Commutative Algebra and Algebraic Geometry: The Fellowship of the Ring: Singular lexicographic points

Seminar | November 19 | 5-6 p.m. | 939 Evans Hall

 Ritvik Ramkumar, UC Berkeley

 Department of Mathematics

The classical Hilbert scheme parameterizes saturated homogenous ideals with a fixed Hilbert polynomial P(t). For each P(t) there is a unique saturated homogeneous ideal, called the lexicographic ideal, that exhibits certain extremal behaviour. A theorem of Reeves and Stillman states that the lexicographic ideal is a smooth point on its Hilbert scheme. In parallel, various authors have conjectured that lexicographic ideals are smooth for more general types of Hilbert schemes. In this talk I will describe these conjectures and give counterexamples to them. On the other hand, I will explain when one can expect the lexicographic ideal to be a smooth point and describe a few results. This is joint work with Alessio Sammartano.