Curve_singularities

---

title: Curve singularities permalink: wiki/Curve_singularities/ layout: wiki

First, I'd like to advance and hopefully complete the MultiplierIdeals package begun at the 2011 IMA workshop. Second, I'd like to add some code to compute multiplier ideals of generic determinantal ideals. Third, I'd like to add code to deal with singularities of plane curves, with the first goal of computing their multiplier ideals (using results of Smith, Thompson, Tucker), and the larger goal of introducing/assembling useful tools for plane curve singularities (e.g., resolution tree, Puiseux pairs).

Background: code by Christine Berkesch and Anton Leykin in the Dmodules package computes multiplier ideals of arbitrary ideals, but is (necessarily) slow; special algorithms for particular classes of ideals can be dramatically better. The HyperplaneArrangements package has code by Graham Denham and Greg Smith to compute multiplier ideals of hyperplane arrangements. I've written a package to compute multiplier ideals of monomial ideals. At the 2011 IMA workshop, Claudiu Raicu, Bart Snapp, and I wrote code to compute multiplier ideals of 1-dimensional binomial ideals (i.e., ideals of monomial curves). As indicated above, I want to add to this list, e.g., plane curve singularities, and in the future, hopefully more. To avoid unwieldliness and a profusion of tiny packages, I'd like to integrate some or all of the existing code into a single MultiplierIdeals package. (Dan Grayson convinced me I should do this --- not that it was a hard argument.)

Multiplier ideals of generic determinantal ideals (i.e., generated by t-by-t minors of a generic matrix) have been found by Amanda Johnson, in her (unfortunately unpublished) dissertation under Karen Smith at University of Michigan in 2003. It is a simple, explicit expression in terms of determinantal ideals generated by minors of size t and smaller. I expect this should be easy to implement, and combine into a MultiplierIdeals package; estimated effort, 1-2 days (after the preparation of the framework for the MultiplierIdeals package).

Thus my proposal is (1) create framework for a MultiplierIdeals package, (2) integrate existing free-floating code (monomial ideals, monomial curves) and make links to existing code in other packages (Dmodules, HyperplaneArrangements), (3) write new code for multiplier ideals of determinantal ideals, (4) (speculatively) write or at least plan code for multiplier ideals of plane curve singularities, possibly leading to new projects to handle plane curve singularities.

Once a package exists, I can imagine just adding code to it to handle more cases such as plane curves. The part that I don't know how to do is how to integrate everything into a single, well-designed package. But hopefully that will be easy for the folks at the workshop.

I have started the integration coding on my own, but it's a little tricky. Specifically, I'm struggling with things like having a "Strategy" optional argument to tell the MultiplierIdeals package whether to use the Monomial algorithm, MonomialCurve, HyperplaneArrangements, etc --- and the possibility of having it "auto-detect" the best algorithm. Part of the problem is that these different algorithms have different inputs; e.g., for HyperplaneArrangements, the input is not an ideal, but rather a list of hyperplanes with multiplicities.

I haven't been able to figure out how to do all of this programming, but I'm sure the workshop participants will be able to tell me what to do (or just help me do it) very easily. Optimistically, this might take as little as 1-2 days, leaving time to participate in other projects, or start playing around with plane curve singularities.

A longer term goal is working toward implementation of resolution of singularities in Macaulay2. This is a truly enormous project, not to be seriously addressed during short-term workshops; however experience with small cases such as plane curve singularities, and Macaulay2 programming in general, is a prerequisite before I can even think about this larger goal; and networking with the Macaulay2 community is also necessary (this project is too much to attack in isolation). So, I don't expect any actual coding for resolution of singularities to happen during the Wake Forest workshop, but perhaps conversations can lay the groundwork for future work --- e.g., thinking about how a package should be designed, what are the goals, what things are already done and what things still need to be done. Of course if any coding actually does happen, great.