Dan Edidin
We introduce the notion of strong regular embeddings of Deligne-Mumford stacks. These morphisms naturally arise in the related contexts of generalized Euler sequences and hypertoric geometry.
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Hypertoric varieties are determined by hyperplane arrangements. In this paper, we use stacky hyperplane arrangements to define the notion of hypertoric Deligne-Mumford stacks. Their orbifold Chow rings are computed. As an application, some examples related to crepant resolutions are discussed.
We study Gromov-Witten theory of hypertoric Deligne-Mumford stacks from two points of view. From the viewpoint of representation theory, we calculate the operator of small quantum product by a divisor, following \cite{BMO}, \cite{MO}, \cite{MS}. From the viewpoint of Lawrence toric geometry, we compare Gromov-Witten invariants of a hypertoric Deligne-Mumford stack with those of its associated Lawrence toric stack.
Any toric Deligne-Mumford stack is a $μ$-gerbe over the underlying toric orbifold for a finite abelian group $μ$. In this paper we give a sufficient condition so that certain kinds of gerbes over a toric Deligne-Mumford stack are again toric Deligne-Mumford stacks.
We define the moduli problem of Hitchin pairs over Deligne-Mumford Stack and prove this moduli problem is represented by a separated and locally finitely presented algebraic space, which is considered as the moduli space of Hitchin pairs over Deligne-Mumford stack.
We prove that the automorphism group of a toric Deligne-Mumford stack is isomorphic to the $2$-group associated to the stacky fan.
We construct a presentation for the Grothendieck group of Deligne-Mumford stacks over a field of characteristic zero. The generators for this presentation are smooth, proper Deligne-Mumford stacks and the relations are expressed in terms of stacky blow-ups. In the process we prove a version of the weak factorization theorem for Deligne-Mumford stacks.
Birational geometry of Deligne-Mumford stacks
本文研究 Deligne-Mumford 栈的双有理几何，并定义了新的双有理不变量。
In this paper, we define $Λ$-quot-functors on Deligne-Mumford stacks. We prove that the $Λ$-quot-functor is representable by an algebraic space. Then, we construct the moduli space of $Λ$-modules on a projective Deligne-Mumford stack. We prove that this moduli space is a quasi-projective scheme.
We prove an Atiyah-Segal isomorphism for the higher $K$-theory of coherent sheaves on quotient Deligne-Mumford stacks over $\C$. As an application, we prove the Grothendieck-Riemann-Roch theorem for such stacks. This theorem establishes an isomorphism between the higher $K$-theory of coherent sheaves on a Deligne-Mumford stack and the higher Chow groups of its inertia stack. Furthermore, this isomorphism is covariant for proper maps between Deligne-Mumford stacks.
We compute the fundamental groups of non-singular analytic Deligne-Mumford curves, classify the simply connected ones, and classify analytic Deligne-Mumford curves by their uniformization type. As a result, we find an explicit presentation of an arbitrary Deligne-Mumford curve as a quotient stack. Along the way, we compute the automorphism 2-groups of weighted projective stacks $\mathcal{P}(n_1,n_2,...,n_r)$. We also discuss connections with the theory of F-groups, 2-groups, and Bass-Serre theory of graphs of groups.
The arithmetic Chow groups and their product structure are extended from the category of regular arithmetic varieties to regular Deligne-Mumford stacks over the ring of integers in a number field.
This paper proves a result on the existence of finite flat scheme covers of Deligne-Mumford stacks. This result is used to prove that a large class of smooth Deligne-Mumford stacks with affine moduli space are quotient stacks, and in the case of quasi-projective moduli space, to reduce the question to one concerning Brauer groups of schemes.
We prove the excision theorem for the $K$-theory of perfect complexes on Deligne-Mumford stacks. This is then used to study the Nisnevich site of such stacks. We prove the Nisnevich descent for the $K$-theory of perfect complexes. We also draw some other consequences.
We give conditions for a n-connective quasicoherent obstruction theory on a Deligne-Mumford stack to come from the structure of a connective spectral Deligne-Mumford stack on the underlying topos.
In this paper we study the integral Chow ring of toric Deligne-Mumford stacks. We prove that the integral Chow ring of a semi-projective toric Deligne-Mumford stack is isomorphic to the Stanley-Reisner ring of the associated stacky fan. The integral orbifold Chow ring is also computed. Our results are illustrated with several examples.
We generalize the Bogomolov-Gieseker inequality for semistable coherent sheaves on smooth projective surfaces to smooth Deligne-Mumford surfaces. We work over positive characteristic $p>0$ and generalize Langer's method to smooth Deligne-Mumford stacks. As applications we obtain the Bogomolov inequality for semistable coherent sheaves on a Deligne-Mumford surface in characteristic zero, and the Bogomolov inequality for semistable sheaves on a root stack over a smooth surface which is equivalent to the Bogomolov inequality for the rational parabolic sheaves on a smooth surface $S$. In a joint a…
Generalizing toric varieties, we introduce toric Deligne-Mumford stacks which correspond to combinatorial data. The main result in this paper is an explicit calculation of the orbifold Chow ring of a toric Deligne-Mumford stack. As an application, we prove that the orbifold Chow ring of the toric Deligne-Mumford stack associated to a simplicial toric variety is a flat deformation of (but is not necessarily isomorphic to) the Chow ring of a crepant resolution.
We study how to use a suitably ample locally free sheaf over a proper Deligne-Mumford stack to furnish an embedding of the stack into a geometric invariant theory (GIT) quotient stack constructed from a finite-dimensional linear representation of the general linear group.
In this paper, we study the geometric invariant theory on algebraic spaces, and construct te moduli spaces of $\mathcal{H}$-semistable sheaves on projective Deligne-Mumford stacks over algebraic spaces $S$. We prove that this moduli space is projective over $S$ as an algebraic space.
We construct full strong exceptional collections of line bundles on smooth toric Fano Deligne-Mumford stacks of Picard number at most two and of any Picard number in dimension two. It is hoped that the approach of this paper will eventually lead to the proof of the existence of such collections on all smooth toric nef-Fano Deligne-Mumford stacks.
We show that the moduli stacks of smooth complete intersections in P^N polarized by O(1) are separated (except for quadrics) and Deligne-Mumford (apart from a few exceptions). ----- On montre que les champs de modules d'intersections completes lisses dans P^N polarisees par O(1) sont separes (sauf dans le cas des quadriques) et de Deligne-Mumford (sauf pour quelques exceptions).
For a smooth Deligne-Mumford stack over $\CC$, we define its associated Kodaira-Spencer differential graded Lie algebra and show that the deformation functor of the stack is isomorphic to the deformation functor of the Kodaira-Spencer algebra if the stack is proper over $\CC$.
We prove the cone theorem for varieties with LCIQ singularities using deformation theory of stable maps into Deligne-Mumford stacks. We also obtain a sharper bound on $-(K_X+D)$-degree of $(K_X+D)$-negative extremal rays for projective $\QQ$-factorial log terminal threefold pair $(X,D)$.
We establish the Airy curve case of a conjecture of Gukov and Sułkowski by reducing to Dijkgraaf-Verlinde-Verlinde Virasoro constraints satisfied by the intersection numbers on moduli spaces of algebraic curves.
We give a new description of the data needed to specify a morphism from a scheme to a toric Deligne-Mumford stack. The description is given in terms of a collection of line bundles and sections which satisfy certain conditions. As applications, we characterize any toric Deligne-Mumford stack as a product of roots of line bundles over the rigidified stack, describe the torus action, describe morphisms between toric Deligne-Mumford stacks with complete coarse moduli spaces in terms of homogeneous polynomials, and compare two different definitions of toric stacks.
The stable reduction theorem of Deligne and Mumford --- The moduli space of smooth projective curves of genus $g$ is a quasi-projective algebraic variety, but is not projective. To understand its geometry, it may be crucial to consider compactifications of this space. By allowing to parameterize as well curves with controlled singularities (the so called stable curves), Deligne and Mumford constructed a projective compactification. The properness of this compactification translates into the stable reduction theorem that they prove, its projectivity is a later theorem of Knudsen and Mumford. Th…
We define Gromov-Witten classes and invariants of smooth proper tame Deligne-Mumford stacks of finite presentation over a Dedekind domain. We prove that they are deformation invariants and verify the fundamental axioms. For a smooth proper tame Deligne-Mumford stack over a Dedekind domain, we prove that the invariants of fibers in different characteristics are the same. We show that genus zero Gromov-Witten invariants define a potential which satisfies the WDVV equation and we deduce from this a reconstruction theorem for genus zero Gromov-Witten invariants in arbitrary characteristic.
We prove the following results for toric Deligne-Mumford stacks, under minimal compactness hypotheses: the Localization Theorem in equivariant K-theory; the equivariant Hirzebruch-Riemann-Roch theorem; the Fourier--Mukai transformation associated to a crepant toric wall-crossing gives an equivariant derived equivalence.
We study strong exceptional collections of line bundles on Fano toric Deligne-Mumford stacks $\mathbb{P}_{\mathbfΣ}$ with rank of Picard group at most two. We prove that any strong exceptional collection of line bundles generates the derived category of $\mathbb{P}_{\mathbfΣ}$, as long as the number of elements in the collection equals the rank of the (Grothendieck) $K$-theory group of $\mathbb{P}_{\mathbfΣ}$.
The goal of this paper is to prove Riemann-Roch type theorems for Deligne-Mumford algebraic stacks. To this end, we introduce a "cohomology with coefficients in representations" and a Chern character, and we prove a Grothendieck-Riemann-Roch theorem for the Riemann-Roch transformation it defines. As a corollary we obtain an Hirzebruch-Riemann-Roch formula for the Euler characteristic of a coherent sheaf, and some formulas for the different topological Euler characteristics of complex algebraic stacks.
In this short note we write down a comparison between the notion of derived Deligne-Mumford stack in the sense of B. Toën and G. Vezzosi [HAG-II] and the one introduced by J. Lurie in [DAG-V]. It is folklore that the two theories yield essentially the same objects, but it is difficult to locate in the literature a precise result, despite being sometimes useful to be able to switch between the two frameworks.
We provide stacky generalizations of classical gauge-theoretic results inspired by Donaldson, the Uhlenbeck-Yau theorem and variants due to Bando and his collaborators. Moreover, we show an application of this machinery in the study of ALE spaces.
Asymptotics of immaculate line bundles on smooth toric Deligne-Mumford stacks
本文研究了光滑环面 Deligne-Mumford 栈上的无瑕线丛，给出了存在无穷多无瑕线丛的判据，肯定回答了 Borisov 和 Wang 的问题，并描述了其渐近行为。
Moduli spaces of semistable pairs on projective Deligne-Mumford stacks
本文把半稳定对模空间构造推广至射影 Deligne–Mumford 栈，研究其形变与障碍理论，并在三维光滑情形定义 PT 不变量。
For a toric Deligne-Mumford (DM) stack, we can consider a certain generalization of the Frobenius endomorphism. For such an endomorphism on a two-dimensional toric DM stack, we show that the push-forward of the structure sheaf generates the bounded derived category of coherent sheaves on the toric DM stack. We also choose a full strong exceptional collection from the set of direct summands of the push-forward in several examples of two dimensional toric DM orbifolds.
The aim of this article is to develop the theory of motivic integration over Deligne-Mumford stacks and to apply it to the birational geometry of stacks.
Given a separated and locally finitely-presented Deligne-Mumford stack $\cX$ over an algebraic space $S$, and a locally finitely-presented $\OO_{\cX}$-module $\cF$, we prove that the Quot functor $\text{Quot}(\cF/\cX/S)$ is represented by a separated and locally finitely-presented algebraic space over $S$. Under additional hypotheses, we prove that the connected components of $\text{Quot}(\cF/\cX/S)$ are quasi-projective over $S$.
Existence of coarse moduli superspaces for Deligne--Mumford superstacks
本文证明在 $\mathbb{Z}[1/2]$ 上具有有限惯性的 Deligne--Mumford 超栈存在粗模超空间，推广 Keel--Mori 定理，并给出反例。
A Quadratically Enriched Pushforward for Deligne--Mumford Stacks
本文利用 Grothendieck--Serre 对偶，为 Deligne--Mumford 栈定义二次加细推前，进而构造欧拉类、欧拉数及基本类，并据此提出二次加细量子 $K$-不变量的定义。
Motivated by the S-duality conjecture of Vafa-Witten, Tanaka-Thomas have developed a theory of Vafa-Witten invariants for projective surfaces using the moduli space of Higgs sheaves. Their definition and calculation prove the S-duality prediction of Vafa-Witten in many cases in the side of gauge group $SU(r)$. In this survey paper for ICCM-2019 we review the S-duality conjecture in physics by Vafa-Witten and the definition of Vafa-Witten invariants for smooth projective surfaces and surface Deligne-Mumford stacks. We make a prediction that the Vafa-Witten invariants for Deligne-Mumford surface…
Laszlo and Olsson constructed Grothendieck's six operations for constructible complexes on Artin stacks in étale cohomology under an assumption of finite cohomological dimension, with base change established on the level of sheaves. In this article we give a more direct construction of the six operations for complexes on Deligne-Mumford stacks without the finiteness assumption and establish base change theorems in derived categories. One key tool in our construction is the theory of gluing finitely many pseudofunctors developed in arXiv:1211.1877. As an application, we prove a Lefschetz-Verdie…
Removing the Torsion-free Hypothesis in a Positivity Theorem on Deligne-Mumford Stacks
本文在Casalaina-Martin与Zhjeqi的正性定理中移除无挠假设，解答其提出的问题，主要结果借助ChatGPT 5.5 Pro、Danus系统等生成式AI获得。
We provide a definition of Vafa-Witten invariants for projective surface Deligne-Mumford stacks, generalizing the construction of Tanaka-Thomas on the Vafa-Witten invariants for projective surfaces inspired by the S-duality conjecture. We give calculations for a root stack over a general type quintic surface, and quintic surfaces with ADE singularities. The relationship between the Vafa-Witten invariants of quintic surfaces with ADE singularities and the Vafa-Witten invariants of their crepant resolutions is also discussed.
There is a well-known stratification of the moduli space $M_g$ of Deligne-Mumford stable curves of genus $g$ by the loci of stable curves with a fixed number $i$ of nodes, where $i \le 3g-3$. The associated moduli stack ${\cal M}_g$ admits an analogous stratification. Our main objects of study are those one-dimensional substacks of the moduli stack, which are the irreducible components of the stratum corresponding to stable curves with exactly $3g-4$ nodes. We describe these substacks explicitely as quotient stacks, and relate them to other, and simpler, moduli stacks of (permutation classes o…
Suppose $F\colon \mathcal{D}(X)\to \mathcal{T}$ is an exact functor from the bounded derived category of coherent sheaves on a smooth projective variety $X$ to a triangulated category $\mathcal{T}$. If $F$ possesses left and right adjoints, then the Bondal-Orlov criterion gives a simple way of determining if $F$ is fully faithful. We prove a natural extension to the case when $X$ is a smooth and proper DM stack with projective coarse moduli space.
Let $X$ be a complex projective algebraic variety with Gorenstein quotient singularities and $\X$ a smooth Deligne-Mumford stack having $X$ as its coarse moduli space. We show that the CSM class $c^{SM}(X)$ coincides with the pushforward to $X$ of the total Chern class $c(T_{I\X})$ of the inertia stack $I\X$. We also show that the stringy Chern class $c_{str}(X)$ of $X$, whenever is defined, coincides with the pushforward to $X$ of the total Chern class $c(T_{II\X})$ of the double inertia stack $II\X$. Some consequences concerning stringy/orbifold Hodge numbers are deduced.
Positivity in the context of Hodge modules and Higgs bundles on Deligne-Mumford stacks
将Popa-Wu和Popa-Schnell关于光滑射影簇上Hodge模和Higgs丛的正性结果推广到具有射影粗模空间的光滑真DM栈上。
Foliations, slope stability, and positivity of log canonical bundles on Deligne-Mumford stacks
将Campana-Păun关于光滑射影簇上叶状结构、斜率稳定性及对数典范丛正性的结果推广到具有射影粗模空间的光滑真DM栈上，作为将Viehweg双曲性结果推广到DM栈及特定KSBA模空间系列论文的第三部分。
Factorizations of Moduli Morphisms and Universal Maps to Deligne-Mumford Stacks
本文研究了模态空间映射的分解，构造了简化堆栈结构但保留更多信息的中间堆栈，并证明了在一定条件下存在到满足良好模性质的堆栈的通用映射，展示了该映射本身是有效的模空间映射。
$E_1$-Degeneration for Irregular Hodge Filtrations on Deligne--Mumford Stacks
针对粗空间拟射影的光滑分离德利涅-芒福德栈，利用Kontsevich复形的有限平展下降，将射影栈情形归约为Esnault–Sabbah–Yu的定理，证明其非正则霍奇滤子在所有有理指标处$E_1$退化，结果适用于各类栈紧化。
We prove a formula for the orbifold Chow ring of semi-projective toric DM stacks, generalizing the orbifold Chow ring formula of projective toric DM stacks by Borisov-Chen-Smith. We also consider a special kind of semi-projective toric DM stacks, the Lawrence toric DM stacks. We prove that the orbifold Chow ring of a Lawrence toric DM stack is isomorphic to the orbifold Chow ring of its associated hypertoric DM stack studied in \cite{JT}.
Movable curve classes and slope stability on Deligne-Mumford stacks
将光滑射影簇上关于可动曲线类和相干层的斜率稳定性的结果推广到具有射影粗模空间的光滑真DM栈上，并建立了Bogomolov-Gieseker不等式。
Descent and Brauer-Manin Obstructions on Deligne-Mumford Stacks
该研究在具有拟射粗模空间的有限型光滑分离德利涅-芒福德堆叠上，证明下降障碍包含于布饶尔-马林障碍，且下降障碍与 étale 布饶尔-马林障碍、迭代下降障碍一致，布饶尔-马林障碍也与迭代布饶尔-马林障碍一致。
The de Rham comparison theorem for varieties, first proved by Faltings, gives the de Rham cohomology of a variety in terms of its p-adic etale cohomology. We extend this theorem to proper, smooth Deligne-Mumford stacks. Two approaches are given, which both in the end reduce the problem to the already known comparison theorem for varieties. The first approach employs the formalism of Weil cohomologies. Unfortunately, this does not result in a complete proof of the comparison theorem, as the author was unable to prove the required compatibility between intersections and cup products. Nevertheles…
A generalized Bondal-Orlov full faithfulness criterion for Deligne-Mumford stacks
本文将 Bondal-Orlov 的 Fourier-Mukai 函子全忠实性判据推广到特征零任意域上的光滑真 Deligne-Mumford 叠，并建立代数 stack 真与驯态射导出范畴的基础结果。
We give a simple proof of the Riemann-Roch theorem for Deligne-Mumford stacks using the equivariant Riemann-Roch theorem and the localization theorem in equivariant K-theory together with some basic commutative algebra of Artin rings.
For any finite abelian group G, the equivariant Gromov-Witten invariants of C^r/G can be viewed as a certain kind of abelian Hurwitz-Hodge integrals. In this note, we use Tseng's orbifold quantum Riemann-Roch theorem to express this kind of abelian Hurwitz-Hodge integrals as a sum over Feynman graphs, where the weight of each graph is expressed in terms of descendant integrals over moduli spaces of stable curves and representations of G. This expression will play a crucial role in the proof of the remodeling conjecture (arXiv:0709.1453, arXiv:0807.0597) for affine toric Calabi-Yau 3-orbifolds …
In casual discussion, a stack is often described as a variety (the coarse space) together with stabilizer groups attached to some of its subvarieties. However, this description does not uniquely specify the stack. Our main result shows that for a large class of stacks one typically encounters, this description does indeed characterize them. Moreover, we prove that each such stack can be described in terms of two simple procedures applied iteratively to its coarse space: canonical stack constructions and root stack constructions. More precisely, if $\mathcal X$ is a smooth separated tame Delign…
We define the Deligne Mumford orbifold axiomatically by a universal mapping property, show that this universal mapping property is equivalent to an infinitessimal universal mapping property, and use the latter to give an existence proof.
In 1969, P. Deligne and D. Mumford compactified the moduli space of curves. Their compactification is a projective algebraic variety, and as such, it has an underlying analytic structure. Alternatively, the quotient of the augmented Teichmueller space by the action of the mapping class group gives a compactification of the moduli space. We put an analytic structure on this compact quotient and prove that with respect to this structure, it is canonically isomorphic (as an analytic space) to the Deligne-Mumford compactification.
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