make affine and projective dehomogenize and homogenize work together

[bhutz]

There are a number of issues currently with the homogenization and dehomogenization functionality. Essentially, what I'd like to see is that dehomogenizing and then homogenizing returns basically the same object. Currently dehomogenization is using AffinePatch, but homogenization is creating a new projective space every time. This causes a number of funny behaviors so should instead be using projective embedding. I'd like to see tests of the following form return True:

P.<x,y,z>=ProjectiveSpace(QQ,2)
H=End(P)
f=H([x*y,y^2,z^2])
n=1
f.dehomogenize(n).homogenize(n)==f

A.<x,y>=AffineSpace(QQ,2)
H=End(A)
f=H([x*y,y^2])
n=0
f.homogenize(n).dehomogenize(n)==f

This also requires having the affine_patch and projective_embedding play nicely together. Fixing this would also allow objects homogenized or dehomogenized from the same space to then live in the same new space. The same for points and currently there is no homogenization function at all for affine points.

I've started working on this and just need to fix a couple more things.

Component: algebraic geometry

Author: Ben Hutz

Branch: 8815855

Reviewer: Joao Alberto de Faria

Issue created by migration from https://trac.sagemath.org/ticket/16838

[bhutz]

Branch: u/bhutz/ticket/16838

[sagetrac-git]

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

c3d007f

16838: fixes for algebraic subschemes

[sagetrac-git]

Commit: c3d007f

[sagetrac-git]

Changed commit from c3d007f to 30e8fde

[sagetrac-git]

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

30e8fde

16838: fix issues with homogenize morphism

[sagetrac-git]

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

16efbbe

16838: slight correction to homogenize

[sagetrac-git]

Changed commit from 30e8fde to 16efbbe

[bhutz]

Author: Ben Hutz

[sagetrac-jdefaria]

comment:6

Outside of a couple of small typos with spacing around equality signs, everything else looks good, ran the long test and confirmed that all tests passed.

[sagetrac-jdefaria]

Reviewer: Joao de Faria

[sagetrac-git]

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

8815855

16838: fixed some typos

[sagetrac-git]

Changed commit from 16efbbe to 8815855

[sagetrac-jdefaria]

comment:9

Everything looks good.

[vbraun]

Changed branch from u/bhutz/ticket/16838 to 8815855

[kcrisman]

Changed commit from 8815855 to none

[kcrisman]

Changed reviewer from Joao de Faria to Joao Alberto de Faria