Multiplier_ideals_of_monomial_space_curves

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title: Multiplier ideals of monomial space curves permalink: wiki/Multiplier_ideals_of_monomial_space_curves/ layout: wiki

Goal: Combinatorially compute multiplier ideals of monomial space curves, following Howard Thompson's theorem in 1.

Why compute multiplier ideals of monomial space curves? There are general questions about multiplier ideals, jumping numbers, and irrelevant exceptional divisors. Computations like this can help generate specific conjectures. One question (suggested by Rob Lazarsfeld) involves computing the multiplier ideal of the generic initial ideal of something, which can be done with existing code for multiplier ideals of monomial ideals (2). But, having done that, I think part of the point is to compare it with the multiplier ideal of the original ideal. This project would fill in that part of the picture.

A computation following Howard's combinatorial theorem should be much faster than a computation by general methods such as Shibuta's method implemented by Anton and Christine in the Dmodules package, or resolution of singularities (not even implemented). And this would be a first step toward computing multiplier ideals of more or less arbitrary binomial ideals, hopefully reaching into higher dimensions where general methods bog down.

How would the computation work? Howard's theorem (on page 2 of the preprint) is pretty explicit. It gives the multiplier ideal (of a monomial space curve) as an intersection of ideals of three types. First, there is the multiplier ideal of the curve's term ideal, which should be possible with the existing code for monomial ideals, linked above. Second, there is a symbolic power of the original binomial ideal. To be honest, I'm not sure how difficult it will be to get a symbolic power of a prime binomial ideal; if anyone has any suggestions, this would be very helpful.

But the main part of the formula (everything except those two terms) is an intersection over some monomial valuation ideals. I think this is where most of the work will be. The intersection is indexed by a certain set G (defined on page 9-10 of the preprint) which is slightly involved. It should be possible to compute both G and the monomial valuation ideals via Bruns's Normaliz software.

Zteitler 07:16, 18 July 2011 (UTC)

It should be possible to compute the symbolic power by just computing an ordinary power, then saturating by the maximal ideal at the origin (the singular point of the curve), since the unwanted primary components concentrate there.

Zteitler 17:50, 20 July 2011 (UTC)

A draft of a paper on Macaulay2 package for computing multiplier ideals of monomial ideals is available here:

3

I had hoped to finish this earlier, to be available for reading ahead of next week's workshop. Oh, well. Still the draft is mostly complete and hopefully will help.

Zteitler 21:09, 21 July 2011 (UTC)

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