#The functions tritangents(d) computes the number of tritangents in a pencil of degree d plane curves, using the SAGE package:
#
# Chow by Lehn Sorger, https://www.math.sciences.univ-nantes.fr/~sorger/chow/html/en/reference/chow/sage/schemes/chow/readme.html
#
#
#The answer is given by the polynomial (d^2+3d-2)(d-3)(d-4)(d-5)
#
#
#sage: for d in range(6,16):
#....:     d, tritangents(d)
#....:     
#(6, 312)
#(7, 1632)
#(8, 5160)
#(9, 12720)
#(10, 26880)
#(11, 51072)
#(12, 89712)
#(13, 148320)
#(14, 233640)
#(15, 353760)
#

from chow.all import *

G=Grass(2,3, chern_class='z', name='G')
z=G.gen()
U=G.sheaves["universal_sub"]

E=ProjBundle( U.symm(3) , 't', name='E')
t=E.gen()

PP=Proj(1, "y")
y=PP.gen()

Y = E*PP

UY=Bundle(Y, 2, U.chern_classes())
X=ProjBundle( UY.dual(), 'h', name='X')  #family of lines over Y
h=X.gen()

OZ=Bundle(X, 1, [1,2*t + 6*h])
O=Bundle(X,1, [1,0])
Z=Bundle(X,1, [1,0]) - OZ**(-1)
f=X.base_morphism()

@cached_function
def tritangents(d):
  L=Bundle(X, 1, [1,d*h + y])
  LL=f.lowerstar( L*Z )
  print LL
  return LL.chern_classes()[6].integral()




#Result
x=var('x')
P=(x**2 + 3*x - 2)*(x - 3)*(x - 4)*(x - 5)