# 	 MAPLE script for computing the number 317.206.375               #
#  of twisted cubics in the general 5ic of P4 using Xavier's space       #
#  cf.                                                                   #
# adapted by A. Meireles from a script by P. Meurer ,                    #
# The number of rational quartics on Calabi-Yau hypersurfaces in         #
# weighted projective space $\ps{2,1^4}$} Math. Scandin, #78, 63-83 1996,#
# http://arXiv.org/abs/math/alg-geom/9409001                             #
##########################################################################
#sf vector space of forms of deg d in variables vars
 sf:=proc(vars,d) local i;
   coeff(expand(series(product((1/(1-t*vars[i])),
   i=1..nops(vars)),t=0,d+1)),t^d)
 end:
 n:=4; d:= 18;
 xx:= seq(x.i, i=0..n); 
 ww:= seq(w.i, i=0..n);
 # F vector space of linear forms  xi, i=0..n
 FF:=0: for i from 0 to n do FF:= FF + x.i od: FF:=FF;
 S2F:=sf([op(FF)],2): S3F:=sf([op(FF)],3): 
 S4F:=sf([op(FF)],4): S5F:=sf([op(FF)],5): 
 ############# 		some useful procedures  	  #############
 # TGo(r)  tangent space of G_o at r
 TGo:= proc(r_) local  res; 
    res:= expand( (F- r_ )* dualrep(r_)  ):
 end:
 # TGrass(idd,E) tangent space of Grass(2,E) at idd 
 TGrass:= proc(idd_,E_) local res;
    res:= expand( (E_ - idd_)*dualrep(idd_) ):
 end:
 # Blow_up(TB , TX ) tangent space and O_B(1)
 # blow up de X along B
 Blow_up:=proc(TB_,TX_) local Tang_, i , N_, TL_, TPN_;
    N_:=expand(TX_-TB_):
    for i from 1 to nops(N_) do
       TL_:=op(i,N_):
       TPN_:=expand( (N_-TL_)/TL_ ):
       Tang_[1][i]:= expand(TB_ + TL_ + TPN_): 
                        # Tang_[1] stores the total tangent  space
       Tang_[2][i]:=TL_:   
                        # Tang_[2] stores  normal direction
    od:
    Tang_[3]:=nops(N_):
    Tang_[4]:=N_:
    Tang_
 end:
 ########## actual computation starts here ################
 fp:= 0;           # fp counts the number of fixpts
 chave=0:
 for j from n to 0 by -1 do    # choice of 3-plane
 F:=FF-x.j: 
 for j1 from 1 to n-1 do 
 for j2 from j1+1 to n do
 X1:=op(j1,F): X2:=op(j2,F):   # choice of line <x1,x2>
 F_:=expand(X1+X2):
 EGo:=expand(X1*F+X2*F-X1*X2): # space of quadratic formas  
                               # zero on <x1,x2>
 TBase:=TGo(X1+X2) + F/x.j:
 q_:=dualrep(X1+X2): 
 cont1:=0: cont2:=0: cont3:=0:
 # at the base 
 for i0 from 1 to n do
  for i1 from i0 to n do
  if member(op(i0,F)*op(i1,F),{op(EGo)}) then
   for i2 from i0 to n do
    for i3 from i2 to n do
    if member(op(i2,F)*op(i3,F),{op(EGo-op(i0,F)*op(i1,F))})
    then
     if ( (i0<>i2) and (i0<>i3) and (i1<>i2) and (i1<>i3) )  
     then
      fp:=fp+1:
      A[fp]:=op(i0,F)*op(i1,F)+op(i2,F)*op(i3,F):
      Q[fp]:=q_: 
      id[fp]:=[op(i0,F)*op(i1,F),op(i2,F)*op(i3,F),
               op(i0,F)*op(i1,F)*op(i2,F)*op(i3,F)/(X1*X2),0,0]: 
      T[fp]:=expand(TGrass(A[fp],EGo) + TBase):
      E1[fp]:=1: E2[fp]:=1: E3[fp]:=1:
      EE[fp]:=S5F-convert({op(expand(id[fp][1]*S3F)), 
            op(expand(id[fp][2]*S3F)),op(expand(id[fp][3]*S3F)),
            op(expand(x.j*S4F))},`+`): 
     fi: fi: od: od: fi: od: od:
 # on X^1
 for k from 1 to 2 do
  for i0 from 1 to n-1 do
   for i1 from (i0+1) to n do
    X:=X.k: Y:=expand(F_-X):
    TB:=expand((F-op(i0,F)-op(i1,F))*dualrep(op(i0,F)+
               op(i1,F))+Y/X):
    Tang:=Blow_up(TB,TGrass(X*op(i0,F)+X*op(i1,F),EGo)):
    T1:=Tang[1]:
    T2:=Tang[2]:
    for i from 1 to Tang[3] do  
           # searching points in Y1 inters B = Y1 inters Y2
           # that appear on Y1' (ou Y2')
    if ((((op(i0,F)=X) and (op(i1,F)=Y)) or ((op(i0,F)=Y) and 
     (op(i1,F)=X))) and (member(op(i0,F)*op(i1,F)*T2[i]/Y,
     {op(F)}))) then 
     cont1:=cont1+1:
     Taux1[1][cont1]:=T1[i]: Taux1[2][cont1]:=T2[i]:
     Taux1[3][cont1]:=[X,Y,op(i0,F),op(i1,F),
                       op(i0,F)*op(i1,F)*T2[i]/Y]:
    elif (((op(i0,F)=X) or (op(i1,F)=X) ) and 
         (member(op(i0,F)*op(i1,F)*T2[i]/Y,{op(F)})))
     then 
     cont3:=cont3+1:   
     Taux3[1][cont3]:=T1[i]: Taux3[2][cont3]:=T2[i]: 
     Taux3[3][cont3]:=1:
     Taux3[4][cont3]:=[X,Y,op(i0,F),op(i1,F),
                       op(i0,F)*op(i1,F)*T2[i]/Y]:
    else fp:=fp+1:  
     A[fp]:=X*op(i0,F)+X*op(i1,F):
     Q[fp]:=q_: 
     id[fp]:=[X*op(i0,F),X*op(i1,F),
              X*op(i0,F)*op(i1,F)*T2[i]/Y,0,0]:  
     E1[fp]:=T2[i]:
     E2[fp]:=1: E3[fp]:=1:
     T[fp]:=expand(T1[i]+TBase):
     EE[fp]:=S5F-convert({op(expand(id[fp][1]*S3F)),
           op(expand(id[fp][2]*S3F)),op(expand(id[fp][3]*S3F)),
           op(expand(x.j*S4F))},`+`): 
    fi: od: od: od: od:
 # on X^2
 # 1st kind: X^2 + XY + XZ
 for k from 1 to cont1 do
  X:=Taux1[3][k][1]: Y:=Taux1[3][k][2]:
  Z1:=Taux1[3][k][3]:Z2:=Taux1[3][k][4]:
  Z:=Taux1[3][k][5]:
  Tang:=Blow_up(Y/X+Z/X+(F-F_-Z)/Z,Taux1[1][k]): 
  T1:=Tang[1]:
  T2:=Tang[2]: 
  for i from 1 to Tang[3] do
  if (member(T2[i]*Y,{op(F)})) then
   cont2:=cont2+1:
   Taux2[1][cont2]:=T1[i]: Taux2[2][cont2]:=Taux1[2][k]: 
   Taux2[3][cont2]:=T2[i]:
   Taux2[4][cont2]:=[X,Y,Z1,Z2,Z,T2[i]*Y]:
  else  fp:=fp+1: 
   A[fp]:=X^2+X*Y:
   Q[fp]:=q_:
   id[fp]:=[X^2,X*Y,X*Z,T2[i]*X*Y*Z,0] :  
   E1[fp]:=Taux1[2][k]:
   E2[fp]:=T2[i]: E3[fp]:=1:
   T[fp]:=expand(T1[i]+TBase):
   EE[fp]:=S5F-convert({op(expand(id[fp][1]*S3F)),
         op(expand(id[fp][2]*S3F)),op(expand(id[fp][3]*S3F)),
         op(expand(id[fp][4]*S2F)),op(expand(x.j*S4F))},`+`): 
  fi: od: od:
 # 2nd kind: WX + WY (+ W^2)
 for k from 1 to n do
 if ((op(k,F)<>X1) and (op(k,F)<>X2)) then
  idd:=op(k,F)*X1+op(k,F)*X2:
  Tang:=Blow_up((F-op(k,F))/op(k,F),TGrass(idd,EGo)):
  T1:=Tang[1]:  T2:=Tang[2]:
  for i from 1 to Tang[3] do
   fp:=fp+1: 
   A[fp]:=idd:
   Q[fp]:=q_: 
   id[fp]:=[op(k,F)*X1,op(k,F)*X2,op(k,F)^2,
            T2[i]*op(k,F)*X1*X2,0]:  
   E1[fp]:=1:
   E2[fp]:=T2[i]: E3[fp]:=1:
   T[fp]:=expand(T1[i]+TBase):
   EE[fp]:=S5F-convert({op(expand(id[fp][1]*S3F)),
         op(expand(id[fp][2]*S3F)),op(expand(id[fp][3]*S3F)),
         op(expand(id[fp][4]*S2F)),op(expand(x.j*S4F))},`+`): 
 od: fi: od:
 # X^3
 # 1st kind: X^2 + XY + XZ + XZK 
 for k from 1 to cont2 do 
  X:=Taux2[4][k][1]: Y:=Taux2[4][k][2]: 
  Z1:=Taux2[4][k][3]: Z2:=Taux2[4][k][4]:
  Z:=Taux2[4][k][5]: K:=Taux2[4][k][6]:
  idd:=X^2+X*Y:
  Tang:=Blow_up((F-F_-K)/K+K/Y+(F-F_-Z)/Z+Y/X,Taux2[1][k]):
  for i from 1 to Tang[3] do 
   fp:=fp+1: 
   A[fp]:= idd:   
   Q[fp]:=q_:  
   id[fp]:= [X^2,X*Y,X*Z,X*Z*K,Tang[2][i]*K*Z*X]: 
   E1[fp]:=Taux2[2][k]:
   E2[fp]:=Taux2[3][k]:
   E3[fp]:=Tang[2][i]:
   T[fp]:=expand(Tang[1][i]+TBase):
   EE[fp]:=S5F-convert({op(expand(id[fp][1]*S3F)),
         op(expand(id[fp][2]*S3F)),op(expand(id[fp][3]*S3F)),
         op(expand(id[fp][4]*S2F)),op(expand(id[fp][5]*S2F)),
         op(expand(x.j*S4F))},`+`): 
  od: od: 
 # 2nd kind: X^2 + XK + XZ  
 for k from 1 to cont3 do 
  X:=Taux3[4][k][1]: Y:=Taux3[4][k][2]: 
  Z1:=Taux3[4][k][3]: Z2:=Taux3[4][k][4]:
  Z:=Taux3[4][k][5]:
  if (Z1=X) then K:=Z2 else K:=Z1 fi: 
  idd:=X^2+X*K:
  Tang:=Blow_up(Y/X+(F-X-K-Z)/Z+(F-X-K)/K,Taux3[1][k]): 
  for i from 1 to Tang[3] do 
   fp:=fp+1: 
   A[fp]:= idd:
   Q[fp]:=q_:  
   id[fp]:=[X^2,X*K,X*Z,X*K*Z,Tang[2][i]*X*Z*K]:  
   E1[fp]:=Taux3[2][k]:
   E2[fp]:=Taux3[3][k]:
   E3[fp]:=Tang[2][i]:
   T[fp]:=expand(Tang[1][i]+TBase):
   EE[fp]:=S5F-convert({op(expand(id[fp][1]*S3F)),
         op(expand(id[fp][2]*S3F)),op(expand(id[fp][3]*S3F)),
         op(expand(id[fp][4]*S2F)),op(expand(id[fp][5]*S2F)),
         op(expand(x.j*S4F))},`+`): 
  od: od: od: od: od:
 # Using Bott 
 WG:=[-2,7,13,17,39]; 
 wg:= seq(WG[i], i=1..n+1);
 resultado:=0:
 for i from 1 to fp do  
   den:=unapply(prodwts(T[i]), ww):
   den:=den(wg):
   f1:=unapply(prodwts(EE[i]), ww):
   f1:=f1(wg):
   g:=unapply(prodwts(Q[i]), ww):
   g:=g(wg):
   f:=(f1*g)/den:                        
   resultado:=simplify(resultado+f):
 od:
  print(fp, resultado);
# should find fp=5550 with resultado=317206375
# I.e., there are 317.206.375 twisted cubics in the general 5ic of P4