|
|
|
The papers below are listed in reverse chronological order.
They have been selected to represent the
range of my 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.
|
Mori theory of moduli spaces of stables curves, 2007.
This pdf file contains the latest working draft of lecture notes that I will be working on
while I am on leave over the 2007-2008 and 2008-2009 academic years.
I used these notes for a set of lectures given in June, 2007 at a workshop
on Moduli spaces and related topics at the Centre de
Recherche Mathematique in Montreal and I will be giving
similar lectures at the Tata Institute in Mumbai, India and the
University of Sydney in Australia while I am on leave.
An introductory survey to the topic of the title.
Stability of Hilbert Points, 2004.
This pdf file contains slides of a survey talk I gave at the Compact Moduli workshop held at the American Institute of Mathematics
in December, 2004.
An evaluation of old and new approaches to verifying stability of Hilbert points of projective
varieties with speculations on where more effective methods might be found.
(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.
Subvarieties of moduli spaces of curves: open problems from an algebro-geometric point of view,
in Mapping class groups and moduli spaces of Riemann surfaces, C. Bodigheimer and R. Hain eds.,
Contemporary Mathematics 150, American Mathematical Society, Providence, RI, 317-344, 1993.
Recent results and open problems about subvarieties of moduli spaces of curves and their properties.
An exposition aimed at those who take an analytic approach to these spaces.
(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.
|
|
|
