Fix normalization in cohomology ring of orbifold toric varieties

[vbraun]

For simplicial toric varieties, the rational cohomology ring and the ratironal Chow group are isomorphic. So if all normalizations are correct, then one should be able to do intersection computations in the cohomology ring. This patch fixes the volume_class() and the constructor of cohomology cycles from cones to make everything work.

It turns out that index_in_saturation does not work for trivial zero-size matrices, this is also fixed.

Apply attachment: trac_12361_fix_toric_cohomology_ring.patch, attachment: trac_12361_index_in_saturation_fix.patch

CC: @novoselt @sagetrac-davideklund

Component: algebraic geometry

Author: Volker Braun

Reviewer: David Eklund, Andrey Novoseltsev

Merged: sage-5.1.beta0

Issue created by migration from https://trac.sagemath.org/ticket/12361

[vbraun]

Attachment: trac_12361_index_in_saturation_fix.patch.gz

Initial patch

[novoselt]

Reviewer: Andrey Novoseltsev

[novoselt]

comment:2

The patchbot complains about added whitespace! Looking at the actual code...

[novoselt]

comment:3

That's a really great expansion of the documentation!

I think I caught some typos:

• In toric_variety line 213 should be "many quantities of interest" without "s" on the end.
• On line 1977 "It gets tricky none of the maximal cones is smooth." does not sound right, some words or punctuation sighs are missing.

[novoselt]

Work Issues: whitespaces and typos

[novoselt]

comment:4

OK, other than the above looks good!

[sagetrac-davideklund]

comment:5

Hi. I looked at this and I agree that it looks good.

One thing:

On line 215 of toric_variety.py it says "For toric varieties with at most orbifold singularities, the rational cohomology ring H(X,\QQ) and the rational Chow ring H(X,\QQ) are isomorphic."

It's a bit confusing that they are both denoted by H(X,\QQ). Maybe A(X,\QQ) could be used for the Chow ring. Or H^*(X,\QQ) for cohomology and H_*(X,\QQ) for the Chow ring (if intersection homology and the Chow ring were already identified somehow).

[vbraun]

comment:7

Oh yes good catch I meant to write A(X,\QQ) of course.

[vbraun]

comment:8

Fixed!

[novoselt]

comment:9

Comments 2&3 are still applicable and the patchbot complains ;-)

[novoselt]

Changed reviewer from Andrey Novoseltsev to David Eklund, Andrey Novoseltsev

[vbraun]

Attachment: trac_12361_fix_toric_cohomology_ring.patch.gz

Updated patch

[vbraun]

comment:10

I've improved the docstrings.

As for the whitespace, I think this is a non-issue. There was some discussion on sage-devel and the consensus seems to be that its not worth the effort. Its a button press away (either with emacs or with the mercurial checkfiles plugin) to strip all trailing whitespace, but that would just break every patch we currently have. Mercurial tells me that there are 170 places in toric_varieties.py with superfluous spaces, so a few more or less doesn't matter. I'd rather not spend an hour fixing all patches that I currently have just to make the whitespace plugin happy.

[novoselt]

Changed work issues from whitespaces and typos to none

[jdemeyer]

Merged: sage-5.1.beta0