#8989: Add support for Fano 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 switch to numbers for easier referencing:
1) I agree that delta/nabla work better in ASCII, but I am also going to
add support for complete intersections where it is customary to use
delta/nabla pair to denote polytopes dual in the nef-partition sense.
While these polytopes will not be accessible at the level of toric variety
(but only at the level of appropriate subschemes), I would prefer to avoid
using nabla here as well and stick with `\Delta^\circ`.
2) Since I don't want to use nabla, I'd rather have delta to denote the
polytope whose face fan we are subdividing. Since in general this polytope
and its parts have more clear relation to the variety than its polar, I
would prefer to use it as "the main one". This is not particularly
customary but is quite natural when working with reflexive polytopes and
that's what this module is about. As an example of similar notation in use
I can give Nill's paper referenced in the module: arXiv:math/0405448v1
(both polytopes and fans live in the M lattice). Another reference is
arXiv:0907.2701v1 [math.CO], but it does not help me to justify my taste
;-) I think I had plans to allow the user to switch from one style to
another, but I don't see any clean way to do it, so probably it was not a
very bright idea.
3) Despite my opposition to other changes, I would like to replace `delta`
with `Delta`. I was avoiding using capitalized methods in general, but it
does make sense here and I agree that for Kaehler/Mori methods capital is
better.
4) `coordinate_indices` - will change and add tests.
5) I see your point against `point_to_variable`, but `nabla_to_variable`
seems to be even more confusing to me... How about just `coordinate`?
{{{
sage: X = toric variety
sage: X.coordinate(9)
z5
sage: X.coordinate((1,-1,1))
z5
}}}
We can even expand it to
{{{
sage: X.coordinate("z5")
z5
sage: X.inject_variables()
sage: X.coordinate(z5)
z5
}}}
i.e. this method will try to convert any input to the appropriate
generator of the coordinate ring. The only subtle point will be that
"plain numbers" will refer to points of Delta rather than generators of
the ring, but that's what one usually wants and there is `X.gen(n)` method
for those who want exactly the n-th variable.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/8989#comment:9>
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