|Location:||MSRI: Simons Auditorium|
Linkage is a theory at the borders between Algebraic Geometry and Commutative Algebra aiming at classifying ideals (varieties) and their properties, based on an operation of "linkage".
Since ideals in the same linkage class share several homological properties, it is important to identify the minimal elements in a linkage class (that is, the `best' and `simplest' elements in it).
For example, ideals in the linkage class of a complete intersection (=licci
ideals) can be very complicated, however, because of linkage they share several interesting (and subtle) properties of complete intersection ideals.
Extensive work by several authors (Ballico, Bolondi, Hartshorne, Lazarsfeld, Martin-Deschamps, Migliore, Nagel, Perrin, Rao, Strano, etc.) provides a well-understood notion of minimality in the case of (homogeneous) ideals of codimension 2, However, in the general case (=non-licci ideals of codimension > 2) it is not even clear what could be a good definition of minimality and, consequently, if minimal elements exist in any linkage class.
Polini and Ulrich found special ideals that are "minimal" in several regards. However, the question of finding a general notion of minimality was wide-open.
In the present talk, we suggest a general notion of minimality for Cohen-Macaulay ideals, which includes the classes of ideals found by Polini-Ulrich . We prove, under reasonable assumptions, that minimal elements exist and are essentially unique.
We then provide several concrete classes of ideals which are minimal.
We will show applications to the question of when two ideals lie in the same even linkage class, and the Buchsbaum-Eisenbud-Horrocks' Conjecture (which suggests lower bounds for the Betti numbers of any Cohen-Macaulay ideal).