weighted projective spaces

[David Eklund]

Hi,

I'm considering opening a ticket for the implementation of weighted projective spaces (in a class of its own). I think it could be quite useful in general but there are also algebraic varieties already in Sage for which weighted projective space is a natural habitat (like hyperelliptic curves).

Does this sound like a good idea? Or is it superfluous?

Are there tickets on this already?

Any ideas of how it can be done? For example, does anyone know how it is done in Magma?

Some technical remarks: it might be that the work is essentially already done in connection with toric varieties. I'm not sure exactly which functionalities I would like, but at least I want to construct them by simply giving the weights and also define subschemes by giving a bunch of weighted homogenous polynomials. Perhaps test smoothness of such subschemes etc. Maybe weighted projective spaces should be constructed as toric varieties. Or perhaps it is better to make an independent implementation of them. Any thoughts?

thanks!

/David Eklund

[Volker Braun]

Hi David,

I think it would be best to construct them as toric varieties. This'll give you lots of functionality. For starters you should probably add a weighted projective space constructor to the toric_varieties library. There is already a toric_varieties.P(int), how about toric_varieties.WP(list of ints). If you want to provide specialized implementations for toric algorithms you can derive from the ToricVarieties class.

Volker

[David Eklund]

Hi Volker,

thanks for the advice! I think basing the implementation on the Cox ring is what I wanted anyway. 

If any number theory people are reading this I think it is worth thinking about making hyperelliptic curves subvarieties of weighted projective planes (whether using the toric variety version I will work on or some other implementation of weighted projective spaces).

/David Eklund

[Marco Streng]

Op 22-05-2012 4:09, David Eklund schreef:

> Hi Volker,
>
> thanks for the advice! I think basing the implementation on the Cox ring
> is what I wanted anyway.
>
> If any number theory people are reading this I think it is worth
> thinking about making hyperelliptic curves subvarieties of weighted
> projective planes

Definitely! That would make it possible to have a smooth projective
model, with the correct points at infinity.

> --
> To post to this group, send an email to sage-...@googlegroups.com
> To unsubscribe from this group, send an email to
> sage-devel+...@googlegroups.com
> For more options, visit this group at
> http://groups.google.com/group/sage-devel
> URL: http://www.sagemath.org

[Volker Braun]

On Tuesday, May 22, 2012 4:16:08 AM UTC-4, Marco Streng wrote:

Definitely! That would make it possible to have a smooth projective
model, with the correct points at infinity.

I don't understand that sentence - a smooth elliptic surface can already be embedded in P^2, right?

Note that the toric variety code assumes that the base ring is a field at various places. So for number theory purposes it might be good to split things into ToricVariety_ring vs. ToricVariety_field. Its mostly my ignorance about toric varieties over general rings that prevented me from doing so... 

[Marco Streng]

Op 22-05-2012 15:26, Volker Braun schreef:

> On Tuesday, May 22, 2012 4:16:08 AM UTC-4, Marco Streng wrote:
>
> Definitely! That would make it possible to have a smooth projective
> model, with the correct points at infinity.
>
>
> I don't understand that sentence - a smooth elliptic surface can already
> be embedded in P^2, right?

That sentence refers to the sentence by David Eklund right above it. So
it has nothing to do with elliptic surfaces, but is about hyperelliptic
curves.

Hyperelliptic curves can be mapped birationally onto a curve in P^2,
just by going via the standard model in A^2 of the form y^2 + h(x)*y =
f(x). But the image is not smooth at infinity for any hyperelliptic
curve (of genus >=2). To make the image smooth, the standard solutions
are to
1) glue two copies of A^2,
2) use a higherdimensional P^n, or
3) use a weighted projective 2-dimensional space.

I think David was aiming at (3), and I was simply welcoming that.

>
> Note that the toric variety code assumes that the base ring is a field
> at various places. So for number theory purposes it might be good to
> split things into ToricVariety_ring vs. ToricVariety_field. Its mostly
> my ignorance about toric varieties over general rings that prevented me
> from doing so...

>
>

[David Eklund]

Hi,

I guess Volker meant Riemann surface when saying smooth elliptic surface in P^2 (that is, an elliptic curve). But as Marco says, this is about hyperelliptic curves. 

Volkers comment that toric varieties in Sage are often assumed to be defined over a field is important and has implications for the issues regarding hyperelliptic curves discussed above. Developing toric varieties over general rings sounds like an intriguing (and elaborate) project. I think I will start small by adding explicit construction of weighted projective spaces to the toric variety library. We'll see what happens after that.

Marco's standard solution number 1) with two glued A^2 also sounds useful. Maybe this is similar to what we would actually be doing with the weighted projective plane P(1,g+1,1) where g is the genus of the curve.

best

/David

> sage-devel+unsubscribe@googlegroups.com

[Marco Streng]

2012/5/24 David Eklund <sven.dav...@gmail.com>:

> Hi,
>
> I guess Volker meant Riemann surface when saying smooth elliptic surface in
> P^2 (that is, an elliptic curve). But as Marco says, this is about
> hyperelliptic curves.
>
> Volkers comment that toric varieties in Sage are often assumed to be defined
> over a field is important and has implications for the issues regarding
> hyperelliptic curves discussed above. Developing toric varieties over
> general rings sounds like an intriguing (and elaborate) project. I think I
> will start small by adding explicit construction of weighted projective
> spaces to the toric variety library. We'll see what happens after that.

Yes, it would be good to have weighted projective spaces over general
rings. But don't worry too much about that for hyperelliptic curves.
First of all, as far as I know, hyperelliptic curves are currently not
really implemented over general rings, such rings are just not
rejected on input. Second, points on elliptic curves over general
rings work in Sage, even though they don't work in the ambient
projective space, so the ambient space apparently does not have to
support everything:

sage: E = EllipticCurve(QQ, [0,0,0,-1,0])
sage: E([1,0])
# point
sage: P = E.ambient_space()
sage: P([1,0])
# point
sage: E = EllipticCurve(Zmod(6), [0,0,0,-1,0])
sage: E([1,0])
# point
sage: P = E.ambient_space()
sage: P([1,0])
# error

sage: Q.<x> = QQ[]
sage: C = HyperellipticCurve(x^6-1)
sage: C([1,0])
# point
sage: Q.<x> = Zmod(6)[]
sage: C = HyperellipticCurve(x^6-1)
sage: C([1,0])
# error

> Marco's standard solution number 1) with two glued A^2 also sounds useful.
> Maybe this is similar to what we would actually be doing with the weighted
> projective plane P(1,g+1,1) where g is the genus of the curve.

The resulting smooth projective hyperelliptic curves are of course
isomorphic, but the two ambient spaces make a difference in
implementation, so there is indeed a choice to be made. I don't know
whether glueing is done anywhere in Sage so far, but neither are
weighted spaces, so I don't know how much work each option would be.

For more on the theory behind (1) and (2), see pages 86 and 87 of
Hindry-Silverman, Diophantine Geometry, An Introduction. Method (1) of
glueing smooth affine curves in two copies of A^2 is A.4.2(a) -- (c).
Method (2) of a higher-dimensional projective space is A.4.2(d), but
is probably not suitable for fast computations due to the high
dimension.

>> > sage-devel+...@googlegroups.com

>> > For more options, visit this group at
>> > http://groups.google.com/group/sage-devel
>> > URL: http://www.sagemath.org
>>

> --
> To post to this group, send an email to sage-...@googlegroups.com
> To unsubscribe from this group, send an email to

> sage-devel+...@googlegroups.com