#15153: Check embedding morphism when comparing two toric varieties
----------------------------------+----------------------------
Reporter: jkeitel | Owner:
Type: defect | Status: new
Priority: major | Milestone: sage-6.0
Component: algebraic geometry | Keywords: toric
Merged in: | Authors:
Reviewers: | Report Upstream: N/A
Work issues: | Branch:
Commit: | Dependencies:
Stopgaps: |
----------------------------------+----------------------------
At the moment two toric varieties are taken to be equal if ''their fans
are the same, names of variables are the same and stored in the same
order, and base fields are the same'', independent of whether
they have the same embedding morphism or not.
What does that imply? Since ToricDivisorGroup inherits from
UniqueRepresentation, two toric varieties that only differ in their
embedding will share the same ToricDivisorGroup object and therefore a
ToricDivisor created by a variety will reference the variety that first
created its ToricDivisorGroup object, independent of whether that's
correct or not.
As a consequence, we have the following behavior:
{{{
sage: P2 = toric_varieties.P(2)
sage: variety1 = P2.affine_patch(2)
sage: variety2 = ToricVariety(variety1.fan())
sage: variety1 == variety2
True
sage: div1 = variety1.divisor(Cone([(1,0)]))
sage: div2 = variety2.divisor(Cone([(1,0)]))
sage: div1.scheme().embedding_morphism()
Scheme morphism:
From: 2-d affine toric variety
To: 2-d CPR-Fano toric variety covered by 3 affine patches
Defn: Defined on coordinates by sending [z0 : z1] to
[z0 : z1 : 1]
sage: div2.scheme().embedding_morphism() # should produce an exception
Scheme morphism:
From: 2-d affine toric variety
To: 2-d CPR-Fano toric variety covered by 3 affine patches
Defn: Defined on coordinates by sending [z0 : z1] to
[z0 : z1 : 1]
sage: div2.scheme() is variety2
False
sage: div2.scheme() is variety1
True
}}}
While this set-up might seem highly artificial, in the context of toric
fibrations in #12892 one naturally gets fiber varieties that only differ
in their embedding.
Unless there are arguments against it, I therefore suggest to extend
{{{ToricVariety_field.__cmp__}}} to include the embedding morphism.
--
Ticket URL: <http://trac.sagemath.org/ticket/15153>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
--
You received this message because you are subscribed to the Google Groups
"sage-trac" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to [email protected].
To post to this group, send email to [email protected].
Visit this group at http://groups.google.com/group/sage-trac.
For more options, visit https://groups.google.com/groups/opt_out.
