Sub-adjoint ideals and hyperplane sections

, Castelnuovo - Mumford Regularity , and Generic Initial Ideals

KOSZUL ALGEBRAS, CASTELNUOVO-MUMFORD REGULARITY, AND GENERIC INITIAL IDEALS Giulio Caviglia The University of Kansas Advisor: Craig Huneke August, 2004 The central topics of this dissertation are: Koszul Algebras, bounds for the Castelnuovo Mumford regularity, and methods involving the use of generic changes of coordinates and generic hyperplane restrictions. We give an introduction to Koszul a...

On the Core of Ideals

The core of ideals was first studied by Rees and Sally [RS], partly due to its connection to the theorem of Briançon and Skoda. Later, Huneke and Swanson [HuS] determined the core of integrally closed ideals in two-dimensional regular local rings and showed a close relationship to Lipman’s adjoint ideal. Recently, Corso, Polini and Ulrich [CPU1,2] gave explicit descriptions for the core of cert...

On One-Sided Ideals of Rings of Linear Transformations

On $z$-ideals of pointfree function rings

Let $L$ be a completely regular frame and $mathcal{R}L$ be the ‎ring of continuous real-valued functions on $L$‎. ‎We show that the‎ ‎lattice $Zid(mathcal{R}L)$ of $z$-ideals of $mathcal{R}L$ is a‎ ‎normal coherent Yosida frame‎, ‎which extends the corresponding $C(X)$‎ ‎result of Mart'{i}nez and Zenk‎. ‎This we do by exhibiting‎ ‎$Zid(mathcal{R}L)$ as a quotient of $Rad(mathcal{R}L)$‎, ‎the‎ ‎...

How Many Hypersurfaces Does It Take to Cut out a Segre Class?

We prove an identity of Segre classes for zero-schemes of compatible sections of two vector bundles. Applications include bounds on the number of equations needed to cut out a scheme with the same Segre class as a given subscheme of (for example) a projective variety, and a ‘Segre-Bertini’ theorem controlling the behavior of Segre classes of singularity subschemes of hypersurfaces under general...