Algebra Seminar

Organizers: Justin Lyle and Paolo Mantero

Speaker Title Abstract
Ben Drabkin
Containment-tight ideals from singular loci of reflection arrangements
Given an ideal I in a commutative Noetherian ring R, the m-th symbolic power of I is denoted I^{(m)}. By results of Ein-Lazarsfeld-Smith, Hochster-Huneke, and Ma-Schwede every ideal I of codimension e in a regular ring satisfies the containment I^{(er)}\subseteq I^r. In many cases, this containment can be improved upon; however, in recent years a number of ideals have been found for which this containment is tight. All known ideals exhibiting tight containments are codimension 2 and satisfy that I^{(3)} is a subset of I^2. Furthermore, most of these ideals define the singular loci of hyperplane arrangements for some complex reflection groups. This talk will aim to classify which complex reflection groups give rise to hyperplane arrangements whose singular loci exhibit the noncontainment I^{(3)} is not a subset of I^2.
Justin Chen
Primary decomposition and Noetherian operators
Primary decomposition is a fundamental problem in computational algebraic geometry. For reduced schemes, numerical irreducible decomposition has been fairly successful, but additional data is needed in the non-reduced case. To this end, one may turn to Noetherian operators, which are polynomial differential operators that encode the multiplicity structure of an arbitrary ideal. I will discuss some theory and algorithms (implemented in Macaulay2) for computing Noetherian operators, which allows for numerical primary decomposition of unmixed ideals. This is joint work with Marc Harkonen, Robert Krone, and Anton Leykin.
Paolo Mantero
Rees-like algebras
Rees-like algebras were recently introduced by McCullough and Peeva to produce the first known counterexamples to the celebrated Eisenbud-Goto conjecture. In this talk we survey two very recent joint papers with J. McCullough and L. E. Miller studying some algebro-geometric properties of Rees-like algebras. For a Rees-like algebra A we discuss its singular locus, its divisor class group and its Picard group. We determine when does A have smooth general hyperplane sections, when is it semi-normal (weakly normal or F-split), and we study its canonical module.