arXiv subjects
Publications and source records attributed to Dan Abramovich.
We define a proper moduli stack for degree $p$ covers $f:Y \to \cX$ where $\cX$ is a twisted stable curve in the sense of [5] and [4], and $Y$ is a stable curve which via $f$ is a torsor over $\cX$ under a finite flat group scheme $\cG \to \cX$.
We discuss the role played by logarithmic structures in the theory of moduli.
This paper is a continuation of our earlier development of a theory of tame Artin stacks. Our main goal here is the construction of an appropriate analogue of Kontsevich's space of stable maps in the case where the target is a tame Artin stack. When the target is a tame Deligne--Mumford stack, the theory was developed by Abramovich and Vistoli, and found a number of applications. The theory for arbitrary tame Artin stacks developed here is very similar, but it is necessary to overcome a number of technical hurdles and to generalize a few questions of foundation.
We describe a new approach to the definition of the moduli functor of stable varieties. While there is wide agreement as to what classes of varieties should appear, the notion of a family of stable surfaces is quite subtle, as key numerical invariants may fail to be constant in flat families. Our approach is to add natural stack structure to stable varieties. For example, given a canonical model we take the global-quotient stack structure arising from its realization as Proj of the canonical ring. Deformations of the stack structure preserve key numerical invariants of the stable variety, including the top self-intersection of the canonical divisor. This approach yields a transparent construction of the moduli stack of stable varieties as a global quotient of a suitable Hilbert scheme of weighted projective stacks.
Long ago, in math.AG/0112004, we pledged more details on the algebraic version of Chen-Ruan's math.AG/0103156. This is it.
We introduce and study a class of algebraic stacks with finite inertia in positive and mixed characteristic, which we call tame algebraic stacks. They include tame Deligne-Mumford stacks, and are arguably better behaved than general Deligne-Mumford stacks. We also give a complete characterization of finite flat linearly reductive schemes over an arbitrary base. Our main result is that tame algebraic stacks are étale locally quotient by actions of linearly reductive finite group schemes. In a subsequent paper we will show that tame algebraic stacks admit a good theory of stable maps.
Awfully idiosyncratic lecture notes from CMI summer school in arithmetic geometry July 31-August 4, 2006. Does not include: rationality problems, techniques of the minimal model problem and much of the rest. Includes: Lecture 0: geometry and arithmetic of curves Lecture 1: Kodaira dimension and properties, rational connectendess, Lang's and Campana's conjectures. Lecture 2: Campana's program; Campana constellations framed in terms of b-divisors, to allow for a definition of Kodaira dimension directly on the base. A speculative notion of firmaments which may deserve further investigation, especially the arithmetic side. Lecture 3: the minimal model program: very short discussion of bend-and-break; even shorter discussion of finite generation and the existence of flip. Lecture 4: Vojta's conjectures, Campana's conjectures, and ABC.
Let $\mathcal{X}$ be a tame proper Deligne-Mumford stack of the form $[M/G]$ where $M$ is a scheme and $G$ is an algebraic group. We prove that the stack $\mathcal{K}_{g,n}(\mathcal{X},d)$ of twisted stable maps is a quotient stack and can be embedded into a smooth Deligne-Mumford stack. When $G$ is finite, we give a more precise construction of $\mathcal{K}_{g,n}(\mathcal{X},d)$ using Hilbert schemes and admissible $G$-covers.
In this work more questions arise than answers given, for which of course we do not apologize. The core of this paper is concerned with the construction of a ``constant'' t-structure on the bounded derived category of coherent sheaves $D(X\times S)$, with $X$ and $S$ smooth varieties, given a nondegenerate t-structure on $D(X)$ with noetherian heart. While we believe this construction and the methods involved should be generally useful in studying derived categories, we present here but one application: we prove a valuative criterion for separation and properness for the collection P(1) of stable objects of phase 1 under a numerical, locally finite and noetherian Bridgeland-Douglas stability condition $(Z,\cP)$ on $D(X)$, where $X$ is a smooth projective variety. As an immediate result, the number of long-exact sequences in the first named author's output is no longer an embarrassment.
The structure of the reduction of an admissible $G$-covering $Y \to X$ at primes $p$ dividing $|G|$ is investigated. Assume $|G|$ is not divisible by $p^2$ and the $p$-Sylow group is normal. Following Raynaud it is shown that there is a group scheme $\cG$ over the smooth locus of $X$ for which $Y$ is still a principal bundle away from the special points. A structure at the nodes involving Artin twisted curves is discussed.
These are lecture notes of a C.I.M.E. course I gave at Cetraro, June 6-11 2005. The theory described is the version of Chen-Ruan's Gromov-Witten theory of orbifolds developed by Graber, Vistoli and me in the algebraic setting, but with introduction beginning in Kontsevich's formula on rational plane curves and through Gromov-Witten theory of algebraic manifolds. As this is not a joint paper, I finally get to give direct credit to Vistoli's beautiful ideas! In the appendix, I include, with permissions, two relevant letters Kontsevich wrote to Borisov in July 1996. People who study motivic integration might want to read and understand something they have found confounding.
Given a birational modification $X \to Y$ of complex projective varieties with fiber dimension 1 and rational singularities, consider the main component of Bridgeland's moduli space $W \to Y$ of perverse point sheaves on $X/Y$. We give criteria for the normalization of $W$ to coincide with the transform (flip/flop) $X^+ \to Y$ of $X \to Y$. First, this holds if the exceptional loci of $X \to Y$ and $W \to Y$ have codimension $>1$. Second, if the ideal sheaf of $X \times_Y X^+$ is flat over $X^+$ then there is at least a morphism $X^+ \to W$. We illustrate the results using a number of examples, including surface modifications and standard Francia flips.
We consider two cases of birational transformations of Q-Gorenstein threefolds: the case of a terminal flop and the case of a Francia flip. In the first case we show that, if one replaces the threefolds by their canonical covering stacks, the the flop coincides with the moduli stack of perverse point sheaves. This adds to a result of Kawamata showing the derived equivalence. Our construction has the drawback that the moduli space is defined using a presentation of the stack. In the second case we show that, if one takes a Francia flip $X \to Y$ and replaces $X$ by its canonical covering stack, then the flip coincides with a version of Bridgelan'd moduli space of pewrverse point sheaves involving a new perversity. This adds to another result of Kawamata. Again there is a drawback - the proof of the result relies on the existence of the flip, in contrast with the results of Bridgeland and of Chen where the relevant flop appears as an outcome of the construction.
A cheap method for constructing canonical models and complete moduli for complex projective varieties with a structure called "rational plurifibration" is given. A result about semistable reduction (whose nature is slightly different from alg-geom/9707012) is deduced. The method involves recursive application of moduli stacks of twisted stable maps (math.AG/9908167) and a systematic circumvention of any difficult MMP issues.
The purpose of this note is to give an overview of our work on defining algebraic counterparts for W. Chen and Y. Ruan's Gromov-Witten Theory of orbifolds. This work will be described in detail in a subsequent paper. The presentation here is generally based on lectures given by two of us at the Orbifold Workshop in Madison, Wisconsin. Following the spirit of the workshop, we give a nontechnical introduction to stacks, a general discussion of twisted stable maps, and a run-through of the issues that arise when generalizing Gromov-Witten theory to Deligne-Mumford stacks. We make a special effort to make our constructions as canonical as we can, systematically using the language of algebraic stacks. Our efforts bear some concrete fruits; in particular, we are able to define the Chen-Ruan product in degree 0 (the so called stringy cohomology of the stack) with integer coefficients. Beware - this note may be vanishing in front of your eyes: first, a somewhat less technical version intended for the workshop proceedings will shortly (we hope) be submitted. Second, a much more detailed paper is in preparation, in which all the technicalities which are swept under the rug here are discussed, as well as some cool stuff about taking roots.
The cherry on top of this stacky paper is the following: for any g>1 we give a finite group G such that the moduli space of connected admissible G-covers of genus g is a smooth, fine moduli space, which is a Galois cover of the moduli space of stable curves. The proof relies on methods introduced by Looijenga and Pikaart-De Jong, and on the theory of twisted G-covers, a theory announced without proofs in math.AG/9811059, section 3, and developed in the first few sections of this paper. This includes a moduli description of the normalization of the Harris-Mumford space of admissible covers, and a study of moduli of stable curves with abelian and non-abelian level structure.
In this note we give a new, natural construction of a compactification of the stack of smooth r-spin curves, which we call the stack of stable twisted $r$-spin curves. This stack is identified with a special case of a stack of twisted stable maps of Abramovich and Vistoli. Realizations in terms of admissible G_m-spaces and Q-line bundles are given as well. The infinitesimal structure of this stack is described in a relatively straightforward manner, similar to that of usual stable curves. We construct representable morphisms from the stacks of stable twisted r-spin curves to the stacks of stable r-spin curves, and show that they are isomorphisms. Many delicate features of r-spin curves, including torsion free sheaves with power maps, arise as simple by-products of twisted spin curves. Various constructions, such as the d bar-operator of Seeley and Singer and Witten's cohomology class go through without complications in the setting of twisted spin curves.
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. Such a factorization exists which is functorial with respect to absolute isomorphisms, and compatible with a normal crossings divisor. The same holds for algebraic and analytic spaces. Another proof of the main theorem by the fourth author appeared in math.AG/9904076.