My area of research is algebraic geometry.
My dissertation is titled "Multiplier Ideals of Line Arrangements".
I computed the multiplier ideals of general arrangements of lines through the origin in C^3,
and even for "most" special arrangements.
My advisor is Rob Lazarsfeld at University of Michigan.
I am interested in certain questions about multiplier ideals.
I am also interested in certain questions related to arrangements.
And I am working with people at Texas A&M University including Frank Sottile
and JM Landsberg on problems involving algebraic geometry, numerical methods,
Schubert calculus, toric varieties, and representation theory.
Papers
- The nef cone volume of generalized Del Pezzo surfaces with Ulrich Derenthal, Michael Joyce, Algebra & Number Theory 2 (2008), no. 2, 157--182
- A note on Mustata's computation of multiplier ideals of hyperplane arrangements, Proc. Amer. Math. Soc. 136 (2008), no. 5, 1575--1579
- On the intersection of the curves through a set of points in P^2,
J. Pure Appl. Algebra 209 (2007), no. 2, 571--581
- Multiplier ideals of general line arrangements in C^3,
Comm. Alg. 35 (2007), no. 6, 1902--1913
Some introductions to multiplier ideals
Sometimes people ask me for references to expository introductions
to multiplier ideals.
I'll add to this list as time goes on.
These are arranged in no particular order,
except that I deliberately put the first two items there.
- Blickle--Lazarsfeld, ``An informal introduction to multiplier ideals'', arxiv:0302.5351.
This covers some properties including vanishing theorems,
Skoda's theorem, and restriction and subadditivity properties;
monomial ideals; jumping numbers;
asymptotic multiplier ideals;
two uniform results in commutative algebra (uniform Artin-Rees numbers
and uniform bounds for symbolic powers).
- Lazarsfeld, Positivity in Algebraic Geometry, volume 2, Springer Ergebnisse.
This has the definitive, complete treatment and the most complete bibliography.
- Grushevsky, ``Multiplier ideals in algebraic geometry'', arxiv:0502.5387.
This discusses vanishing theorems and general properties;
singularities of theta divisors;
asymptotic multiplier ideals;
Siu's proof of deformation invariance of plurigenera of varieties of general type;
multiplier ideals in the analytic setting;
analytic proofs of vanishing theorems and invariance of plurigenera.
- Ein, ``Multiplier Ideals, Vanishing Theorem and Applications'', arxiv:9709.5015.
This paper surveys both vanishing theorems
and multiplier ideals, and discusses several applications,
including singularities of theta divisors; adjoint linear series;
a theorem of Levine on invariance of plurigenera (following an idea of Siu);
a theorem of Esnault and Viehweg on zeros of polynomials.
Being over ten years old, it does not always reflect the current notation.
- Demailly, Analytic methods in algebraic geometry, http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/eem2007.pdf.
This is a textbook on complex geometry and analytic methods such as L^2 estimates.
- Siu, ``Multiplier ideal sheaves in complex and algebraic geometry'', arxiv:0504259.
(From the Arxiv abstract: This article discusses the geometric application
of the method of multiplier ideal sheaves. It first briefly describes its
application to effective problems in algebraic geometry and then presents
and explains its application to the deformational invariance of plurigenera
for general compact algebraic manifolds. Finally its application to the
conjecture of the finite generation of the canonical ring is explored and
the use of complex algebraic geometry in complex Neumann estimates is
discussed.)
This is not elementary but gives an idea of how multiplier ideals bridge
between algebraic geometry and complex geometry.
- Ein--Mustata, ``Invariants of singularities of pairs'', arxiv:0604.5601.
An update as of August 2006.
Discusses different approaches to multiplier ideals including
the definitions by local integrability, by resolution of singularities,
by arc spaces, and by characteristic p methods.
Discusses a large number of applications, including some very recent ones.
- Dissertations by, for example, Jason Howald (a short survey on multiplier ideals,
computation for monomial ideals), Amanda Johnson (a [different] short survey on multiplier
ideals, computation for generic determinantal ideals),
myself (a short very elementary discussion of how to compute multiplier ideals,
computation for line arrangements),
and an increasing number of other students
who have written dissertations on multiplier ideals.
Please send me suggestions or comments.
In addition you may be interested in these:
- Some expository notes based on
talks I gave in the TAMU Several Complex Variables seminar
in February 2008 on multiplier ideals, aimed at relating
resolution of singularities to the problem of
simplifying integrals.
(July 16, 2008: v0.2. Numerous minor improvements.)
- A talk I gave
at UT Austin in April 2008 on multiplier ideals
of hyperplane arrangements.
Information about the
Special Session
on Algebraic Geometry of Matrices and Determinants
held at the AMS Spring 2008 Southeastern Sectional Meeting in Baton Rouge, LA, March 2008.
AGIL
Information about the regional conference AGIL: Algebraic Geometry In Louisiana:
- Fall 2006: October 7, 2006, at Southeastern Louisiana University.
- Spring 2007: April 14, 2007, at Tulane University.
Some time ago, I began work on a short expository set of notes
on Chern classes in algebraic geometry, particularly in the context
of enumerative problems.
The notes are not quite done. I hope to finish them as soon as possible;
in the meantime, I have uploaded a draft, in PDF format (21 pages):
An informal introduction to computing with Chern classes
(Oct. 31, 2004).
Here is a modest collection of links to pages which in turn link to
interesting sources for algebraic geometry.
A few other sites that I go to sometimes when I need general math information:
zteitler@tamu.edu
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