Topics in combinatorial commutative algebra and algebraic geometry.

摘要

This thesis consists of results that involve several topics in commutative algebra and algebraic geometry. Most of the proofs have combinatorial nature. Here, I summarize the main results. • In chapter 1, I show that the local cohomology modules of toric algebras have finite length as D-modules, generalizing the classical case of polynomial algebras. As an application, I compute the characteristic cycles of certain local cohomology modules. • In chapter 2, I characterize the complete intersection matrix Schubert varieties, generalizing the result on one-sided ladder determinantal varieties. Also, I give a new proof of the F-rationality of matrix Schubert varieties that doesn't rely on the results of Schubert varieties. As a consequence, this provides an alternative proof of the following well known facts: Schubert varieties in flag varieties are normal and have rational singularities. • In chapter 3, I construct a three-dimensional complete intersection toric variety on which the subadditivity formula of multiplier ideals doesn't hold, answering a question of S. Takagi and K.-i. Watanabe. • In chapter 4, I compute the multiplier ideals (in the sense of T. De Fernex and C. D. Hacon) on determinantal varieties, generalizing a result of A. Johnson. As a consequence, this shows that determinantal varieties are log terminal and provides a supportive example to a question of N. Hara concerning test ideals.