General curve constructors¶
AUTHORS:
 William Stein (20051113)
 David Kohel (200601)
 Grayson Jorgenson (20166)

sage.schemes.curves.constructor.
Curve
(F, A=None)¶ Return the plane or space curve defined by
F
, whereF
can be either a multivariate polynomial, a list or tuple of polynomials, or an algebraic scheme.If no ambient space is passed in for
A
, and ifF
is not an algebraic scheme, a new ambient space is constructed.Also not specifying an ambient space will cause the curve to be defined in either affine or projective space based on properties of
F
. In particular, ifF
contains a nonhomogenous polynomial, the curve is affine, and ifF
consists of homogenous polynomials, then the curve is projective.INPUT:
F
– a multivariate polynomial, or a list or tuple of polynomials, or an algebraic scheme.A
– (default: None) an ambient space in which to create the curve.
EXAMPLES: A projective plane curve
sage: x,y,z = QQ['x,y,z'].gens() sage: C = Curve(x^3 + y^3 + z^3); C Projective Plane Curve over Rational Field defined by x^3 + y^3 + z^3 sage: C.genus() 1
EXAMPLES: Affine plane curves
sage: x,y = GF(7)['x,y'].gens() sage: C = Curve(y^2 + x^3 + x^10); C Affine Plane Curve over Finite Field of size 7 defined by x^10 + x^3 + y^2 sage: C.genus() 0 sage: x, y = QQ['x,y'].gens() sage: Curve(x^3 + y^3 + 1) Affine Plane Curve over Rational Field defined by x^3 + y^3 + 1
EXAMPLES: A projective space curve
sage: x,y,z,w = QQ['x,y,z,w'].gens() sage: C = Curve([x^3 + y^3  z^3  w^3, x^5  y*z^4]); C Projective Curve over Rational Field defined by x^3 + y^3  z^3  w^3, x^5  y*z^4 sage: C.genus() 13
EXAMPLES: An affine space curve
sage: x,y,z = QQ['x,y,z'].gens() sage: C = Curve([y^2 + x^3 + x^10 + z^7, x^2 + y^2]); C Affine Curve over Rational Field defined by x^10 + z^7 + x^3 + y^2, x^2 + y^2 sage: C.genus() 47
EXAMPLES: We can also make nonreduced nonirreducible curves.
sage: x,y,z = QQ['x,y,z'].gens() sage: Curve((xy)*(x+y)) Projective Conic Curve over Rational Field defined by x^2  y^2 sage: Curve((xy)^2*(x+y)^2) Projective Plane Curve over Rational Field defined by x^4  2*x^2*y^2 + y^4
EXAMPLES: A union of curves is a curve.
sage: x,y,z = QQ['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 Plane Curve over Rational Field defined by x^7 + x^4*y^3 + x^3*y^4 + y^7 + x^4*z^3 + y^4*z^3 + x^3*z^4 + y^3*z^4 + z^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: x^3 + y^3 + z^3, x^4 + y^4 + z^4
Note that the intersection has dimension \(0\).
sage: X.dimension() 0 sage: I = X.defining_ideal(); I Ideal (x^3 + y^3 + z^3, x^4 + y^4 + z^4) of Multivariate Polynomial Ring in x, y, z over Rational Field
EXAMPLES: In three variables, the 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 = QQ['x,y,z'].gens() sage: Curve(x^2+y^2) Projective Conic Curve over Rational Field defined by x^2 + y^2 sage: Curve(x^2+y^2+z) Traceback (most recent call last): ... TypeError: x^2 + y^2 + z is not a homogeneous polynomial
The defining polynomial must always be nonzero:
sage: P1.<x,y> = ProjectiveSpace(1,GF(5)) sage: Curve(0*x) Traceback (most recent call last): ... ValueError: defining polynomial of curve must be nonzero
sage: A.<x,y,z> = AffineSpace(QQ, 3) sage: C = Curve([y  x^2, z  x^3], A) sage: A == C.ambient_space() True