Tue, November 3, 2020
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6:00pm [6:30pm] Claudia Polini, University of Notre Dame, IN, USA
Description:
Date and Time: 3 November 2020, 6:30pm IST/ 1:00pm GMT/ 09:00am EDT (joining time: 6:15 pm IST - 6:30 pm IST) Speaker: Claudia Polini, University of Notre Dame, IN, USA Google meet link: meet.google.com/urk-vxwh-nri Title: Core of ideals - Part 1 Abstract: Let I be an ideal in a Noetherian commutative ring. Among all the closures of I, the integral closure plays a central role. A reduction of I is a subideal with the same integral closure. We can think of reductions as simplifications of the given ideal, which carry most of the information about I itself but, in general, with fewer generators. Minimal reductions, reductions minimal with respect to inclusion, are loosely speaking the counterpart of the integral closure. However, unlike the integral closure, minimal reductions are not unique. For this reason we consider their intersection, called the core of I. The core is related to adjoint and multiplier ideals. A motivation for studying this object comes from the Briancon-Skoda theorem. Furthermore a better understanding of the core could lead to solving Kawamata's conjecture on the non-vanishing of sections of certain line bundle. In this talk I will discuss the importance of the core, its ubiquity in algebra and geometry, and some effective formulas for its computation.