#10529: dimension() and is_smooth() for algebraic subschemes of toric varieties
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Reporter: vbraun | Owner: AlexGhitza
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-4.7
Component: algebraic geometry | Keywords:
Author: Volker Braun | Upstream: N/A
Reviewer: Andrey Novoseltsev | Merged:
Work_issues: |
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Changes (by vbraun):
* status: needs_work => needs_review
Comment:
1. Thats just the 0-dimensional case. I don't care about `Jacobian` vs.
`Jacobian_ideal`, but since you thought of the right thing I would prefer
to keep the shorter name.
2. In projective coordinates the CI equations are implied by the minors of
the partial derivatives, which is part of why it is usually not spelled
out. For speed, it is important to add the equations so that the minors
(which often have high degree) can be reduced quickly. Also, in the
context of subschemes it is dangerous to not add the equations; For
example if you compute the Milnor number you really do only want the
critical points on the subvariety. So I very much prefer it the way it is.
3. If you have a faster algorithm then you should be writing a paper on
it. But note that you need the rank '''everywhere''' on the subvariety :-)
4. Computing the Jacobian ideal is fast. Eventually you'll have to compute
Groebner basis, this is the slow and memory intensive part. But we can't
cache all of these with whatever term ordering the user is going to use. I
think its better to cache the final result (whether the variety is smooth)
instead of the Jacobian ideal.
5. OK
6. in #10540, the `affine_neighbourhood` method is removed. There are two
different methods `affine_patch(int)` and `neighbourhood(point)`. The
neighborhood may or may not be centered.
7. Eventually the `_best_affine_patch` should be the "best" patch in the
same sense as for projective varieties. But we first need some way to
easily access the homogeneous rescalings. So I'll leave this as a TODO.
8. In #10540, this is refactored into `subscheme.embedding_morphism()` and
`subscheme.embedding_origin` methods. No more public attributes. I think
it is better to have un-shifted affine patches for toric varieties as it
makes the embedding map easier. But it would be nice to have a `shift()`
method, feel free to write a patch on top of #10540 :-)
9. Implemented in #10540
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/10529#comment:22>
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