A266322 Genera of complete intersection curves.

0, 1, 3, 4, 5, 6, 9, 10, 13, 15, 16, 17, 19, 21, 25, 28, 31, 33, 36, 37, 41, 45, 46, 49, 51, 55, 61, 64, 65, 66, 73, 76, 78, 81, 85, 91, 97, 99, 100, 101, 105, 106, 109, 113, 120, 121, 129, 136, 141, 144, 145, 148, 153, 161, 163, 166, 169, 171, 176, 181, 190

Offset

1,3

Links

Pieter Belmans, Table of n, a(n) for n = 1..215

P. Deligne and N. Katz, Groupes de monodromie en géométrie algébrique (SGA 7 II), Springer-Verlag, 1973, pages 39-61.

Prog

(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 realize a curve
        # print (g, degrees)
      # 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))
      # add (d_1, ..., d_{n-1}, 2): with , 1 at the end genus is constant
      queue.append(degrees + (2, ))
  return sorted(genera)

Crossrefs

Sequence in context: A129948 A050414 A342469 * A136681 A206330 A104373

Adjacent sequences: A266319 A266320 A266321 * A266323 A266324 A266325

Keyword

nonn

Author

Pieter Belmans, Dec 27 2015

Status

approved