Doctor of Philosophy
Dr. Graham Denham
This thesis is on algebraic and algebraic geometry aspects of complex hyperplane arrangements and multiarrangements. We start by examining the basic properties of the logarithmic modules of all orders such as their freeness, the cdga structure, the local properties and close the first chapter with a multiarrangement version of a theorem due to M. Mustata and H. Schenck.
In the next chapter, we obtain long exact sequences of the logarithmic modules of an arrangement and its deletion-restriction under the tame conditions. We observe how the tame conditions transfer between an arrangement and its deletion-restriction.
In chapter 3, we use some tools from the intersection theory and show that the intersection cycle of a certain projective variety has a closed answer in terms of the characteristic polynomial. This result is used to compute the leading parts of the Hilbert polynomial and Hilbert series of the logarithmic ideal. As a consequence, we recover some of the classical results of the theory such as the Solomon-Terao formula for tame arrangements. This is done by computing the Hilbert series in two different ways. We also introduce the notion of logarithmic Orlik-Terao ideal and show that the intersection lattice parametrizes a primary decomposition. The chapter is closed by a generalization of logarithmic ideals to higher orders. It is shown that these ideals detect the freeness of the corresponding logarithmic modules.
The last chapter is a generalization of the notion of logarithmic ideal to multiarrangements. Some of the basic properties of these ideals are investigated. It is shown that one obtains a natural resolution of this ideal by logarithmic modules under the tame condition. In the final section it is shown that the intersection cycle of the logarithmic ideal of a free multiarrangement is obtained from its characteristic polynomial, similar to simple arrangements.
Garrousian, Mehdi, "Characteristic Polynomial of Arrangements and Multiarrangements" (2011). Electronic Thesis and Dissertation Repository. 142.