Seminar Fellowship of the Ring Jugal Verma IIT Gandhinagar Source скачать в хорошем качестве

Seminar Fellowship of the Ring Jugal Verma IIT Gandhinagar Source

Seminar: Fellowship of the Ring - Jugal Verma (IIT Gandhinagar) Title: Adjoint of powers of ideals in regular domains Abstract: The Adjoint of an ideal $I$ in a regular domain, denoted by $adj(I)$, was introduced by J. Lipman. It is an integrally closed ideal that contains $I$. It is closely related to multiplier ideals in algebraic geometry and test ideals for tight closure. Using this ideal, Lipman improved the Brian\{c}con-Skoda theorem for integral closure of ideals. He conjectured that $adj(I^n)=I adj(I^{n-1})$ for all $n\geq \ell(I)$ where $\ell(I)$ is the analytic spreadof $I$.This conjecture was proved by Lipman in dimension 2 and by Dale Cutkosky for all local rings of smooth projective varieties in characteristic 0. A related problem is the subadditivity of adjoints: $adj(IJ)\subseteq adj(I) ad(J).$ In the first half of this talk, I will introduce the adjoint of an ideal and explain what is known about them. The second half of the talk is devoted to a unification of Lipman's formula for the adjoint of powers and the subadditivity of adjoints. We employ a formula of MA Hoskin and P Deligne, mixed multiplicities and joint reductions of ideals. If time permits, I will discuss the connection between the adjoint of a monomial ideal in a polynomial ring and the normal Hilbert polynomial of $I.$ This talk is based on joint work with Clare D'Cruz, Saipriya Dubey, Dan Katz and Vijay Kodiyalam.