Opened 10 years ago
Last modified 9 years ago
#11599 closed enhancement
Wrap fan moriphism in toric morphism — at Version 8
Reported by: | vbraun | Owned by: | AlexGhitza |
---|---|---|---|
Priority: | major | Milestone: | sage-5.0 |
Component: | algebraic geometry | Keywords: | |
Cc: | davideklund, fschulze, mmarco | Merged in: | |
Authors: | Volker Braun | Reviewers: | |
Report Upstream: | N/A | Work issues: | |
Branch: | Commit: | ||
Dependencies: | Stopgaps: |
Description (last modified by )
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.
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:
Change History (9)
comment:1 Changed 10 years ago by
comment:2 Changed 10 years ago by
P.S. Ordered dictionaries seem very appropriate for this approach, but we need to upgrade Python for them.
comment:3 Changed 10 years ago by
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 ;)
comment:4 Changed 10 years ago by
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.
comment:5 Changed 10 years ago by
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.
comment:6 Changed 10 years ago by
- Description modified (diff)
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 :-)
comment:7 Changed 10 years ago by
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.
comment:8 Changed 10 years ago by
- Description modified (diff)
- Status changed from new to needs_review
We should. I have been using the following code so far:
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