- Moduli of twisted spin curves, with Tyler Jarvis. In this note we give a new, natural construction of a compactification of the stack of smooth $r$-spin curves.
- Compactifying the space of stable maps, with Angelo Vistoli. We define two equivalent notions of twisted stable map from a curve to a Deligne-Mumford stack with projective moduli space, and we prove that twisted stable maps of fixed degree form a complete Deligne-Mumford stack with projective moduli space. The results were announced in math.AG/9811059; we refer to this last preprint for a discussion of the ideas behind our construction.
- Torification and
Factorization of Birational Maps with
Kalle Karu,
Kenji Matsuki, and
Jaroslaw Wlodarczyk. Building on the
work of the fourth author in math.AG/9904074, we prove the
weak factorization conjecture for birational maps in characteristic zero: a
birational map between complete nonsingular varieties over an algebraically
closed field K of characteristic zero is a composite of blowings up and
blowings down with smooth centers.

Another proof of the same theorem appeared independently by the fourth author in math.AG/9904076. - Complete moduli for families over semistable curves with Angelo Vistoli. This note is but a research announcement, summarizing and explaining results proven and detailed in forthcoming papers. When one studies families of objects over curves, and the objects are parametrized by a Deligne-Mumford stack M, then the families are equivalent to morphisms of curves into M. In order to have complete moduli for such families, one needs to compactify the stack of stable maps into M. It turns out that in the boundary, the curve must acquire extra structure and you'd better read the paper to see what that structure is. Applications to fibered surfaces (see math.AG/9804097), admissible covers, and level structures are discussed. http://xxx.lanl.gov
- Uniformity of stably integral points on principally polarized abelian surfaces, with Kenji Matsuki. We prove, assuming that the conjecture of Lang and Vojta holds true, that there is a uniform bound on the number of stably integral points in the complement of the theta divisor on a principally polarized abelian surface defined over a number field.
- Stable maps and Hurwitz schemes in mixed characteristic, with Frans Oort. We define complete Hurwitz schemes in mixed characteristics, using stable maps.
- The formula 12 = 10 + 2 x 1 and its generalizations, with Aaron Bertram. Counting rational curves on F2.
- Alterations and Resolutions of singularities, with Frans Oort, This is our contribution submitted to the working week on resolution of singularities, an expository introduction to de Jong's work on alterations.
- Complete moduli for fibered surfaces, with Angelo Vistoli. This is a first installment of our work on stable maps into Deligne-Mumford stacks, where we consider the case of stable maps into the stack of stable pointed curves.
- Weak semistable reduction in characteristic 0, with Kalle Karu. In this paper we consider the general case of semistable reduction in characteristic 0. First we define what we mean by a semistable morphism in terms of toroidal embeddings. Then we reduce the varieties to toroidal embeddings and solve a slightly weaker version of semistable reduction. We also state the full semistable reduction problem in terms of combinatorics of the associated polyhedral complexes.
- Extending Triangulations and Semistable Reduction, with Maurice Rojas. We prove a result about extending triangulations of a subcomplex, and deduce a refinement of the result in the paper "Weak semistable reduction in characteristic 0".
- Equivariant Resolution of Singurlarities in Characteristic 0, with Jianhua Wang. A new proof of equivariant resolution of singularities under a finite group action in characteristic 0 is provided. Math. Research Letters 4, 427-433 (1997).
- A linear lower bound on the gonality of modular curves. The result in the title is proven and applications discussed. This is a piece of my thesis, somewhat expanded. IMRN 1996 No 20, 1005-1011.
- Smoothness, Semistability, and Toroidal Geometry, with Johan de Jong. We provide a new proof of resolution of singularities in characteristic 0. To appear in Journal of Algebraic Geometry.
- A high fibered power of a family of varieties of general type dominates a variety of general type. The result in the title is proven. A dried out version of this appeared on Inventionnes Math. 128, 481-494 (1997).
- Lang maps and Harris's conjecture. Motivated by a conjecture of Harris generalizing both Lang's and Manin's conjectures, we define and briefly discuss the universal dominant rational map to a variety of general type. To appear in Israel Journal of Mathematics.
- Uniformite' des points rationnels des courbes alge'briques sur les extensions quadratiques et cubiques. We refine a result of L. Caporaso, J. Harris and B. Mazur, who showed that Lang's conjecture implies that the number of rational points on a curve of genus g>1 is uniformly bounded. We show that the conjecture implies that this number is bounded uniformly over all quadratic and cubic extensions of a given number field. Appeared in C. R. Paris 321, p.755, now superseded by Pacelli.
- Lang's conjectures, fibered powers, and uniformity, with Felipe Voloch. Following the results of L. Caporaso, J. Harris and B. Mazur, we study the implication of Lang's conjectures together with conjecture H for uniform bounds on rational points on varieties in higher dimensions, and a few surprising geometric implications.
- Uniformity of stably integral points on elliptic curves. We demonstrate an analogue of the result of L. Caporaso, J. Harris and B. Mazur, showing that the Lang - Vojta conjecture implies a uniform bound on the number of stably integral points on an elliptic curve over a number field, as well as the uniform boundedness conjecture (Merel's theorem). Inventionnes Math. 127, 307-317 (1997).
- Formal finiteness and the torsion conjecture on elliptic curves. A proof of the uniform boundedness conjecture up to degree 14. Disappeared in an infamous Asterisque volume (228), now superseded by Merel.
- Toward a proof of the Mordell-Lang conjecture in characteristic p, with Felipe Voloch. This paper is concerned with an analogue in positive characteristic of the conjecture known as the Mordell - Lang conjecture. Appeared in Duke IMRN, June 1991. Now superseded by Hrushovsky.
- Subvarieties of semiabelian varieties. Part of my thesis, published in Compositio Math. 1994.