O ideal jacobiano de um arranjo de hiperplanos

Abstract

In this work, we are interested in exploring properties of the Jacobian ideal of a form f defined by a hyperplane arrangement A in n-dimensional affine space over a field of characteristic zero. We aim to present two main results: the Jacobian ideal Jf as a minimal reduction of the ideal I, defined by the (m− 1)-products of the linear forms defined by A, when this is an almost generic arrangement, and the Rose- Terao- Yuzvinski theorem, a result which gives us the homological dimension of the module of logarithmic derivations of f, in the case where A is generic. To this end, we introduce important concepts from Commutative Algebra, such as Rees Algebra, special fiber, and reduction of an ideal I, as well as the relevant algebraic invariants: saturation index of an ideal and the Castelnuovo-Mumford regularity of a module.