Module: sage.schemes.plane_curves.constructor
Author Log:
Module-level Functions
F) |
Return the plane or space curve defined by
, where
can be
either a multivariate polynomial, a list or tuple of polynomials,
or an algebraic scheme.
If
is in two variables the curve is affine, and if it is
homogenous in
variables, then the curve is projective.
A projective plane curve
sage: x,y,z = Q['x,y,z'].gens() sage: C = Curve(x^3 + y^3 + z^3); C Projective Curve over Rational Field defined by z^3 + y^3 + x^3 sage: C.genus() 1
Affine plane curves
sage: x,y = GF(7)['x,y'].gens() sage: C = Curve(y^2 + x^3 + x^10); C Affine Curve over Finite Field of size 7 defined by y^2 + x^3 + x^10 sage: C.genus() 0 sage: x, y = Q['x,y'].gens() sage: Curve(x^3 + y^3 + 1) Affine Curve over Rational Field defined by 1 + y^3 + x^3
A projective space curve
sage: x,y,z,w = Q['x,y,z,w'].gens() sage: C = Curve([x^3 + y^3 - z^3 - w^3, x^5 - y*z^4]); C Projective Space Curve over Rational Field defined by -1*w^3 - z^3 + y^3 + x^3 sage: C.genus() 13
An affine space curve
sage: x,y,z = Q['x,y,z'].gens() sage: C = Curve([y^2 + x^3 + x^10 + z^7, x^2 + y^2]); C Affine Space Curve over Rational Field defined by z^7 + y^2 + x^3 + x^10 sage: C.genus() 47
We can also make non-reduced non-irreducible curves.
sage: x,y,z = Q['x,y,z'].gens() sage: Curve((x-y)*(x+y)) Projective Curve over Rational Field defined by -1*y^2 + x^2 sage: Curve((x-y)^2*(x+y)^2) Projective Curve over Rational Field defined by y^4 - 2*x^2*y^2 + x^4
A union of curves is a curve.
sage: x,y,z = Q['x,y,z'].gens() sage: C = Curve(x^3 + y^3 + z^3) sage: D = Curve(x^4 + y^4 + z^4) sage: C.union(D) Projective Curve over Rational Field defined by z^7 + y^3*z^4 + y^4*z^3 + y^7 + x^3*z^4 + x^3*y^4 + x^4*z^3 + x^4*y^3 + x^7
The intersection is not a curve, though it is a scheme.
sage: X = C.intersection(D); X Closed subscheme of Projective Space of dimension 2 over Rational Field defined by: z^3 + y^3 + x^3 z^4 + y^4 + x^4
Note that the intersection has dimension 0 .
sage: X.dimension() 0 sage: I = X.defining_ideal(); I Ideal (z^3 + y^3 + x^3, z^4 + y^4 + x^4) of Polynomial Ring in x, y, z over Rational Field
Defining equation must be homogeneous. If the parent polynomial ring is in three variables, then the defining ideal must be homogeneous.
sage: x,y,z = Q['x,y,z'].gens() sage: Curve(x^2+y^2) Projective Curve over Rational Field defined by y^2 + x^2 sage: Curve(x^2+y^2+z) Traceback (most recent call last): ... TypeError: defining polynomials (= z + y^2 + x^2) must be homogeneous