Topics on perfect ideals of codimension two - OLIVEIRA, Geisa Gama (2025)
In this work, we are interested in perfect ideals I of codimension two in a polynomial ring R over a field of characteristic zero. The overall interest is on the homological nature of the main algebras related to the Hilbert-Burch syzygy matrix associated to I, in particular on the Cohen-Macaulay property of the Rees algebra and the special fiber of I. We develop this study in three contexts: monomial ideals, ideals defined by the 2-minors of homogeneous 3 × 2 matrices, and the defining ideal of a finite set of reduced points in projective 2-space. In addition, we investigate geometric aspects related to these ideals, focusing on how their algebraic properties are reflected in the associated rational maps.