Resolution of singularities

[Dan Abramovich]

Hi all - 

a student of mine, Jonghyun Lee, will be coding the resolution of singularities algorithm of the

paper

https://arxiv.org/abs/1906.07106

as part of his senior thesis. He is working in sage (that's what he chose!).

two questions:

1. Are there packages in sage parallel to those available in Singular or Macaulay?

2. Is there any existing resolution of singularities package? Anyone else working on similar projects?

Best,

- Dan Abramovich

[Dima Pasechnik]

On Fri, Aug 23, 2019 at 3:55 AM Dan Abramovich <dan_abr...@brown.edu> wrote:
> a student of mine, Jonghyun Lee, will be coding the resolution of singularities algorithm of the
>
> paper
>
>
> https://arxiv.org/abs/1906.07106
>
>
> as part of his senior thesis. He is working in sage (that's what he chose!).
>
>
> two questions:
>
>
> 1. Are there packages in sage parallel to those available in Singular or Macaulay?

Singular has an efficient interface to Sage, so normally one would
create a Sage interface to a Singular
package rather than an independent implementation.

The interface to Macaulay2 is on the other hand pretty basic and slow,
so it might be
different for Macaulay2.

What kind of functionality of Sage is needed for the project which is
not in Singular?

A more efficient way might be to implement the algorithm in Singular
(or Macaulay2) and then
expose in in Sage, rather than spend time fighting the interfaces.

>
>
> 2. Is there any existing resolution of singularities package? Anyone else working on similar projects?

I know about

1) resolution of some kind of toric singularites package in Macaulay2

2) resolution of surface singularites in Singular
https://www.singular.uni-kl.de/Manual/latest/sing_934.htm#SEC986

HTH
Dima
>
> Best,
>
>
> - Dan Abramovich
>
>
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[Volker Braun]

The toric version is also natively in Sage, e.g. http://doc.sagemath.org/html/en/reference/schemes/sage/schemes/toric/morphism.html

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[Dan Abramovich]

Thanks, Dima!

The SINGULAR package does the classical resolution algorithm well. Jonghyun, the student, is reading the developer Anne Frühbis Krüger's extensive texts about it. The new algorithm uses in part the same objects. 

So as you say, it might make sense to move over to SINGULAR. 

- Dan  

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[Dan Abramovich]

Thanks - this might be relevant for another project of another student, coding this

https://arxiv.org/abs/1905.00872

One thing which may be missing in any of the existing systems is a Deligne-Mumford stack class extending "scheme".

- Dan

[mmarco]

Jayson Jorgenson implemented resolution of singularities for plane curves as a GSoC project a few years ago:

https://doc.sagemath.org/html/en/reference/curves/sage/schemes/curves/affine_curve.html

But I guess you are talking about the general case.

IIRC, singular returns the output of the resolution in a complicated way (defining several rings, and some ideals inside them). Maybe it would make sense to use the Sage schemes infraestructure to wrap that in a more user-friendly way (although the schemes code itself could use some work too).

[Dan Abramovich]

Jayson - 

I see your point regarding Sage schemes. The implementation in SINGULAR is what's known as embedded resolution, which returns a scheme (the ambient variety) and a subscheme (the resolved subvariety). 

I gather in SINGULAR as it stands the scheme must be presented explicitly as a bunch of rings and the subscheme as a bunch of ideals. I think SINGULAR was originally conceived with affine schemes in mind, which require just one ring, so the infrastructure was not really necessary.

- Dan

[Kwankyu Lee]

IIRC, singular returns the output of the resolution in a complicated way (defining several rings, and some ideals inside them). Maybe it would make sense to use the Sage schemes infrastructure to wrap that in a more user-friendly way (although the schemes code itself could use some work too).

I think this ticket

https://trac.sagemath.org/ticket/28389

could be an initial step toward that direction. That is needed to develop sage schemes with Singular background.

[mmarco]

Ups, I wrote it wrong. His name is Grayson Jorgenson, not Jayson.