pbelmans

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pbelmans

\documentclass[10pt,a4paper,landscape]{article}
\usepackage[T1]{fontenc}
\usepackage{amsmath, amsfonts}
\usepackage[charter]{mathdesign}
\usepackage[scaled]{beramono,berasans}
\usepackage{ifthen}
\usepackage{tikz}
\usetikzlibrary{decorations.pathmorphing, arrows}

\begin{document}

pbelmans

# determine the linear equation of a hypersurface under the d-uple embedding
def getLinearEquation(equation):
  d = equation.degree()
  monomials = getMonomials(equation.parent(), d)
  coefficients = [equation.monomial_coefficient(monomial) for monomial in monomials]
  ring = PolynomialRing(QQ, 'x', len(coefficients))

  return sum(c * x for c, x in zip(coefficients, ring.gens()))

# determine the maximum degree of a list of (homogeneous) equations

pbelmans

def mA(n):
  if n % 2 == 1: return (n - 1) / 2
  else:          return (n - 2) / 2

def mD(n):
  if n % 2 == 1: return (n - 3) / 2
  else:          return (n - 2) / 2

def u(n):
  if n % 2 == 1: return mD(n) + 1

pbelmans

# TODO write code

pbelmans

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

pbelmans

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)

pbelmans

\documentclass[12pt]{standalone}
\usepackage{tikz-cd}
\usetikzlibrary{decorations.markings}
\makeatletter
\tikzcdset{
open/.code={\tikzcdset{hook, circled};},
closed/.code={\tikzcdset{hook, slashed};},
open'/.code={\tikzcdset{hook', circled};},
closed'/.code={\tikzcdset{hook', slashed};},
circled/.code={\tikzcdset{markwith={\draw (0,0) circle (.375ex);}};},

pbelmans

from sage.schemes.chow.all import *

# dimension 6
X = ChowScheme(6, ["c2", "c4", "c6"], [2, 4, 6])
X.chowring().inject_variables()
td = Sheaf(X, 6, 1 + sum(X.gens())).todd_class()
integrand = (td._logg()/2)._expp().by_degrees()[6]

# O'Grady 6
values = {c2^3: 30720, c2*c4: 7680, c6: 1920}