Understanding normalisation and basic properties of weighted projective spaces

wps.sage

import itertools

def reduce(Q):
  return tuple([qi / gcd(Q) for qi in Q])

def normalise(Q):
  Q = reduce(Q)
  D = [gcd(Q[:i] + Q[i+1:]) for i in range(len(Q))]
  A = [lcm(D[:i] + D[i+1:]) for i in range(len(Q))]
  a = lcm(A)

  return tuple([qi / ai for (qi, ai) in zip(Q, A)])

def isNormalised(Q):
  return Q == normalise(Q)

def isGorenstein(Q):
  return sum(Q) % lcm(Q) == 0


bound = 20
dimension = 2

#bound = 25
#dimension = 3


weights = itertools.product(range(1, bound), repeat=dimension+1)
weights = filter(lambda Q: Q == tuple(sorted(Q)), weights)

normalised = filter(isNormalised, weights)
Gorenstein = filter(isGorenstein, normalised)

print len(normalised)
print len(Gorenstein), Gorenstein