Commutative Algebra Seminar

Title

When is an ideal monomializable?

Abstract

We give a characterization of ideals which are monomial under some system of coordinates. Our motivation is to understand when does a homogeneous ideal I, in a polynomial ring S over an infinite field, admit a monomializable Artinian reduction. That is, when does there exist a choice of general hyperplanes which cut down the affine cone over Proj(S/I) to a zero-dimensional scheme which is defined, in some coordinates, by a monomial ideal. This question is difficult in general; an affirmative answer for any class of ideals is extremely useful. For example, if S/I is Cohen-Macaulay, its graded betti table is preserved on passing to an Artinian reduction. If any such Artinian reduction is monomializable, then many invariants of the betti table of S/I (type, multiplicity, regularity, etc.) become readily computable and have strong constraints.

We discuss the subtleties and obstructions involved in the choice of these general hyperplanes and some surprising applications to linkage theory. This is joint work with Alessandro De Stefani and Matteo Varbaro.