My mathematical works are listed below in reverse chronological order. 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. Many items are followed by a telegraphic technical abstract in smaller blue type and/or by an even briefer informal description for non-specialists. Where possible, the title links to a copy of the article or book. For further details, see my mathscinet author page https://mathscinet.ams.org/mathscinet/MRAuthorID/194372.
(with Dave Bayer) The poset of zero-one factorizations, in preparation.
Gives a global description of the poset of 0-1 factorizations of (1-xn)/(1-x) in terms of chains of divisors and then reviews 135 years of study of this problem and applications of its solution in 40 papers.
Sacks of dice with fair totals, Amer. Math. Monthly 125 (2018), no. 7, 579–592.
The question of when rolling a set of loaded dice yields all totals with equal probability goes back 60 years. This paper provides a canonical construction of all such sets.
The previous paper with Bayer provides a much more elegant solution that extends to infinite cases.
(with David Swinarski) Can you play a fair game of craps with a loaded pair of dice?, Amer. Math. Monthly 123 (2016), no. 2, 136–148
Studies the question of whether the distribution of the totals obtained by rolling pair of loaded dice can mimic that of a fair pair and generalizations and offshoots of this problem.
The dual of the preceding problems turned out to have a far messier answer.
(with Dawei Chen and Gavril Farkas) Effective cones of moduli spaces of curves and abelian varieties, in A Celebration of Algebraic Geometry, Clay Mathematics Proceedings, Volume 18 (Providence, RI: American Mathematical Society, 2013).
Surveys what is known---or more often, not---about these cones and reviews the applications of these results to other birational questions about these spaces.
(co-editor with Gavril Farkas) Handbook of Moduli, I, II and III, Advanced Lectures in Mathematics, Volumes 24--26 (Beijing: Higher Education Press, 2013) and (Somerville, MA: International Press, 2013);
The Handbook is a three-volume survey totaling some 1750 pages of all areas of current research activity into moduli spaces in algebraic geometry with 35 contributions from some 50 of the leading experts in the subject.
(with David Swinarski) Gröbner Techniques for Low-Degree Hilbert Stability, Exp. Math. 20 (2011), 34–56.
Combines ideas about state polytopes from my paper Dave Bayer cited below and Kempf's theory of worst one-parameter subgroups to identify situations in which it is possible to deduce stability for low values of the degree by a machine-assisted symbolic calculation in Macaulay2. Examples are used to confirm predictions in the log minimal model program.
An idea that waited 30 years for the right application.
Mori theory of moduli spaces of stable curves, unpublished draft (2009).
Based on lectures given at the Université de Montréal Summer school in 2007 that establish foundational material, review the history of the subject, and conclude by describing recent work on the effective and nef cones of these spaces.
These incomplete notes are occasionally cited because they are still the only introduction to some topics.
Math4Life, (New York: Projective Press, 2010).
A hypertext for a one-semester, terminal mathematics course for intending humanities and social science majors that tried to ask questions that might interest such students and to fully explain their answers.
Sadly, most students in this population are not interested in any mathematical questions.
GIT constructions of moduli spaces of stable curves and maps, Surv. Diff. Geom. XIV, Geometry of Riemann surfaces and their moduli spaces, (Somerville: International Press, 2010), 315–369.
This survey reviews Gieseker's classical construction in detail, explains the new ideas in recent constructions for pointed curves and maps and in the Hassett–Keel log minimal model program .
(with Donghoon Hyeon) Stability of elliptic tails and 4-canonical models, Math. Res. Lett. 17 (2010), 721--729.
Shows that the GIT quotient of the locus of 4-canonical stable curves in the corresponding Hilbert or Chow scheme is Schubert's moduli space of pseudo-stable curves, and answers a thirty year old question from my doctoral thesis by giving examples of varieties that are Chow strictly semi-stable but either Hilbert stable or Hilbert unstable.
A curious example of a GIT wall crossing for which the moduli problems on the two sides are different but the moduli spaces are the same.
(with Angela Gibney and Sean Keel) Towards the ample cone of $\overline{\mathcal M}}_{g,n}$, J. Amer. Math. Soc. 15 (2002), 273--294.
Conjectures that the Mori cones of these spaces are generated by curve strata and reduces the general conjecture to the genus 0 case, verifies this in some cases and draws various consequences for the birational geometry of these spaces.
My most influential paper, one that inspired a great deal of activity in the decade following its publication. The strongest conjectures are known to be false for large g but those for the nef cone have been checked up to g=44 and remain open in general.
Stability of Hilbert points of generic K3 surfaces, Centre de Recerca Mathemática, Publicación 401 (1999)
A short proof that a K3 surface with Picard number 1 embedded by a primitive divisor class has an asymptotically stable Hilbert point.
By showing that the Hilbert semistable loci for K3s are non-empty, this makes semistable reduction constructions of GIT moduli possible in principle.
(with Joe Harris) Moduli of Curves, Graduate Texts in Mathematics 187, (New York: Springer-Verlag, 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.
Still the text which introduces most graduate students to the subject, translated into Russian and Chinese.
Subvarieties of moduli spaces of curves: open problems from an algebro-geometric point of view, Mapping class groups and moduli spaces of Riemann surfaces (G\"ottingen, 1993/Seattle, WA, 1993), 317--343, Contemp. Math., 150, Amer. Math. Soc., Providence, RI.
An expository survey of then recent but now somewhat dated results and open problems about subvarieties of moduli spaces of curves and their properties.
(with Joe Harris) Slopes of effective divisors on the moduli space of stable curves, Inv. Math. 99 (1990) 321–355.
Makes a conjecture on the shape of the cone of effective divisors on this space that would imply a converse to the Harris-Mumford General Type Theorem and gives estimates for this cone in all genera which prove the conjecture for genus at most~6.
My second most influential paper. Farkas and Popa (MR2123229) gave counterexamples in 2005 but known examples still do not disprove a version asymptotic in g and Farkas' 2005 claim that $\overline{M}_{22}$ and $\overline{M}_{23}$ are of general type was only fully proved (with Jensen and Payne, https://arxiv.org/abs/2005.00622) in 2025.
Constructing the moduli space of stable curves, Lectures on Riemann surfaces (Trieste, 1987), M. Cornalba, X. Gomez-Mont and A. Verjovsky eds., 201–244 (World Sci. Publ., Teaneck, NJ).
Expository outline of the construction via G.I.T. with background and applications based on lectures given at an ITCP graduate school.
(with Dave Bayer) Standard bases and geometric invariant theory, I: State polytopes and initial ideals, J. Symb. Comp. 6 (1988), 209--217.
Relates the initial forms of the ideal of a projective variety to the geometric invariant theory of its Hilbert point(s). The paper was later republished as part of a collection of research articles on computational issues.
(with Shigefumi Mori and David Morrison) On four dimensional terminal quotient singularities, Math. of Comp. 5 (1988), 769–786.
Conjectures a classification of such singularities, proves the terminality of the candidate singularities, and deduces geometric consequences (known in dimension 3 by a case analysis) with computer based evidence.
I had as much fun working on this paper as on any I have written. The conjecture was proved by Sankaran (MR1021875) in 1990. A paper of Borisov (MR2438068) establishes an amazing connection to the Riemann hypothesis.
(with Henry Pinkham) Galois Weierstrass points and Hurwitz characters, Annals of Math. 124 (1986), 591--625.
Characterizes gap sequences of Galois Weierstrass points and includes a generalization of Iwasawa's computation of the index of the Stickelberger ideal from prime to composite orders and a determination of the positive cone in this ideal.
Another paper that was a pleasure to write. The appearance of class numbers of imaginary quadratic fields would surely have stumped us has I not seen the Maillet determinants we were then trying to show to be 1 while browsing new journals in the Columbia math library.
(with John Morgan) A Van Kampen theorem for weak joins, J. London Math. Soc., 53 (1986) no. 3, 562--576.
Calculation of the fundamental group of a weak join in terms of the fundamental groups of its components.
I tackled this problem because, at a Columbia colloquium dinner, Eilenberg challenged the table to show that the proof of Griffiths (MR0080103) was wrong.
(with Mark Green) The equations defining Chow varieties, Duke J. Math. 53 (1986), 733–747.
Characterizes Chow forms amongst all hypersurfaces in grassmannians and deduces equations for Plücker embeddings of Chow varieties.
Submitted in 1979! By far my longest delay from submission to press.
(with Ulf Persson) Numerical sections on elliptic surfaces, Compositio Math. 59 (1986), 323–338.
Generalization to arbitrary elliptic surfaces of earlier work with Persson yielding a refinement of the Shioda-Tate formula taking account of torsion.
Steinberg was in the audience when I spoke on this at UCLA and at tea explained how to eliminate a lengthy case analysis by a favorite trick.
(with Tom Evans) Sensitivity to retinal defocus with aspheric soft lenses: predictions and clinical validation,
Am. J. Optometry and Physiological Optics 61 (1984), 729–736.
Clinical data from presbyopic patients explained using a computer simulation of a mathematical model of the human eye.
Tom was a childhood friend whose need for some mathematical help was solved when I ran into him while in Toronto on an NSERC Junior Fellowship.
(with David Gieseker) Hilbert stability of rank two bundles on curves, J. Differential Geometry 19 (1984) 1–29.
Compactifies the moduli space of such bundles by combining geometric invariant theory methods of Gieseker with ideas from my thesis.
(with Ulf Persson) The group of sections on a rational elliptic surface, in Algebraic geometry---open problems (Ravello, 1982), 321–347, Lecture Notes in Math. 997, (Springer, Berlin).
Gives a geometric proof of a formula of Manin relating the group law on the sections to the addition in the Neron-Severi group, with applications.
Greatly generalized in the sequel above, also written with Ulf.
Projective Stability of ruled surfaces, Inv. Math. 50 (1980), 269–304.
For a vector bundle E over a curve C, shows that slope-stability of E and stability of suitable projective models of the projectivization of E are equivalent and studies other relations between various notions of stability of such models.
My Harvard Ph.D. thesis.
(notes for David Mumford) Stability of projective varieties, L'Ens. Math 23 (1977), 39–110.
Basic reference for GIT constructions, by Gieseker's method, of moduli of projective varieties.
Mumford had arranged for me to be at I.H.E.S. when he was giving his Fields Medal lectures and I had the great fortune to be able to redact these notes.