Publications and Work in Progress
1. The Signature of the Chern Coefficients of Local Rings (with L. Ghezzi and W. Vasconcelos),
Submitted.
2. Full ideals. (with H. Lee, S. Noh, and D. Rush), Communications in Algebra, To appear.
Contractedness of $m$--primary integrally closed ideals played a
central role in the development of Zariski's theory of integrally
closed ideals
in two-dimensional regular local rings $(R, m)$. In
such rings, the contracted $m$-primary ideals are known to be
characterized by the property that
$I : m$ = $I : x$ for some $x
\in m \setminus m^2$. We call the ideals with this property full ideals and compare this class of ideals with the classes
of
$m$-full ideals, basically full ideals, and contracted ideals in
higher dimensional regular local rings.
The $m$--full ideals are
easily seen to be full. In this paper, we find a sufficient
condition for a full ideal to be $m$--full.
We also show the
equivalence of the properties full, $m$--full, contracted,
integrally closed and normal, for the class of parameter ideals.
We
then find a sufficient condition for a basically full parameter
ideal to be full.
3. On the homology of two-Dimensional elimination. (with A. Simis and W. Vasconcelos),
Journal of Symbolic Computation 43(2008) 275--292.
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 of their calculation in terms of certain Hilbert coefficients.
In dimension two, after observing that
the structure of its irreducible ideals (always complete
intersections by a classical theorem of Serre) leads to 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 $\leq 5$,
in producing the full sets of equations of the ideals. In the
process, it answers affirmatively some questions raised by D. Cox
.
4. Normalization of Modules (with B. Ulrich and W. Vasconcelos), J. Algebra 303(2006) 133--145.
We introduce techniques to derive estimates for the degrees of the
generators of the integral closure of several classes of Rees
algebras of modules, and to bound the length of normalization
processes. In the case of regular base rings, the bounds are
expressed in terms of Buchsbaum--Rim multiplicities and a module
version of Brian\c{c}on--Skoda numbers.
5. Simple valuation ideals of order 2 in two-dimensional regular local rings (with H. Lee and S. Noh),
Communications of Korean Mathematical society 20(3)(2005) 427--436.
6. Integrally Closed Modules and their Divisors (with S. Noh and W. Vasconcelos),
Communications in Algebra 33(2005) 4719--4733.
There is a beautiful theory of integrally closed 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 closures of some class of modules and to
study the completeness of the modules of K\"ahler differentials.
7. Rees Algebras of Conormal Modules Journal of Pure and Applied Algebra, 193 (2004) 231--249.
We deal with classes of prime ideals whose associated
graded ring is isomorphic to the Rees algebra of the conormal
module. With such prime ideals we describe the divisor class
groups of the associated graded rings. Furthermore we study the
relationship between the normality of the conormal module and the
completeness of components of the associated graded ring.
8. Specialization and integral closure (with B. Ulrich), Preprint.
9. Rees algebras of conormal modules. Ph.D. thesis, Rutgers University, 2003.
10. The Equations of Almost Complete Intersections , (with A. Simis and W. Vasconcelos), In preparation.
Seminars
Commutative Algebra and Algebraic Geometry Seminar (CUNY Graduate Center and Rutgers University).
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