A first step towards linear systems of hypersurfaces in Sage

[mminzlaff]

In Magma, one can do the following:

> Q := RationalField();
> P<x,y,z> := ProjectiveSpace(Q,2);
> L := LinearSystem(P,2);
> L;
Linear system on Projective Space of dimension 2
Variables : x, y, z
with 6 sections: x^2 x*y x*z y^2 y*z z^2
> p := P ! [3,2,1];
> L1 := LinearSystem(L,p);
> L1;
Linear system on Projective Space of dimension 2
Variables : x, y, z
with 5 sections:
x^2 - 9*z^2
x*y - 6*z^2
x*z - 3*z^2
y^2 - 4*z^2
y*z - 2*z^2

Sage does not have this functionality. This patch will be a first step towards adding a class LinearSystem to Sage.

The goal is to add a method _linear_system_as_kernel to projective spaces that returns a matrix whose kernel can be identified with the degree d hypersurfaces with multiplicity at least m at pt.

(I actually need this method in the context of equimultiple liftings of plane curves over finite fields for which I will open a separate ticket.)

Apply:

• attachment: trac_12995_initial.patch
• attachment: trac_12995_review2.patch

Component: algebraic geometry

Author: Moritz Minzlaff

Reviewer: David Eklund

Merged: sage-5.5.beta0

Issue created by migration from https://trac.sagemath.org/ticket/12995

[mminzlaff]

Author: Moritz Minzlaff

[sagetrac-davideklund]

Reviewer: David Eklund

[sagetrac-davideklund]

comment:3

All tests pass on Sage 5.2 and Mac OS X lion (except two tests that fail on an unpatched Sage 5.2).

Let me look through the details and test the functionality and get back here.

[sagetrac-davideklund]

comment:4

In the documentation part of the patch it says that the output is a matrix of size

{m-1+n choose n}x{d+1 choose n}

but is the number of columns really {d+1 choose n} and not {d+n choose n}?

[sagetrac-davideklund]

comment:5

The syntax

raise Error, 'error message'

has been deprecated in Python. http://docs.python.org/release/2.6.8/tutorial/errors.html#raising-exceptions

We should use

raise Error('error message')

instead.

[sagetrac-davideklund]

comment:6

Tests pass on Sage 5.3 (Mac OS X lion) also.

[mminzlaff]

comment:7

Dear David,

thanks for your remarks and your time to review this patch! I have uploaded a new version of the patch moments ago. You are quite right about the d+1 vs. d+n thing. I have also changed the syntax for raising Errors as you suggested.

[sagetrac-davideklund]

comment:8

Ok, looks good! I will put this to positive review soon.

[sagetrac-davideklund]

Attachment: trac_12995_review.patch.gz

Fixed typos and line continuation of strings.

[sagetrac-davideklund]

comment:11

Hi!

I have a question: It says in the documentation part that the output is with respect to reverse lexicographic ordering of monomials. But it seems to me that, for example in the case of degree 2 in projective 2-space the monomials are ordered as:

[x<sup>2,xy,xz,y</sup>2,yz,z^2]

which seems to be just the lexicographic ordering rather than reverse lexicographic ordering.

If you agree I can incorporate this change into the review patch.

[mminzlaff]

comment:12

I don't have time tonight to answer your question, but I'll try to look at it tomorrow!

[sagetrac-davideklund]

comment:13

Ok, this looks good!

If you agree with the changes in the review patch and we resolve the issue about the monomial ordering, then we are good to go.

[mminzlaff]

Attachment: trac_12995_review2.patch.gz

fixes two more things in documentation/comments

[mminzlaff]

comment:15

Hi David,

please excuse my slow response. I've recently left academia and can only find time to look at Sage in the evenings (but I am happy to do so!).

Thanks again for your comments. You are quite right, I've changed the documentation accordingly. I also saw where I got confused and changed one line of comment, too. :)

[sagetrac-davideklund]

comment:16

Thanks!

Actually, I think you are responding very quickly. I will try to get the patch bot to apply the right patches so that we can get rid of the ugly red blob at the top.

[sagetrac-davideklund]

comment:17

For the patchbot:

Apply trac_12995_initial.patch, trac_12995_review2.patch

[jdemeyer]

Merged: sage-5.5.beta0

[jdemeyer]

comment:22

Could you please make the commit message more descriptive than "incorporates David's remarks".

[sagetrac-davideklund]

comment:23

Yes, a more descriptive message would be better. Moritz, do you think you can take care of this? Thanks!

[mminzlaff]

comment:24

Hi,

I will get to it later tonight/tomorrow.

[mminzlaff]

Attachment: trac_12995_initial.patch.gz

[mminzlaff]

comment:25

Ok, done. :)

[sagetrac-davideklund]

comment:26

Thanks!