#8989: Add support for Fano toric varieties
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Reporter: novoselt | Owner: AlexGhitza
Type: enhancement | Status: needs_work
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):
A polytope and its dual are not interchangeable when you are talking about
one particular toric variety, but if you are working with two varieties
associated to these polytopes, things do become somewhat
interchangeable...
My point is that if I have created a standalone polytope, called it
`Delta`, and cooked up a toric variety `X` from it one way or another,
then it is cool if `X.Delta()` gives me exactly this `Delta` back.
However, if I now used the same construction to create a toric variety `Y`
associated to `Delta.polar()` or maybe some other polytope `nabla`, it is
confusing that `Delta.polar() == Y.Delta()` or `nabla == Y.Delta()`.
In mathematics it is more or less OK to abuse notation and use the same
names describing different objects, or to say "everything will be the same
up to relabelling." When it is not enough, one can also start putting
tildes, primes, daggers, etc. on similar but different objects to make
them a little bit different. But method names cannot be slightly altered
or "relabeled in an obvious way." So it is not that I don't want to use
standard notation, I just don't think that it is usable in tasks involving
more than one object - if you want to work with them at the same time,
they must have different names/notation.
The polytope whose face fan is used has the advantage of being uniquely
determined (with the agreement of using primitive vectors along the rays),
while many different polytopes may give the same normal fan. So it seems
to me that using just `polytope` makes sense. If you don't like it, how
about `fan_polytope`? (`spanned_polytope`? `face_polytope`?..)
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/8989#comment:13>
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