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