cartesian products of projective space #15448

bhutz · 2013-11-23T16:29:10Z

While there is some functionality for cartesian products of projective space in Toric Varieties, it does not mesh well with the functionality of Projective Space. This ticket is to implement a general notion of cartesian products of projective space building off of the projective space functionality.

My main interest in this is dynamical systems on products of projective space and subvarieties of products of projective space.

Component: algebraic geometry

Author: Ben Hutz

Branch: 85d88d0

Reviewer: Volker Braun, Joao Alberto de Faria

Issue created by migration from https://trac.sagemath.org/ticket/15448

The text was updated successfully, but these errors were encountered:

bhutz · 2013-11-23T16:31:16Z

comment:1

Attachment: trac_15448_projective_product.patch.gz

The current patch is still needs work, mainly lacking in more testing. However, it does implement the majority of what is needed for cartesian products of projective space and their subschemes.

nbruin · 2013-11-23T18:35:24Z

comment:2

Have you tried working with multiply graded rings instead? Computing with bihomogeneous polynomials in
k[x0,x1,x2,y0,y1,y2] tends to be MUCH more efficient than working in the Segre embedding, because the rationality is much better expressed in the bihomogeneous case. Having some functionality to obtain the Segre embedding is of course useful, but I suspect that people having to work in products of projective spaces would be better off with multiply graded coordinate rings most of the time.

bhutz · 2013-11-23T18:46:47Z

comment:3

As far as I'm aware there is no native way to use multiply graded rings in Sage. I'd be happy to look into another way to do things, that's why I've put this up here as 'mostly done'.

Currently, everything but the subscheme dimension computation does just use K[x,y] and ensure multi-homgeneity. For the dimension, it seemed simpler to just use the Segre-embedding since I had already implemented that.

nbruin · 2013-11-23T20:02:37Z

comment:4

Replying to @bhutz:

As far as I'm aware there is no native way to use multiply graded rings in Sage. I'd be happy to look into another way to do things, that's why I've put this up here as 'mostly done'.

The arithmetic in a multiply graded ring only depends on the ring structure, so in that respect nothing extra is necessary. You just need to interpret the results properly, i.e., that the ideal
(x-y,u-v) in k[x,y,u,v] describes a point in P1xP1 and not a line in P3. I guess knowing you're working with bihomogeneously generated ideals might affect your choice of term ordering if you need to compute groebner bases.

Basically what I expect is that it's possible to use/adapt the toric variety framework for dynamic purposes as well. I haven't looked into it myself. I'm just sharing my experience that in cases where I needed products of projective varieties, I found using multiple gradings initially daunting but eventually not bad at all and much more convenient and efficient.

bhutz · 2014-09-03T17:54:23Z

Branch: u/bhutz/ticket/15448

sagetrac-git · 2014-09-08T19:33:46Z

Commit: 6d82bd9

sagetrac-git · 2014-09-08T19:33:46Z

Branch pushed to git repo; I updated commit sha1. New commits:

`6d82bd9`	`15448: clean up and error fixing`

sagetrac-git · 2014-09-12T13:03:38Z

Branch pushed to git repo; I updated commit sha1. New commits:

`1a84aa7`	`15448: finish segre embedding`

sagetrac-git · 2014-09-12T13:03:38Z

Changed commit from 6d82bd9 to 1a84aa7

bhutz · 2014-09-12T13:06:19Z

Changed author from bhutz to Ben Hutz

vbraun · 2014-09-15T16:10:26Z

comment:12

Just found your ticket... I've been working on products of projective spaces, mostly for complete intersections in them. See #16987...

We should think about how to merge the two branches. I would prefer to have a separate directory and not use mixed camelcase / underscores in the global namespace (inside of modules its fine). Which is why I used ProductProjectiveSpaces and sage.schemes.product_projective.

bhutz · 2014-09-15T18:02:56Z

comment:13

hmm...yes I looked through your branch some and it seems like most of our basic premises are the same in how we are implementing the products although I seem to be using the underlying structures from projective_space a little bit more. However, the two branches do seem reasonably compatible.

I have no issue with your naming conventions and separate directory, but I'm not sure the best way to go about merging these two sets of changes. You have more experience on the workflow side of things, so I'm happy to entertain suggestions.

bhutz · 2014-09-15T18:30:52Z

comment:14

After some more looking, the biggest difference is that I've based everything off of projective_space functionality. I really do prefer that approach (this makes affine patches, morphisms, etc. much easier to work with) and I don't think it will have much, if any, affect on how you've done complete intersections.

Here is what I'd like to do in an ideal world:

take your first commit and basically move it into my branch. However, I'd still like to build off of projective space. I can integrate any of your additional functionality and merge the differences in our approaches.
then you could have your ticket be just the complete intersection portion with this ticket as a dependency. You'll probably have to make a few minor adjustments, but I don't think it will be too much.

As I said above, I'm also open to other suggestions.

vbraun · 2014-09-15T18:36:00Z

Reviewer: Volker Braun

vbraun · 2014-09-15T18:36:00Z

comment:15

Sounds good to me. Can you also adjust your docstrings to always have OUTPUT: on a single line? See http://www.sagemath.org/doc/developer/coding_basics.html#docstring-markup-with-rest-and-sphinx

bhutz · 2014-09-15T18:40:07Z

comment:16

ok. I won't get to it today or tomorrow, but will asap after that.

sagetrac-jdefaria · 2014-10-22T16:51:36Z

comment:20

A couple of issues....
The following need better error messages

PP.<x0,x1,y0,y1> = ProductProjectiveSpaces([1,1],QQ)
H = Hom(PP,PP)
f = H([x0*x1*y1^3,x0*x1*y1*y0^2])

Proj.<x0,x1,y0,y1> = ProductProjectiveSpaces(1,QQ)

One should not be able to define a product with only one projective space

Prod.<x0,x1> = ProductProjectiveSpaces([1],QQ)
Prod

Have an extra input for the gens of the new PS, following error highlights why:

PP=ProductProjectiveSpaces([1,2],QQ,'u')
PP.segre_embedding()

Not as big a deal, but when return_embedding == True, it should return only the embedding instead of a tuple

PP=ProductProjectiveSpaces([1,2],QQ,'u')
PP.affine_patch([1,0],True)

Saw some small spacing issues scattered throughout the documentation as well, make sure to go back and double check

sagetrac-jdefaria · 2014-10-22T16:51:36Z

Changed reviewer from Volker Braun to Volker Braun, Joao de Faria

sagetrac-git · 2014-11-02T17:16:43Z

Changed commit from 76cbaed to 806cd01

sagetrac-git · 2014-11-02T17:16:43Z

Branch pushed to git repo; I updated commit sha1. New commits:

`b982b51`	`Merge branch 'master' into ticket/15448`
`806cd01`	`15448: merged in beta6 and fixes from review`

bhutz · 2014-11-02T17:18:54Z

comment:22

Fixes the issues mentioned. With a quick look through the documentation I wasn't sure what spacing issues you were referring to. Could you be more specific.

sagetrac-jdefaria · 2014-11-05T17:11:31Z

comment:23

Looked over the code, everything looks fine now, setting to positive review

sagetrac-jdefaria · 2014-11-05T18:08:05Z

comment:24

After speaking to the author, I am switching back to needs review so that the other reviewer can also look over it

vbraun · 2014-11-05T19:29:36Z

comment:25

Looks good to me. Sorry, don't have much time this semester. It would be nice if you could follow the docstring style guide a little bit closer:

http://www.sagemath.org/doc/developer/coding_basics.html#docstring-markup-with-rest-and-sphinx

always capitalize EXAMPLES::
always newline after OUTPUT: (no "OUTPUT: foo" in a single line), EXAMPLES::, ...

fchapoton · 2014-11-05T20:38:17Z

Changed branch from u/bhutz/ticket/15448 to public/15448

fchapoton · 2014-11-05T20:38:17Z

comment:26

I cleaned the doc a little bit.

And also a few things in the code. There remains a pyflakes warning about an used variables "dims" in the morphism.py file.

New commits:

`8cf318c`	`Merge branch 'u/bhutz/ticket/15448' into 6.4.rc1`
`abdb87a`	`trac #15448 doc cleaning patch`

fchapoton · 2014-11-05T20:38:17Z

Changed commit from 806cd01 to abdb87a

bhutz · 2014-11-08T15:32:27Z

Changed branch from public/15448 to u/bhutz/ticket/15448

bhutz · 2014-11-08T15:37:58Z

Changed commit from abdb87a to 85d88d0

bhutz · 2014-11-08T15:37:58Z

comment:28

yes, sorry, those docs could have been cleaner. Thanks Frederic for fixing them. I've now removed the dims variable in morphism.py as it became unnecessary after I implemented _factors(), but didn't get removed at that time.

New commits:

`85d88d0`	`15448: minor code fix`

sagetrac-jdefaria · 2014-11-11T14:28:35Z

comment:29

Looked at everything once more, doc test came back clean, setting to positive review

vbraun · 2014-11-14T21:01:35Z

Changed branch from u/bhutz/ticket/15448 to 85d88d0

jdemeyer · 2015-02-23T16:24:46Z

Changed reviewer from Volker Braun, Joao de Faria to Volker Braun, Joao Alberto de Faria

jdemeyer · 2015-02-23T16:24:46Z

Changed commit from 85d88d0 to none

bhutz added this to the sage-6.1 milestone Nov 23, 2013

bhutz added c: algebraic geometry labels Nov 23, 2013

bhutz self-assigned this Nov 23, 2013

sagetrac-vbraun-spam mannequin modified the milestones: sage-6.1, sage-6.2 Jan 30, 2014

sagetrac-vbraun-spam mannequin modified the milestones: sage-6.2, sage-6.3 May 6, 2014

sagetrac-vbraun-spam mannequin modified the milestones: sage-6.3, sage-6.4 Aug 10, 2014

bhutz added the s: needs review label Sep 12, 2014

bhutz added s: needs work and removed s: needs review labels Sep 15, 2014

sagetrac-jdefaria mannequin added s: needs work and removed s: needs review labels Oct 22, 2014

bhutz added s: needs review and removed s: needs work labels Nov 2, 2014

sagetrac-jdefaria mannequin added s: positive review and removed s: needs review labels Nov 5, 2014

sagetrac-jdefaria mannequin added s: needs review and removed s: positive review labels Nov 5, 2014

sagetrac-jdefaria mannequin added s: positive review and removed s: needs review labels Nov 11, 2014

vbraun removed the s: positive review label Nov 14, 2014

vbraun closed this as completed in ac0c18a Nov 14, 2014

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cartesian products of projective space #15448

cartesian products of projective space #15448

bhutz commented Nov 23, 2013

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bhutz commented Sep 3, 2014

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cartesian products of projective space #15448

cartesian products of projective space #15448

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bhutz commented Nov 23, 2013

bhutz commented Nov 23, 2013

nbruin commented Nov 23, 2013

bhutz commented Nov 23, 2013

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bhutz commented Sep 3, 2014

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