Claudia Polini, University of Notre Dame, IN, USA
Description
Date and Time: 6 November 2020, 6:30pm IST/ 1:00pm GMT/ 08:00am 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: The core of monomial ideals
Abstract: Let $I$ be a monomial ideal. Even though there may not exist any
proper reduction of $I$ which is monomial (or even homogeneous), the
intersection of all reductions, the core, is again a monomial ideal. The
integral closure and the adjoint of a monomial ideal are again monomial
ideals and can be described in terms of the Newton polyhedron of $I$. Such
a description cannot exist for the core, since the Newton polyhedron only
recovers the integral closure of the ideal, whereas the core may change
when passing from $I$ to its integral closure. When attempting to derive
any kind of combinatorial description for the core of a monomial ideal
from the known colon formulas, one faces the problem that the colon
formula involves non-monomial ideals, unless $I$ has a reduction $J$
generated by a monomial regular sequence. Instead, in joint work with
Ulrich and Vitulli, we exploit the existence of such non-monomial
reductions to devise an interpretation of the core in terms of monomial
operations. This algorithm provides a new interpretation of the core as
the largest monomial ideal contained in a general locally minimal
reduction of $I$. In recent joint work with Fouli, Montano, and Ulrich, we
extend this formula to a large class of monomial ideals and we study the
core of lex-segment monomial ideals generated in one-degree.