Defining affine curves in 3D space

[wjp]

Reported by Ronald van Luijk:

because the Curve constructor automatically interprets a homogeneous polynomial in 3 variables as a projective curve, the following doesn't work:

A.<x,y,z>=AffineSpace(QQ,3)
C=Curve([x-y,x-z])

CC: @kwankyu

Component: algebraic geometry

Issue created by migration from https://trac.sagemath.org/ticket/7954

[wjp]

Attachment: 7954_curve_constructor.patch.gz

[wjp]

comment:1

The attached patch adds an optional space='affine'/'projective' keyword argument to the Curve constructor.

[williamstein]

Attachment: trac_7954-part2.patch.gz

the previous patch was part 1. apply this after that one.

[williamstein]

comment:2

Hi,

I made some important improvements to the patch while refereeing it. However, it is clear after looking at this code for a bit that there is a MAJOR design flaw (which Willem pointed out to me in person). The flaw is this in plane_curve/affine_curve.py:

class AffineSpaceCurve_generic(Curve_generic, AlgebraicScheme_subscheme_affine):
    def _repr_type(self):
        return "Affine Space"

However, Curve_generic is very much a plane curve:

class Curve_generic(AlgebraicScheme_subscheme):
...
    def defining_polynomial(self):
        return self.defining_polynomials()[0]

Thus the isa relationship that must be satisfied by derivation of objects is broken.

The space curve code must be completely moved out of this directory to the appropriate place.

[williamstein]

comment:3

I'm going to wait to talk to willem about this instead of just trying to do it myself.

[kwankyu]

comment:9

The issue seems to have been solved now by adding Ambient space argument to the Curve constructor.

sage: A.<x,y,z>=AffineSpace(QQ,3)
sage: C = Curve([x-y,y-z],A)
sage: C
Affine Curve over Rational Field defined by x - y, y - z