User question about toric varieties

[kcrisman]

On the sagenb github site (https://github.com/sagemath/sagenb/issues/402):

Hi there,

I tried to verify with Sage that P^2 is projective. Unfortunately, Sage replied with 'false'. Here is what I tried:

P2=toric_varieties.P2();
P2.is_projective()

The last line returned 'false'. The same happens for e.g. P1xP1.

[I am running SageMath version 7.2, Release Date: 2016-05-15]

Is this a bug? Thank you!

[kcrisman]

Hi there,

I tried to verify with Sage that P^2 is projective. Unfortunately, Sage replied with 'false'. Here is what I tried:

P2=toric_varieties.P2();
P2.is_projective()

The last line returned 'false'. The same happens for e.g. P1xP1.

This seems to be about an ambient space, and apparently only returns True if you are in a certain subclass of schemes, regardless.

sage: A.ambient_space()

2-d CPR-Fano toric variety covered by 3 affine patches

sage: A.is_projective??

Signature: A.is_projective()

Source:

    def is_projective(self):

        """

        Return whether this ambient space is projective n-space.

        EXAMPLES::

            sage: AffineSpace(3,QQ).is_projective()

            False

            sage: ProjectiveSpace(3,QQ).is_projective()

            True

        """

        # overloaded in the projective space derived class

        return False

 

[Andrey Novoseltsev]

This is a loooong standing bug that nobody cared about enough to fix:
https://trac.sagemath.org/ticket/11394