Coverage for local/lib/python2.7/site-packages/sage/schemes/hyperelliptic_curves/kummer_surface.py : 0%

""" 

Kummer surfaces over a general ring 

""" 

#***************************************************************************** 

# Copyright (C) 2006 David Kohel <kohel@maths.usyd.edu> 

# Distributed under the terms of the GNU General Public License (GPL) 

# http://www.gnu.org/licenses/ 

#***************************************************************************** 

from sage.schemes.projective.projective_space import ProjectiveSpace 

from sage.schemes.projective.projective_subscheme\ 

import AlgebraicScheme_subscheme_projective 

from sage.categories.homset import Hom 

from sage.categories.all import Schemes 

# The generic genus 2 curve in Weierstrass form: 

# 

# y^2 + (c9*x^3 + c6*x^2 + c3*x + c0)*y = 

# a12*x^6 + a10*x^5 + a8*x^4 + a6*x^3 + a4*x^2 + a2*x + a0. 

# 

# Transforms to: 

# 

# y^2 = (c9^2 + 4*a12)*x^6 + (2*c6*c9 + 4*a10)*x^5 

# + (2*c3*c9 + c6^2 + 4*a8)*x^4 + (2*c0*c9 + 2*c3*c6 + 4*a6)*x^3 

# + (2*c0*c6 + c3^2 + 4*a4)*x^2 + (2*c0*c3 + 4*a2)*x + c0^2 + 4*a0 

class KummerSurface(AlgebraicScheme_subscheme_projective): 

def __init__(self,J): 

""" 

""" 

R = J.base_ring() 

PP = ProjectiveSpace(3,R,["X0","X1","X2","X3"]) 

X0, X1, X2, X3 = PP.gens() 

C = J.curve() 

f, h = C.hyperelliptic_polynomials() 

a12 = f[0]; a10 = f[1]; a8 = f[2]; 

a6 = f[3]; a4 = f[4]; a2 = f[5]; a0 = f[6] 

if h != 0: 

c6 = h[0]; c4 = h[1]; c2 = h[2]; c0 = h[3] 

a12, a10, a8, a6, a4, a2, a0 = \ 

(4*a12 + c6**2, 

4*a10 + 2*c4*c6, 

4*a8 + 2*c2*c6 + c4**2, 

4*a6 + 2*c0*c6 + 2*c2*c4, 

4*a4 + 2*c0*c4 + c2**2, 

4*a2 + 2*c0*c2, 

4*a0 + c0**2) 

F = \ 

(-4*a8*a12 + a10**2)*X0**4 + \ 

-4*a6*a12*X0**3*X1 + \ 

-2*a6*a10*X0**3*X2 + \ 

-4*a12*X0**3*X3 + \ 

-4*a4*a12*X0**2*X1**2 + \ 

(4*a2*a12 - 4*a4*a10)*X0**2*X1*X2 + \ 

-2*a10*X0**2*X1*X3 + \ 

(-4*a0*a12 + 2*a2*a10 - 4*a4*a8 + a6**2)*X0**2*X2**2 + \ 

-4*a8*X0**2*X2*X3 + \ 

-4*a2*a12*X0*X1**3 + \ 

(8*a0*a12 - 4*a2*a10)*X0*X1**2*X2 + \ 

(4*a0*a10 - 4*a2*a8)*X0*X1*X2**2 + \ 

-2*a6*X0*X1*X2*X3 + \ 

-2*a2*a6*X0*X2**3 + \ 

-4*a4*X0*X2**2*X3 + \ 

-4*X0*X2*X3**2 + \ 

-4*a0*a12*X1**4 + \ 

-4*a0*a10*X1**3*X2 + \ 

-4*a0*a8*X1**2*X2**2 + \ 

X1**2*X3**2 + \ 

-4*a0*a6*X1*X2**3 + \ 

-2*a2*X1*X2**2*X3 + \ 

(-4*a0*a4 + a2**2)*X2**4 + \ 

-4*a0*X2**3*X3 

AlgebraicScheme_subscheme_projective.__init__(self, PP, F) 

X, Y, Z = C.ambient_space().gens() 

if a0 ==0: 

a0 = a2 

phi = Hom(C,self)([0,Z**2,X*Z,a0*X**2],Schemes()) 

C._kummer_morphism = phi 

J._kummer_surface = self