Rees-like Algebras

Abstract: Given their importance in constructing counterexamples to the Eisenbud-Goto Conjecture, it is reasonable to study the algebra and geometry of Rees-like algebras further. Given a graded ideal I of a polynomial ring S, its Rees-like algebra is S[It, t^2], where t is a new variable. Unlike the Rees algebra, whose defining equations are difficult to compute in general, the Rees-like algebra has a concrete minimal generating set in terms of the generators and first syzygies of I. Moreover, the free resolution of this ideal is well understood. While it is clear that the Rees-like algebra of an ideal is never normal and only Cohen-Macaulay if the ideal is principal, I will explain that it is often seminormal, weakly normal, or F-pure. I will also discuss the computation of the singular locus, how the singular locus is affected by homogenization, and the structure of the canonical module, class group, and Picard group. This talk is joint work with Paolo Mantero and Lance E. Miller.

Mathematics

Audience: advanced learners

IIT Bombay Virtual Commutative Algebra Seminar

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