Book a Demo!
CoCalc Logo Icon
StoreFeaturesDocsShareSupportNewsAboutPoliciesSign UpSign In
sagemath
GitHub Repository: sagemath/sagelib
 Path: blob/master/sage/schemes/hyperelliptic_curves/kummer_surface.py
4102 views
1
"""

2
Kummer surfaces over a general ring

3
"""

4


5
#*****************************************************************************

6
#  Copyright (C) 2006 David Kohel <[email protected]>

7
#  Distributed under the terms of the GNU General Public License (GPL)

8
#                  http://www.gnu.org/licenses/

9
#*****************************************************************************

10


11
from sage.schemes.generic.all import ProjectiveSpace

12
from sage.schemes.generic.algebraic_scheme \

13
     import AlgebraicScheme_subscheme_projective

14
from sage.categories.homset import Hom

15
from sage.categories.all import Schemes

16


17
# The generic genus 2 curve in Weierstrass form:

18
#

19
# y^2 + (c9*x^3 + c6*x^2 + c3*x + c0)*y =

20
#       a12*x^6 + a10*x^5 + a8*x^4 + a6*x^3 + a4*x^2 + a2*x + a0.

21
#

22
# Transforms to:

23
# 

24
# y^2 = (c9^2 + 4*a12)*x^6 + (2*c6*c9 + 4*a10)*x^5 

25
#     + (2*c3*c9 + c6^2 + 4*a8)*x^4 + (2*c0*c9 + 2*c3*c6 + 4*a6)*x^3 

26
#     + (2*c0*c6 + c3^2 + 4*a4)*x^2 + (2*c0*c3 + 4*a2)*x + c0^2 + 4*a0

27


28
class KummerSurface(AlgebraicScheme_subscheme_projective):

29
    def __init__(self,J):

30
        """

31
        """

32
        R = J.base_ring()

33
        PP = ProjectiveSpace(3,R,["X0","X1","X2","X3"])

34
        X0, X1, X2, X3 = PP.gens()

35
        C = J.curve()

36
        f, h = C.hyperelliptic_polynomials()

37
        a12 = f[0]; a10 = f[1]; a8 = f[2];

38
        a6 = f[3]; a4 = f[4]; a2 = f[5]; a0 = f[6]

39
        if h != 0:

40
            c6 = h[0]; c4 = h[1]; c2 = h[2]; c0 = h[3]

41
            a12, a10, a8, a6, a4, a2, a0 = \

42
                 (4*a12 + c6**2,

43
                  4*a10 + 2*c4*c6,

44
                  4*a8 + 2*c2*c6 + c4**2,

45
                  4*a6 + 2*c0*c6 + 2*c2*c4,

46
                  4*a4 + 2*c0*c4 + c2**2,

47
                  4*a2 + 2*c0*c2,

48
                  4*a0 + c0**2)

49
        F = \

50
          (-4*a8*a12 + a10**2)*X0**4 + \

51
          -4*a6*a12*X0**3*X1 + \

52
          -2*a6*a10*X0**3*X2 + \

53
          -4*a12*X0**3*X3 + \

54
          -4*a4*a12*X0**2*X1**2 + \

55
          (4*a2*a12 - 4*a4*a10)*X0**2*X1*X2 + \

56
          -2*a10*X0**2*X1*X3 + \

57
          (-4*a0*a12 + 2*a2*a10 - 4*a4*a8 + a6**2)*X0**2*X2**2 + \

58
          -4*a8*X0**2*X2*X3 + \

59
          -4*a2*a12*X0*X1**3 + \

60
          (8*a0*a12 - 4*a2*a10)*X0*X1**2*X2 + \

61
          (4*a0*a10 - 4*a2*a8)*X0*X1*X2**2 + \

62
          -2*a6*X0*X1*X2*X3 + \

63
          -2*a2*a6*X0*X2**3 + \

64
          -4*a4*X0*X2**2*X3 + \

65
          -4*X0*X2*X3**2 + \

66
          -4*a0*a12*X1**4 + \

67
          -4*a0*a10*X1**3*X2 + \

68
          -4*a0*a8*X1**2*X2**2 + \

69
          X1**2*X3**2 + \

70
          -4*a0*a6*X1*X2**3 + \

71
          -2*a2*X1*X2**2*X3 + \

72
          (-4*a0*a4 + a2**2)*X2**4 + \

73
          -4*a0*X2**3*X3

74
        AlgebraicScheme_subscheme_projective.__init__(self, PP, F)

75
        X, Y, Z = C.ambient_space().gens()

76
        if a0 ==0:

77
            a0 = a2

78
        phi = Hom(C,self)([0,Z**2,X*Z,a0*X**2],Schemes())

79
        C._kummer_morphism = phi

80
        J._kummer_surface = self

81
    

82


83
    

84


85