Below are some of my papers, in postscript or d.v.i.:
Partial Support for research provided by NSF and Sloan foundations
- 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
- Torification and
Factorization of Birational Maps with
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
- Complete moduli for
families over semistable curves
Angelo Vistoli. This note is but a
research announcement, summarizing and explaining results
proven and detailed in forthcoming papers. When one studies families of
over curves, and the objects are parametrized by a Deligne-Mumford stack
then the families are equivalent to morphisms of curves into M. In order
complete moduli for such families, one needs to compactify the stack of
maps into M. It turns out that in the boundary, the curve must acquire
structure and you'd better read the paper to see what that structure is.
Applications to fibered surfaces (see math.AG/9804097), admissible
level structures are discussed.
- 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
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
- 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
- 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
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
- Lang's conjectures, fibered powers, and
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
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
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
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.