[sage-devel] Toric varieties in Sage

[Andrey Novoseltsev]

Hi Volker (and sage-devel),

I guess face_lattice trac ticket is not the best place for this discussion ;-)

What are your goals for toric varieties, at what stage is your work, and when do you want/need to finish it?

My answers to these questions.

GOAL: have a nice framework for doing all the computations that I need for my work. I actually have quite a bit of code related to computation in toric varieties and their subvarieties, but a lot of it was done sort of "using polytopes and polynomial rings directly" to represent these subvarieties etc. This makes it hard to adjust for new projects and almost impossible to use by people other than me. I also have some speed/size goals: I should be able to work with a 5-dimensional toric variety with 800 generating rays and 12 000 generating cones.

CURRENT STAGE: It is possible to create cones/fans/(fano) toric varieties and their subschemes. There are a few places in the construction which still need polishing (in particular, I need to have face lattices and that's what I am working on right now). I will also need to include suitable classes for morphisms related to toric varieties. It took me a while to figure out how they work in Sage, but I think that now I can do it soon. Once this is done, "integration into standard framework" will be finished and it would be possible to make derived work based on these classes (e.g. I also have quite a bit of code for working with fibrations).

DEADLINE: Due to a number of reasons and promises, the first version of patches will definitely be on trac in three weeks (May 30). It may lack doctests, but otherwise should be OK.

Andrey

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[Andrey Novoseltsev]

From http://trac.sagemath.org/sage_trac/ticket/8656#comment:6

Comment(by vbraun):

I've implemented all the fan/lattice basics, cohomology and Chern
clases,
Chow ring, divisors, Mori cone.

The current status is at [http://www.stp.dias.ie/~vbraun/Sage/],
documentation is at
[http://www.stp.dias.ie/~vbraun/Sage/html/en/reference/sage/geometry/
toricvariety.html]

[Andrey Novoseltsev]

I've posted my current files on

http://sage.math.washington.edu/home/novoselt/toric_varieties/

From a glance at Volker's code above, the most apparent difference in
approaches is that I was designing cones and fans to be standalone
(with the plan that they should go to sage/geometry eventually), but
ToricVariety and FanoToricVariety are AmbientSpaces in the sense of
sage/schemes/generic and should eventually go there.

[Volker Braun]

Hi Andrey (and sage-devel)

I'm using toric varieties for my research, mainly for Calabi-Yau
manifolds in string theory. So my goal for the toric varieties package
is to implement everything that is known ;-)


My current status is that I've implemented the following:

* The basic fan construction is done incrementally (subcones are only
computed when necessary)

* Classes for N-lattice, M-lattice, Cones, Divisors, Chow cycles.

* Those pesky sublattices of N that are associated to a cone.

* Cone-orbit correspondence

* Cohomology ring, Chern classes, integration.

* Chow group and intersection.

* Toric divisors and sections=H^0

* Mori/Kahler cone

* simple constructors from LatticePolytopes, etc.

* (almost) 100% doctest coverage


In progress:

* sheaf cohomology

* toric morphisms (mostly done, subclasses domain/range Cone and
ToricVariety). Probably includes already everything you are thinking
about for toric fibrations.


Documentation (probably the best introduction) is at:

http://www.stp.dias.ie/~vbraun/Sage/html/en/reference/sage/geometry/toricvariety.html

The current code is at

http://www.stp.dias.ie/~vbraun/Sage/

Best wishes,
Volker