, 48 ],
[ 1, 2, 12, 15, 30, 60 ],
[ 1, 2, 12, 16, 29, 60 ],
[ 1, 2, 12, 18, 21, 54 ],
[ 1, 2, 12, 18, 33, 66 ],
[ 1, 2, 12, 28, 41, 84 ],
[ 1, 2, 12, 30, 45, 90 ],
[ 1, 2, 13, 17, 20, 53 ],
[ 1, 2, 13, 18, 33, 67 ],
[ 1, 2, 13, 30, 46, 92 ],
[ 1, 2, 13, 31, 46, 93 ],
[ 1, 2, 14, 18, 21, 56 ],
[ 1, 2, 14, 19, 22, 58 ],
[ 1, 2, 14, 32, 49, 98 ],
[ 1, 2, 14, 33, 50, 100 ],
[ 1, 2, 15, 20, 37, 75 ],
[ 1, 2, 15, 35, 52, 105 ],
[ 1, 2, 16, 21, 24, 64 ],
[ 1, 2, 16, 22, 25, 66 ],
[ 1, 2, 16, 22, 41, 82 ],
[ 1, 2, 16, 37, 56, 112 ],
[ 1, 2, 16, 38, 57, 114 ],
[ 1, 2, 17, 23, 26, 69 ],
[ 1, 2, 17, 23, 43, 86 ],
[ 1, 2, 17, 40, 60, 120 ],
[ 1, 2, 18, 24, 27, 72 ],
[ 1, 2, 18, 24, 45, 90 ],
[ 1, 3, 3, 3, 5, 15 ],
[ 1, 3, 3, 4, 4, 15 ],
[ 1, 3, 3, 4, 8, 19 ],
[ 1, 3, 3, 5, 6, 18 ],
[ 1, 3, 3, 5, 9, 21 ],
[ 1, 3, 3, 5, 11, 23 ],
[ 1, 3, 3, 7, 7, 21 ],
[ 1, 3, 3, 7, 11, 25 ],
[ 1, 3, 3, 7, 13, 27 ],

[ 1, 1, 2, 3, 3, 10 ],
The curve Ga of 1/3(1,2) of deg 1/9 contrib (2*t^3 + t^4 + 2*t^5)/PP(1,1,3,3).

[ 1, 1, 2, 3, 6, 13 ],
Curve of 1/3(1,2) of deg 1/(3*6) with emb 1/6(1,2,3)
plus curve of 1/2 of deg 1/(2*6) with emb 1/6(1,2,3)
together contribute
(t^9 + 2*t^8 + 2*t^7 + 3*t^6 + 2*t^5 + 2*t^4 + t^3)/PP(1,2,3,6)

[ 1, 1, 2, 6, 9, 19 ],
Curve of 1/3 plus curve of 1/2

[ 1, 1, 3, 3, 5, 13 ], a1 := -2; a2 := 1; B := [[5,1,1,3]];
Curve Ga of 1/3(1,2) of degree 1/9 contrib
 (3*t^3 + 2*t^4 + 3*t^5)/PP(1,1,3,3).
NB different from Y(10) in PP(1,1,2,3,3) above.

[ 1, 1, 3, 3, 8, 16 ], a1 := -2; a2 := 1;
Curve Ga of 1/3(1,2) of degree 1/9 contrib
 (4*t^3 + 3*t^4 + 4*t^5)//PP(1,1,3,3).

[ 1, 1, 3, 4, 4, 13 ],
Curve Ga of 1/4(1,3) degree 1/4^2 contrib
 2*t^7 + 4*t^6 + 3*t^5 + 4*t^4 + 2*t^3

[ 1, 1, 3, 4, 8, 17 ],
Curve of 1/4(1,3) of degree 1/(4*8) with emb 1/8(1,3,4)
contrib
(2*t^11+3*t^10+3*t^9+4*t^8+3*t^7
 +4*t^6+3*t^5+3*t^4+2*t^3)/PP(1,1,4,8)

[ 1, 1, 3, 5, 9, 19 ],
Curve of 1/3(1,2) of degree 1/(3*9) with emb 1/9(1,3,5)
contrib
(2*t^11 + 2*t^10 + 3*t^9 + 2*t^8 + 2*t^7
 + 2*t^6 + 3*t^5 + 2*t^4 + 2*t^3)/PP(1,1,3,9)

[ 1, 1, 3, 6, 8, 19 ],
Curve of 1/3 plus curve of 1/2

[ 1, 1, 3, 7, 12, 24 ],
[ 1, 1, 3, 9, 13, 27 ],
[ 1, 1, 3, 9, 14, 28 ],
[ 1, 1, 3, 10, 15, 30 ],
[ 1, 1, 4, 5, 5, 16 ],
[ 1, 1, 4, 5, 10, 21 ],
[ 1, 1, 4, 7, 12, 25 ], curve of 1/4
[ 1, 1, 4, 8, 10, 24 ],
[ 1, 1, 4, 8, 14, 28 ],
[ 1, 1, 4, 10, 15, 31 ],
[ 1, 1, 4, 11, 16, 33 ],
[ 1, 1, 4, 12, 18, 36 ],
[ 1, 1, 5, 6, 12, 25 ], curve of 1/6
[ 1, 1, 5, 7, 7, 21 ],
[ 1, 1, 5, 7, 13, 27 ],
[ 1, 1, 5, 8, 10, 25 ],
[ 1, 1, 5, 8, 14, 29 ],
[ 1, 1, 5, 12, 18, 37 ],
[ 1, 1, 5, 13, 19, 39 ],
[ 1, 1, 5, 13, 20, 40 ],

[ 1, 1, 6, 8, 10, 26 ],

[ 1, 1, 6, 8, 15, 31 ],
[ 1, 1, 6, 8, 16, 32 ],
[ 1, 1, 6, 9, 16, 33 ],
[ 1, 1, 6, 10, 12, 30 ],
[ 1, 1, 6, 10, 18, 36 ],
[ 1, 1, 6, 15, 22, 45 ],
[ 1, 1, 6, 16, 24, 48 ],
[ 1, 1, 7, 9, 11, 29 ],
[ 1, 1, 7, 9, 17, 35 ],
[ 1, 1, 7, 9, 18, 36 ],
[ 1, 1, 7, 10, 12, 31 ],
[ 1, 1, 7, 16, 24, 49 ],
[ 1, 1, 7, 17, 26, 52 ],
[ 1, 1, 8, 10, 12, 32 ],
[ 1, 1, 8, 10, 20, 40 ],
[ 1, 1, 8, 11, 20, 41 ],
[ 1, 1, 8, 12, 14, 36 ],
[ 1, 1, 8, 12, 22, 44 ],
[ 1, 1, 8, 19, 28, 57 ],
[ 1, 1, 8, 20, 30, 60 ],
[ 1, 1, 9, 12, 14, 37 ],
[ 1, 1, 9, 12, 22, 45 ],
[ 1, 1, 9, 21, 31, 63 ],
[ 1, 1, 9, 21, 32, 64 ],
[ 1, 1, 10, 13, 15, 40 ],
[ 1, 1, 10, 23, 35, 70 ],
[ 1, 1, 11, 15, 17, 45 ],
[ 1, 1, 11, 15, 28, 56 ],
[ 1, 1, 11, 26, 39, 78 ],
[ 1, 1, 12, 16, 18, 48 ],
[ 1, 1, 12, 16, 30, 60 ],
[ 1, 2, 2, 2, 3, 10 ],
[ 1, 2, 2, 2, 5, 12 ],
[ 1, 2, 2, 2, 7, 14 ],
[ 1, 2, 2, 3, 3, 11 ],
[ 1, 2, 2, 3, 4, 12 ],
[ 1, 2, 2, 3, 6, 14 ],
[ 1, 2, 2, 3, 7, 15 ],
[ 1, 2, 2, 3, 8, 16 ],
[ 1, 2, 2, 4, 5, 14 ],

==============
Summary of jobs done:
==============

Isolated B only. These are now routine

[ 1, 1, 1, 1, 1, 5 ], nonsingular, a1 := 1; a2 := 1; B := [];
[ 1, 1, 1, 1, 2, 6 ], nonsingular, a1 := 0; a2 := 1; B := [];
[ 1, 1, 1, 1, 3, 7 ], has a1 := 0; a2 := 0; B := [ 1/3(1,1,1) ];
[ 1, 1, 1, 1, 4, 8 ], a1 := 0; a2 := 0; B := [];
[ 1, 1, 1, 2, 3, 8 ], a1 := -1; a2 := 1; B := [[3,1,1,1]];
[ 1, 1, 1, 2, 5, 10 ], a1 := -1; a2 := 1; B := [];
[ 1, 1, 1, 3, 3, 9 ], a1 := -1; a2 := 0; B := 3 x [3,1,1,1];
[ 1, 1, 1, 3, 5, 11 ], a1 := -1; a2 := 0; B := [[3,1,1,1],[5,3,1,1]];
[ 1, 1, 1, 3, 6, 12 ], a1 := -1; a2 := 0; B := 3 x [3,1,1,1];
[ 1, 1, 1, 5, 7, 15 ], a1 := -1; a2 := 0; B := [[7,1,1,5]];
[ 1, 1, 1, 5, 8, 16 ], a1 := -1; a2 := 0; B := [[5,1,1,3]];
[ 1, 1, 1, 6, 9, 18 ], a1 := -1; a2 := 0; B := [[3,1,1,1]];
[ 1, 1, 2, 3, 5, 12 ], a1 := -2; a2 := 2; B := [[5,1,1,3]];
[ 1, 1, 2, 3, 7, 14 ], B := [[3,1,1,1]];
[ 1, 1, 2, 5, 7, 16 ], B := [[5,1,2,2], [7,1,1,5]];
[ 1, 1, 2, 7, 11, 22 ],
[ 1, 1, 3, 3, 4, 12 ],
[ 1, 1, 3, 3, 7, 15 ],
[ 1, 1, 3, 5, 10, 20 ],
[ 1, 1, 3, 5, 5, 15 ], // 0
[ 1, 1, 3, 5, 7, 17 ], // 0
[ 1, 1, 3, 7, 9, 21 ], B := [[9,1,1,7],[3,1,1,1],[3,1,1,1]];

Isolated B + Curve of 1/2 + emb. These are rapidly becoming routine.

[ 1, 1, 1, 2, 2, 7 ],
Curve of 1/2(1,1) as only singularity, contrib -t^3/PP(1,1,2,2)

[ 1, 1, 1, 2, 4, 9 ],
Curve of 1/2 of degree 1/(2*4) with emb 1/4(1,1,2) contrib t^4/PP(1,1,2,2)

[ 1, 1, 1, 4, 6, 13 ],
Y(13) in PP(1, 1, 1, 4, 6)
 A curve Ga of 1/2 of degree 1/24 with emb 1/4(1,1,2) and 1/6(1,1,4)
a1 := -1; a2 := 0;
Pa := (1 + a1*t+ a2*t^2 + a1*t^3 + t^4)/(1-t)^4;
P := (1-t^13)/&*[1-t^i : i in [1,1,1,4,6]];
(P-Pa)*&*[1-t^i : i in [1,1,4,6]]; // t^9 + 2*t^8 + 2*t^7 + 3*t^6 + 2*t^5 + 2*t^4 + t^3
 Thus the curve contributes (t^9 + 2*t^8 + 2*t^7 + 3*t^6 + 2*t^5 + 2*t^4 + t^3)/PP(1,1,4,6)
 ?? This has degree 13/24 whereas Ga only has deg 1/24. ??
 ?? Note that Pa itself only gives deg 0.

Curve of 1/2 of degree 1/(4*6) with emb 1/4(1,1,2) and 1/6(1,1,4) contrib
(t^9+2*t^8+2*t^7+3*t^6+2*t^5+2*t^4+t^3)/PP(1,1,4,6)

[ 1, 1, 2, 2, 2, 8 ], a1 := -2; a2 := 4;
Curve Ga of 1/2 of degree 8/2^3 = 1 and genus 3 contrib -4*t^3/PP(1,1,2,2)

[ 1, 1, 2, 2, 3, 9 ], a1 := -2; a2 := 3;
Here the curve Ga of 1/2 of deg 1/4 and genus 0 contrib -t^3/PP(1,1,2,2)

[ 1, 1, 2, 2, 4, 10 ], a1 := -2; a2 := 3;
The curve Ga of 1/2 of deg 5/8 and genus 2 (probably?) with emb
1/4(1,1,2) contrib (-2*t^3 + t^4 - 2*t^4)/PP(1,1,2,4)

[ 1, 1, 2, 2, 5, 11 ],
The curve Ga of 1/2 of deg 1/4 contrib -t^3/PP(1,1,2,2).

[ 1, 1, 2, 2, 6, 12 ], a1 := -2; a2 := 3;
The curve Ga of 1/2 of deg 1/2 contrib -2*t^3/PP(1,1,2,2).

[ 1, 1, 2, 3, 4, 11 ],
Curve Ga of 1/2 of deg 1/8 with one emb 1/4(1,1,2) contrib
 t^4/PP(1,1,2,4)

[ 1, 1, 2, 4, 4, 12 ],
Curve of 1/2 of deg 12/(2*4*4) with 3 x emb 1/4(1,1,2) contrib
3*t^4/PP(1,1,2,4)

[ 1, 1, 2, 4, 6, 14 ],
Curve of 1/2 of deg 14/(2*4*6) = 7/24 with emb 1/4(1,1,2)
and 1/6(1,1,4) contrib (2*t^8 + 3*t^6 + 2*t^4)/PP(1,1,4,6)

[ 1, 1, 2, 4, 7, 15 ],
Curve of 1/2 of deg 1/(2*4) with emb 1/4(1,1,2) contrib (t^4)/PP(1,1,2,4)

[ 1, 1, 2, 4, 8, 16 ],
Curve of 1/2 of deg 16/(2*4*8) = 1/4 with 2 x emb 1/4(1,1,2)
contrib (2*t^4)/PP(1,1,2,4)

[ 1, 1, 2, 5, 8, 17 ],
Curve of 1/2 of deg 1/(2*8) with emb 1/8(1,2,5)
contrib (-t^9 - t^6 - t^3)/PP(1,1,2,8)

[ 1, 1, 2, 6, 8, 18 ],
Curve of 1/2 of deg 18/(2*6*8) = 3/16 with emb 1/8(1,1,6)
contrib (t^8 + t^6 + t^4)/PP(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) contrib t^10 + t^7 + t^4

[1, 1, 2, 6, 10, 20]
Curve of 1/2 of deg 20/(2*6*10) = 1/6 with emb 1/6(1,1,4)
contrib t^6 + t^4

[ 1, 1, 3, 6, 10, 21 ], B := [[3,1,1,1] : i in [1..3]];
Curve of 1/2 deg 1/(6*10) with emb 1/10(1,3,6)

[ 1, 1, 4, 4, 6, 16 ],
Curve of 1/2 of deg 16/(4*4*6) = 1/16 with emb 4 x 1/4(1,1,2) and 1/6(1,1,4)

[ 1, 1, 4, 4, 10, 20 ],
Curve of 1/2 of deg 1/16 with emb 5 x 1/4(1,1,2)

[ 1, 1, 4, 6, 6, 18 ], OK
[ 1, 1, 4, 6, 8, 20 ], OK
[ 1, 1, 4, 6, 12, 24 ], OK

[ 1, 1, 6, 8, 8, 24 ],
Curve of 1/2 deg 24/(6*8*8) = 1/16 with emb 3 x 1/8(1,1,6) contrib
(2*t^9 + 3*t^8 + 2*t^7 + 3*t^6 + 2*t^5 + 3*t^4 + 2*t^3)/PP(1,1,2,8)

[ 1, 1, 6, 8, 16, 32 ],
Curve of 1/2 of deg 32/(6*8*16) = 1/24 with emb 1/6(1,1,4) and
2 x 1/8(1,1,6)

// My precious functions
// ===============

/* This function computes the PERIODIC contribution to Hilbert series of an
isolated orbifold point 1/r(a,b..). It assumes that the univariate polynomial
ring Q[t] is defined and that the input list L = [a1,a2,.. ] consists of
coprime elements. */

function Pc(r,L)
  if &or[GreatestCommonDivisor([r,a]) ne 1 : a in L] // other mugtrap
     then error "Error: Not Coprime";
  end if;
A := (t^r-1) div (t-1);
B := t*&*[1-t^i : i in L];
h_throwaway, al_throwaway, be := XGCD(A,B);
 return r*t*be;
end function;

/* This function computes the GROWING contribution to Hilbert series of an
isolated orbifold point 1/r(a,b..). It assumes that the univariate polynomial
ring Q[t] is defined and that the input list L = [a1,a2,.. ] consists of
coprime elements. */

function Pd(r,L)
n := #L; f := (1-t)^(n+1)*Pc(r,L);
return
&+ [-Coefficient(f,k)*t^i where k is Min(i,n+1-i) : i in [1..n] ]
 / (1-t)^(n+1);
end function;

function Pa(n1,n2)  /* Part I of Hilbert series of CY 3-fold
 depending on n1 = number of 1, n2 = number of 2.
 (1-t)^4*(1 + P1*t + P2*t^2) = (1 + a1*t + a2*t^2) mod t^3.
 where P1 is n1 and P2 is (n1 + 1 chews 2) + n2. */
a1 := n1-4;
a2 := Binomial(n1+1,2) + n2 - 4*a1 - 10;
 return (1 + a1*t+ a2*t^2 + a1*t^3 + t^4)/(1-t)^4;
end function;

function PA(A) /* the same as Pa */
n1:= #[a : a in A | a eq 1];
n2:= #[a : a in A | a eq 2];
 return Pa(n1,n2);
end function;

function Pb(B)
S := [ FieldOfFractions(R) |
 1/r*(1/(1-t^r)*Pc(r,L) + Pd(r,L))
 where r is b[1] where L is b[2..#b]
 : b in B];
return &+S;
end function;

function P(d,A)
return (1-t^d)/&*[1-t^i : i in A];
end function;

/* Contr of a 1/r(1,r-1) curve Ga is
a * t^3 * ((1-t^(r-2)) div (1-t)) * ((1-t^r) div (1-t))
- b * t^{r+1}
all over the standard denom 1/&*[1-t^i : i in [1,1,r,r]]. Note the funny way
of putting in the parameters: b = deg Ga and
 deg_Ga(E1-E{r-1}) = 2*r*a - 2*b.  Comment. */

function Ppc(r,a,b)
X := t^3*((1-t^(r-2)) div (1-t)) * ((1-t^r) div (1-t));
Y := -t^(r+1);
denom := &*[1-t^i : i in [1,1,r,r]];
return (a*X+b*Y)/denom;
end function;

/* The contr of a curve of 1/2 of deg 1. (Multiply by deg Ga.) */
function Phalfc()
return (-4*t^3)/&*[1-t^i : i in [1,1,2,2]];
end function;

/* The contr of an emb 1/2r(1,1,2r-2). (Note the parameter is r.) */
function Pemb2(r)
return -(r-1)/(4*r)*Phalfc()
   + &+[t^(2*i) : i in [2..r]]/&*[1-t^i: i in [1,1,2,2*r]];
end function;

c2 :=  (-4*t^3)/&*[1-t^i : i in [1,1,2,2]];

c4 := (1/2*t^5 + t^4 + 1/2*t^3)/&*[1-t^i : i in [1,1,2,4]];
c6 := (2/3*t^7 + t^6 + 2/3*t^5 + t^4 + 2/3*t^3)/&*[1-t^i : i in [1,1,2,6]];
c8 := (3/4*t^9 + t^8 + 3/4*t^7 + t^6 + 3/4*t^5 + t^4 + 3/4*t^3)/
                           &*[1-t^i : i in [1,1,2,8]];
c10 := 1/10*(8*t^10+2*t^9+6*t^8+4*t^7
                +4*t^6+6*t^5+2*t^4+8*t^3)/&*[1-t^i : i in [1,1,1,10]];
c12 := (5/6*t^13 + t^12 + 5/6*t^11 + t^10 + 5/6*t^9 + t^8
 + 5/6*t^7 + t^6 + 5/6*t^5 + t^4 + 5/6*t^3)/&*[1-t^i : i in [1,1,2,12]];

c8125 := (-3/4*t^9+1/4*t^7-t^6+1/4*t^5-3/4*t^3)/&*[1-t^i : i in [1,1,1,8]];
c8125 eq 3/16*c2*(1+t) + (t^5-t^6+t^7)/&*[1-t^i : i in [1,1,1,8]];
c8233 := - 9/16*Phalfc() + (t^8-2*t^7+3*t^6-2*t^5+t^4)/&*[1-t^i : i in [1,1,2,8]];

c10136 := -1/10*(6*t^10+4*t^9-8*t^8+8*t^7+8*t^6
                 -8*t^5+4*t^4+6*t^3)/&*[1-t^i: i in [1,1,1,10]];
c10136 eq 3/20*c2 + (-t^4+t^5-t^7+t^9-t^10)/&*[1-t^i : i in [1,1,2,10]];

c10136 := 9/52*Phalfc() + (-t^10+t^9-t^7+t^5-t^4)/&*[1-t^i : i in [1,1,2,10]];

c10127 := (t^4+t^7+t^10)/&*[1-t^i : i in [1,1,2,10]];

c141310 := (t^15+t^13+t^12+2*t^9+t^6+t^5+t^3)/&*[1-t^i : i in [1,1,2,10]];

c14158 := (t^14+t^13+t^10+t^8+t^5+t^4)/&*[1-t^i : i in [1,1,2,14]]);

c221912 := 5/11*Phalfc() + (-t^24+2*t^23+t^21-t^20+2*t^19+t^17-t^16+t^15
 +t^13-t^12+t^11+2*t^9-t^8+t^7+2*t^5-t^4)/&*[1-t^i: i in [1,2,3,22]];

c201217 := - 11/40*Phalfc() + (-2*t^21-t^19-t^18-t^17-2*t^15-t^13-t^12
 -t^11-2*t^9-t^7-t^6-t^5-2*t^3)/&*[1-t^i : i in [1,1,2,20]];

c3411023 := (t^34+t^33+t^32+t^30+t^29+t^28+t^26+t^23+t^22+t^20+t^19+
 t^18+t^16+t^15+t^12+t^10+t^9+t^8+t^6+t^5+t^4)&*[1-t^i : i in [1,1,2,34]];

c4 := -1/8*c2 + (t^4)/&*[1-t^i: i in [1,1,2,4]];
c6 := -1/6*c2 + (t^4+t^6)/&*[1-t^i: i in [1,1,2,6]];
c8 := -3/8*c2 + (t^4+t^6+t^8)/&*[1-t^i: i in /[1,1,2,8]];
c10 := -1/5*c2 + (t^4+t^6+t^8+t^10)/&*[1-t^i: i in [1,1,2,10]];
c12 := -5/24*c2 + (t^4+t^6+t^8+t^10+t^12)/&*[1-t^i: i in [1,1,2,12]];
c18 := -2/9*c2 + (t^4 + t^6 + t^8 + t^10 + t^12 + t^14 + t^16 + t^18)/&*[1-t^i: i in [1,1,2,18]];

c2r :=  -(r-1)/(4*r)*c2 + &+[t^(2*i) : i in [2..r]]/&*[1-t^i: i in [1,1,2,2*r]];

-(r-1)/(4*r)*Phalfc() + (t^4-t^(2*r+2))/&*[1-t^i: i in [1,1,2,2,2*r]];