My primary research interests currently lie in moduli and classification
problems in algebraic geometry and in connections between these
areas and computational algebraic geometry.
Moduli of curves
My main research interest is the study of moduli spaces of stable curves.
I am currently editing with Gavril Farkas at Humboldt a multi-volume Handbook of Moduli that aims to provide open active areas of research to wider audiences (details are on my
selected publications page) and I am a co-organizer WOMB, an annual Workshop on Moduli and Birational Geometry, held each summer in Korea under the auspices of POSTECH.
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, described in more detail on my
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
My most recent work has involved trying to find way to verify predictions that the F-conjecture
makes about log minimal model programs for these moduli spaces. These programs are currently
a very active area of research, outlined in my lecture notes on the Mori theory of these spaces and
in my survey of GIT constructions. My recent papers with Hyeon and Swinarski apply and extend techniques that were the focus of
my thesis-see the section on stability of Hilbert ponts below- to check some of these predictions.
Recently, I have also been 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 also recently
become the focus of much interesting work by several young algebraic geometers.
Stability of Hilbert points
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.
I served co-director of the 2008 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, he 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
Recently I have been collaborating with Dave to
refine his work and to apply it to a new class of problems. The applications involve other old results from work with Dave Bayer in the 1980s.
We 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.
Swinarski and I have been able to extend these results
in a way that, for the first time, allows examples
to be treated by computer assisted calculations. The examples we treat are special but have significant applications
to the predictions in the log minimal model discussed above by showing the existence of the necessary GIT quotients. Our methods have recently been applied in a beautiful paper of Alper, Fedorchuk and Smyth to prove that such quotients exist in all the relevant GIT setups, although much work remains to be done to verify that these quotients have all the properties predicted for them.