Conic parametrization broken

[kliem]

The final output here is wrong:

sage: c = Conic(GF(2), [1,1,1,1,1,0])
....: 
sage: c.parametrization()
(Scheme morphism:
   From: Projective Space of dimension 1 over Finite Field of size 2
   To:   Projective Conic Curve over Finite Field of size 2 defined by x^2 + x*y + y^2 + x*z + y*z
   Defn: Defined on coordinates by sending (x : y) to
         (x*y + y^2 : x^2 + x*y : x^2 + x*y + y^2),
 Scheme morphism:
   From: Projective Conic Curve over Finite Field of size 2 defined by x^2 + x*y + y^2 + x*z + y*z
   To:   Projective Space of dimension 1 over Finite Field of size 2
   Defn: Defined on coordinates by sending (x : y : z) to
         (y : x))
sage: f, g = c.parametrization()
sage: (g*f).is_one()
False

The same here:

sage: R.<x,y,z> = QQ[]                                                                                                                                                              
sage: C = Curve(7*x^2 + 2*y*z + z^2)                                                                                                                                                
sage: f, g = C.parametrization(); f,g                                                                                                                                               
(Scheme morphism:
   From: Projective Space of dimension 1 over Rational Field
   To:   Projective Conic Curve over Rational Field defined by 7*x^2 + 2*y*z + z^2
   Defn: Defined on coordinates by sending (x : y) to
         (-2*x*y : x^2 + 7*y^2 : -2*x^2),
 Scheme morphism:
   From: Projective Conic Curve over Rational Field defined by 7*x^2 + 2*y*z + z^2
   To:   Projective Space of dimension 1 over Rational Field
   Defn: Defined on coordinates by sending (x : y : z) to
         (-1/2*x : 1/7*y + 1/14*z))
sage: (g*f).is_one()                                                                                                                                                                
False
sage: g([0, -1, 2])                                                                                                                                                                 
...
ValueError: [0, 0] does not define a valid point since all entries are 0
sage: p = g.domain().defining_polynomial()                                                                                                                                          
sage: p([0, -1, 2])                                                                                                                                                                 
0

Depends on #33953

CC: @mstreng @JohnCremona

Component: algebraic geometry

Author: Kwankyu Lee

Branch/Commit: b2af690

Reviewer: Marco Streng

Issue created by migration from https://trac.sagemath.org/ticket/31892

[JohnCremona]

comment:2

There is an actual mathematical issue here. The map from P^2 to the conic is defined everywhere by a single triple of polynomials, but the map from the conic to P^1 cannot be defined by a single pair.

For example the familiar easiest example X^2 + Y^2 = Z^2

is parametrized by mapping (u : v) to

(x : y : z) = (u^2 - v^2 : 2 u v : u^2 + v^2),

but the reverse map is given by mapping (x : y : z) to

(u : v) = (x + z : y) = (y : z - x)

where

• the first formula (x + z : y) is defined away from (-1 : 0 : 1)
• and the second (y : z - x) away from (1 : 0 : 1),
• and where they are both defined they agree.

That's the way maps between curves works.

Maps betwee (Sage) Schemes should be capable of being defined on patches like this.

The docstring for the method parametrization() does have the

Warning "The second map is currently broken
and neither the inverse nor well-defined."

which while not being grammatical does give a warning.

[mstreng]

comment:5

The inverse is correct in both examples at this ticket description, but the way it works in SageMath is a little bit broken.

Here's what I get in version 9.5. I'm using paramatrization(P) in order to replicate the ticket description's example, because parametrization is randomized if you don't specify a point.

sage: C = Conic(GF(2), [1,1,1,1,1,0])                                           
sage: P = C([0,0,1])                                                            
sage: (f, g) = C.parametrization(P); (f, g)                                     
(Scheme morphism:
   From: Projective Space of dimension 1 over Finite Field of size 2
   To:   Projective Conic Curve over Finite Field of size 2 defined by x^2 + x*y + y^2 + x*z + y*z
   Defn: Defined on coordinates by sending (x : y) to
         (x*y + y^2 : x^2 + x*y : x^2 + x*y + y^2),
 Scheme morphism:
   From: Projective Conic Curve over Finite Field of size 2 defined by x^2 + x*y + y^2 + x*z + y*z
   To:   Projective Space of dimension 1 over Finite Field of size 2
   Defn: Defined on coordinates by sending (x : y : z) to
         (y : x))
sage: g.representatives()                                                                       
[Scheme morphism:
   From: Projective Conic Curve over Finite Field of size 2 defined by x^2 + x*y + y^2 + x*z + y*z
   To:   Projective Space of dimension 1 over Finite Field of size 2
   Defn: Defined on coordinates by sending (x : y : z) to
         (y : x),
 Scheme morphism:
   From: Projective Conic Curve over Finite Field of size 2 defined by x^2 + x*y + y^2 + x*z + y*z
   To:   Projective Space of dimension 1 over Finite Field of size 2
   Defn: Defined on coordinates by sending (x : y : z) to
         (x + y + z : y + z)]
sage: g*f                                                                       
Scheme endomorphism of Projective Space of dimension 1 over Finite Field of size 2
  Defn: Defined on coordinates by sending (x : y) to
        (x^2 + x*y : x*y + y^2)
sage: f*g                                                                       
Scheme endomorphism of Projective Conic Curve over Finite Field of size 2 defined by x^2 + x*y + y^2 + x*z + y*z
  Defn: Defined on coordinates by sending (x : y : z) to
        (x^2 + x*y : x*y + y^2 : x^2 + x*y + y^2)
sage: P1 = f.domain()
sage: P1.hom(P1.gens(), P1) == g*f                             
True

You can see from the output of representatives that g really knows that the first formula is not defined at all points and hence there is a second formula stored inside g, which does work in (0:0:1). This agrees with what John explained. Moreover, we have (x^2+xy : xy + y^2) = (x(x+y) : y(x+y)) = (x : y), so the output of g*f really is the identity map, and the same holds for f*g. So this is correct so far. Then the following is all incorrect:

g(P)
ValueError: [0, 0] does not define a valid point since all entries are 0
(g*f).is_one()
False
(f*g).is_one()
False
sage: C.Hom(C).one() == f*g
False
sage: (x,y,z) = C.gens(); C.hom([x,y,z],C) == f*g                                                   
False

Something indeed needs to be done about this. The following indicates that we are very close.

sage: g.representatives()[1](P)
(1 : 1)
sage: (f*g).representatives()[0]                                                                
Scheme endomorphism of Projective Conic Curve over Finite Field of size 2 defined by x^2 + x*y + y^2 + x*z + y*z
  Defn: Defined on coordinates by sending (x : y : z) to
        (x : y : z)

Here are some other things that don't work:

sage: (f*g).representatives()[0].is_one()
False
sage: C.Hom(C).one().is_one()
TypeError: ...
sage: (g*f).representatives()                                                                   
AttributeError: ...

But I don't know enough about the inner workings of SageMath schemes to fix it.

ps. The warning mentioned by John refers to this ticket, so there seems to be no reason to doubt the output of parametrization.

[mstreng]

comment:7

See #33603: Fix Conic doctests.

[kwankyu]

Branch: u/klee/31892

[kwankyu]

Author: Kwankyu Lee

[sagetrac-git]

Commit: 9b0ba8f

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

9b0ba8f

Refactor identity morphism

[kwankyu]

comment:11

Part of the issue is fixed incidentally by #33953.

The branch fixes other issues related with the identity morphisms.

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

7e9b7d3

Fix some doctests

[sagetrac-git]

Changed commit from 9b0ba8f to 7e9b7d3

[kwankyu]

comment:14

With #33953, this g([0, -1, 2]) gives the correct answer.

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

f89842e

Fix more doctest errors

[sagetrac-git]

Changed commit from 7e9b7d3 to f89842e

[sagetrac-git]

Changed commit from f89842e to 78ba3a9

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

78ba3a9

Fix morphism of conics

[kwankyu]

comment:17

The last commit contains fixes so that g([0, -1, 2]) gives the correct answer.

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

d9982f3

Extend to morphisms to affine spaces

[sagetrac-git]

Changed commit from c95d429 to d9982f3

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

e66bb26

Minor edits

[sagetrac-git]

Changed commit from d9982f3 to e66bb26

[kwankyu]

comment:33

Added Cremona's example :) Needs review.

[sagetrac-git]

Changed commit from e66bb26 to 527c3a4

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

527c3a4

Merge branch 'develop' into t/31892/31892

[kwankyu]

comment:35

Rebased onto the latest develop branch, with the side purpose of checking trac after recent maintenance.

[mstreng]

comment:36

Tests pass. Code looks good. Just one thing before giving this a positive review:

Could you change the following mathematically incorrect example?

sage: C = Curve(x^2 + y^2 - z^2)
sage: A.<u> = AffineSpace(QQ, 1)
sage: f = C.hom([(x + z)/y], A)
sage: g = C.hom([y/(z - x)], A)
sage: f == g
True

The reason: it relies on mathematically incorrect behaviour of SageMath. There are no non-constant morphisms from C to A. Indeed, all non-constant morphisms from C to P¹ are surjective, hence do not land inside A. For example, f((1:0:1)) is not defined as an element of A. SageMath incorrectly claims that f is a morphism, as follows:

sage: R.<x,y,z> = QQ[]                                                                                                      
sage: C = Curve(x^2 + y^2 - z^2)                                                                                            
sage: A.<u> = AffineSpace(QQ, 1)                                                                                            
sage: f = C.hom([(x + z)/y], A)                                                                                             
sage: f                                                                                                                     
Scheme morphism:
  From: Projective Conic Curve over Rational Field defined by x^2 + y^2 - z^2
  To:   Affine Space of dimension 1 over Rational Field
  Defn: Defined on coordinates by sending (x : y : z) to
        ((x + z)/y)
sage: f.parent()                                                                                                            
Set of morphisms
  From: Projective Conic Curve over Rational Field defined by x^2 + y^2 - z^2
  To:   Affine Space of dimension 1 over Rational Field

This behaviour is not introduced at this ticket, but was already there in e.g. SageMath 9.5, so you do not need to fix it. But we should not add examples to the documentation that rely on this incorrect behaviour.
Perhaps you could change the example as follows:

sage: C = Curve(x^2 + y^2 - z^2)
sage: P.<u,v> = ProjectiveSpace(QQ, 1)
sage: f = C.hom([x + z, y], P)
sage: g = C.hom([y, z - x], P)
sage: f == g
True

And/or you could give constant examples with codomain A. And perhaps you could also add an h for which f == h is False.

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

c789509	Merge branch 'develop' into t/31892/31892
b2af690	Fix an errorneous example

[sagetrac-git]

Changed commit from 527c3a4 to b2af690

[kwankyu]

comment:38

Replying to Marco Streng:

Could you change the following mathematically incorrect example?

sage: C = Curve(x^2 + y^2 - z^2)
sage: A.<u> = AffineSpace(QQ, 1)
sage: f = C.hom([(x + z)/y], A)
sage: g = C.hom([y/(z - x)], A)
sage: f == g
True

The reason: it relies on mathematically incorrect behaviour of SageMath. There are no non-constant morphisms from C to A. Indeed, all non-constant morphisms from C to P¹ are surjective, hence do not land inside A. For example, f((1:0:1)) is not defined as an element of A.

You are right. I wonder how I transformed John's correct example to a wrong one!

Perhaps you could change the example as follows:

sage: C = Curve(x^2 + y^2 - z^2)
sage: P.<u,v> = ProjectiveSpace(QQ, 1)
sage: f = C.hom([x + z, y], P)
sage: g = C.hom([y, z - x], P)
sage: f == g
True

and perhaps you could also add an h for which f == h is False.

I did this. Thank you.

[JohnCremona]

comment:39

Thanks to all (both) of you for working on this.

[mstreng]

comment:40

Replying to Marco Streng:

Just one thing before giving this a positive review

Thanks for fixing this. Looks good to me.

[mstreng]

Reviewer: Marco Streng

[kwankyu]

comment:42

Thanks for the review!

[vbraun]

Changed branch from u/klee/31892 to b2af690