#14353: Factor toric morphism into surjective and generically injective
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Reporter: vbraun | Owner: AlexGhitza
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-5.9
Component: algebraic geometry | Resolution:
Keywords: | Work issues:
Report Upstream: N/A | Reviewers:
Authors: Volker Braun | Merged in:
Dependencies: | Stopgaps:
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Comment (by novoselt):
Hey Volker, thanks for opening this ticket!
As I was thinking more about it, toric morphisms naturally factor into
THREE: if `f: Sigma --> Sigma'`, then we have
`Sigma --f_s--> Sigma_s --f_b--> Sigma_i --f_i--> Sigma'`
where `f_s` is surjection onto the fan which is the image of Sigma, `f_i`
is injection of Sigma' intersected with the image sublattice, and `f_b` is
a birational map between the intermediate toric varieties which does not
have to be surjective or injective. The map from a single blowup chart
into the original variety is precisely `f_b` with the other two being
identities.
I would like to do the following:
- adjust factoring to return three morphisms instead of two;
- make intermediate fans live in the `image_sublattice`;
- name them, perhaps, `domain_fan_image` (under the map of lattices) and
`codomain_fan_restriction` (to the `image_sublattice`);
- get rid of `restrict_to_image` method as it is misleading, if not
wrong: restriction to the image of a toric morphism is not necessarily a
toric morphism (it is not for the blow up chart). It does not seem like it
is used anywhere but one place and basically it was trying to do the
factoring in a not very good way.
I am working on it now on top of your patches, so let me know if you have
severe objections!
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/14353#comment:4>
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