arXiv subjects
Publications and source records attributed to Dan Abramovich.
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.
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.
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. This gives a uniform version, in the spirit of a result of Caporaso-Harris-Mazur, of an unconditional theorem of Faltings. We utilize recent results of Alexeev and Nakamura on complete moduli for quasi-abelian varieties. We expect that a thorough understanding of current work of Alexeev should give a more general result for abelian varieties of an arbitrary dimension with a polarizing divisor of an arbitrary degree - a proposed approach for such a generalization is given at the end of the paper.
In this short note, we propose a definition of complete Hurwitz schemes (and stacks) in mixed characteristic. We follow an idea of R. Pandharipande, and define the complete Hurwitz stack as a substack of stable maps of degree d of nodal pointed curves of genus g to the universal curve over the moduli stack of stable pointed curves of genus h. This substack is simply the closure of the locus of Hurwitz covers in characteristic 0. We describe in detail the reduction modulo 2 of the complete Hurwitz stack of genus 1 double covers of the projective line. The result is wilder than you would imagine in your wildest dreams. We end by speculating about a possible relationship with the work of Abramovich-Vistoli about stable maps into Deligne-Mumford stacks.
On July 26, 1995, at the University of California, Santa Cruz, a young Dutch mathematician by the name Aise Johan de Jong made a revolution in the study of the arithmetic, geometry and cohomology theory of varieties in positive or mixed characteristic. The talk he delivered, first in a series of three entitled "Dominating Varieties by Smooth Varieties", had a central theme: a systematic application of fibrations by nodal curves. Among the hundreds of awe struck members of the audience, participants of the American Mathematical Society Summer Research Institute on Algebraic Geometry, many recognized the great potential of Johan de Jong's ideas even for complex algebraic varieties, and indeed soon more results along these lines began to form. This paper is an outgrowth of our course material prepared for the Working Week on Resolution of Singularities, which was held during September 7-14, 1997 in Obergurgl, Tirol, Austria. As we did in the workshop, we intend to explain Johan de Jong's results in some detail, and give some other results following the same paradigm, as well as a few applications, both arithmetic and in characteristic zero. We hope that the reader will come to share some of the excitement we felt on that beautiful July day in Santa Cruz.
Families of stable curves of genus $γ$ over a smooth curve $C$ correspond to morphisms from $C$ to the moduli stack of stable curves $\bar{\cal M}_γ$. It is natural to compactify the corresponding moduli problem using stable maps into the stack. In order to get a complete moduli problem, the source curves must acquire extra structure, and you will have to read the paper to find what that is. In this paper, we define these stable maps into $\bar{\cal M}_γ$ in characteristic 0, and show that they form a proper Deligne-Mumford stack admitting a projective coarse moduli space. A comparison with Alexeev's work is given. Natural generalizations for stable maps into other Deligne-Mumford stacks will appear in a follow up paper, along with applications to areas such as admissible covers, level structures and higher dimensional families.
Let X->B be a morphism of varieties in characteristic zero. Semistable reduction has been proved for dim(B)=1 (Kempf, Knudsen, Mumford, Saint-Donat), dim(X)=dim(B)-1 (de Jong) and dim(X)=dim(B)+2 (Alexeev, Kollar, Shepherd-Barron). In this paper we consider the general case. 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.
The result in the title is proven, using the Selberg estimate on the leading eigenvalue of the non-Euclidean Laplacian, and the method of conformal volumes of Li and Yau.
A new proof of equivariant resolution of singularities under a finite group action in characteristic 0 is provided. We assume we know how to resolve singularities without group action. We first prove equivariant resolution of toroidal singularities. Then we reduce the general case to the toroidal case.
We prove the following theorem: Fibered Power Theorem: Let $X\rar B$ be a smooth family of positive dimensional varieties of general type, with $B$ irreducible. Then there exists an integer $n>0$, a positive dimensional variety of general type $W_n$, and a dominant rational map $X^n_B \das W_n$.
We provide a new proof of the following result: Let $X$ be a variety of finite type over an algebraically closed field $k$ of characteristic 0, let $Z\subset X$ be a proper closed subset. There exists a modification $f:X_1 \rar X$, such that $X_1$ is a quasi-projective nonsingular variety and $Z_1 = f^{-1}(Z)_\red$ is a strict divisor of normal crossings. Needless to say, this theorem is a weak version of Hironaka's well known theorem on resolution of singularities. Our proof has the feature that it builds on two standard techniques of algebraic geometry: semistable reduction for curves, and toric geometry. Another proof of the same result was discovered independently by F. Bogomolov and T. Pantev. The two proofs are similar in spirit but quite different in detail.
The Lang map, namely the universal dominant rational map to a variety of general type, is constructed and briefly discussed in relation with arithmetic conjectures of Harris, Lang and Manin. Existence of the Lang map follows from the additivity of Kodaira dimension, but the fine structure depends on conjectures on birational classification of algebraic varieties. Serious applications of the Lang map are still being searched.
A common practice in arithmetic geometry is that of generalizing rational points on projective varieties to integral points on quasi-projective varieties. Following this practice, we demonstrate an analogue of a 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).
The purpose of this note is to wish a happy birthday to Professor Lucia Caporaso.* We prove that Conjecture H of Caporaso et. al. ([CHarM], sec. 6) together with Lang's conjecture implies the uniformity of rational points on varieties of general type, as predicted in [CHarM]; a few applications in arithmetic and geometry are stated. Let X be a variety of general type defined over a number field K. It was conjectured by S. Lang that the set of rational points X(K) is not Zariski dense in X. In the paper [CHarM] of L. Caporaso, J. Harris and B. Mazur it is shown that the above conjecture of Lang implies the existence of a uniform bound on the number of K-rational points of all curves of fixed genus g over K. The paper [CHarM] has immediately created a chasm among arithmetic geometers. This chasm, which often runs right in the middle of the personalities involved, divides between loyal believers of Lang's conjecture, who marvel in this powerful implication, and the disbelievers, who try (so far in vain) to use this implication to derive counterexamples to the conjecture. In this paper we will attempt to deepen this chasm, using the techniques of [CHarM] and continuing [aleph], by proving more implications, some of which very strong, of various conjectures of Lang. Along the way we will often use a conjecture donned by Caporaso et. al. Conjecture H (see again [CHarM], sec. 6) about Higher dimensional varieties, which is regarded very plausible among experts of higher dimensional algebraic geometry. In particular, we will show
We refine a result of L. Caporaso, J. Harris and B. Mazur, and prove: Supposons que la conjecture de Lang soit vraie. Soit $K$ un corps des nombres et $g>1$ un entier. Il existe un nombre $N(K,g)$ tel que si $L$ est une extension de degré $\leq 3$ de $K$ et $C$ est une courbe lisse projective connèxe, de genre $g$ définie sur $L$ on a $$\# C(L) < N(K,g). $$