Indeterminacy locus and image of morphisms of schemes

[klee]

Implemented indeterminacy locus and image of morphisms of affine and projective schemes.

Along the way, cleaned up some files in schemes module.

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The author was supported by NRF of Korea 2019, 2020

[git]

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

​17aec4e

Added indeterminacy_locus() and image()

[git]

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

​721b23e

Fixes for doctest failures

[git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

​5c49204

Added indeterminacy_locus() and image()

[git]

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

​aded9e3

Added a reference

[git]

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

​4ea25f2

A pyflakes fix

[git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

​383b618

Added indeterminacy_locus() and image()

[tscrim]

A few things:

It would be good to have a more clear indication for this list here

             sage: g.representatives()
             [Scheme morphism:
                From: Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
                x^2 - y^2 - y
                To:   Affine Space of dimension 1 over Rational Field
                Defn: Defined on coordinates by sending (x, y) to
-                     (x/y), Scheme morphism:
+                     (x/y),
+             Scheme morphism:
                From: Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
                x^2 - y^2 - y
                To:   Affine Space of dimension 1 over Rational Field
                Defn: Defined on coordinates by sending (x, y) to
                      ((y + 1)/x)]

and similar changes for the other tests in this method.

Do you need to move the SchemeMorphism.__init__(self, parent) in SchemeMorphism_polynomial.__init__? I feel it would be better to do it after the possible error messages for when those are raised so as to not do unnecessary work (granted, if it is needed to be initialized for the tests, then this comment is moot).

PEP8:

-p = source_ring(poly.numerator())/source_ring(poly.denominator())
+p = source_ring(poly.numerator()) / source_ring(poly.denominator())

I don't understand this bit of code:

        if not self.codomain().is_projective():
            pass

There is no else statement attached, so I feel it can be removed.

Is the Chow_form still usable by instances of AlgebraicScheme_subscheme_projective? Are they suppose to be?

[klee]

Replying to tscrim:

A few things:

It would be good to have a more clear indication for this list here

             sage: g.representatives()
             [Scheme morphism:
                From: Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
                x^2 - y^2 - y
                To:   Affine Space of dimension 1 over Rational Field
                Defn: Defined on coordinates by sending (x, y) to
-                     (x/y), Scheme morphism:
+                     (x/y),
+             Scheme morphism:
                From: Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
                x^2 - y^2 - y
                To:   Affine Space of dimension 1 over Rational Field
                Defn: Defined on coordinates by sending (x, y) to
                      ((y + 1)/x)]

and similar changes for the other tests in this method.

OK. Done.

Do you need to move the SchemeMorphism.__init__(self, parent) in SchemeMorphism_polynomial.__init__? I feel it would be better to do it after the possible error messages for when those are raised so as to not do unnecessary work (granted, if it is needed to be initialized for the tests, then this comment is moot).

No. You are right. I wrongly thought it is needed. Moved back.

PEP8:

-p = source_ring(poly.numerator())/source_ring(poly.denominator())
+p = source_ring(poly.numerator()) / source_ring(poly.denominator())

Done.

I don't understand this bit of code:

        if not self.codomain().is_projective():
            pass

There is no else statement attached, so I feel it can be removed.

This part was "to do", and then I somehow forgot.. Now I filled the missing part, which is for morphisms from projective subscheme to affine space. Thank you for spotting this!

Is the Chow_form still usable by instances of AlgebraicScheme_subscheme_projective? Are they suppose to be?

Chow_form method assumes the base ring is a field. Methods of AlgebraicScheme_subscheme_projective should assume just rings. So it would be usable when the assumption is met.

[git]

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

​213ff8f

Fixes for reviewer comments

[git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

​7ad46c0

Fixes for reviewer comments

[tscrim]

Thank you for the changes and explanations. LGTM.