#8988: Add support for toric varieties
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Reporter: novoselt | Owner: AlexGhitza
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-4.5
Component: algebraic geometry | Keywords:
Author: Andrey Novoseltsev | Upstream: N/A
Reviewer: Volker Braun | Merged:
Work_issues: |
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Comment(by novoselt):
I think (strongly ;-)) that `Kaehler_cone` method should return a cone in
the class group in some basis, since such cones with some additions form a
complete fan of the GKZ decomposition.
We should, however, have a clear way of going from divisors associated to
rays to this basis and, perhaps, a way to go back. The first is easy, the
i-th ray is represented by the i-th column of the Gale transform matrix.
However, it may not be obvious and does not feel natural, since one has to
involve functions of the fan, rather than toric variety directly.
For `Mori_cone` I would prefer to get the "traditional dual" of the
`Kaehler_cone`, i.e. just the cone formed by facet normals, because it is
confusing otherwise.
Again, there should be a clean way to go from any vector in the ambient
space of `Mori_cone` to the longer vector with clear interpretation of
each entry. I am OK with having a special function for the generators of
the Mori cone and your proposed name is fine (although I have always seen
this abbreviation as GLSM and would prefer all-capital version).
'''Regarding this ticket''', my main point is that these functions require
some more work and since they operate with divisors, perhaps we can move
them to the ticket implementing divisors? Or, in order to avoid adding
more stuff there and therefore potentially delaying it, I can create a new
ticket for implementing all of the above (including, finally, `cone.dual`
to make `Mori_cone` trivial).
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/8988#comment:28>
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