All genera of complete intersection curves up to a certain cutoff

genera.sage

def genus(degrees):
  n = len(degrees) + 1
  return 1 + 1 / 2 * prod(degrees) * (sum(degrees) - n - 1)

"""
Generate a list of all genera of complete intersection curves up to a cutoff

Observe that the genus strictly increases if we increase the degree of a
defining equation, while adding a hyperplane section keeps the degree fixed.
So we can obtain all low genera starting from the line in P^2, and increasing
the number of equations and the degrees of the defining equations
"""
def listOfGenera(cutoff):
  queue = [(1,)]
  genera = []

  while len(queue) > 0:
    degrees = queue.pop()
    g = genus(degrees)

    if g < cutoff:
      # if we haven't found this one yet we add it to the list
      if g not in genera:
        genera.append(g)
        # use this to get information on how to realise a curve
        # print (g, degrees)

      # add (d_1,...,d_{n-1},2): with ,1 at the end genus is constant
      queue.append(degrees + (2,))

      # add all valid (d_1,...,d_i+1,...,d_{n-1})
      for i in range(len(degrees)):
        new = list(degrees)
        new[i] = new[i] + 1
        
        # we only look at increasing lists of degrees
        if sorted(new) == new: 
          queue.append(tuple(new))

  return sorted(genera)