| 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. |
