Integrally Closed Ideals on Log Terminal Surfaces are Multiplier Ideals

Integrally closed ideals in two-dimensional regular local rings are multiplier ideals

There has arisen in recent years a substantial theory of "multiplier ideals'' in commutative rings. These are integrally closed ideals with properties that lend themselves to highly interesting applications. But how special are they among integrally closed ideals in general? We show that in a two-dimensional regular local ring with algebraically closed residue field, there is in fact no difference between "multiplier" and "integrally closed" (or "complete.") However, among multiplier ideals arising from an integer multiplying constant (also known as adjoint ideals) the only simple complete one…

Local syzygies of multiplier ideals

In recent years, multiplier ideals have found many applications in local and global algebraic geometry. Because of their importance, there has been some interest in the question of which ideals on a smooth complex variety can be realized as multiplier ideals. Other than integral closure no local obstructions have been known up to now, and in dimension two it was established by Favre-Jonsson and Lipman-Watanabe that any integrally closed ideal is locally a multiplier ideal. We prove the somewhat unexpected result that multiplier ideals in fact satisfy some rather strong algebraic properties inv…

A note on Mather-Jacobian multiplier ideals

By using Mather-Jacobian multiplier ideals, we first prove a formula on comparing Grauert-Riemenschneider canonical sheaf with canonical sheaf of a variety over an algebraically closed field of characteristic zero. Then we turn to study Mather-Jacobian multiplier ideals on algebraic curve, in which case the definition of Mather-Jacobian multiplier ideal can be extended to a ground field of any characteristic. We show that Mather-Jacobian multiplier ideal on curves is essentially the same as an integrally closed ideal. Finally by comparing conductor ideal with Mather-Jacobian multiplier ideal, …

When does the subadditivity theorem for multiplier ideals hold?

Demailly, Ein and Lazarsfeld \cite{DEL} proved the subadditivity theorem for multiplier ideals, which states the multiplier ideal of the product of ideals is contained in the product of the individual multiplier ideals, on non-singular varieties. We prove that, in two-dimensional case, the subadditivity theorem holds on log-terminal singularities. However, in higher dimensional case, we have several counter-examples. We consider the subadditivity theorem for monomial ideals on toric rings, and construct a counter-example on a three-dimensional toric ring.

特征零下的奇点阈值

Thresholds of singularities in characteristic zero

该研究在特征零情形下构造了Frobenius阈值的两个类似物Hironaka阈值与Koszul-Hironaka阈值，证明其对参数理想一致且为有理数，揭示了其与乘子理想跳跃数、模p约化Frobenius阈值极限的关联。