The papers below are listed in reverse chronological order, and are a selection, representing my
main mathematical interests. My specialty
is algebraic geometry, especially moduli theory but I
have also done a lot of work with a computational
flavor and have taken a few excursions into number
theory and topology. Each bibliographic reference is followed by a telegraphic
technical abstract in smaller type and then by an
even briefer informal description for non-specialists.
(with Gavril Farkas) Handbook of Moduli,
to appear in three volumes in the series Advanced Lectures in Mathematics of International Press (Somerville, MA).
A collection of some 35 articles assembled with the goal of introducing non-experts interested to active areas in the subject.
(with David Swinarski) New Gröbner Approaches to Hilbert Stability,
Exp. Math. 20, 34-56, 2011.
Recent predictions of the log minimal model program and the paper with Gibney and Keel verified by combining symbolic calculations with some theoretical refinements.
An idea I had 30 years ago, finally realized through Dave's skill in combining a variety of symbolic software tools.
GIT constructions of moduli spaces of stable curves and maps,
pp. 315-370 in Surveys in Differential Geometry, XIV: Geometry of Riemann surfaces and their moduli spaces, 2010.
An exposition of the classical contructions of the title and of several recent ones with applications to log minimal models and a list of open problems.
Commissioned for a volume celebrating the 40th anniversary of the great collaboration of Fields medalists Pierre Deligne and David Mumford.
(with Donghoon Hyeon) Stability of Tails and 4-canonical models, Mathematics Research Letters, 17, 721-729, 2010.
Fills in a gap from a now classic paper of Schubert in a surprising way.
A short paper, prompted by questions in Swinarski's thesis, that almost wrote itself.
Mori theory of moduli spaces of stable curves, in preparation 2011.
This pdf file contains notes based on lectures I gave in June, 2007 at the Centre de
Recherches Mathématiques Program on Moduli Spaces in Montréal and, in 2008, while on leave in Asia.
A final chapter covering recent developments not discussed in the lectures is planned but the current draft is fairly self-contained.
A history of progress in understanding birational questions about moduli spaces of curves
leading to a disussion of current work and open problems.
(with Angela Gibney and Sean Keel) Towards the ample cones of moduli spaces of stable curves,
J. Amer. Math. Soc. 15, 273-294, 2002.
Gives a conjectural description of the extremal rays of these spaces, deduces as a consequence inequalities
describing their ample cones, reduces the case of general genus to that of genus 0 and gives other
evidence for and consequences of the conjecture.
O so near (much nearer than I hoped to get a few years ago) and yet so far.
If you have an online subscription to JAMS, you can download the published version.
If not, you can download the substantially identical preprint math.AG/0006208.
Stability of Hilbert points of generic K3 surfaces, Centre de Recerca Matemática Publication 401, 1999.
A short proof that K3 surfaces embedded by a primitive divisor class have stable Hilbert points.
Back to the ideas in my thesis: see the last entry below.
(with Joe Harris) Moduli of curves, Graduate Texts in Mathematics 187, Springer-Verlag, New York, NY, 1998.
A book aimed at providing a broad introduction to the main theorems, techniques and
open problems in the theory of moduli of algebraic curves with on emphasis on accessible treatments of basic results and important examples.
My magnum opus. You can read a major review of the book in the
Bulletin of the American Mathematical Society.
For ordering information, contact Springer-Verlag.
(with Joe Harris) Slopes of effective divisors on the moduli space of stable curves, Inv. Math. 99 321-355, 1990.
Conjectures the shape of the cone of effective divisors on this space and gives estimates
for this cone in all genera which prove the conjecture for genus at most 6.
I continue to work on this and related questions.
(with Dave Bayer) Standard bases and geometric invariant theory, I: State polytopes and initial ideals, J. Symb. Comp. 6, 209-217, 1988.
Relates the initial forms of the ideal of a projective variety to the geometric invariant theory of its Hilbert point(s).
Pure mathematics used to shed light on a standard computational method.
(with Shigefumi Mori and David Morrison) On four dimensional terminal quotient singularities, Math. of Comp. 51, 769-786, 1988.
Conjectures a classification of such singularities, proves the terminality of the candidate singularities,
and outlines various geometric consequences of the conjectures with computer based evidence.
Experimental mathematics which used a computer investigation to discover the asymptotic order in an initially chaotic problem.
(with Henry Pinkham) Galois Weierstrass points and Hurwitz characters, Annals of Math. 124, 591-625, 1986.
Completely describes gap sequences of Galois Weierstrass points and gives related applications.
A personal favorite where a geometric question had surprising number theoretic ramifications.
(with John Morgan) A Van Kampen theorem for weak joins, J. London Math. Soc., 3rd Series, 53, 562-576, 1986.
Calculation of the fundamental group of a weak join in terms of the fundamental groups of its components.
An excursion into topology to answer a question posed by Sammy Eilenberg.
(with David Gieseker) Hilbert stability of rank two bundles on curves, J. Differential Geometry 19, 1-29, 1984.
Gives a compactification of the moduli space of such bundles via geometric invariant theory.
An extension of the ideas in my thesis.
(with Tom Evans) Sensitivity to retinal defocus with aspheric soft lenses: predictions and clinical validation,
Am. J. Optometry and Physiological Optics 61, 729-736, 1984.
Clinical data from presbyopic patients explained using a computer simulation of a mathematical model of the human eye.
Straightforward applied modelling.
Projective Stability of ruled surfaces, Inv. Math. 50, 269-304, 1980.
For a vector bundle E over a curve C, shows that bundle-stability of E and stability of suitable projective models of P(E) are equivalent.
Also studies the relation between different notions of stability of such models.
My doctoral thesis in algebraic geometry.