Kummer surfaces over a general ring¶

class sage.schemes.hyperelliptic_curves.kummer_surface.KummerSurface(J)[source]¶

Bases: AlgebraicScheme_subscheme_projective

EXAMPLES:

Sage

sage: R.<x> = QQ[]
sage: f = x^5 + x + 1
sage: X = HyperellipticCurve(f)
sage: J = Jacobian(X)
sage: K = KummerSurface(J); K
Closed subscheme of Projective Space of dimension 3 over Rational Field defined by:
X0^4 - 4*X0*X1^3 + 4*X0^2*X1*X2 - 4*X0*X1^2*X2 + 2*X0^2*X2^2 + X2^4 - 4*X0^3*X3 - 2*X0^2*X1*X3 - 2*X1*X2^2*X3 + X1^2*X3^2 - 4*X0*X2*X3^2

Python

>>> from sage.all import *
>>> R = QQ['x']; (x,) = R._first_ngens(1)
>>> f = x**Integer(5) + x + Integer(1)
>>> X = HyperellipticCurve(f)
>>> J = Jacobian(X)
>>> K = KummerSurface(J); K
Closed subscheme of Projective Space of dimension 3 over Rational Field defined by:
X0^4 - 4*X0*X1^3 + 4*X0^2*X1*X2 - 4*X0*X1^2*X2 + 2*X0^2*X2^2 + X2^4 - 4*X0^3*X3 - 2*X0^2*X1*X3 - 2*X1*X2^2*X3 + X1^2*X3^2 - 4*X0*X2*X3^2