Wrap fan morphism in toric morphism

[vbraun]

Since we have the very nice fan morphism class, we should use it to define toric morphisms of toric varieties.

A big part of the patch is porting the scheme morphisms / hom sets to new-style parents and coercion. Categories should be better, too. Fixes #7946 and #6810 as a side effect.

The first two patches bring some sanity to the scheme morphisms. The 3rd patch changes names of methods/classes to something more reasonable and adds documentation. The 4th patch actually adds toric morphisms defined by polynomials or fan morphisms.

Apply:

• attachment: trac_11599_no_circular_imports.patch
• attachment: trac_11599_homset_new_coercion_model.patch
• attachment: trac_11599_rename_morphisms.patch
• attachment: trac_11599_toric_morphisms.patch
• attachment: trac_11599_reviewer.patch
• attachment: trac_11599_remove_class_suffix.patch
• attachment: trac_11599_remaining_fixes.patch
• attachment: trac_11599_numerical_noise.patch

CC: @sagetrac-davideklund @sagetrac-fschulze @miguelmarco

Component: algebraic geometry

Author: Volker Braun

Reviewer: Andrey Novoseltsev

Merged: sage-5.0.beta9

Issue created by migration from https://trac.sagemath.org/ticket/11599

[novoselt]

comment:1

We should. I have been using the following code so far:

def toric_morphism_list(phi, X, Y):
    r"""
    Given a fan morphism phi from X.fan() -> Y.fan(), return the list representation of the corresponding toric morphism.
    """
    x = [SR.var(x_i, domain="positive") for x_i in X.coordinate_ring().variable_names()]
    result = [1] * Y.fan().nrays()
    for rho, x_rho in zip(X.fan(1), x):
        sigma_prime = phi.image_cone(rho)
        degrees = sigma_prime.ray_matrix() \ phi(rho.ray(0))
        for i, d in zip(sigma_prime.ambient_ray_indices(), degrees):
            result[i] *= x_rho^d
    return result

def toric_morphism_dictionary(phi, X, Y):
    r"""
    Given a fan morphism phi from X.fan() -> Y.fan(), return the dictionary representation of the corresponding toric morphism.
    """
    x = [SR.var(x_i, domain="positive") for x_i in X.coordinate_ring().variable_names()]
    y = [SR.var(y_i, domain="positive") for y_i in Y.coordinate_ring().variable_names()]
    result = dict((y_i, 1) for y_i in y)
    for rho, x_rho in zip(X.fan(1), x):
        sigma_prime = phi.image_cone(rho)
        degrees = sigma_prime.ray_matrix() \ phi(rho.ray(0))
        for i, d in zip(sigma_prime.ambient_ray_indices(), degrees):
            result[y[i]] *= x_rho^d
    return result

Using the dictionary representation it is quite easy to compute pullbacks, the problem here is that the underlying map of total coordinate rings is not a ring homomorphism, since it is likely to involve roots. The following paper may be useful for "correct and general" implementation:
http://arxiv.org/abs/1004.4924

[novoselt]

comment:2

P.S. Ordered dictionaries seem very appropriate for this approach, but we need to upgrade Python for them.

[vbraun]

comment:3

Thanks for pointing out the reference. Its fairly obvious that one has to use roots to write the maps but still its good to see that somebody worked out all the details. Though from a quick browse it seems like they don't elaborate on the relation with fan morphisms. E.g. the embedding of the obit closure can be written as a polynomial map in homogeneous coordinates but is not a toric morphism (given by a fan morphism).

My plan is to implement maps by homogeneous coordinate polynomials and maps by fan morphisms separately, with conversion methods from one to the other if it exists.

Eventually we should also have maps involving roots. I'm not sure how we should implement them; Just using symbolic ring variables would be simple but not play nice with compositions. At one point it would be good to write our own "Homogeneous coordinate ring" class that knows about the homogeneous rescalings. This would then allow for fractional powers in some nicer way. But I'll leave it for another ticket ;)

[novoselt]

comment:4

They don't elaborate the relation with fan morphisms because they consider arbitrary maps of toric varieties as varieties, including those that don't care about toric structure at all. In particular it is applicable to equivariant morphisms and orbit inclusions. And even in these cases it is necessary to use roots if one of the varieties is not smooth. E.g. a resolution of a singular variety would correspond to the identity map of lattices, but would involve roots in homogeneous coordinates.

[vbraun]

comment:5

I know. But without roots you can still have homogeneous polynomial maps and toric morphisms. Neither of the two is contained in the other. So I'm planning on implementing toric (equivariant) morphism separately.

[vbraun]

Description changed:

--- 
+++ 
@@ -1 +1,7 @@
 Since we have the very nice fan morphism class, we should use it to define toric morphisms of toric varieties.
+
+A big part of the patch is porting the scheme morphisms / hom sets to new-style parents and coercion. Categories should be better, too. Fixes #7946 as a side effect.
+
+Apply:
+* [attachment: trac_11599_no_circular_imports.py](https://github.com/sagemath/sage-prod/files/10653334/trac_11599_no_circular_imports.py.gz)
+* [attachment: trac_11599_homset_new_coercion_model.patch](https://github.com/sagemath/sage-prod/files/10653338/trac_11599_homset_new_coercion_model.patch.gz)

[vbraun]

comment:6

Ok so far no new functionality. But I think now the groundwork is done and we can actually do some work on top of it. I'm sorry for the giant patch :-)

[vbraun]

comment:7

Pickling is broken for some complex object like EllipticCurveTorsionSubgroup that contain circular self-references. This is a bug in unpickling the category framework, and will be addressed elsewhere. I've marked the offending unpickling doctests with # optional - pickling so we can fix them later.

[vbraun]

Updated patch

[vbraun]

Author: Volker Braun