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My primary research interests currently lie in moduli and classification
problems in algebraic geometry and in connections between these
areas and computational algebraic geometry.
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Moduli of curves
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My main research interest is the study of moduli spaces of stable curves.
In additional my own research projects, I devoted several years to writing, jointly with Joe Harris at Harvard,
a research monograph Moduli of Curves on these spaces: for details see my
selected publications
page. The audience for this book ranges from graduate students in algebraic geometry,
to researchers in cognate specialties in mathematics and other disciplines, especially physics.
While the book is primarily expository - almost all the theorems proved can be found in the
literature - most of our effort was devoted not to harmonizing or
polishing such treatments but to guiding the reader as to how these results are
used in practice and where their limitations lie. Our goal was to develop a "user-friendly"
book which makes a subject whose study has traditionally had very steep prerequisites as accessible
as possible. In addition, the book contains hundreds of examples and exercises to allow the reader
to explore the range of techniques and problems which make up the subject.
In the course of writing the book, I became interested in several new problems concerning
moduli spaces. I am currently trying to understand the structure of the cone of curves on the
moduli space of stable curves, especially Mori's "extremal rays". In 1997, I found a beautiful conjectural
description of all such rays (it later turned out that other had made the similar conjectures) and using results of Cornalba and Harris on the ample cone, was able to check a few
cases. Angela Gibney, Sean Keel and I were able to make substantial
progress on theis F-conjecture which is described in more detail on my
selected publications
page. While I am on leave, I plan to carry out a new computational study of a related
combinatorial conjecture that would complete the proof of the F-conjecture.
Recently, I have begun thinking about slopes of effective divisors on these same spaces.
This was a subject I studied with Joe Harris almost 20 years ago that has recently
become the focus of much interesting work by several young algebraic geometers.
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Stability of Hilbert points
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This was the area in which I wrote my Harvard doctoral thesis and in which I
concentrated in the first few years of my career.
In the last few years, I have been an unofficial co-director of the doctoral
thesis of Dave Swinarski at Columbia. Dave came to Columbia interested in
Hilbert stability problems and we have been working on developing new
criteria for verifying this stability. In particular, Dave has been able to
use some old ideas of mine to come up with a new stability criterion and to
apply to to prove Hilbert stability in situations where known criteria were
not adequate. One of my projects while I am on leave is to find ways to
refine his work and to apply to a new class of problems that arises in work
on so-called log minimal models, a topic that has been getting a lot of attention
in the past year or two.
Going further back, in 1986, Dave Bayer and I showed that the state polytopes of a
for projective variety X (which arises in the study of its geometric invariant
theory) and the monomial specializations of the ideal I(X) of X mutually determine
each other: see the joint paper listed in my
selected publications. This gives a combinatorial method for
studying monomial limits of X and, by extension, the Gröbner bases of I(X) which are the fundamental
tools in computations with such ideals. Recently, new computational tools have been
developed that make using these ideas much more practical for stability problems
that are a focus of current interest.
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