Jooyoun Hong

On the homology of two-dimensional elimination

We study birational maps with empty base locus defined by almost complete intersection ideals. Birationality is shown to be expressed by the equality of two Chern numbers. We provide a relatively effective method for their calculation in terms of certain Hilbert coefficients. In dimension 2 the structure of the irreducible ideals-always complete intersections by a classical theorem of Serre-leads by a natural approach to the calculation of Sylvester determinants. We introduce a computer-assisted method (with a minimal intervention by the computer) which succeeds, in degree @?5, in producing the full sets of equations of the ideals. In the process, it answers affirmatively some questions raised by Cox [Cox, D.A., 2006. Four conjectures: Two for the moving curve ideal and two for the Bezoutian. In: Proceedings of Commutative Algebra and its Interactions with Algebraic Geometry, CIRM, Luminy, France, May 2006 (available in CD media)].

Specialization and integral closure

We prove that the integral closedness of any ideal of height at least two is compatible with specialization by a generic element. This opens the possibility for proofs using induction on the height of an ideal. Also, with additional assumptions, we show that an element is integral over a module if it is integral modulo a generic element of the module. This turns questions about integral closures of modules into problems about integral closures of ideals, by means of a construction known as Bourbaki ideal.

Extremal Rees algebras

We study almost complete intersections ideals whose Rees algebras are extremal in the sense that some of their fundamental metrics---depth or relation type---have maximal or minimal values in the class. The focus is on those ideals that lead to almost Cohen--Macaulay algebras and our treatment is wholly concentrated on the nonlinear relations of the algebras. Several classes of such algebras are presented, some of a combinatorial origin. We offer a different prism to look at these questions with accompanying techniques. The main results are effective methods to calculate the invariants of these algebras.

Variation of Hilbert coefficients

For a Noetherian local ring (R,m), the first two Hilbert coefficients, e0 and e1, of the I-adic filtration of an m-primary ideal I are known to code for properties of R, of the blowup of Spec(R) along V (I), and even of their normalizations. We give estimations for these coefficients when I is enlarged (in the case of e1 in the same integral closure class) for general Noetherian local rings.

The positivity of the first coefficients of normal Hilbert polynomials

Let R be an analytically unramified local ring with maximal ideal m and d = dim R > 0. If R is unmixed, then e 1 I (R) ≥ 0 for every m-primary ideal I in R, where e 1 I (R) denotes the first coefficient of the normal Hilbert polynomial of R with respect to I. Thus the positivity conjecture on e 1 I (R) posed by Wolmer V. Vasconcelos is settled affirmatively.

SIMPLE VALUATION IDEALS OF ORDER TWO IN 2-DIMENSIONAL REGULAR LOCAL RINGS

Let (R, m) be a 2-dimensional regular local ring with algebraically closed residue field R/m. Let K be the quotient field of R and v be a prime divisor of R, i.e., a valuation of K which is birationally dominating R and residually transcendental over R. Zariski showed that there are finitely many simple v-ideals and all the other v-ideals are uniquely factored into a product of those simple ones. It then was also shown by Lipman that the predecessor of the smallest simple v-ideal P is either simple (P is free) or the product of two simple v-ideals (P is satellite), that the sequence of v-ideals between the maximal ideal and the smallest simple v-ideal P is saturated, and that the v-value of the maximal ideal is the m-adic order of P. Let m = (x, y) and denote the v-value difference |v(x) - v(y)| by . In this paper, if the m-adic order of P is 2, we show that . We also show that when w is the prime divisor associated to a simple v-ideal of order 2 and that w(R) = v(R) as well.

Integrally Closed Modules and Their Divisors

ABSTRACT There is a beautiful theory of integral closure of ideals in regular local rings of dimension two, due to Zariski, several aspects of which were later extended to modules. Our goal is to study integral closures of modules over normal domains by attaching divisors/determinantal ideals to them. They will be of two kinds: the ordinary Fitting ideal and its divisor, and another ‘determinantal’ ideal obtained through Noether normalization. They are useful to describe the integral closure of some class of modules and to study the completeness of the modules of Kähler differentials.